DE102023201147A1 - Novel coherent lidar system for environmental detection - Google Patents

Abstract

Coherently operating lidar system for environmental detection, which emits a phase-modulated signal, in particular an irregular change over discrete phase values, preferably over only two phase values that differ by approximately 180°, characterized in that the frequency to which the phase is modulated changes continuously at least in sections, preferably with at least approximately a linear progression.

Description

The invention relates to a coherent lidar system, in particular for environmental detection for motor vehicle applications. The novel coherent lidar system works with phase modulation and, according to the invention, additionally has frequency modulation, in which the frequency changes continuously at least in sections, preferably with an at least approximately linear progression. The lidar system advantageously uses the continuous frequency change to change the beam direction in a first spatial direction with the aid of at least one waveguide and/or for a sign-correct determination of the reception frequency and thus for a clear determination of the relative speed of objects when using a real-valued mixer. Different devices can be used for the beam direction change in a second spatial direction orthogonal to the first, in particular an element with controllable optical material properties. The novel lidar system can be implemented with a high level of semiconductor integration.

State of the art

Motor vehicles are increasingly being equipped with driver assistance systems that use sensor systems to detect the environment and, based on the traffic situation detected, derive automatic reactions from the vehicle and/or instruct the driver, particularly by warning them. A distinction is made between comfort and safety functions.

However, developments are now moving in an even more advanced direction. The driver is no longer just assisted, but the driver's task is increasingly being carried out autonomously by the vehicle, i.e. the driver is increasingly being replaced; this is known as autonomous driving.

Autonomous driving in particular requires sensors with highly accurate environmental information that can be easily evaluated by machines. Radar systems are limited in their angular accuracy and separation capability and cannot yet satisfactorily meet these high detection requirements on their own or in combination with camera systems. For this reason, lidar systems are also used in parallel, which have a similarly high angular resolution (horizontal and vertical) as a camera, but also provide distance information and separation capability in each pixel. Today, so-called time-of-flight lidar systems are mostly used, which treat electromagnetic radiation in the sense of particles and can therefore only directly measure the distance but not the relative speed. However, coherent lidar systems are now increasingly coming into focus, which treat electromagnetic radiation in the sense of waves (like radar systems) and can therefore also directly measure the relative speed of objects via the Doppler effect. Further advantages of coherent lidar systems are that they are robust against interference from other sources (e.g. from other lidar systems or sunlight) and that they have a higher sensitivity at greater distances and thus allow greater ranges. In addition, coherent lidar systems are said to have a higher potential for high semiconductor integration, which promises lower manufacturing costs.

In coherent lidar systems, the emitted electromagnetic wave is modulated, i.e. it changes in at least one of the parameters amplitude, frequency or phase over time - otherwise distance measurement would not be possible. The most commonly used modulation in coherent lidar systems is linear frequency modulation (FMCW = frequency modulated continuous wave), which usually consists of two frequency ramps whose slopes have opposite signs. However, this modulation has ambiguity problems, particularly with multiple reflections in the same beam direction, and generating a highly linear frequency change is also complex. These disadvantages do not occur or are less common with phase modulation (e.g. with pseudorandom changes over discrete phase values at a fixed transmission frequency), but the digital evaluation of the received signals is more complex and the approaches proposed in the prior art are associated with disadvantages, particularly with regard to sensitivity and thus range. To date, no modulation forms are known that include or allow a continuous frequency change within the data recording of a pixel and across pixels, which would be advantageous for a beam direction change using a waveguide. The beam direction change is often still implemented mechanically in both spatial directions, which is complex, leads to large designs and has disadvantages in terms of robustness. Real-valued mixers are usually used (since they require significantly less effort than complex-valued mixers); however, determining the sign of the reception frequency is generally problematic or not possible, so that the relative speed and, if applicable, the distance of objects There are two hypotheses. The potential of semiconductor integration is not fully exploited in current coherent lidar systems.

Task, solution and advantages of the invention

The object of the invention is to provide a novel coherent lidar system which has a high performance, but nevertheless allows a cost-effective implementation and a small installation space.

This object is basically achieved by a lidar system according to claim 1. Appropriate embodiments of the invention are claimed in the subclaims. A core idea is that a phase modulation with superimposed frequency modulation is used, which in particular advantageously enables a change in beam direction with the aid of at least one waveguide and allows the use of a real-valued mixer.

The advantages of the invention arise in particular from the fact that a high-performance lidar system can be realized through high semiconductor integration at a low price and with a small installation space.

The coherent lidar system for environmental detection according to the invention emits a phase-modulated signal and is characterized in that the frequency to which the phase is modulated changes continuously at least in sections, preferably with an at least approximately linear progression. The phase-modulated signal contains an irregular change over discrete phase values; there are preferably only two phase values that differ by approximately 180°.

Preferably, the lidar system receives the signals reflected back from objects, which are delayed compared to the transmitted signal by the distance-dependent transit time and are shifted in frequency both by the relative speed-dependent Doppler effect and by the distance-dependent transit time caused by the linear frequency change, converts them into a low-frequency signal by mixing and digitizes them into a reception sequence. According to the invention, a two-dimensional correlation filtering is carried out in digital signal processing means to determine the a priori unknown dimensions of time shift and frequency shift of signals reflected from objects, and the object distance is determined from the determined time shift and the radial relative speed of the object is determined from the determined frequency shift minus its contribution caused by the time shift. The correlation filtering represents an optimal filtering in order to ensure the most accurate determination possible (i.e. high sensitivity and range) of the lidar system.

Preferably, at least part of the two-dimensional correlation filter is implemented by a hard-wired (i.e. implemented in the hardware and not changeable) digital circuit designed as a pipeline (i.e. a structure with several calculation stages separated by buffers), wherein several or all output values in one of the two dimensions are determined per clock cycle of the digital circuit and via a sequence of clock cycles in the other dimension.

The hard-wired digital circuit can expediently have a front stage in which a signal sequence or its conjugate complex values and a phase modulation sequence or its conjugate complex values are multiplied by a position shifted relative to one another, optionally followed by a decimation of the sequence resulting from this multiplication and/or optionally followed by an expansion with zeros and followed by a fast Fourier transformation (FFT) implemented in several stages, wherein each individual arithmetic operation is implemented in a dedicated circuit and from cycle to cycle in the first stage the shift between signal and modulation sequence is changed and the result of a Fourier transformation is produced at the output of the rear stage, wherein this result relates in each case to output data of the front stage generated several cycles previously.

Advantageously, the hard-wired digital circuit can be used for several or preferably all pixels of the lidar system due to its high throughput rate, whereby these pixels can be generated in particular by scanning light beams and/or parallel reception paths and each pixel relates to data acquisition and evaluation, in particular for generating a detection.

Furthermore, a binary phase modulation, i.e. consisting of only two phase values that differ by approximately 180°, can be used, whereby the multiplications with the values of the modulation sequence can be realized by switchable inverters.

According to an advantageous embodiment of the invention, the signal multiplications with rotation factors required for the fast Fourier transformation are realized with a few additions and/or subtractions of shifted signal values, wherein preferably a maximum of one addition or subtraction is used to realize a real-valued multiplication.

Preferably, the bit length used varies across the pipeline stages of the hardwired digital circuit and is preferably only so large that the quantization noise generated in the digital circuit does not significantly increase the system noise generated in the analog part of the receiver.

Advantageously, the fast Fourier transform can be performed in the form of a structure with decimation in frequency in order to avoid reordering the input data in the form of long lines, as well as having the longest lines of the structure and the non-trivial multiplications in the front stages with their smaller bit length.

Conveniently, truncation can be used for quantization and/or purely bit-wise inversion for inversion, and the effects of the resulting mean errors are compensated by adding correction values in a stage of the digital circuit.

Furthermore, the hard-wired digital circuit can be extended by one or more further stages for evaluating the result of the two-dimensional correlation, in particular for magnitude or power formation and subsequent summation and/or maximum search.

Advantageously, the components of couplings and reflections contained in the digitized received signal within the lidar system or from its immediate surroundings, in particular a cover, can be largely eliminated by adding or subtracting a compensation sequence.

The components of couplings and reflections contained in the digitized received signal within the lidar system or from its immediate surroundings, in particular a cover, can expediently be compensated by at least one of the following approaches: One or more average values are determined using the product of the received sequence and the modulation sequence at the same or similar time, with several average values being formed by considering several sections and/or different phase value combinations of the phase modulation sequence. To form an average over all values of the product of the received sequence and the modulation sequence, the output value of the hard-wired digital circuit, which belongs to the zero frequency of the Fourier transformation and to the zero shift between the signal and modulation sequence in the first stage, is used, preferably by suspending the clocking of the first stage during the calculation of this value. Average values are formed over several acquisition cycles and/or different, preferably closely spaced pixels.

According to an advantageous embodiment of the invention, the components of couplings and reflections contained in the digitized received signal within the lidar system or from its immediate surroundings, in particular a cover, are compensated by a sequence which is formed in the hard-wired digital circuit by assigning, with the correct sign, one or more mean values determined via the product of the received sequence and the modulation sequence.

The inventive approaches for determining and realizing the correction values by which the effects of couplings and reflections within the lidar system or from its immediate surroundings, in particular a cover, are compensated, can also be used without superimposed frequency modulation.

According to an advantageous embodiment of the invention, the reception sequence or its conjugate complex values and the phase modulation sequence or its conjugate complex values are multiplied by a position shifted relative to one another, the respective shift corresponding to a runtime. The resulting product sequence is multiplied by a sequence of complex rotation factors with at least approximately linearly changing phase, the rotation factors for the respective runtime at least partially compensating the frequency contribution of the linear frequency modulation, with a frequency offset also being implemented if necessary. If necessary, a decimation of the sampling rate of the sequence is then carried out. This is followed by a discrete Fourier transformation, preferably implemented via a fast Fourier transformation. This procedure is carried out for all travel times of interest, whereby the sequence of the complex rotation factors changes over the respective time shift.

In the case of a frequency modulation that is not exactly linear, the linearity error can be taken into account by multiplying the reception sequence or its conjugate complex values and the phase modulation sequence or its conjugate complex values by a position shifted relative to one another during two-dimensional correlation filtering, with the respective shift corresponding to a running time. The resulting product sequence is multiplied by a sequence of complex rotation factors, with the rotation factors for the respective running time at least approximately correcting the non-constant frequency contribution of the non-linear frequency modulation. This is followed by a discrete Fourier transformation, preferably implemented via a fast Fourier transformation. This procedure is carried out for all running times of interest, with the sequence of complex rotation factors for correcting the linearity errors of the frequency modulation changing over the respective time shift.

Preferably, the multiplications with the complex rotation factors for correcting the linearity errors of the frequency modulation and/or for the runtime-dependent frequency shift are also implemented in the hard-wired digital circuit, whereby the two rotation factors can be combined if necessary. Advantageously, these multiplications are implemented via individual programmable structures with low accuracy in the range of a few bits and thus with little effort.

In a further advantageous embodiment of the invention, the hard-wired digital circuit has a front stage in which a Fourier transform of a signal sequence or its conjugate complex values and a Fourier transform of the modulation sequence or its conjugate complex values are multiplied by a position shifted relative to one another, optionally followed by a decimation of the sequence resulting from this multiplication and/or optionally followed by an expansion with zeros and followed by a fast inverse Fourier transform implemented in several stages, wherein each individual arithmetic operation is implemented in a dedicated circuit, the shift between the two Fourier transforms is changed from cycle to cycle in the first stage and the result of an inverse Fourier transform is produced at the output of the rear stage, wherein this result relates in each case to several cycles of the previously generated output data of the front stage.

According to an advantageous embodiment of the invention, when using a real-valued mixer, a determination of the reception frequency with the correct sign and thus the unambiguous determination of the relative speed of objects can be realized based on the fact that, particularly for more distant objects, only a relative speed hypothesis is possible or at least more plausible due to the known propagation time-dependent component of the frequency shift.

According to a further advantageous embodiment of the invention, when using a real-valued mixer, a determination of the reception frequency with the correct sign and thus the unambiguous determination of the relative speed of objects can be realized by using a real-valued mixer in that the steepness of the frequency change is varied and data of different steepness are recorded and evaluated for an object and it is used that the two frequency shift effects of relative speed and transit time are then in different relations to one another.

Conveniently, the sign, but not the amount of the steepness of the frequency change can be varied, so that the two frequency shift effects of relative velocity and propagation time add up with different signs and thus the sign of the reception frequency can be determined.

According to an advantageous embodiment of the invention, the frequency change is used to change the beam direction, in particular in a continuous manner at least in sections, in order to be able to record data for several pixels in different directions.

Advantageously, the beam direction change by frequency change can be realized by using a waveguide with several coupling points, preferably in an equidistant grid on a line, for transmission and reception.

Preferably, the beam direction change by frequency change is used for a first spatial direction and there is a further device for beam direction change for a second spatial direction, which is at least approximately perpendicular to the first, wherein preferably one of the two spatial directions is horizontal and the other vertical.

In an advantageous embodiment of the invention, a change in beam direction is realized by irradiating material or reflecting it from material in which at least one optical material property is changed by a control variable, in particular by applying voltage or passing current through it, wherein the control variable and thus the change in the optical material property can be locally different.

Preferably, for beam direction change for the second spatial direction, a body made of preferably monolithic material is irradiated, which has a constant, in particular triangular, cross-section in the orientation direction of the waveguide and can optionally be combined with a lens also having a constant cross-section in this dimension for beam bundling for the second spatial direction, wherein the electrical control variable is designed to change the optical property with respect to the orientation direction of the body with a constant body cross-section.

Furthermore, a flat transparent or reflective liquid crystal element with a one-dimensional lattice structure and control via a voltage can be used to change the beam direction for a second spatial direction.

Advantageously, a transparent or reflective liquid crystal array can be used for beam direction change for the second spatial direction, wherein this liquid crystal array has only a few elements in the orientation direction of the waveguide, preferably only a single, then rod-shaped element, preferably this liquid crystal array is also used for beam bundling alone or to support the beam bundling of a lens for the second spatial direction and preferably with the help of the liquid crystal array, manufacturing tolerances of the optically relevant components and their arrangement relative to one another are also compensated.

Expediently, there can be several transmit-receive paths operating in parallel, each with a waveguide for changing the beam direction in the first spatial direction by changing the frequency, preferably fed by a common laser source with subsequent phase modulation, whereby these several transmit-receive paths open up different beam directions in parallel in at least one of the two spatial directions.

According to an advantageous embodiment of the invention, different beam directions in the second spatial direction are generated both by several parallel transmitting/receiving paths with preferably similarly radiating waveguides lying parallel to one another and by changing the optical properties of a material being irradiated or reflected via an electrical control variable, and the beam directions through several parallel transmitting/receiving paths for each control of the material being irradiated or reflected address either a group of directly adjacent pixels or a group with pixels that are at least partially further apart, wherein the control of the material being irradiated or reflected is successively changed so that the entire beam direction range of interest is covered, optionally with a larger angular distance between the pixels towards the outside via a non-equidistant arrangement of the waveguides lying next to one another, in particular with spaced-apart groups of waveguides, wherein the distance within a group increases towards the outside, and with the optical beam width in the second spatial direction then preferably increasing towards the outside.

Preferably, waveguides arranged parallel and close to one another can be different in order to open up different beam direction ranges in the first spatial direction by means of the same frequency change, so that only a part of the beam direction range in the first spatial direction is covered by a single waveguide.

Advantageously, the beam direction change in the second spatial direction can be realized in a continuous manner with the aid of an electronic or mechanical approach, whereby a slow scanning via frequency change in the first spatial direction is made possible in particular by the fact that during a complete scan in the second spatial direction, the scanning via continuous frequency change in the first spatial direction only progresses by approximately one pixel at a time.

In a further advantageous embodiment of the invention, there are several transmit-receive paths operating in parallel and preferably fed by a common laser source, each with a switching matrix for sequential switching within a respective group of waveguides, wherein the preferably similar waveguides are each used for beam direction change in the first spatial direction by frequency change and are all arranged next to one another in such a way that they completely open up the second spatial direction, optionally with a larger angular distance of the pixels towards the outside via a non-equidistant arrangement of the adjacent waveguides and with then preferably increasing optical beam width towards the outside in the second spatial direction.

In a further lidar system according to the invention, there is a transmission-reception path with a switching matrix for sequential switching between several waveguides that are preferably parallel and close to one another, wherein the waveguides are different in order to open up different beam direction ranges in the first spatial direction by means of a frequency change of the same type, so that only a part of the beam direction range in the first spatial direction is covered by a single waveguide and the required frequency tuning range of the laser source is reduced, and the beam direction change in the second spatial direction is realized in a continuous manner with the aid of an electronic or mechanical approach, whereby slow scanning via frequency change in the first spatial direction is made possible in particular by the fact that during a complete scan in the second spatial direction, the scanning via continuous frequency change in the first spatial direction only progresses by about one pixel.

Advantageously, the beam direction change by frequency change can be of different speeds, in particular characterized in that it is slower in a central region than in outer regions.

Conveniently, the frequency change for changing the beam direction can be generated by one or more frequency-tunable laser sources, whereby when using several laser sources each one only covers a part of the frequency range required for changing the beam direction.

The inventive approaches with a transmit-receive unit with one or more waveguides for scanning via frequency in a first spatial direction and a scanner for a second spatial direction can also be used in combination with other forms of modulation, in particular when the frequency is changed stepwise, or even for non-coherent lidar systems.

In a further advantageous embodiment of the invention, when using a real-valued mixer, a determination of the reception frequency with the correct sign and thus the unambiguous determination of the relative speed of objects is realized in that the frequency change for the beam direction change in the first spatial direction has a different, in particular alternating sign across different beam directions of the second spatial direction and this different sign in different, in particular adjacent beam direction planes in which the same object is detected leads to the two frequency shift effects of relative speed and transit time adding up with different signs and thus the sign of the reception frequency can be determined.

Advantageously, in a lidar system in which deviations in the beam directions can occur, in particular due to misalignment, due to frequencies that are not precisely known and/or due to the dependence of the optical property on the control variable that is not precisely known, this deviation can be determined from the measured radial relative velocities of stationary objects in order to later take these deviations into account and/or correct them.

For stationary objects, the road surface can be used, preferably by determining its angle in the vertical direction from the measured distance and the sensor installation height.

The inventive approach for determining deviations in the beam directions can also be used with other forms of modulation, since it is essentially based only on the inherent Doppler measurement capability of coherent lidar systems.

Preferably, the devices for changing the beam direction are used to compensate for a misalignment and/or to adapt the detection range adaptively, in particular depending on the traffic situation.

For a further embodiment of the lidar system according to the invention, a phase-modulated signal is emitted, the phase of this signal being generated by switching between discrete values, i.e. from a sequence of phase values, and the times of this phase switching forming a subset of an equidistant time grid. The phase modulation sequence contains two sequentially arranged or two periodically nested sequences, the first sequence having constant or periodic, in particular alternating phase values, while the second sequence essentially changes irregularly between phase values. According to the invention, the frequency to which the phase is modulated changes continuously, at least in sections, preferably with an at least approximately linear progression.

Preferably, a binary phase modulation is used, i.e. consisting of only two phase values that differ by approximately 180°, and the second sequence either alternates pseudorandomly between these two phase values or consists of a code whose autocorrelates have low side lobes.

The signals reflected back from objects, which are delayed compared to the transmitted signal by the distance-dependent propagation time and are shifted in frequency both by the relative speed-dependent Doppler effect and by the distance-dependent propagation time caused by the linear frequency change, can be received and converted into a low-frequency signal by mixing and digitized into a reception sequence. From this reception sequence, the variable dimensions of time shift and frequency shift of signals reflected from objects are determined in digital signal processing means, whereby the first phase modulation sequence with constant or periodic phase values is primarily used to determine the frequency shift and the second phase modulation sequence with irregular changes between phase values is used to determine the time shift. According to the invention, the object distance is calculated from the determined time shift and the radial relative speed of the object is calculated from the determined frequency shift minus the contribution caused by the time shift.

Preferably, the two phase modulation sequences are nested alternately, i.e. with period two related to the modulation rate.

Advantageously, a phase modulation sequence consisting of two periodically nested sequences is repeated periodically, whereby the cyclical property of the modulation and reception sequence is exploited, particularly in the case of continuous scanning of the laser beam, and detection is realized in different directions.

Preferably, the laser beam is scanned continuously and overlapping sections of a long, possibly periodic phase modulation sequence, which includes two periodically nested sequences, are used for the successive detection directions.

In a further advantageous embodiment of the invention, the following steps are carried out, whereby the nesting of the two sequences with respect to the sampling time of the reception sequence should have the period N ₁ : To determine the frequency shift of objects, N ₁ discrete Fourier transformations are calculated, preferably with the aid of fast Fourier transformations, over values of the reception sequence from the periodic grid of the first sequence, possibly interrupted by zeros or extended with zeros, as well as from N ₁ -1 shifts of this grid, whereby the shift increases by one grid value in each case. In each of these N ₁ discrete Fourier transforms, the respective frequencies of magnitude peaks that lie above a detection threshold are determined. At the respective frequencies and the respective grid shifts, the reception sequence in the periodic and correspondingly shifted grid of the second sequence is rotated back in frequency. A correlation is determined between the sequence generated in this way and the second modulation sequence. And from values of this correlation, in particular from magnitude peaks above a detection threshold, as well as the respective grid shift, the respective time shift and thus object distance are determined, as well as from the respective frequency minus its contribution caused by the time shift, the radial relative velocity of the respective object, whereby any ambiguities in the frequency are resolved with the help of the phase relationship between values of the discrete Fourier transform and the correlation.

Advantageously, a first detection threshold can be used to determine the magnitude peaks of the discrete Fourier transform; at the magnitude peaks, the complex value of the discrete Fourier transform and the respective complex value of the associated correlation at their magnitude peaks are added together, if necessary several times while compensating for possible phase shifts, and these sums are checked for a second detection threshold, whereby the first detection threshold is less high above the noise.

In a further advantageous embodiment of the invention, the position of the road surface is determined, in particular at greater distances, in that the power of a correlation or the complex-valued product between a first correlation and the conjugate complex of a second correlation is summed up for better differentiation from system noise at the same vertical angle across several pixels at different, in particular adjacent, horizontal angles and/or within a pixel over different, in particular adjacent distances, wherein in the case of the product of two correlations, their received signals originate at least partially from the same reflection points on the road, in particular by alternating assignment of the values of the received sequence.

This inventive approach for determining the position of the road surface at greater distances can also be used with other forms of modulation, for example pure linear frequency modulation (usually consisting of two frequency ramps whose slopes have opposite signs). The correlation values used are then based on a correspondingly different correlation calculation, with pure linear frequency modulation in the form of one FFT per frequency ramp.

When using a real-valued mixer, a determination of the reception frequency with the correct sign and thus the unambiguous determination of the relative speed of objects can be conveniently achieved by having at least two of the discrete phase values used for the phase modulation in irregular sequence that are neither in phase nor out of phase, calculating the correlation between the frequency-shifted reception sequence and the complex-valued modulation sequence or its conjugate complex for different reception frequencies and using the fact that this correlation has different levels for positive and negative reception frequencies to identify the sign of the reception frequency.

Advantageously, a set of at least three phase values is used which are nominally evenly distributed over one revolution, but may also deviate significantly from these nominal positions.

Preferably, the phase values are realized by switching between lines of slightly different lengths or by a combination of switching between lines of slightly different lengths and a switchable inverter.

The inventive approach for determining the sign of the reception frequency in a real-valued mixer using a phase modulation which includes at least two phase values which are neither in phase nor in antiphase can also be used without superimposed frequency modulation.

In a further advantageous embodiment of the invention, monitoring of the devices for changing the beam direction, in particular for ensuring eye safety, is realized by checking received signals for changes in the respective spatial direction with regard to object reflections or with regard to the proportions of internal reflections and couplings as well as reflections of a cover.

The lidar system according to the invention comprises in particular the following three main components, the electronic parts of which are preferably located on the same circuit board: firstly, a photonic chip, in particular containing a frequency-tunable laser source, a phase modulation unit consisting of a switchable inverter, parallel transmission-reception paths and waveguides, secondly, a digital chip, in particular containing analog-digital converters, a hard-wired computing logic for determining a two-dimensional correlation with downstream evaluation, one or more microcontrollers and/or DSPs and means for generating control signals and thirdly, a scanner for a spatial direction realized by material with electrically controllable optical properties, in particular a liquid crystal element or a liquid crystal array, or by a switching matrix for each of the parallel transmission-reception paths or by a mechanical approach.

Short description of the drawings

• 1 shows the coherent lidar system with binary phase modulation.
• 2 shows the course of the pseudorandom binary phase modulation.
• In 3 the real part of the low-frequency analog received signal for an object is shown.
• 4 shows the two-dimensional correlation of the received signal for two objects.
• In 5 the linear change of the transmission frequency during phase modulation is shown.
• 6 shows the received frequency shifted in time to the transmitted frequency in the case of no Doppler shift (i.e. at zero relative velocity).
• 7 shows the two-dimensional correlation of the received signal for two objects for a linear frequency change superimposed on the phase modulation.
• 8 shows the effort-optimized hard-wired digital circuit for the realization of the two-dimensional correlation filtering; 8a an overview and in 8b-8d the three blocks are shown in detail.
• In 9 The effort-optimized realization of a complex-valued multiplier for an exemplary rotation factor is shown.
• 10 shows the effort-optimized hard-wired digital circuit for the realization of the two-dimensional correlation filtering when applying a frequency shift and decimation before the FFT; 10a an overview and in 10b-10d the three blocks are shown in detail.
• In 11 A realization of the rotation factors for frequency shifting is shown.
• 12 shows a curve of the transmission frequency with squared error to the ideal linear change.
• 13 shows the reception frequency range over the object distance and shades the area where ambiguity occurs in a real-valued mixer due to an unknown sign.
• 14 shows the reception frequency range over the object distance with inverted modulation bandwidth.
• 15 shows the reception frequency range over the object distance with inverted and doubled modulation bandwidth.
• In 16 A waveguide with equidistant coupling points is shown, which is used for focusing and scanning via frequency change in the first spatial direction.
• 17 shows a temporal linear progression of the beam angle and the corresponding transmission frequency over which the beam angle is swung.
• 18 shows a temporal progression of the beam angle with different scanning speeds and the corresponding transmission frequency.
• 19 shows in two different perspectives an arrangement with a waveguide and a lens for focusing in a second spatial direction.
• 20 shows the arrangement according to 19 with an additional prism-shaped element made of material with electrically controllable dielectric constant for scanning in a second spatial direction.
• 21 shows in three different perspectives an arrangement with a waveguide and a transparent one-dimensional liquid crystal array for focusing and scanning in a second spatial direction.
• 22 shows in three different perspectives an arrangement with a waveguide and a reflective one-dimensional liquid crystal array for focusing and scanning in a second spatial direction.
• In 23 A two-dimensional liquid crystal array is shown.
• 24 shows an arrangement with 32 parallel and equidistant waveguides.
• 25 shows a two-dimensional pixel array for 32 similar equidistant waveguides with a small spacing.
• 26 shows a two-dimensional pixel array for 32 similar equidistant waveguides with a 32-fold larger spacing.
• 27 shows a two-dimensional pixel array for 32 identical waveguides with different spacing.
• 28 shows a two-dimensional pixel array for 32 different equidistant waveguides with a small spacing and further stepwise scanning in a second spatial direction.
• 29 shows a two-dimensional pixel array for 32 different equidistant waveguides with a small spacing and continuous scanning in a second spatial direction.
• 30 shows the derivation for the radial component of the relative velocity of a stationary object for elevation angle 0°.
• 31 shows the derivation for the radial component of the relative velocity of a stationary object for any angular position as well as the derivation of the angle in the vertical direction under which the road surface is seen.
• In 32 shows an area of the road surface on which a beam hits.
• 33 shows an implementation for three equidistant phase values with a switch between three different length line sections.
• 34 shows an implementation for four equidistant phase values with a switchable inverter and a switch between two different length cable sections.
• 35 shows a state-of-the-art modulation sequence, which is sequentially composed of two subsequences.
• In 36 A new modulation sequence with two periodically nested subsequences is shown.
• 37 shows the two FFTs for the first modulation subsequence for the example of two objects.
• 38 shows the two correlations to the second modulation subsequence for the example of two objects.
• In 39 An alternative new modulation sequence with two periodically nested subsequences is shown.
• 40 shows according to the invention a new modulation sequence with two periodically nested subsequences and superimposed linear frequency modulation.

Examples of implementation

wobei c = 3·10⁸m/s die Lichtgeschwindigkeit ist, und mit einer von der radialen Relativgeschwindigkeit v abhängigen und damit variablen, durch den Dopplereffekt erzeugten Frequenzverschiebung $${\text{f}}_{\text{D}}=2\text{v}/\mathrm{\lambda }$$

wird dieses Signal dann von der Sende-Empfangs-Einheit 1.6 aufgenommen und über den Zirkulator 1.5 in den weiteren Empfangspfad geleitet. In einem komplexwertigen Mischer 1.8 (auch als IQ-Mischer bezeichnet) wird das modulierte empfangene Signal mit dem unmodulierten Lasersignal überlagert und mit Hilfe der Photodiodeneinheit 1.9 in ein komplexwertiges niederfrequentes Signal umgesetzt; die Frequenz f_e dieses Signals entspricht der Dopplerverschiebung f_D nach Bez. (1b): $${\text{f}}_{\text{e}}={\text{f}}_{\text{D}}=2\text{v}/\mathrm{\lambda },$$

die Modulation dieses Signals ist um die Signallaufzeit gegenüber der des Sendesignals verzögert. In 3 ist der Realteil dieses niederfrequenten analogen Empfangssignals e_a(t) im Falle eines Objektes dargestellt. Anschließend wird das Signal in einer Analog-Digital-Wandler-Einheit 1.10 mit der Abtastfrequenz f_s = 300MHz, d. h. alle T_s = 3.33ns abgetastet und digitalisiert - die resultierenden Werte des Realteils sind in 3 als Punkte gekennzeichnet; weil hier Abtastzeit T_s = 3.33ns und Modulationszeit T_m = 3.33ns gleich sind, ist auch die Zahl N_s der Abtastwerte pro Modulationsperiode gleich deren Rasterlänge N_m, also N_s = N_m = 4096, so dass nachfolgend für beide ein N = 4096 ohne Index benutzt wird. Das komplexwertige Abtastsignal e(n), im Folgenden auch als Empfangsfolge bezeichnet, kann wie folgt beschrieben werden: $$\text{e}\left(\text{n}\right)=\text{a}\cdot \text{b}\left(\text{n}-{\text{m}}_0\right)\cdot \text{exp}\left(2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\text{k}}_0\right),$$

wobei hier angenommen wird, dass die Laufzeit t₀ ein ganzzahliges Vielfaches m₀ der Modulationszeit T_m = 3.33ns und damit auch der Abtastzeit T_s = 3.33ns ist: $${\text{m}}_0={\text{t}}_0/{\text{T}}_{\text{m}}=2\text{r}/\text{c}/{\text{T}}_{\text{m}}$$

und das zur Dopplerverschiebung korrespondierende $${\text{k}}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot {\text{f}}_{\text{D}}=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\text{v}/\mathrm{\lambda }$$

ebenfalls ganzzahlig ist; a ist die komplexwertige Amplitude der Empfangsfolge, „exp“ bezeichnet die Exponentialfunktion und ĵ ist die imaginäre Einheit. 1 shows a schematic of a coherent lidar system 1.1. The laser source 1.2 generates a coherent signal in the wavelength range of approximately λ = 1550nm; the coherence length is at least several microseconds and the frequency is constant according to the state of the art. The signal then passes into a switchable inverter 1.3, with which the sign of the signal can be changed, which corresponds to a phase shift of 180°. Sign changes only take place in a fixed grid of e.g. 3.33ns and are, according to the state of the art, pseudorandom, ie the sign is only changed after T _m = 3.33ns with a frequency or probability of 50%. In 2 a course of this modulation sequence b(n), which consists of the values +1 and -1 and is also referred to as binary, is shown; it repeats itself with the period N _m = 4096, i.e. every 13.6µs. The modulated signal passes through an amplifier 1.4, a circulator 1.5 (i.e. a transmit-receive switch), is emitted via a transmit-receive unit 1.6 and partially reflected back by an object 1.7; with a delay that depends on the object distance r and is therefore variable $${\text{t}}_0=2\text{r}/\text{c},$$

where c = 3·10 ⁸ m/s is the speed of light, and with a frequency shift dependent on the radial relative velocity v and thus variable, generated by the Doppler effect $${\text{e}}_{\text{D}}=2\text{v}/\mathrm{\lambda }$$

This signal is then received by the transmit-receive unit 1.6 and passed through the circulator 1.5 into the further receiving path. In a complex-valued mixer 1.8 (also called IQ mixer net) the modulated received signal is superimposed with the unmodulated laser signal and converted into a complex-valued low-frequency signal using the photodiode unit 1.9; the frequency f _e of this signal corresponds to the Doppler shift f _D according to equation (1b): $${\text{e}}_{\text{e}}={\text{e}}_{\text{D}}=2\text{v}/\mathrm{\lambda },$$

the modulation of this signal is delayed by the signal propagation time compared to that of the transmission signal. In 3 the real part of this low frequency analogue received signal e _a (t) is shown in the case of an object. The signal is then sampled and digitized in an analogue-digital converter unit 1.10 with the sampling frequency f _s = 300MHz, ie every T _s = 3.33ns - the resulting values of the real part are in 3 marked as points; because the sampling time T _s = 3.33ns and the modulation time T _m = 3.33ns are the same, the number N _s of samples per modulation period is also equal to their grid length N _m , i.e. N _s = N _m = 4096, so that N = 4096 without an index is used for both below. The complex-valued sampling signal e(n), also referred to as the reception sequence below, can be described as follows: $$\text{e}\left(\text{n}\right)=\text{a}\cdot \text{b}\left(\text{n}-{\text{m}}_0\right)\cdot \text{ex}\left(2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\text{k}}_0\right),$$

where it is assumed that the running time t ₀ is an integer multiple m ₀ of the modulation time T _m = 3.33ns and thus also of the sampling time T _s = 3.33ns: $${\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_0={\text{<figure-callout id="0" label="t" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">t</figure-callout>}}_0/{\text{T}}_{\text{m}}=2\text{r}/\text{c}/{\text{T}}_{\text{m}}$$

and the Doppler shift corresponding $${\text{k}}_{}$$$${}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot {\text{e}}_{\text{D}}=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\text{v}/\mathrm{\lambda }$$

is also an integer; a is the complex-valued amplitude of the received sequence, “exp” denotes the exponential function and ĵ is the imaginary unit.

wobei „sum_i=1,...,l“ die Summenfunktion über den Index i = 1,...,l der I nicht longitudinal ausgedehnten Einzelobjekte darstellt.The complex-valued reception sequence e(n) according to (3) refers to a single object without longitudinal extension and to an ideal receiver. In fact, there may be several and/or extended objects and an additional noise r _e (n) is generated in the receiver, in particular by thermal noise; then the reception sequence results $$\text{e}\left(\text{n}\right)={\text{sum}}_{\text{i}=\mathrm{1,}\dots ,\text{I}}\left[{\text{a}}_{\text{i}}\cdot \text{b}\left(\text{n}-{\text{m}}_{\mathrm{0,}\text{i}}\right)\cdot \text{ex}\left(2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\text{k}}_{\mathrm{0,}\text{i}}\right)\right]+{\text{r}}_{\text{e}}\left(\text{n}\right),$$

where “sum _i=1,...,l ” represents the sum function over the index i = 1,...,l of the I non-longitudinally extended individual objects.

dabei korrespondiert M-1 zu der größten anzunehmenden bzw. interessierenden Objektentfernung und für die Dopplerverschiebung k wird angenommen, dass sie alle Werte annehmen kann. Damit ergibt sich die zweidimensionale Korrelation E_m,k zu $$\begin{array}{c}{\text{E}}_{\text{m},\text{k}}={\text{sum}}_{\text{n}=\mathrm{0,}\dots ,\text{N}-1}\left(\text{e}\left(\text{n}\right)\cdot \text{conj}\left({\widehat{\text{e}}}_{\text{m},\text{k}}(\text{n}\right)\right)\\ ={\text{sum}}_{\text{n}=\mathrm{0,}\dots ,\text{N}-1}\left(\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\cdot \text{exp}\left(-\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{k}\right)\right),\text{ m}=\mathrm{0,}\dots ,\text{M}-1\text{ und k}=\mathrm{0,}\dots ,\text{N}-\mathrm{1,}\end{array}$$

wobei „conj“ die Konjugiert-Komplex-Bildung bezeichnet und die Modulationsfolge b(n) wegen ihrer Reellwertigkeit sich dabei nicht ändert. Die Korrelation E_m,k besitzt Betragsspitzen (oft auch als Leistungsspitzen bezeichnet) an den Stellen (m,k) = (m_0,i,k₀,_i) von Objekten; 4 zeigt den Betrag der zweidimensionalen Korrelation für zwei Objekte gleicher Empfangsamplitude bei (m_0,1,k_0,1) = (300,3846) und (m_0,2,k_0,2) = (101,1000), was zur ihren Entfernungen r₁ = 150m und r₂ = 50.5m und den radialen Relativgeschwindigkeiten v₁ = -7.1m/s und v₂ = 56.8m/s korrespondiert (beim oben für k unsymmetrisch gewählten Bereich k = 0,...,N-1 werden k > N/2 negative Dopplerfrequenzen und damit negative Relativgeschwindigkeiten zugeordnet; eine negative Rekativgeschwindigkeit bedeutet, dass sich das Objekt relativ wegbewegt).The discrete travel times m _0,i and the discrete Doppler shifts k _0,i of the I objects are to be determined from the reception sequence e(n) of the time period n = 0,1,...,N-1. For the most accurate determination, i.e. the best possible separation of signal and noise and thus maximum sensitivity and range of the lidar system, the so-called optimal filtering is to be used, i.e. filtering by correlation between the reception sequence e(n) and the two-dimensional space ê _m,k (n) of the possible ideal amplitude-normalized reception sequences of a single object: $${\widehat{\text{e}}}_{\text{m},\text{k}}\left(\text{n}\right)=\text{b}\left(\text{n}-\text{m}\right)\cdot \text{ex}\left(\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{k}\right)\text{with m}=\mathrm{0,}\dots ,\text{M}-1\text{and k}=\mathrm{0,}\dots ,\text{N}-1;$$

where M-1 corresponds to the largest assumed or interesting object distance and the Doppler shift k is assumed to be able to take on all values. This gives the two-dimensional correlation E _m,k as $$\begin{array}{c}{\text{E}}_{\text{m},\text{k}}={\text{sum}}_{\text{n}=\mathrm{0,}\dots ,\text{N}-1}\left(\text{e}\left(\text{n}\right)\cdot \text{conj}\left({\widehat{\text{e}}}_{\text{m},\text{k}}(\text{n}\right)\right)\\ ={\text{sum}}_{\text{n}=\mathrm{0,}\dots ,\text{N}-1}\left(\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\cdot \text{ex}\left(-\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{k}\right)\right),\text{m}=\mathrm{0,}\dots ,\text{M}-1\text{and k}=\mathrm{0,}\dots ,\text{N}-\mathrm{1,}\end{array}$$

where “conj” denotes the conjugated complex formation and the modulation sequence b(n) does not change due to its real value. The correlation E _m,k has magnitude peaks (often also referred to as power peaks denoted) at the positions (m,k) = (m _0,i ,k ₀ , _i ) of objects; 4 shows the magnitude of the two-dimensional correlation for two objects of the same reception amplitude at (m _0.1 ,k _0.1 ) = (300.3846) and (m _0.2 ,k _0.2 ) = (101.1000), which corresponds to their distances r ₁ = 150m and r ₂ = 50.5m and the radial relative velocities v ₁ = -7.1m/s and v ₂ = 56.8m/s (for the range k = 0,...,N-1 chosen asymmetrically above for k, k > N/2 are assigned negative Doppler frequencies and thus negative relative velocities; a negative reactive velocity means that the object is moving away relatively).

$${\text{v}}_{\text{i}}={\text{k}}_{\text{0}\text{,i}}\cdot \mathrm{\lambda }/\left(2\text{N}\cdot {\text{T}}_{\text{s}}\right).$$

In order to determine the distance r _i and the radial relative speed v _i of objects, the magnitude peaks of the two-dimensional correlation E _m,k must be determined, using only magnitude peaks that are above a detection threshold in order to distinguish them from system noise. According to equations (3b) and (3c), r _i and v _i can be calculated from the positions of the magnitude peaks, i.e. the discrete travel times m _0,i and the discrete frequency shifts k _0,i as follows: $${\text{r}}_{\text{i}}={\text{m}}_{\mathrm{0,}\text{i}}\cdot \text{c}\cdot {\text{T}}_{\text{m}}/\mathrm{2,}$$

$${\text{v}}_{\text{i}}={\text{<figure-callout id="0" label="k" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">k</figure-callout>}}_{\text{0}\text{,i}}\cdot \mathrm{\lambda }/\left(2\text{N}\cdot {\text{T}}_{\text{s}}\right).$$

The distance and relative speed of several objects can therefore be determined directly and unambiguously from a modulation sequence. This is a great advantage over the linear frequency modulation frequently used in coherent lidar systems with two frequency ramps whose slopes have opposite signs - in this case, ambiguities are unavoidable when there are several objects.

wobei k = 0,...,N-1 die Ausgangsdimension der FFT, also die diskrete Frequenz ist, so reduziert sich der Rechenaufwand auf die Ordnung M·N·log₂(N).The calculation of this two-dimensional correlation and its subsequent evaluation take place in the digital signal processing unit 1.11. It represents a high effort with order N·M·N. The above relationship (6) can also be viewed as a discrete Fourier transform via the product e(n)·b(nm), n = 0,...,N-1, which is to be determined for each m = 0,...,M-1; the discrete Fourier transform (DFT) is calculated via the fast Fourier transform (FFT): $${\text{E}}_{\text{m},\text{k}}={\text{FFT}}_{\text{k}}\left(\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\right)\text{with m}=\mathrm{0,}\dots ,\text{M}-\mathrm{1,}$$

where k = 0,...,N-1 is the output dimension of the FFT, i.e. the discrete frequency, the computational effort is reduced to the order M·N·log ₂ (N).

Phase modulation with superimposed linear frequency modulation

Up to now, according to the state of the art, it was assumed that the frequency to which the phase modulation sequence b(n) is impressed is constant, i.e. does not change. In the following, the inventive approach is considered that the frequency changes continuously, whereby an ideal linear change is first assumed; this is in 5 where the frequency of the laser source 1.2 and thus the transmission frequency f _TX (t) increases linearly by B = 800MHz (i.e. with a gradient of B/T _pm = 58.6MHz/µs) within a modulation period of duration T _pm = N·T _m = 13.7µs. In general, the following applies to the transmission frequency f _TX (t): $${\text{e}}_{\text{TX}}\left(\text{t}\right)={\text{e}}_{\text{TX0}}+\text{t}\cdot \text{B}/{\text{T}}_{\text{pm}}{\text{with T}}_{\text{pm}}=\text{N}\cdot {\text{T}}_{\text{m}}.$$

und damit gegenüber der Sendefrequenz f_TX(t) nach unten verschoben. Diese durch die Laufzeit bedingte Frequenzverschiebung f_r beträgt $${\text{f}}_{\text{r}}=-{\text{t}}_0\cdot \text{B}/{\text{T}}_{\text{pm}}$$

und mit Laufzeit t₀ nach Bez. (1a): $${\text{f}}_{\text{r}}=-2\text{r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}};$$

für das Beispiel einer Objektentfernung r = 150m und damit einer Laufzeit t₀ = 1µs sowie den obigen Modulationswerten ergibt sich die laufzeitgedingte Frequenzverschiebung zu f_r = 73.2MHz. Gemäß Bez. (11b) ist die laufzeitgedingte Frequenzverschiebung zu f_r proportional zur Objektentfernung r.Without Doppler shift (i.e. with initially assumed relative velocity zero) and as in 6 As shown, the frequency f _RX (t) of the received signal is delayed due to the propagation time t ₀ : $${\text{e}}_{\text{RX}}\left(\text{t}\right)={\text{e}}_{\text{TX}}\left(\text{t}-{\text{t}}_0\right)={\text{e}}_{\text{TX0}}+\left(\text{t}-{\text{t}}_0\right)\cdot \text{B}/{\text{T}}_{\text{pm}},$$

and thus shifted downwards compared to the transmission frequency f _TX (t). This frequency shift f _r caused by the runtime is $${\text{e}}_{\text{r}}=-{\text{<figure-callout id="0" label="t" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">t</figure-callout>}}_0\cdot \text{B}/{\text{T}}_{\text{pm}}$$

and with running time t ₀ according to (1a): $${\text{e}}_{\text{r}}=-2\text{r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}};$$

For the example of an object distance r = 150m and thus a propagation time t ₀ = 1µs and the above modulation values, the propagation time-related frequency shift is f _r = 73.2MHz. According to equation (11b), the propagation time-related frequency shift to f _r is proportional to the object distance r.

The total frequency shift and thus the frequency f _e of the received signal after mixing is now composed of the Doppler shift f _D according to (1b) and the above delay-related component f _r according to (11b): $${\text{e}}_{\text{e}}={\text{e}}_{\text{D}}+{\text{e}}_{\text{r}}=2\text{v}/\mathrm{\lambda }-2\text{r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}}.$$

und unter Verwendung der diskreten Laufzeit m₀ nach (3b) sowie T_pm = N·T_m: $${\text{k}}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{f}}_{\text{D}}-{\text{f}}_{\text{r}}\right)=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \text{v}/\mathrm{\lambda }-{\text{m}}_0\cdot {\text{T}}_{\text{s}}\cdot \text{B},$$

wobei diese diskrete Empfangsfrequenz k₀ zuerst auch als ganzzahlig angenommen wird. Der Ansatz der Optimalfilterung, also die Filterung durch Korrelation zwischen der Empfangsfolge e(n) und dem zweidimensionalen Raum ê_m,k(n) der möglichen idealen amplitudennormierten Empfangsfolgen eines Einzelobjekts bleibt somit weiterhin gültig, gekennzeichnet durch die Bez. (6) und (8) für die zweidimensionale Korrelation E_m,k; dabei eignet sich Bez. (8): $$\begin{array}{cc}{\text{E}}_{\text{m},\text{k}}={\text{FFT}}_{\text{k}}\left(\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\right)& \text{mit m}=\mathrm{0,}\dots ,\text{M}-\mathrm{1,}\end{array}$$

also die schnellen Fouriertransformationen über jeweiliges Produkt zwischen Empfangsfolge e(n) und jeweilig verschobener Modulationsfolge b(n-m), für eine aufwandsgünstige Berechnung. 7 zeigt den Betrag der zweidimensionalen Korrelation E_m,k für das obige Beispiel zweier Objekte mit den Entfernungen r₁ = 150m und r₂ = 50.5m sowie den radialen Relativgeschwindigkeitenv₁ = -7.1m/s und v₂ = 56.8m/s; die diskreten Zeitverschiebungen m_0,1 = 300 und m_0,2 = 101 bleiben unverändert, während sich die diskreten Frequenzverschiebungen gemäß Bez. (13b) um die laufzeitbedingten Anteile -m_0,1·T_s·B = -800 und -m_0,2·T_s·B = -264 auf k_0,1 = 3846-800 = 3046 und k_0,2 = 1000-264 = 736 ändern.This is a particular difference to the case of a constant transmission frequency according to the state of the art considered initially. The phase modulation sequence shifted by the propagation time is unchanged on this reception frequency, so that for the reception sequence e(n) after sampling and digitization, the relation (3a) still applies, whereby for the discrete reception frequency k ₀ instead of relation (3c) now applies: $${\text{k}}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{e}}_{\text{D}}+{\text{e}}_{\text{r}}\right)=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \text{v}/\mathrm{\lambda }-\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \text{r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}},$$

and using the discrete running time m ₀ according to (3b) and T _pm = N·T _m : $${\text{k}}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{e}}_{\text{D}}-{\text{e}}_{\text{r}}\right)=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \text{v}/\mathrm{\lambda }-{\text{m}}_0\cdot {\text{T}}_{\text{s}}\cdot \text{B},$$

where this discrete reception frequency k ₀ is also initially assumed to be an integer. The approach of optimal filtering, i.e. filtering by correlation between the reception sequence e(n) and the two-dimensional space ê _m,k (n) of the possible ideal amplitude-normalized reception sequences of a single object, thus remains valid, characterized by the terms (6) and (8) for the two-dimensional correlation E _m,k ; here, term (8) is suitable: $$\begin{array}{cc}{\text{E}}_{\text{m},\text{k}}={\text{FFT}}_{\text{k}}\left(\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\right)& \text{with m}=\mathrm{0,}\dots ,\text{M}-\mathrm{1,}\end{array}$$

i.e. the fast Fourier transforms via the respective product between the reception sequence e(n) and the respective shifted modulation sequence b(nm), for a cost-effective calculation. 7 shows the magnitude of the two-dimensional correlation E _m,k for the above example of two objects with the distances r ₁ = 150m and r ₂ = 50.5m and the radial relative velocities _{v 1} = -7.1m/s and v ₂ = 56.8m/s; the discrete time shifts m _0.1 = 300 and m _0.2 = 101 remain unchanged, while the discrete frequency shifts according to (13b) change by the transit time-related components -m _0.1 ·T _s ·B = -800 and -m _0.2 ·T _s ·B = -264 to k _0.1 = 3846-800 = 3046 and k _0.2 = 1000-264 = 736.

während für die Bestimmung der radialen Relativgeschwindigkeiten v_i der laufzeitbedingte Anteil an den diskreten Frequenzverschiebungen k_0,i zu berücksichtigen ist - mit Hilfe der Bez. (13b) ergibt sich: $${\text{v}}_{\text{i}}=\mathrm{\lambda }/\left(2\text{N}\cdot {\text{T}}_{\text{s}}\right)\cdot \left({\text{k}}_{\mathrm{0,}\text{i}}+{\text{m}}_{\mathrm{0,}\text{i}}\cdot {\text{T}}_{\text{s}}\cdot \text{B}\right);$$

im Unterschied zur ursprünglichen Bez. (7b) ist also von der diskreten Frequenzverschiebung k_0,i der laufzeitbedingte Beitrag -m_0,i·T_s·B zu subtrahieren, welcher proportional zur diskreten Laufzeit m_0,i ist.For the determination of the object distances r _i from the positions (m _0,i, k _0,i ) of the magnitude peaks of the correlation, the following equation (7a) still applies: $${\text{r}}_{\text{i}}={\text{m}}_{\mathrm{0,}\text{i}}\cdot \text{c}\cdot {\text{T}}_{\text{m}}/\mathrm{2,}$$

while for the determination of the radial relative velocities v _i the transit time-related part of the discrete frequency shifts k _0,i has to be taken into account - with the help of the equation (13b) it results: $${\text{v}}_{\text{i}}=\mathrm{\lambda }/\left(2\text{N}\cdot {\text{T}}_{\text{s}}\right)\cdot \left({\text{k}}_{\mathrm{0,}\text{i}}+{\text{m}}_{\mathrm{0,}\text{i}}\cdot {\text{T}}_{\text{s}}\cdot \text{B}\right);$$

In contrast to the original equation (7b), the delay-related contribution -m _0,i ·T _s ·B, which is proportional to the discrete delay m 0,i, has to be subtracted from the discrete frequency shift k _{0 ,i} .

The system approaches and the advantages that enable the combination of phase modulation and a changing frequency are explained in more detail later.

Calculation logic for realizing the two-dimensional correlation

First, it is discussed how the calculation of the two-dimensional correlation E _m,k can be realized in the digital signal processing unit 1.11. The reception sequence e(n) of the time period n = 0,1,...,N-1 considered so far and the associated correlation E _m,k refer to a single detection direction, i.e. related to the horizontal and vertical direction of a pixel. In fact, in each Acquisition cycle, for which a duration of 100ms is initially assumed, covers about 160,000 acquisition directions, i.e. pixels; this is typically realized by a combination of parallel transmitter and receiver, i.e. parallel acquisition of pixels, and scanning, i.e. sequential acquisition of pixels. Parallel transmitter and receiver means that all elements 1.4-1.10 in 1 (if necessary apart from a common focusing and beam steering device in the transmitting-receiving unit 1.6) multiple times, e.g. 32 times. Scanning can be done e.g. by continuous mechanical movement (e.g. of a mirror) or electronically, by continuous change or by sequential switching (possibilities of electronic scanning are explained later). With continuous scanning it is also possible that pixels partially overlap, i.e. that the rear of the N values of the reception sequence e(n) of a pixel are also used as the front values of the next pixel. If it is now assumed that M = 500 (corresponds to a maximum distance of 249.5m in the above design), then the FFT of length 4096 from equation (8) is to be calculated 800 million times per second. It is not possible to implement this many FFT calculations using microprocessors or DSPs (digital signal processors); the clock speed of such processors is typically in the range of 1 GHz, i.e. almost a complete FFT of length 4096 would have to be calculated per clock cycle, but modern processors can usually only perform up to 100 multiplications and additions per clock cycle, even when using parallel vector processing units, which is orders of magnitude less than the requirement for an FFT of length 4096. Therefore, these many FFTs can only be implemented using special computing logic implemented in hardware. Since the FFT algorithm consists of many sub-elements known as butterflies, several butterflies containing a programmable multiplier are often implemented for a special computing logic of the FTT implemented in hardware, since the rotation factor to be multiplied changes over the sequence of butterflies. However, implementing programmable multipliers is complex.

Since the number of 800 million FFTs to be calculated per second corresponds approximately to the realizable clock speed of 1GHz of such a calculation logic, one can dispense with programmable multipliers by implementing each butterfly of the FFT and thus each adder and multiplier contained therein (for the corresponding rotation factor) directly in a calculation logic. This is in 8 shown in block 8.4; an FFT of length 4096 consists of log ₂ (4096) = 12 sequential stages, and in each of these stages there are 2048 butterflies, each of which determines two output values from two complex-valued input values via a complex-valued addition and subtraction as well as a complex-valued multiplication (in 8 In the first FFT stage, a butterfly is highlighted by thick lines as an example). Since the many sequential arithmetic operations cannot be calculated in one cycle, registers must be inserted for intermediate storage; in 8 Between each of the 12 FFT stages there are buffers, symbolized by blocks z ^-1 - in fact there can be more, since the supply lines of the first stages are very long and additional buffers may be required (the buffers take on new values at their input every clock cycle). The calculation therefore takes place in a so-called pipeline - the calculation of an FFT takes place over several clock cycles and the arithmetic circuit contains data from several FFTs; every clock cycle the input data of an FFT is fed into the arithmetic circuit designed as a pipeline, the result, i.e. the output data of this FFT, is then available at the output of the arithmetic circuit after several clock cycles, so that the result of a new FFT is obtained every clock cycle.

also Einheitszeigern (Betrag = 1) gebildet; im Allgemeinen sind dazu vier reellwertige Multiplizierer nötig. Jeder dieser reellwertigen Multiplizierer wird typischerweise durch zahlreiche Additionen geschobener Werte realisiert. Allerdings wird hier keine hohe Genauigkeit der Faktoren benötigt; beispielsweise kann ein Fehler von bis zu 1/32 toleriert werden, d. h. man kann die quantisierten Werte $${\text{w}}_{\text{p},\text{q}}=1/16\cdot \text{round}\left(16\cdot \text{cos}\left(\mathrm{\pi }\cdot 2^{\text{p}}\cdot \text{q}/\text{N}\right)\right)+\widehat{\text{j}}\cdot 1/16\cdot \text{round}\left(-16\cdot \text{sin}\left(\text{p}\cdot 2^{\text{p}}\cdot \text{q}/\text{N}\right)\right)$$

verwenden; dabei bezeichnet „round“ die Rundungsfunktion. Das durch diese Rundung generierte Rauschen am Ausgang der Korrelation E_m,k liegt unter dem benötigten Dynamikbereich und auch typischerweise unter dem Effekt des Empfängerrauschens; und der Signalverlust durch dieses Rauschen ist auch vernachlässigbar. Damit muss man nur noch Multiplizierer mit den Faktoren ±1/16, ±2/16, ..., ±15/16 realisieren. Als Beispiel werde der Multiplizierer mit dem Faktor 7/16 betrachtet; wegen $$7/16=8/16-1/16=1/2-1/16=2^{-1}-2^{-4}$$

kann dieser durch Subtraktion des um vier Stellen nach rechts geschobenen Eingangswertes von dem um eine Stelle nach rechts geschobenen Eingangswert realisiert werden - dabei wird von einer binären Zahlendarstellung ausgegangen; die obige Darstellung von 7/16 wird CSD-Code (Canonic-Signed-Digit Code) bezeichnet. Bis auf die Faktoren ±11/16 und ±13/16 kann man alle der obigen Faktoren nach Bez. (16) mit höchstens einer Addition oder Subtraktion realisieren; um für diese Faktoren ±11/16 und ±13/16 nicht zwei Additionen bzw. Subtraktionen einsetzen zu müssen, werden sie durch ±10/16 und ±14/16 angenähert, was immer noch zu akzeptablem Quantisierungsrauschen führt.The main effort for the realization of such a calculation logic are the multipliers. In them, the product between complex-valued signals and the rotation factors $$\begin{array}{l}{\text{w}}_{\text{p},\text{q}}=\text{ex}\left(-\widehat{\text{j}}\mathrm{\pi }\cdot {\text{2}}^{\text{p}}\cdot \text{q}/\text{N}\right)\\ \begin{array}{cc}=\text{cos}\left(\mathrm{\pi }\cdot 2^{\text{p}}\cdot \text{q}/\text{N}\right)+\widehat{\text{j}}\cdot \text{sin}\left(-\mathrm{\pi }\cdot 2^{\text{p}}\cdot \text{q}/\text{N}\right)& \text{with p}=\mathrm{1,}\dots ,{\text{log}}_2\left(\text{N}\right)-1\end{array}\text{and q}=\mathrm{1,}\dots ,\text{N}/2^{\text{p}}-\mathrm{1,}\end{array}$$

i.e. unit pointers (magnitude = 1); in general, four real-valued multipliers are required for this. Each of these real-valued multipliers is typically implemented by numerous additions of shifted values. However, high precision of the factors is not required here; for example, an error of up to 1/32 can be tolerated, ie the quantized values can be $${\text{w}}_{\text{p},\text{q}}=1/16\cdot \text{round}\left(16\cdot \text{cos}\left(\mathrm{\pi }\cdot 2^{\text{p}}\cdot \text{q}/\text{N}\right)\right)+\widehat{\text{j}}\cdot 1/16\cdot \text{round}\left(-16\cdot \text{sin}\left(\text{p}\cdot 2^{\text{p}}\cdot \text{q}/\text{N}\right)\right)$$

where “round” denotes the rounding function. The noise generated by this rounding at the output of the correlation E _m,k is below the required dynamic range and also typically below the effect of the receiver noise; and the signal loss due to this noise is also negligible. This means that one only has to implement multipliers with the factors ±1/16, ±2/16, ..., ±15/16. As an example, consider the multiplier with the factor 7/16; because $$7/16=8/16-1/16=1/2-1/16=2^{-1}-2^{-4}$$

This can be achieved by subtracting the input value shifted four places to the right from the input value shifted one place to the right - this assumes a binary number representation; the above representation of 7/16 is called CSD code (Canonic-Signed-Digit Code). With the exception of the factors ±11/16 and ±13/16, all of the above factors according to (16) can be achieved with at most one addition or subtraction; in order to avoid having to use two additions or subtractions for these factors ±11/16 and ±13/16, they are approximated by ±10/16 and ±14/16, which still leads to acceptable quantization noise.

dargestellt; aus dem binären Eingangswert i = i_Re + ĵ·i_lm mit Länge 9bit entsteht nach Multiplikation der Ausgangswert o = o_Re + ĵ·o_lm gleicher Länge, wobei als Beispiel Zahlenwerte eingetragen sind. Für den Real- und Imaginärteil des Ausgangswertes sind jeweils zwei reellwertige Multiplizierer durch jeweils einen Addierer zu realisieren; danach sind jeweils die beiden Teilergebnisse für Real- und Imaginärteil aufzuaddieren. Da auch das Negative von Real- und Imaginärteil des Eingangswertes benötigt wird, findet eine Invertierung statt. Diese Invertierung wird hier einfach durch bitweises Invertieren realisiert, d. h. die zusätzliche Addition von 1, also eines sogenannten LSBs (Least-Significant-Bit) wird weggelassen; der dadurch entstehende Fehler kann durch Aufaddieren von Korrekturwerten zu den Eingangswerten der FFT kompensiert werden - dazu kann der Effekt dieser bei der Invertierung fehlenden Werte 1 am Ausgang der FFT bestimmt und über eine inverse DFT auf den Eingang umgerechnet werden. Auch bei dem Rechtsschieben der binären Werte (für Realisierung der Multiplizierer) wird einfach der hintere Teil weggelassen, also keine Rundung durchgeführt; die dadurch entstehenden Fehler sind mittelwertbehaftet und ihre mittleren Fehler können auch wieder über Aufaddieren von Korrekturwerten zu den Eingangswerten der FFT kompensiert werden. Beim Rechtsschieben bleibt die Bitlänge des Wertes unverändert; dazu muss bei der hier betrachteten Zweikomplementdarstellung nur das oberste Bit des Eingangswertes entsprechend erweitert, also kopiert werden. Das Rechtsschieben selber wird einfach über entsprechende Verdrahtung realisiert und benötigt somit keinen Aufwand. Da die aufzumultiplizierenden Drehfaktoren immer den Betrag 1 haben, haben Ein- und Ausgangswerte der Multiplizierer selben Wertebereich; es müssen also keine zusätzlichen Bits nach oben hin erweitert werden.In 9 is the complex-valued multiplier for the rotation factor $${\text{w}}_{\mathrm{2,740}}=1/16\cdot \left(-10-\widehat{\text{j}}\cdot \text{12}\right)=\left(-2^{-1}-2^{-3}\right)+\widehat{\text{j}}\cdot \left(-2^{-1}-2^{-2}\right)$$

shown; the binary input value i = i _Re + ĵ·i _lm with a length of 9 bits produces, after multiplication, the output value o = o _Re + ĵ·o _lm of the same length, with numerical values entered as an example. For the real and imaginary parts of the output value, two real-valued multipliers must be implemented using an adder each; then the two partial results for the real and imaginary parts must be added together. Since the negative of the real and imaginary parts of the input value is also required, an inversion takes place. This inversion is implemented here simply by bit-by-bit inversion, i.e. the additional addition of 1, i.e. a so-called LSB (least significant bit), is omitted; the resulting error can be compensated by adding correction values to the input values of the FFT - for this purpose, the effect of these missing values 1 during inversion can be determined at the output of the FFT and converted to the input using an inverse DFT. When shifting the binary values to the right (to implement the multipliers), the rear part is simply omitted, i.e. no rounding is carried out; the resulting errors are averaged and their average errors can also be compensated by adding correction values to the input values of the FFT. When shifting to the right, the bit length of the value remains unchanged; for this, in the two's complement representation considered here, only the top bit of the input value needs to be extended accordingly, i.e. copied. The right shift itself is simply implemented using appropriate wiring and therefore requires no effort. Since the rotation factors to be multiplied always have the value 1, the input and output values of the multipliers have the same value range; so no additional bits need to be extended upwards.

In the butterflies, a complex-valued addition and subtraction of two complex-valued values takes place; the amount of the result can therefore be twice as large as the amount of the input values, so that the value range must be extended upwards by one bit. This would increase the bit length by 12 over the 12 stages of the FFT. However, the noise component of the values originating from the receiver noise also increases over the additions and subtractions, on average by √2 in terms of amplitude. After every two stages, the noise amplitude doubles. Therefore, the least significant bit, i.e. the LSB, can be omitted in every second stage (in this case, scaling is done by a factor of 0.5); the quantization noise generated in this way is less than the effect of the receiver noise, since the value range at the input of the FFT is selected so that the receiver noise already has the amplitude of several LSBs there. The effect of simply omitting the LSBs (i.e. without rounding), i.e. the resulting mean errors, can be compensated by adding correction values to the input values of the FFT. In the circuit according to 8 This scaling is omitted in the last stage, since there are no further calculation steps within the FFT that would benefit from a reduction in the bit length; thus, the bit length increases over the FFT from 8 bits at the input to 15 bits at the output.

According to 8 the FFT is implemented in a decimation-infrequency FFT structure in order to have the longest lines of the structure and the nontrivial multiplications in the front stages with their smaller bit length (there are no multiplications in the back two stages; the factor -ĵ only represents corresponding wiring). In addition, this structure avoids reordering the input data in the form of long lines; reordering the output data in their natural order is not required for further processing here. This structure therefore requires less implementation effort than the alternative decimation-in-time FFT structure.

die Werte dieser Korrelation unterscheiden sich von denen der Bez. (8) in der Phase, sind aber im Betrag identisch und nur dieser ist für die weitere Auswertung relevant, so dass hier vereinfachend dasselbe Formelzeichen benutzt wird (dieser Zusammenhang ergibt sich aus dem Zeitverschiebungsversatz der Fouriertransformation). Das Produkt zwischen zyklisch verschobener Empfangsfolge e(mod_N(n+m)) und Modulationsfolge b(n) wird in 8 im Block 8.2 gebildet und über schaltbare Invertierer realisiert; für Werte 1 von b(n) bleibt der Eingangswert unverändert, für Werte -1 wird er einfach bitweise invertiert - der Effekt des Weglassens der beim Invertieren eigentlich nötigen Addition von einem LSB wird durch Aufaddieren von Korrekturwerten zu den Eingangswerten der FTT im Block 8.3 kompensiert. Es sei bemerkt, dass die schaltbaren Invertierer nur dann nötig sind, wenn sich die Modulationsfolge ändern kann - falls nicht, dann kann man die Invertierung hart implementieren und natürlich nur dort, wo sie stattfindet. Das zyklische Verschieben der Empfangsfolge wird über die Kette der Register z^-1 realisiert, in welche initial die Werte der Empfangsfolge geladen werden.According to (8), to determine the correlation E _m,k, the FFT is applied to the product between the reception sequence e(n) and the shifted modulation sequence b(nm). Due to the cyclic nature of the modulation sequence b(n) (it has a period of N), one can also form the product between the unshifted modulation sequence b(n) and the cyclically shifted reception sequence e(mod _N (n+m)), where "modn" represents the modulo function for the module N, and apply the FFT to it: $$\begin{array}{cc}{\text{E}}_{\text{m},\text{k}}={\text{FFT}}_{\text{k}}\left(\text{e}\left({\text{mod}}_{\text{N}}\left(\text{n}+\text{m}\right)\right)\cdot \text{b}\left(\text{n}\right)\right)& \text{with m}=\mathrm{0,}\dots ,\text{M}-1\end{array};$$

the values of this correlation differ from those of the relation (8) in phase, but are identical in magnitude and only this is relevant for further evaluation, so that the same formula symbol is used here for simplicity (this relationship results from the time shift offset of the Fourier transformation). The product between the cyclically shifted reception sequence e(mod _N (n+m)) and the modulation sequence b(n) is given in 8 formed in block 8.2 and implemented using switchable inverters; for values of 1 of b(n) the input value remains unchanged, for values of -1 it is simply inverted bit by bit - the effect of omitting the addition of one LSB that is actually required for inversion is compensated for by adding correction values to the input values of the FTT in block 8.3. It should be noted that the switchable inverters are only necessary if the modulation sequence can change - if not, then the inversion can be hard-implemented and of course only where it takes place. The cyclic shifting of the receive sequence is implemented using the chain of registers z ^-1 , into which the values of the receive sequence are initially loaded.

Before this, correction values c ₁ (n) are added to the reception sequence in block 8.1, which serves to compensate for the effects of couplings and reflections within the lidar system or from its immediate surroundings, in particular a cover; this will be discussed in more detail later.

As already explained above, pure truncation is used for quantization and pure bit-wise inversion is used for inversion to simplify calculations; the effects of the resulting mean-value errors are compensated for by adding correction values c ₂ (n) in block 8.3 before the FFT. This correction stage could also be implemented after the FFT instead of before the FFT.

wobei „max“ und „min“ die Maximums- und Minimumsfunktion bezeichnen; diese Berechnung lässt sich mit geringem Logikaufwand implementieren.After the FFT, i.e. after forming the correlation E _m,k, the result is processed further. First, in block 8.5, the absolute value for each of the N=4096 complex values is formed. Since no high level of accuracy is required here, the following approximation can be used for the absolute value |i| of the complex value i = i _Re + ĵ·i _lm : $$\begin{array}{cc}\left|\text{i}\right|=\text{max}\left(\text{u}+\text{v}/\mathrm{8,}\text{u}-\text{u}/8+\text{v}/2\right)& \text{with u}=\text{max}\left(\left|{\text{i}}_{\text{re}}\right|,\left|{\text{i}}_{\text{In the}}\right|\right)\end{array}\text{and v}=\text{min}\left(\left|{\text{i}}_{\text{re}}\right|,\left|{\text{i}}_{\text{In the}}\right|\right),$$

where “max” and “min” denote the maximum and minimum functions; this calculation can be implemented with little logic effort.

The amounts of the N=4096 values calculated in this way go both to block 8.6 for summation and to block 8.7 for maximum calculation. Both blocks are designed in cascaded form; in each of the 12 stages, the sums or maxima of value pairs are calculated. Registers required between the stages are not shown.

The summation is required to estimate the noise level in order to be able to distinguish peaks in the correlation generated by objects from noise peaks. Since there are very few peaks in the correlation generated by objects, i.e. most of the values are just noise, the sum after division by 4096, i.e. shifting to the right by 12 bits, provides a good estimate of the noise level.

The maximum determination determines the maximum value and the associated index k of the N=4096 FFT output values for the respective time shift m (which corresponds to the distance), i.e. in the frequency shift dimension (which, in addition to the Doppler component caused by the relative speed, also has a transit time component). If this maximum is at least a factor of 3 above the estimated noise, it is considered to be generated by an object; from the associated time shift m=m _0.1 and the associated frequency shift index k=k _0.i, the distance and relative speed of the respective object i can be determined using the equations (7a) and (14), and from the level its reflectivity. If only the absolute maximum is determined as shown in block 8.7, for a distance If the highly reflective object in the respective pixel is to be covered, only the object with the strongest reflection is to be determined. If the very unlikely case is to be covered that there are two objects with different relative speeds at a distance (i.e., in the range of about half a meter) in a pixel, the respective maximum of several value blocks could also be output - because of the cascaded structure of the maximum search, e.g. of 8 blocks of equal length. If the input data of the maximum search are arranged accordingly, several blocks can also be used to interpolate the magnitude peak in the FFT for a more precise determination of the frequency shift; because this magnitude peak is typically seen in two neighboring FFT values (since it is not - as previously considered - at an integer index k ₀ ) and if these are in different blocks of the maximum search by arranging the input data accordingly, both values are then obtained.

It should also be noted that no window function was used for the FFT here, i.e. no multiplication of the FFT input values with a kind of bell curve; this would only be necessary or useful if two objects with similar relative speed and significantly different reflection strengths can occur at the same distance in a pixel and are to be separated. In particular, if no window function is used at the input of the FFT, the sensitivity at the output of the FFT is reduced (i.e. the detection capability of objects with weak reflectivity and a long distance) if the frequency shift index k ₀ is not an integer, i.e. the magnitude peak is divided between two neighboring FFT values. This effect can be reduced by choosing the length of the FFT to be longer than that of its input signal, i.e. zeros are appended to the input signal, which is known as zero padding.

With regard to the index determination in the maximum search, it should be noted that, due to the cascaded implementation, this can be built up very easily bit by bit, starting with the LSB; at the output of each comparison of two values, in addition to the current maximum value, there is also an index value whose bit length corresponds to the number of the stage. The index created in this way refers to the linear numbering at the input of the maximum search; since the numbering at the output of the FFT is scrambled with regard to the frequency shift index k, a conversion/mapping must be carried out later.

For the output dimension of the FFT, i.e. the discrete frequency, the non-symmetrical range k = 0,...,N-1 has so far been considered - as is generally the case; the actual frequency shift k can, however, take on either sign and is typically limited by previous low-pass filtering, e.g. as part of the analog-digital converter, whereby for this limitation the range k = -N/2,...,+N/2 is assumed here for the sake of simplicity, so that the upper half of k = 0,...,N-1 can be mapped to negative values by subtracting N.

In the design considered here (sampling time T _s = 3.33 ns and modulation bandwidth B = 800 MHz of the linear frequency modulation), the relative speed range at the output of the FFT for an object distance of zero is approximately ±419 km/h (from reference (14) with k _0,i = ±N/2) and for a maximum object distance of 249.5 m approximately -147...+690 km/h (from reference (14) with k _0,i = ±N/2 and m _0,i = M-1 = 499); thus, the possible or functionally interesting range of relative speeds is completely covered and, except for negative speeds at large distances, even significantly overfilled - later it will be shown how a uniform coverage of the relative speed range, i.e. for all distances, can be achieved in order to realize a reduction in the FFT length with the help of a subsequent decimation. In general, decimation can be used before the FFT if the range of possible frequency shifts is smaller than the frequency range of the FFT, i.e. if, for example, the sampling and modulation time is shorter and/or the length N of the modulation sequence is longer than previously considered; in the simplest case, such decimation is carried out by forming partial sums of the product sequence of the shifted reception sequence multiplied by the modulation sequence.

As already mentioned above, the received signal is low-pass filtered after the mixer - this can be done in a dedicated filter or as part of the analog-digital converter (especially if this is implemented as a delta-sigma converter). For optimal sensitivity (i.e. optimal signal-to-noise ratio), an optimal filter is used based on the modulation form: in the case of a rectangular transmit-side modulation signal, which also retains its shape in the received signal (i.e. after mixing), the impulse response of the low-pass also has this rectangular shape. After the low-pass, a triangular shape of twice the length is obtained for the elements of the received signal - but only exactly when the frequency shift is zero. For other frequency shifts, filtering with such a low-pass filter is even less optimal (in the sense of maximum sensitivity). the larger the frequency shift; by using a smaller sampling time T _s (i.e. higher sampling frequency), the relative range of the frequency shifts and thus the loss of signal-to-noise ratio in the low-pass filtering can be reduced, whereby the modulation time T _m can then also be selected to be higher than the sampling time in order not to significantly increase the required computational effort in combination with decimation before the FFT. Due to the low-pass filtering, m magnitude peaks typically occur in the correlation E _m,k at two consecutive discrete distances, so that an interpolation can be carried out using their values to determine the distance more precisely.

At the exit of the 8 The hard-wired digital circuit shown generates information every clock cycle, i.e. approximately every nanosecond, as to whether and at what relative speed there is an object in the respective pixel and at the respective distance being considered. The reception sequence e(n) of a pixel is loaded into the register once and is then retained for M = 500 clock cycles with cyclic shifting (for the 500 different shifts m and thus different distances). After M = 500 clock cycles, the reception sequence of the next pixel is then loaded. In this way, all 160,000 pixels, for example, are successively evaluated from a detection cycle with a duration of 100 ms.

The logic of the digital circuit according to 8 consists mainly of adding - around 300,000 adders with an average length of 12 bits are required. The ever smaller structure sizes of semiconductor technologies for digital circuits make it possible to implement such a large hard-wired circuit - both in terms of costs and power consumption. Compared to state-of-the-art frequency modulation, phase modulation shifts the implementation effort for coherent lidar systems more into the digital area (since the analogue part becomes simpler, since - as explained later - no high-precision linearity of the frequency change and no complex-value receivers are required); due to the permanent and rapid progress in semiconductor technologies for digital circuits, this leads to a more cost-optimised, i.e. cheaper, solution.

wobei „CC_m“ die zyklische Korrelation zwischen den beiden Folgen der Länge N bedeutet und dabei m = 0,...,N-1 die Dimension am Ausgang der Korrelation ist; da in der betrachteten Auslegung N>M ist, werden durch die zyklische Korrelation also mehr diskrete Entfernungen m als benötigt prozessiert.Now another one opposite 8 alternative structure. The two-dimensional correlation E _m,k according to (6) can also be seen as a one-dimensional temporal correlation between reception sequence e(n) and sequence b(n)·exp(-j2π·n/N·k), which is carried out for each k: $$\begin{array}{cc}{\text{E}}_{\text{m},\text{k}}={\text{CC}}_{\text{m}}\left(\text{e}\left(\text{n}\right),\text{b}\left(\text{n}\right)\cdot \text{ex}\left(-\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{k}\right)\right)& \text{with k}=\mathrm{0,}\dots ,\text{N}-\mathrm{1,}\end{array}$$

where “CC _m ” means the cyclic correlation between the two sequences of length N and m = 0,...,N-1 is the dimension at the output of the correlation; since in the design considered N>M, more discrete distances m are processed by the cyclic correlation than required.

wobei IFFT_m die inverse schnelle Fouriertransformation bedeutet und m = 0,...,N-1 ihre Ausgangsdimension ist (hier ist schon angenommen, dass die DFT über eine FFT realisiert wird). Gemäß dem Frequenzverschiebungssatz der Fouriertransformation bedeutet der auf die Modulationsfolge b(n) im Zeitbereich angewendete Faktor exp(-j2π·n/N·k) eine Verschiebung im Frequenzbereich, also der Fouriertransformierten: $$\begin{array}{l}{\text{E}}_{\text{m},\text{k}}={\text{IFFT}}_{\text{m}}\left(\text{E}\left(\text{I}\right)\cdot \text{B}\left(\text{I}+\text{k}\right)\right)\\ \text{mit E}\left(\text{I}\right)={\text{FFT}}_{\text{I}}\left(\text{e}\left(\text{n}\right)\right),\text{ B}\left(\text{I}\right)={\text{FFT}}_{\text{I}}\left(\text{b}\left(\text{n}\right)\right)\text{ und k}=\mathrm{0,}\dots ,\text{N}-1;\end{array}$$

wegen dem Frequenzverschiebungssatz der Fouriertransformation und der zyklischen Natur der diskreten Fouriertransformierten können für den Betrag der Korrelation folgende weitere Umformungen vorgenomen werden: $$\begin{array}{c}\left|{\text{E}}_{\text{m},\text{k}}\right|=\left|{\text{IFFT}}_{\text{m}}\left(\text{E}\left(\text{I}-\text{k}\right)\cdot \text{B}\left(\text{I}\right)\right)\right|\\ =\left|{\text{IFFT}}_{\text{m}}\left(\text{E}\left({\text{mod}}_{\text{N}}\left(\text{I}-\text{k}\right)\right)\cdot \text{B}\left(\text{I}\right)\right)\right|\end{array}$$

$$\begin{array}{cccc}\text{mit}& \text{E}\left(\text{l}\right)={\text{FFT}}_{\text{l}}\left(\text{e}\left(\text{n}\right)\right),& \text{B}\left(\text{l}\right)={\text{FFT}}_{\text{l}}\left(\text{b}\left(\text{n}\right)\right)& \text{und k}=\mathrm{0,}\dots ,\text{N}-1\end{array}.$$

A cyclic correlation in the time domain corresponds to a multiplication of the discrete Fourier transform in the frequency domain: $$\begin{array}{cc}{\text{E}}_{\text{m},\text{k}}={\text{IFFT}}_{\text{m}}\left(\text{FFT}\left(\text{e}\left(\text{n}\right)\right)\cdot \text{FFT}\left(\text{b}\left(\text{n}\right)\cdot \text{ex}\left(-\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{k}\right)\right)\right)& \text{with k}=\mathrm{0,}\dots ,\text{N}-1\end{array},$$

where IFFT _m means the inverse fast Fourier transform and m = 0,...,N-1 is its output dimension (here it is already assumed that the DFT is realized via an FFT). According to the frequency shift theorem of the Fourier transform, the factor exp(-j2π·n/N·k) applied to the modulation sequence b(n) in the time domain means a shift in the frequency domain, i.e. of the Fourier transform: $$\begin{array}{l}{\text{E}}_{\text{m},\text{k}}={\text{IFFT}}_{\text{m}}\left(\text{E}\left(\text{I}\right)\cdot \text{B}\left(\text{I}+\text{k}\right)\right)\\ \text{with E}\left(\text{I}\right)={\text{FFT}}_{\text{I}}\left(\text{e}\left(\text{n}\right)\right),\text{B}\left(\text{I}\right)={\text{FFT}}_{\text{I}}\left(\text{b}\left(\text{n}\right)\right)\text{and k}=\mathrm{0,}\dots ,\text{N}-1;\end{array}$$

Due to the frequency shift theorem of the Fourier transform and the cyclic nature of the discrete Fourier transform, the following further transformations can be made for the magnitude of the correlation: $$\begin{array}{c}\left|{\text{E}}_{\text{m},\text{k}}\right|=\left|{\text{IFFT}}_{\text{m}}\left(\text{E}\left(\text{I}-\text{k}\right)\cdot \text{B}\left(\text{I}\right)\right)\right|\\ =\left|{\text{IFFT}}_{\text{m}}\left(\text{E}\left({\text{mod}}_{\text{N}}\left(\text{I}-\text{k}\right)\right)\cdot \text{B}\left(\text{I}\right)\right)\right|\end{array}$$

$$\begin{array}{cccc}\text{with}& \text{E}\left(\text{l}\right)={\text{FFT}}_{\text{l}}\left(\text{e}\left(\text{n}\right)\right),& \text{B}\left(\text{l}\right)={\text{FFT}}_{\text{l}}\left(\text{b}\left(\text{n}\right)\right)& \text{and k}=\mathrm{0,}\dots ,\text{N}-1\end{array}.$$

This connection can be translated into a structure analogous to that in 8 shown structure. The input values of the structure are the FFT of the reception sequence, which is to be calculated in advance. It is multiplied in a form cyclically shifted by k with the previously determined FFT of the modulation sequence b(n). This is followed by an inverse FFT (IFFT), which differs from the FFT only in the sign of the rotation factors; the output dimension of this IFFT is the distance dimension m (in the structure according to 8 it is the frequency shift dimension k), i.e. for a discrete frequency shift k, the two-dimensional correlation E _m,k is present over the distance dimension m = 0,...,N-1. This is followed by a magnitude calculation and then a sum and maximum calculation, this time over the distance dimension. In a pixel there can be several reflections with different distances, but the probability that these have the same frequency shift (which is made up of contributions from relative speed and distance) is low, so that here too the determination of the absolute maximum is sufficient; as an example, consider the case of fog plus a possibly stationary object in a pixel: although all reflections then have the same relative speed, they have a different frequency shift due to different distances. From the maxima above the detection threshold with indices m=m _0,i and k=k _0,i the distance and relative speed of the respective object i can again be determined using the equations (7a) and (14).

If the same modulation sequence b(n) is always used, the multipliers for the realization of the product E(mod _N (Ik))·B(I) can be implemented in hard-wired form - through approximation and CSD representation, only a small implementation effort is then required. If the modulation sequence changes, programmable multipliers may be necessary, which mean significantly more implementation effort.

As explained above, with the assumed N>M (N ≈ 8M) many more discrete distances m = 0,..., N-1 are usually processed than required. This can be avoided by carrying out a decimation before the IFFT - with a decimation by a factor of 8, the N = 4096 values then only result in 512 values, which are fed into the IFFT; in the simplest case, the decimation is implemented by adding 8 neighboring values. This means that the IFFT only has the dimension 512 instead of the original dimension 4096 and therefore requires much less implementation effort; with its length 512 it also covers the full distance range of the length M = 500.

However, it must be taken into account that such a structure must be clocked N = 4096 times per pixel (this is how many shifts must be calculated for the FFT of the input signal); this is a good factor of 8 more than in the structure according to 8 , which, however, is already at the maximum of what is feasible with a clock rate of about 1 GHz. Therefore, this alternative structure for reference (22) would have to be constructed several times, which more than negates the advantage of the shorter length of the IFFT. Such an alternative structure is useful when the product of the sampling length N (and thus according to the design of the modulation length considered so far) and the number of pixels is so small that the structure is only needed a few times, preferably once.

Optimized calculation logic using rotation factors and decimation

und für die diskrete Frequenz k₀ der Empfangsfolge e(n) gilt: $${\text{k}}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{f}}_{\text{D}}\cdot {\text{f}}_{\text{r}}\right)=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \text{v}/\mathrm{\lambda }-{\text{m}}_{\text{0}}\cdot {\text{T}}_{\text{s}}\cdot \text{B}.$$

As explained above, the total frequency shift and thus the frequency f _e of the received signal is composed of the Doppler shift and the delay-related component generated by the linear frequency modulation: $${\text{e}}_{\text{e}}={\text{e}}_{\text{D}}+{\text{e}}_{\text{r}}=\text{2v}/\mathrm{\lambda }-\text{2r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}},$$

and for the discrete frequency k ₀ of the reception sequence e(n) the following applies: $${\text{<figure-callout id="0" label="k" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">k</figure-callout>}}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{e}}_{\text{D}}\cdot {\text{e}}_{\text{r}}\right)=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \text{v}/\mathrm{\lambda }-{\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\text{0}}\cdot {\text{T}}_{\text{s}}\cdot \text{B}.$$

die Ausgangsdimension der FFT ist also nun die diskrete Frequenz k, so dass gemäß Bez. (13a) und (24) die Position k₀ von FFT-Betragsspitzen nur noch zur Relativgeschwindigkeit des jeweiligen Objekts korrespondiert: $${\underset{\_}{\text{k}}}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \text{v}/\mathrm{\lambda }.$$

The transit time-related component (in the above relationships the second component to distance r or discrete transit time m ₀ ) means that the frequency range relevant for the evaluation depends on the distance and increases with it. This means that for the calculation structure according to 8 effectively a longer FFT and thus more effort is required compared to pure phase modulation, i.e. no additional linear frequency modulation. This problem can be solved by eliminating the runtime-related frequency component -m ₀ ·T _s ·B before the FFT; this will now be briefly derived, for which the two-dimensional correlation E _m,k according to equation (6) is unformed as follows: $$\begin{array}{c}{\text{E}}_{\text{m},\text{k}}={\text{sum}}_{\text{n}=\mathrm{0,}\dots ,\text{N}-1}\left(\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\cdot \text{ex}\left(-\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{k}\right)\right),m=\mathrm{0,}\dots ,\text{M}-1\text{and k}=\mathrm{0,}\dots ,\text{N}-\mathrm{1,}\left(\left(6\right)\right)\\ ={\text{sum}}_{\text{n}=\mathrm{0,}\dots ,\text{N}-1}\left(\left[\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\cdot \text{ex}\left(+\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}\right)\right]\cdot \text{ex}\left(-\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{k}+\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}\right]\right)\right)\\ ={\text{sum}}_{\text{n}=\mathrm{0,}\dots ,\text{N}-1}\left(\left[\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\cdot \text{ex}\left(+\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}\right)\right]\cdot \text{ex}\left(-\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{k}\right)\right)\\ \text{with}\underset{\_}{\text{k}}=\text{k}+\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}\left(23\right)\\ ={\text{FFT}}_{\underset{\_}{\text{k}}}\left(\left[\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\right]\cdot \text{ex}\left(+\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}\right)\right);\left(24\right)\end{array}$$

The output dimension of the FFT is now the discrete frequency k, so that according to (13a) and (24) the position k ₀ of FFT magnitude peaks only corresponds to the relative velocity of the respective object: $${\underset{\_}{\text{k}}}_0=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \text{v}/\mathrm{\lambda }.$$

zu multiplizieren. Diese Multiplikation findet im Block 10.8 der in 10 dargestellten Berechnungsstruktur statt, also vor Beginn der FFT mit der im Block 10.3 stattfindenden Addition von Korrekturwerten c₂(n) für mittelwertbehaftete Fehler innerhalb der FFT durch Abschneiden und rein bitweises Invertieren; Blöcke in 10, welche gegenüber der Berechnungsstruktur nach 8 im Wesentlichen unverändert sind (sich primär nur in der Dimension ändern), sind durch gleiche Subnummern .1-.7 gekennzeichnet.The product between reception sequence e(n) and shifted modulation sequence b(nm) is before FFT with the rotation factors $${\text{d}}_{\text{n},\text{m}}=\text{ex}\left(+\widehat{\text{j}}2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}\right)$$

This multiplication takes place in block 10.8 of the 10 shown calculation structure, i.e. before the start of the FFT with the addition of correction values c ₂ (n) for mean errors within the FFT by truncation and purely bit-wise inversion in block 10.3; blocks in 10 , which compared to the calculation structure according to 8 are essentially unchanged (changing primarily only in dimension) are identified by identical subnumbers .1-.7.

in die Frequenz $$\underset{\_}{\underset{\_}{\text{k}}}=\underset{\_}{\text{k}}-{\underset{\_}{\text{k}}}_{\text{offset}}$$

mit symmetrischem Bereich transferiert werden; mit Bez. (23) und (27) erhält man für diese laufzeitbereinigte und zentrierte Frequenz: $$\underset{\_}{\underset{\_}{\text{k}}}=\text{k}+\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{v}}_{\text{max}}+{\text{v}}_{\text{min}}\right)/\mathrm{\lambda }.$$

With unchanged dimension N = 4096 of the FFT, which for the discrete frequency k opens up the range k = -N/2,..., N/2, the relative speed v results in approximately the range ±419km/h (from v = λ/(2N·T _s )·k with k = ±N/2), which is significantly above the relevant speed range of, for example, V _min = -80km/h to v _max = +280km/h - for objects moving towards each other (positive sign of v), higher relative speeds are functionally relevant than for objects moving away (negative sign). The corresponding asymmetrical range for the frequency k can be determined by appropriate shifting by the offset frequency. $${\underset{\_}{\text{k}}}_{\text{offset}}=\text{N}\cdot {\text{T}}_{\text{s}}\cdot 2\cdot \left({\text{v}}_{\text{max}}+{\text{v}}_{\text{min}}\right)/2/\mathrm{\lambda }=\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{v}}_{\text{max}}+{\text{v}}_{\text{min}}\right)/\mathrm{\lambda }$$

in the frequency $$\underset{\_}{\underset{\_}{\text{k}}}=\underset{\_}{\text{k}}-{\underset{\_}{\text{k}}}_{\text{offset}}$$

with symmetrical range; using (23) and (27) one obtains for this runtime-corrected and centered frequency: $$\underset{\_}{\underset{\_}{\text{k}}}=\text{k}+\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{v}}_{\text{max}}+{\text{v}}_{\text{min}}\right)/\mathrm{\lambda }.$$

so dass vor der FFT die Produktfolge e(n)·b(n-m) mit den modifizierten Drehfaktoren $$\begin{array}{c}{\text{d}}_{\text{n},\text{m}}=\text{exp}\left(+\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]\right)\\ =\text{exp}\left(+\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{v}}_{\text{max}}+{\text{v}}_{\text{min}}\right)/\mathrm{\lambda }\right]\right)\end{array}$$

zu multiplizieren ist.Using this frequency k for the FFT, the two-dimensional correlation E _m,k is obtained analogously to the derivation of its representation according to (24): $$\begin{array}{c}{\text{E}}_{\text{m},\text{k}}={\text{FFT}}_{\underset{\_}{\underset{\_}{\text{k}}}}\left(\left[\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\right]\cdot \text{ex}\left(+\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]\right)\right)\\ ={\text{FFT}}_{\underset{\_}{\underset{\_}{\text{k}}}}\left(\left[\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)\right]\cdot \text{ex}\left(+\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{v}}_{\text{max}}+{\text{v}}_{\text{min}}\right)/\mathrm{\lambda }\right]\right)\right),\end{array}$$

so that before the FFT the product sequence e(n)·b(nm) with the modified rotation factors $$\begin{array}{c}{\text{d}}_{\text{n},\text{m}}=\text{ex}\left(+\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]\right)\\ =\text{ex}\left(+\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left({\text{v}}_{\text{max}}+{\text{v}}_{\text{min}}\right)/\mathrm{\lambda }\right]\right)\end{array}$$

to be multiplied.

For the example relative speed range from v _min = -80km/h to v _max = +280km/h, the symmetrical range k = -881,...,+881 results for this frequency k at the output of the FFT. This range is smaller than half the FFT length N/2 = 2048, so that a decimation by a factor of 2 can be carried out before the FFT. In the simplest case, this can be done - as in block 10.9 of 10 shown - by adding two consecutive values of the total N values. The N/2 sum values then form the input values of the FFT, which is preceded by block 10.3 for adding correction values c ₂ (n). Due to the halved length N/2 = 2048, the FFT has one stage less, i.e. only 11 stages (see FFT block 10.4 in calculation structure according to 10 ). The frequency k at the output of the FFT only extends over the range k = 0,..., N/2-1. The following blocks 10.5-10.7 for calculating magnitude, sum and maximum also only have half the dimension N/2 = 2048.

womit dann Abtastzeit T_s und Modulationszeit T_m weiterhin gleich sind, und damit auch weiterhin die Empfangsfolge e(n) und die Modulationsfolge b(n) sich auf die gleiche diskrete Zeit n beziehen. Einziger Unterschied in den obigen Betrachtungen und Beziehungen ist, das für die Verschiebungen m nun nur jeder zweite Wert, also nur geradzahlige Werte m = 0,2,4,...,M-2 zu betrachten sind. Die den Berechnungsstrukturen zugrundeliegende Darstellung der zweidimensionalen Korrelation E_m,k, bei welcher statt der Modulationsfolge die Empfangsfolge verschoben wird, lautet dann analog zu Bez. (17) unter Verwendung der obigen Bez. (30) sowie unter Berücksichtigung der Dezimation: $$\begin{array}{l}{\text{E}}_{\text{m},\text{k}}={\text{FFT}}_{\underset{\_}{\text{k}}}\left(\text{s}\left(\text{n}\right)\cdot \text{exp}\left(+\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]\right)\right)\\ \text{mit s}\left(\text{n}\right)=\text{e}\left({\text{mod}}_{\text{N}}\left(\text{n}+\text{m}\right)\right)\cdot \text{b}\left(\text{n}\right)+\text{e}\left({\text{mod}}_{\text{N}}\left(\text{n}+1+\text{m}\right)\right)\cdot \text{b}\left(\text{n}+1\right),\\ \text{n}=\mathrm{0,2,4,}\dots ,\text{N}-2,\text{ m}=\mathrm{0,2,4,}\dots ,\text{M}-2,\underset{\_}{\underset{\_}{\text{k}}}=\mathrm{0,}\dots ,\text{N}/2-1\end{array}$$

As explained above, the calculation structure can cover around 160,000 pixels; according to previous considerations, M = 500 shifts m between the reception sequence and the modulation sequence must be calculated per pixel, which requires M = 500 clock cycles. With a maximum realizable clock rate of the calculation logic of around 1GHz, only a detection cycle of around 100ms is possible (a certain overhead is still taken into account, e.g. for loading data and reconfiguring the system). In reality, however, a cycle time of 50ms is often required, which would then require the calculation structure to be implemented twice. To avoid this, the modulation time T _m can be effectively doubled in order to only have to calculate half as many shifts per pixel; this effective doubling can also be defined by the fact that, with T _m = 3.33ns unchanged, two consecutive values of the modulation sequence b(n) are equal: $$\begin{array}{cc}\text{b}\left(\text{n}+1\right)=\text{b}\left(\text{n}\right)& \text{with n}=\mathrm{0,2,4,}\dots ,\text{N}-2\end{array},$$

whereby the sampling time T _s and the modulation time T _m are still the same, and thus the reception sequence e(n) and the modulation sequence b(n) continue to refer to the same discrete time n. The only difference in the above considerations and relationships is that for the shifts m now only every second value, i.e. only even-numbered values m = 0,2,4,...,M-2, are to be considered. The representation of the two-dimensional correlation E _m,k underlying the calculation structures, in which the reception sequence is shifted instead of the modulation sequence, is then analogous to equation (17) using the above equation (30) and taking decimation into account: $$\begin{array}{l}{\text{E}}_{\text{m},\text{k}}={\text{FFT}}_{\underset{\_}{\text{k}}}\left(\text{s}\left(\text{n}\right)\cdot \text{ex}\left(+\widehat{\text{j}}\text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]\right)\right)\\ \text{with s}\left(\text{n}\right)=\text{e}\left({\text{mod}}_{\text{N}}\left(\text{n}+\text{m}\right)\right)\cdot \text{b}\left(\text{n}\right)+\text{e}\left({\text{mod}}_{\text{N}}\left(\text{n}+1+\text{m}\right)\right)\cdot \text{b}\left(\text{n}+1\right),\\ \text{n}=\mathrm{0,2,4,}\dots ,\text{N}-2,\text{m}=\mathrm{0,2,4,}\dots ,\text{M}-2,\underset{\_}{\underset{\_}{\text{k}}}=\mathrm{0,}\dots ,\text{N}/2-1\end{array}$$

The product formation between cyclically shifted reception sequence e(mod _N (n+m)) and modulation sequence b(n) according to (33) is shown in block 10.2; the preceding addition of the correction values c ₁ (n) according to block 10.1 remains the same as the original structure according to 8 unchanged.

Since the structure according to 10 per pixel only M/2 = 250 times, i.e. half as often as the original structure after 8 to be clocked through, half the cycle time of 50ms is possible; in addition, the implementation effort is only about half as large, since the data dimension is halved for the essential blocks.

The simplest implementation of decimation so far has been considered by adding two consecutive values. The first-order low-pass filter thus created (i.e. length 2) has a large transition area with a relatively low slope; this leads on the one hand to a significant change in the level in the relevant frequency range k = -881,...,+881 (by 2.15dB between k = 0 and k = ±881), and on the other hand to a loss of sensitivity after decimation due to noise folding in from higher frequencies (up to 2.15dB at k = ±881). A higher-order low-pass filter can make its transition area sharper; with a third-order low-pass filter with the four coefficients [0.5, 1, 1, -0.5] the level difference is only 0.91dB and the maximum loss of sensitivity is 1.88dB. This third-order low-pass filter requires three additions, but still no multiplications, since the coefficients with the value 0.5 can be realized by shifting them to the right by one bit, i.e. with pure wiring (for the negative sign of the coefficient -0.5, a bit-wise inversion can again be used to simplify things). With high-order low-pass filters and coefficients that are not in a power-of-two grid (i.e. that require one or more additions themselves), even sharper edges can be achieved. It should also be noted that frequencies k above ±881 can also be considered and determined (and thus relative speeds outside the corresponding range -80km/h,...,+280km/h); however, level and sensitivity losses then increase and above k = ±1024 ambiguities arise (since frequencies are reflected in the range below ±1024).

Now the implementation of the multiplications with the rotation factors d _n,m described in reference (33) is explained, i.e. unit vectors with phases from the full angle range 0...2π. The phases of the rotation factors do not need to be implemented with arbitrary precision, but the phase that is closest to the real phase of d _n,m can be used from a limited set of phases. The simplest approach would be to implement only the four phases 0, π/2, π and 3π/2, i.e. the rotation factors 1, ĵ, -1 and -ĵ, which does not require multiplication; however, the resulting effects are then not negligible, i.e. the effective loss of sensitivity as well as noise or pseudo-magnitude peaks generated at the output of the FFT. Therefore, the approach with the 8 phase values 0, π/4, ..., 7π/4 is considered below, i.e. the realized rotation factors are $${\underset{\_}{\text{d}}}_{\text{n},\text{m}}=\text{ex}\left(+\widehat{\text{j}}\mathrm{\pi }/4\cdot {\text{mod}}_8\left(\text{round}\left(8\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]\right)\right)\right).$$

Thus, in addition to the simple rotation factors ±1 and ±j, there are also rotation factors (±1±ĵ)/√2. The factor 1/√2 = 0.707 can be approximately realized by 1-2 ^-2 = 0.75, i.e. effectively by an addition (in addition to inverting and shifting to the right by 2 bits, which can be implemented easily or without effort).

zu verwenden ist, werden die 6 Umschalter (zwischen dem Wert 0 oder dem jeweiligen Eingangswert) und die 4 schaltbaren Bitinvertierer aus der Logik 11.21 angesteuert. In dieser Logik werden die 10 benötigten binären Schaltsignale aus dem jeweiligen am Eingang liegenden diskreten Phasenwert p_n,m = 0,...,7, also aus 3 Bit, berechnet. Der jeweilige Wert p_n,m, also gemäß Bez. (34): $${\text{p}}_{\text{n},\text{m}}={\text{mod}}_8\left(\text{round}\left(8\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]\right)\right)\text{ mit m}=\mathrm{0,2,4}\dots ,\text{M}-\mathrm{2,}$$

wird im Block 11.1 mit Hilfe eines Integrators (also einer rekursiven Struktur erster Ordnung) bestimmt; der bzgl. der Verschiebung m lineare Anteil (welcher sich also beim Durchtakten der Berechnungsstruktur sukzessive erhöht) wird durch rekursives Aufaddieren des Wertes im Registers R2 realisiert, der konstante Anteil (zu -k_offset) durch initiales Laden des Integrator-Registers RI aus dem Register R1. Die beiden Werte round(-2¹⁴·n/N·k_offset) und round(2¹⁴·2·n/N·T_s·B) der Register R1 und R2 sind gegenüber denen in Bez. (36) um den Faktor 2¹¹ skaliert, also effektiv wird mit 11 Nachkommastellen gerechnet, um insbesondere bei der sukzessiven Integration des Wertes von Register 11.12 eine genügende Genauigkeit zu erzielen (dieser Wert wird ja M/2 = 250 mal aufintegriert, d. h. dass sich der initiale Rundungsfehler dieses Registerwertes um den Faktor 250 verstärkt; und es sei bemerkt, dass wegen der Verwendung von nur geradzahligen Werten von m noch der zusätzliche Faktor 2 in round(2¹⁴·2·n/N·T_s·B) nötig ist). Da der Integrator mit 14 Bit in Zweierkomplementarithmetik realisiert ist und somit Überlauf über einen effektiven Wert 8 ignoriert, führt er inhärent die Modulo-Bildung nach Bez. (36) aus. Das Quantisieren auf den Bereich p_n,m = 0,...,7 wird durch Verwendung der oberen 3 Bits, also der 3 MSBs realisiert; tatsächlich wird dadurch aber nicht das Runden nach Bez. (36), sondern ein Abschneiden implementiert, was sich vom Runden im Mittel um 0.5 unterscheidet - da dieser mittlere Fehler über alle N Drehfaktoren (n = 0,...,N-1) gemacht wird, bedeutet er nur eine konstante Phasenverschiebung, was aus funktionaler Sicht nicht relevant ist (eine konstante Phasenverschiebung der Eingangsdaten der FFT bewirkt nur eine konstante Phasenverschiebung der FFT-Ausgangsdaten, womit sich deren Betrag, also deren einzig relevante Größe nicht ändert).Since the rotation factors d _n,m change over the shift m, i.e. when the calculation logic is clocked, a programmable structure is required for their implementation, which is shown for example in block 11.2 of the 11 for an n (in total, this structure exists N = 4096 times). From a complex input value with real part i _Re (n,m) and imaginary part i _lm (n,m), the complex output value with real part o _Re (n,m) and imaginary part o _lm (n,m) is formed by multiplying with a rotation factor d _n,m ; depending on which of the 8 rotation factors $${\underset{\_}{\text{d}}}_{\text{p}=\mathrm{0,}\dots \mathrm{,7}}=\mathrm{1,}0.75\cdot \left(+1+\widehat{\text{j}}\right),\widehat{\text{j}},0.75\cdot \left(-1+\widehat{\text{j}}\right),-\mathrm{1,}0.75\cdot \left(-1-\widehat{\text{j}}\right),-\widehat{\text{j}},0.75\cdot \left(+1-\widehat{\text{j}}\right)$$

is to be used, the 6 switches (between the value 0 or the respective input value) and the 4 switchable bit inverters are controlled from the logic 11.21. In this logic, the 10 required binary switching signals are calculated from the respective discrete phase value p _n,m = 0,...,7 at the input, i.e. from 3 bits. The respective value p _n,m , i.e. according to reference (34): $${\text{p}}_{\text{n},\text{m}}={\text{mod}}_8\left(\text{round}\left(8\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]\right)\right)\text{with m}=\mathrm{0,2,4}\dots ,\text{M}-\mathrm{2,}$$

is determined in block 11.1 using an integrator (i.e. a first-order recursive structure); the linear part with respect to the shift m (which therefore increases successively as the calculation structure is clocked through) is realized by recursively adding the value in register R2, the constant part (to -k _offset ) by initially loading the integrator register RI from register R1. The two values round(-2 ¹⁴ ·n/N·k _offset ) and round(2 ¹⁴ ·2·n/N·T _s· B) of the registers R1 and R2 are scaled by a factor of 2 ¹¹ compared to those in (36), so effectively 11 decimal places are used for the calculation in order to achieve sufficient accuracy, particularly in the successive integration of the value of register 11.12 (this value is integrated M/2 = 250 times, i.e. the initial rounding error of this register value is increased by a factor of 250; and it should be noted that because only even values of m are used, the additional factor 2 in round(2 ¹⁴ ·2·n/N·T _s ·B) is necessary). Since the integrator is implemented with 14 bits in two's complement arithmetic and thus ignores overflow over an effective value of 8, it inherently carries out the modulo formation according to (36). Quantization to the range p _n,m = 0,...,7 is implemented by using the upper 3 bits, i.e. the 3 MSBs; in fact, however, this does not implement rounding according to (36), but rather truncation, which differs from rounding on average by 0.5 - since this average error is made over all N rotation factors (n = 0,...,N-1), it only means a constant phase shift, which is not relevant from a functional point of view (a constant phase shift of the FFT input data only causes a constant phase shift of the FFT output data, which means that their absolute value, i.e. their only relevant size, does not change).

The realization of the multiplication with the rotation factors d _n,m according to 11 requires only a small amount of effort. With a little more effort, a normal complex-valued multiplier (best based on 4 real-valued multipliers) for programmable factors of length 3 bits (i.e. for values -1, -0.75, ..., 0.5, 0.75); because the value +1 cannot be realized, scaling the rotation factors, e.g. with a factor of 0.875, can be advantageous. The factors to be used for the four real-valued multipliers can be determined analogously to 11 can be derived from the respective value p _n,m using logic, whereby this can also be longer than 3 bits (since more than 8 different rotation factors can be realized with such a multiplier).

Compensating a non-linear frequency response

So far, an ideally linear curve has been assumed for the transmission frequency f _TX (t): $$\begin{array}{cc}{\text{e}}_{\text{TX}}\left(\text{t}\right)={\text{e}}_{\text{TX0}}+\text{t}\cdot \text{B}/{\text{T}}_{\text{pm}}& {\text{with T}}_{\text{pm}}=\text{N}\cdot {\text{T}}_{\text{m}}.\end{array}$$

auf und für die die Sendefrequenz ergibt sich: $${\text{f}}_{\text{TX}}\left(\text{t}\right)={\text{f}}_{\text{TX0}}+\text{t}\cdot \text{B}/{\text{T}}_{\text{pm}}+\text{Q}\cdot {\left(\text{t}-{\text{T}}_{\text{pm}}/2\right)}^2.$$

In reality, however, the modulation will not be perfect; often it has - as in 12 illustrated - a square error $${\text{e}}_{\text{TX},\text{Q}}\left(\text{t}\right)=\text{Q}\cdot {\left(\text{t}-{\text{T}}_{\text{pm}}/2\right)}^2$$

and for which the transmission frequency is: $${\text{e}}_{\text{TX}}\left(\text{t}\right)={\text{e}}_{\text{TX0}}+\text{t}\cdot \text{B}/{\text{T}}_{\text{pm}}+\text{Q}\cdot {\left(\text{t}-{\text{T}}_{\text{pm}}/2\right)}^2.$$

die beiden hinteren Terme beschreiben den Fehler $${\text{f}}_{\text{r},\text{Q}}\left(\text{t}\right)={\text{f}}_{\text{TX},\text{Q}}\left(\text{t}-{\text{t}}_{\text{0}}\right)-{\text{f}}_{\text{TX},\text{Q}}\left(\text{t}\right)$$

der laufzeitbedingten Frequenzverschiebung (welche in der Frequenzverschiebung nach Bez. (11a) für ideale Linearität nicht enthalten sind): $${\text{f}}_{\text{r},\text{Q}}\left(\text{t}\right)=-\text{Q}\cdot 2\cdot \left(\text{t}-{\text{T}}_{\text{pm}}/2\right)\cdot {\text{t}}_0+\text{Q}\cdot {\text{t}}_0{}^2.$$

Without Doppler shift, the reception frequency f _RX (t) corresponds to the transmission frequency f _TX (tt ₀ ) shifted by the transit time t ₀ ; the transit time-related frequency shift f _r (t) between the reception and transmission frequencies is then: $$\begin{array}{c}{\text{e}}_{\text{r}}\left(\text{t}\right)={\text{e}}_{\text{RX}}\left(\text{t}\right)-{\text{T}}_{\text{TX}}\left(\text{t}\right)={\text{t}}_{\text{TX}}\left(\text{t}-{\text{t}}_{\text{0}}\right)-{\text{e}}_{\text{TX}}\left(\text{t}\right)=\\ ={\text{e}}_{\text{TX0}}+\left(\text{t}-{\text{t}}_{\text{0}}\right)\cdot \text{B}/{\text{T}}_{\text{pm}}+\text{Q}\cdot {\left(\text{t}-{\text{t}}_{\text{0}}-{\text{T}}_{\text{pm}}/2\right)}^2-\left[{\text{e}}_{\text{TX0}}+\text{t}\cdot \text{B}/{\text{T}}_{\text{pm}}+\text{Q}\cdot {\left(\text{t}-{\text{T}}_{\text{pm}}/2\right)}^2\right]\\ =-{\text{t}}_{\text{0}}\cdot \text{B}/{\text{T}}_{\text{pm}}-\text{Q}\cdot 2\left(\text{t}-{\text{T}}_{\text{pm}}/2\right)\cdot {\text{t}}_{\text{0}}+\text{Q}\cdot {\text{t}}_{\text{0}}{}^{\text{2}};\end{array}$$

the two last terms describe the error $${\text{e}}_{\text{r},\text{Q}}\left(\text{t}\right)={\text{e}}_{\text{TX},\text{Q}}\left(\text{t}-{\text{t}}_{\text{0}}\right)-{\text{e}}_{\text{TX},\text{Q}}\left(\text{t}\right)$$

the time-dependent frequency shift (which is not included in the frequency shift according to (11a) for ideal linearity): $${\text{e}}_{\text{r},\text{Q}}\left(\text{t}\right)=-\text{Q}\cdot 2\cdot \left(\text{t}-{\text{T}}_{\text{pm}}/2\right)\cdot {\text{<figure-callout id="0" label="t" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">t</figure-callout>}}_0+\text{Q}\cdot {\text{<figure-callout id="0" label="t" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">t</figure-callout>}}_0{}^2.$$

und mit t = n·Ts (mit n = 0,_...,N-1), t₀ = m₀·T_m (siehe auch Bez. (3b)) und T_pm = N·T_m: $$\begin{array}{r}{\text{k}}_{\mathrm{0,}\text{Q}}\left(\text{n}\right)=\text{Q}\cdot {\text{N}}^{\text{2}}\cdot {\text{T}}_{\text{m}}{}^{\text{2}}\cdot {\text{T}}_{\text{s}}\cdot {\text{m}}_{\text{0}}+\text{Q}\cdot \text{N}\cdot {\text{T}}_{\text{m}}{}^2\cdot {\text{m}}_0{}^2-\text{Q}\cdot 2\cdot \text{N}\cdot {\text{T}}_{\text{m}}\cdot {\text{T}}_{\text{s}}{}^2\cdot {\text{m}}_{\text{0}}\cdot \text{n}\\ \text{mit n}=\mathrm{0,}\dots ,\text{N}-1.\end{array}$$

After mixing and digitization, this error is transferred directly to the reception sequence e(n), i.e. to the error k _0,Q of the discrete reception frequency and analogous to (13a) this results in: $${\text{k}}_{\mathrm{0,}\text{Q}}=\text{N}\cdot {\text{T}}_{\text{s}}\cdot {\text{e}}_{\text{r},\text{Q}}\left(\text{t}\right)=\text{N}\cdot {\text{T}}_{\text{s}}\cdot \left(-\text{Q}\cdot 2\cdot \text{t}\cdot {\text{t}}_0+\text{Q}\cdot {\text{T}}_{\text{pm}}\cdot {\text{t}}_0+\text{Q}\cdot {\text{t}}_0{}^2\right),$$

and with t = n·Ts (with n = 0, _... ,N-1), t ₀ = m ₀ ·T _m (see also equation (3b)) and T _pm = N·T _m : $$\begin{array}{r}{\text{<figure-callout id="0" label="k" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">k</figure-callout>}}_{\mathrm{0,}\text{Q}}\left(\text{n}\right)=\text{Q}\cdot {\text{<figure-callout id="2" label="N" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">N</figure-callout>}}^{\text{2}}\cdot {\text{<figure-callout id="2" label="T m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">T</figure-callout>}}_{\text{m}}{}^{\text{2}}\cdot {\text{T}}_{\text{s}}\cdot {\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\text{0}}+\text{Q}\cdot \text{N}\cdot {\text{<figure-callout id="2" label="T m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">T</figure-callout>}}_{\text{m}}{}^2\cdot {\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_0{}^2-\text{Q}\cdot 2\cdot \text{N}\cdot {\text{T}}_{\text{m}}\cdot {\text{T}}_{\text{s}}{}^2\cdot {\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\text{0}}\cdot \text{n}\\ \text{with n}=\mathrm{0,}\dots ,\text{N}-1.\end{array}$$

Over the N values of the reception sequence e(n), the first two terms are constant, i.e. they represent a constant frequency error and in the two-dimensional correlation E _m,k only shift the position of the magnitude peak in the frequency dimension k; in contrast, the last term changes linearly over the discrete time n, i.e. the frequency changes during the reception sequence, which leads to a broadened magnitude peak in the frequency dimension k in the two-dimensional correlation E _m,k - the width is approximately Q·2·N ² ·T _s ² ·T _m ·m ₀ . Shifting and broadening of the magnitude peak decrease with the discrete running time m ₀ , i.e. increases with the object distance. While the effect of the shift can easily be taken into account when converting frequency k ₀ into relative speed, the broadening leads to a loss of sensitivity (the height of the magnitude peak is reduced), to a less precise determination of the position of the magnitude peak and thus of the relative speed, and to a poorer separation of two objects with similar relative speed at the same distance.

bei dieser Aufleitung ist die Integrationskonstante weggelassen, weil ein konstanter Phasenanteil funktional nicht relevant ist. In Darstellung für die diskrete Zeit n ergibt sich mit t = n·Ts, t₀ = m₀·T_m und T_pm = N·T_m: $$\begin{array}{r}{\mathrm{\phi }}_{\text{r},\text{Q}}\left(\text{n}\right)=-2\mathrm{\pi }\cdot \text{Q}\cdot {\text{T}}_{\text{m}}\cdot {\text{T}}_{\text{s}}{}^2\cdot {\text{m}}_{\text{0}}\cdot {\text{n}}^{\text{2}}+2\mathrm{\pi }\cdot \text{Q}\cdot \text{N}\cdot {\text{T}}_{\text{m}}{}^{\text{2}}\cdot {\text{T}}_{\text{s}}\cdot {\text{m}}_{\text{0}}\cdot \text{n}+2\mathrm{\pi }\cdot \text{Q}\cdot {\text{T}}_{\text{m}}{}^2\cdot {\text{T}}_{\text{s}}\cdot {\text{m}}_{\text{0}}{}^2\cdot \text{n}\\ \text{mit n}=\mathrm{0,}\dots ,\text{N}-1.\end{array}$$

In the following, it is explained how this broadening of the magnitude peak can be avoided. The phase error results from the frequency error f _r,Q (t) according to (39). $$\begin{array}{c}{\mathrm{\phi }}_{\text{r},\text{Q}}\left(\text{t}\right)=2\mathrm{\pi }\cdot {\dot{\text{e}}}_{\text{r},\text{Q}}\left(\text{t}\right)=2\mathrm{\pi }\cdot \left({\dot{\text{e}}}_{\text{TX},\text{Q}}\left(\text{t}-{\text{t}}_0\right)-{\dot{\text{e}}}_{\text{TX},\text{Q}}\left(\text{t}\right)\right)\\ =-2\mathrm{\pi }\cdot \text{Q}\cdot {\text{t}}_0\cdot {\text{t}}^{\text{2}}+2\mathrm{\pi }\cdot \text{Q}\cdot {\text{T}}_{\text{pm}}\cdot {\text{t}}_{\text{0}}\cdot \text{t}+2\mathrm{\pi }\cdot \text{Q}\cdot {\text{t}}_0{}^2\cdot \text{t;}\end{array}$$

In this derivation, the integration constant is omitted because a constant phase component is not functionally relevant. In representation for the discrete time n, with t = n·Ts, t ₀ = m ₀ ·T _m and T _pm = N·T _m , we get: $$\begin{array}{r}{\mathrm{\phi }}_{\text{r},\text{Q}}\left(\text{n}\right)=-2\mathrm{\pi }\cdot \text{Q}\cdot {\text{T}}_{\text{m}}\cdot {\text{T}}_{\text{s}}{}^2\cdot {\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\text{0}}\cdot {\text{<figure-callout id="2" label="n" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">n</figure-callout>}}^{\text{2}}+2\mathrm{\pi }\cdot \text{Q}\cdot \text{N}\cdot {\text{<figure-callout id="2" label="T m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">T</figure-callout>}}_{\text{m}}{}^{\text{2}}\cdot {\text{T}}_{\text{s}}\cdot {\text{m}}_{\text{0}}\cdot \text{n}+2\mathrm{\pi }\cdot \text{Q}\cdot {\text{<figure-callout id="2" label="T m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">T</figure-callout>}}_{\text{m}}{}^2\cdot {\text{T}}_{\text{s}}\cdot {\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\text{0}}{}^2\cdot \text{n}\\ \text{with n}=\mathrm{0,}\dots ,\text{N}-1.\end{array}$$

stattfindet (es sei bemerkt, dass bei einem Ansatz wie in 10 für m nur die Untermenge der geradzahlige Werte zu benutzen ist). Die Realisierung dieser Drehfaktoren kann mit der Realisierung den obigen Drehfaktoren d_n,m nach Bez. (27) oder (31) zur Kompensierung der laufzeitabhängigen Frequenzverschiebung und eines gegebenenfalls unsymmetrischen Dopplerfrequenzbereichs kombiniert werden - es sind dann Drehfaktoren mit der Summe der Phasen der beiden einzelnen Drehfaktoren zu implementieren. Die ersten beiden Phasenanteile in d_n,m,Q nach Bez. (43) sind linear in der Verschiebung m, so dass sie bei einer Realisierung der Drehfaktoren analog zu 11 als zusätzlicher Teil des Registers R2, welches Eingang des Integrators ist, implementierbar sind: $$\text{R2}=\text{round}\left(2^{14}\cdot 2\cdot \left[\text{n}/\text{N}\cdot {\text{T}}_{\text{s}}\cdot \text{B}+\text{Q}\cdot {\text{T}}_{\text{m}}\cdot {\text{T}}_{\text{s}}{}^2\cdot \text{m}\cdot {\text{n}}^2-\text{Q}\cdot \text{N}\cdot {\text{T}}_{\text{m}}{}^2\cdot {\text{T}}_{\text{s}}\cdot \text{m}\cdot \text{n}\right]\right);$$

der zusätzliche Teil sind die beiden hinteren Anteile proportional zur Stärke Q des quadratischen Frequenzfehlers. Der hintere, zur Verschiebung m quadratische Anteil in Bez. (44) ist häufig vernachlässigbar, da er wegen M << N deutlich kleiner als die anderen Anteile ist; falls er aber doch realisiert werden soll, benötigt er einen Integrator zweiter Ordnung - es gibt also dann eine drittes Register, das in einem vorgeschalteten Integrator aufintegriert wird, und der Ausgang dieses Integrators geht dann als weiterer Eingang in den Integrator nach 11.This phase error for an object with the discrete transit time m ₀ can be corrected by adding the negative of the above phase error for each displacement m when calculating the two-dimensional correlation E _m,k by multiplying the product sequence e(n)·b(nm) by the rotation factor before the FFT. $$\begin{array}{r}\hfill \begin{array}{c}{\text{d}}_{\text{n},\text{m},\text{Q}}=\text{ex}\left(-\widehat{\text{j}}\cdot {\mathrm{\phi }}_{\text{r},\text{Q}}\left(\text{n}\right)\right)\\ =\text{ex}\left(\widehat{\text{j}}\cdot \text{2}\mathrm{\pi }\cdot \left[\text{Q}\cdot {\text{T}}_{\text{m}}\cdot {\text{T}}_{\text{s}}{}^{\text{2}}\cdot \text{m}\cdot {\text{<figure-callout id="2" label="n" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">n</figure-callout>}}^{\text{2}}-\text{Q}\cdot \text{N}\cdot {\text{<figure-callout id="2" label="T m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">T</figure-callout>}}_{\text{m}}{}^2\cdot {\text{T}}_{\text{s}}\cdot \text{m}\cdot \text{n}-\text{Q}<figure-callout\; id="2"\; label="\cdot \; T\; m"\; filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png"\; state="\{\{state\}\}">\cdot </figure-callout>{\text{T}}_{\text{m}}{}_{}{}^2\cdot {\text{T}}_{\text{s}}\cdot {\text{m}}^2\cdot \text{n}\right]\right)\end{array}\\ \hfill \text{with n}=\mathrm{0,}\dots ,\text{N}-1\text{and m}=\mathrm{0,}\dots ,\text{M}-1\end{array}$$

takes place (it should be noted that with an approach such as in 10 for m only the subset of even-numbered values is to be used). The realization of these rotation factors can be combined with the realization of the above rotation factors d _n,m according to ref. (27) or (31) to compensate for the runtime-dependent frequency shift and a possibly asymmetrical Doppler frequency range - rotation factors with the sum of the phases of the two individual rotation factors are then to be implemented. The first two phase components in d _n,m,Q according to ref. (43) are linear in the shift m, so that when the rotation factors are realized analogously to 11 can be implemented as an additional part of the register R2, which is the input of the integrator: $$\text{R2}=\text{round}\left(2^{14}\cdot 2\cdot \left[\text{n}/\text{N}\cdot {\text{T}}_{\text{s}}\cdot \text{B}+\text{Q}\cdot {\text{T}}_{\text{m}}\cdot {\text{T}}_{\text{s}}{}^2\cdot \text{m}\cdot {\text{n}}^2-\text{Q}\cdot \text{N}\cdot {\text{T}}_{\text{m}}{}^2\cdot {\text{T}}_{\text{s}}\cdot \text{m}\cdot \text{n}\right]\right);$$

the additional part are the two rear parts proportional to the strength Q of the squared frequency error. The rear part, which is squared to the shift m in reference (44), is often negligible, since it is significantly smaller than the other parts due to M <<N; if it is to be implemented, however, it requires a second order integrator - so there is then a third register that is integrated in an upstream integrator, and the output of this integrator then goes as a further input to the integrator after 11 .

The correction of the frequency error can in principle take place after the FFT instead of before it; the multiplication with the rotation factors d _n,m,Q , n=0,...,N-1 before the FTT corresponds to a convolution of the spectrum (i.e. the DFT or FFT) of this rotation factor sequence. In the case of the quadratic frequency error considered, this spectrum has a broad magnitude peak around zero (the width of the magnitude peak increases with shift m); this means that the convolution can be restricted to the summation over a few values, weighted with complex factors - nevertheless, such an implementation would be significantly more complex than the approach presented above with correction before FFT, also because these weighting factors depend on the shift m.

So far, a squared error of the transmission frequency according to (37) has been considered. The above considerations can also be applied to general errors f _TX,Q (t) of the transmission frequency, i.e. general deviation The rotation factors by which the product sequence e(n)·b(nm) is to be multiplied to correct the transmission frequency error are in general form using equation (41a) and t = n·T _s and t ₀ = m _0· T _m : $$\begin{array}{c}{\text{d}}_{\text{n},\text{m},\text{Q}}=\text{ex}\left(-\widehat{\text{j}}\cdot {\mathrm{\phi }}_{\text{r},\text{Q}}\left(\text{t}\right)\right)=\text{ex}\left(-\widehat{\text{j}}\cdot \text{2}\mathrm{\pi }\cdot {\dot{\text{e}}}_{\text{r},\text{Q}}\left(\text{t}\right)\right)=\text{ex}\left(-\widehat{\text{j}}\cdot \text{2}\mathrm{\pi }\cdot \left[{\dot{\text{e}}}_{\text{TX},\text{Q}}\left(\text{t}-{\text{t}}_{\text{0}}\right)-{\dot{\text{e}}}_{\text{TX},\text{Q}}\left(\text{t}\right)\right]\right).\\ =\text{ex}\left(-\widehat{\text{j}}\cdot \text{2}\mathrm{\pi }\cdot \left[{\dot{\text{e}}}_{\text{TX},\text{Q}}\left(\text{n}\cdot {\text{T}}_{\text{s}}-{\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\text{0}}\cdot {\text{T}}_{\text{m}}\right)-{\dot{\text{e}}}_{\text{TX},\text{Q}}\left(\text{n}\cdot {\text{T}}_{\text{s}}\right)\right]\right).\end{array}$$

These rotation factors can be combined with the rotation factors d _n,m according to (31) to compensate for the time-dependent frequency shift and an asymmetrical Doppler frequency range. For a realization analogous to 11 The discrete phase values p _n,m with value range 0,1,...,7 (i.e. length 3 bits) are then in extension of the relation (36), which referred to an ideal linear frequency modulation: $${\text{p}}_{\text{n},\text{m}}={\text{mod}}_8\left(\text{round}\left(8\cdot \text{n}/\text{N}\cdot \left[\text{m}\cdot {\text{T}}_{\text{s}}\cdot \text{B}-{\underset{\_}{\text{k}}}_{\text{offset}}\right]-8\cdot \left({\dot{\text{e}}}_{\text{TX},\text{Q}}\left(\text{n}\cdot {\text{T}}_{\text{s}}-{\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\text{0}}\cdot {\text{T}}_{\text{m}}\right)-{\dot{\text{e}}}_{\text{TX},\text{Q}}\left(\text{n}\cdot {\text{T}}_{\text{s}}\right)\right)\right)\right).$$

Block 11.1 in 11 to generate these discrete phase values can be replaced by a register into which the respective pre-calculated phase value is written. Instead of a register for just one 3-bit long phase value, a long register can also be used into which the discrete phase values are written across all m, i.e. in the example considered above, M/2 = 250 values of length 3 bits, whereby this register is then clocked through, i.e. shifted through, m; this has the advantage that no writing process has to be carried out during the calculation of a pixel and that if the transmission frequency errors are the same across several pixels, writing only has to be done once (in this case, cyclic shifting must be used). This approach with pre-calculated register values can also be combined with an approach according to 11 be combined with an integrator circuit, so that only the part that cannot be realized with an integrator circuit has to be in these registers and this may lead to the fact that one register can be used for the generation of several phase values p _n,m , i.e. for different n and/or m, which reduces the number of registers required and the pre-calculation effort (also for a microcontroller).

In order to compensate for a non-linear frequency curve, you have to know the errors, i.e. deviations from the linear curve. As explained above, uncompensated errors in the frequency curve lead in particular to broadened magnitude peaks in the frequency dimension k; for the assumption of a quadratic error, there is a fixed relationship between the width of the magnitude peak and the error strength Q (see also above). A non-linear frequency curve across pixels also leads to angular errors (this will be discussed in more detail later, including how such angular errors can be determined); if these angular errors are known, they can be used to determine the error in the frequency curve - not only across pixels, but also within a pixel, since the error curve usually has a square shape at least in sections.

Compensation of internal couplings and reflections from cover

Above it was explained that in block 8.1 or 10.1 of the hard-wired calculation logic according to 8 or 10 Correction values c ₁ (n) are added to the reception sequence in order to compensate for the effects of couplings and reflections within the lidar system or from its immediate surroundings, in particular a cover. It should be mentioned that radar sensors that are based on the same concept of coherent operation as coherent lidar sensors, just in a different electromagnetic frequency range, are known to be advantageous in compensating for such effects. We will now explain how these correction values can be determined in a coherent lidar system with phase modulation and how they can be easily implemented.

If one assumes that, on the one hand, the discrete distance m = 0 is exactly at the distance zero and, on the other hand, that the distance of these couplings and reflections is also negligibly small (therefore, the reception frequency f _e = 0; the Doppler component is also zero), then the level P of these couplings and reflections generated in the reception sequence can be determined by averaging over the product of the reception sequence e(n) and the unshifted modulation sequence b(n) (the signal component in the reception sequence originating from real objects has a lower level and, on the other hand, it is largely averaged out because it corresponds to a shifted modulation sequence and also generally has a reception frequency f _e ≠ 0). The correction values c ₁ (n) then result as the thus determined Level P multiplied by the unshifted modulation sequence b(n) with the values ±1. The determination of this mean value (by summing over the N = 4096 values e(n)·b(n) and shifting the result by 12 bits to the right) can be carried out either outside or inside the hard-wired computing logic. When implemented within the computing logic, either an additional block can be implemented for this purpose or the value of the two-dimensional correlation E _m,k at the discrete distance m = 0 and the discrete frequency k = 0, which corresponds to the sum of e(n)·b(n), is used directly in the existing computing logic. If the correlation result is used, then it must be taken into account that the computing logic is designed in the form of a pipeline and therefore the calculation takes several cycles; If the level value calculated via the two-dimensional correlation is to be used in the same pixel, the shifting of the input sequence in block 8.2 or 10.2, i.e. the clocking of this block, must be suspended for the time being. Alternatively, the level value from the previous cycle or from previously processed neighboring pixels could be used (provided that the value P does not change significantly from pixel to pixel). For the level value P determined in this way, the hard-wired calculation logic assigns the values ±P to the correction values c ₁ (n); if the same modulation sequence b(n) is always used, the respective sign can be implemented hard-wired (negative sign preferably only by bit-by-bit inversion); if b(n) can change, switchable inverters are required.

Due to the smoothing of the received pulse shape described above and because the reflection of a cover in particular can have a not entirely negligible transit time, a signal component may still be visible even at the discrete distance m = 1. The level in the received sequence e(n) is then not only dependent on the respective value of b(n), but also on the previous value b(n-1), so that two average values must be determined: one over the time values n for which b(n) and b(n-1) have the same sign, and one for unequal signs of b(n) and b(n-1). These two level values are then assigned to the correction values c ₁ (n) in the hard-wired logic with the correct sign, depending on whether b(n) and b(n-1) have the same or opposite sign to the respective n. In principle, these two level values could also be determined from the two sums of e(n)·b(n) and e(n)·b(n-1) over all n, whereby these sums are either determined in an explicit calculation or taken from the two-dimensional correlation. If the received pulse is smoothed and/or shifted to such an extent that a value of the received sequence e(n) is influenced by three neighboring values of b(n), the method described above must be extended accordingly.

Due to the linear frequency change superimposed on the phase modulation, the reception sequence can have a low frequency, particularly from the reflection of a cover; even if only a fraction of a period is passed through during the N = 4096 values of a pixel, the compensation described above with its assumption of a reception frequency of zero and thus a constant phase would no longer be completely effective. To reduce the effect of the frequency, level values could be determined in sections, e.g. over 4 sections of length N/4 = 1024. Alternatively, if the small reception frequency is known, it could be eliminated before determining the level values by multiplying it by the corresponding rotation factors and then these rotation factors could be used again when determining the correction values c ₁ (n); however, this is quite complex. The level values can also be determined again from the two-dimensional correlation; an unknown frequency can also be determined using interpolation.

If the effects of coupling and reflection are at least approximately constant over time and/or over small pixel areas, an average can be taken over acquisition cycles and/or pixel areas. However, it is generally not the case that the effects are constant across all pixels, since, for example, different beam directions result in different phase positions of the reflections from a cover.

Determination of the sign of the reception frequency in a real-valued mixer using linear frequency modulation

The following sections will explain the advantages and system approaches that enable the combination of phase modulation and a changing frequency.

setzt sich die Empfangsfrequenz f_e nach Mischung und auch nach Digitalisierung aus der durch die Relativgeschwindigkeit v generierte Dopplerverschiebung f_D und der durch die lineare Frequenzmodulation bewirkten Frequenzverschiebung f_r zusammen, wobei f_r proportional zur Objektentfernung r ist, die mit Hilfe der Phasenmodulation bestimmt wird. Der zum relevanten Relativgeschwindigkeitsbereich v_min,...,v_max korrespondierende Bereich der Empfangsfrequenz f_e verschiebt sich somit über die Objektentfernung r; dies ist in 13 für v_min = -80km/h und v_max = +280km/h sowie die oben betrachtete Auslegung der linearen Frequenzmodulation, d. h. B = 800MHz und T_pm = 13.7µs dargestellt. Als Beispiel werde die Objektentfernung r = 200m betrachtet; der relevante Frequenzbereich erstreckt sich über f_e = -106.,...,22.3MHz. Da bei einem reellwertigen Mischer nur der Betrag, aber nicht das Vorzeichen der Empfangsfrequenz bestimmt werden kann, kommt es für die Empfangsfrequenz im Bereich f_e = -22.3,...,22.3MHz, welcher zum Relativgeschwindigkeitsbereich v = 156,...,280km/h korrespondiert, zu Mehrdeutigkeit - es wird also z. B. nicht zwischen den zwei Relativgeschwindigkeiten v = 156km/h und v = 280km/h unterschieden, oder auch nicht zwischen v = 186km/h und v = 250km/h; für alle Empfangsfrequenzen mit gemessenem Betrag |f_e| > 22.3MHz ist nur das negative Vorzeichen möglich, so dass sich die Relativgeschwindigkeit eindeutig bestimmen lässt. In 13 ist der mehrdeutige Bereich der Empfangsfrequenz schraffiert dargestellt. Für die maximale Objektentfernung 249.5m liegt fast der gesamte relevante Frequenzbereich f_e = -126.2,...,2.9MHz im Negativen, so dass es abgesehen vom Relativgeschwindigkeitsbereich v = 265,...,280km/h zu keinen Mehrdeutigkeiten kommt.In the lidar system considered so far, 1 the mixer is designed as a complex value, which represents a significant additional effort (almost twice) for the reception path compared to a real-valued mixer (with only one real-valued output). When using a real-valued mixer, however, only the magnitude of the reception frequency can be determined, not the sign, since there are two magnitude peaks in the correlation E _m,k at (m ₀ ,+k ₀ ) and (m ₀ ,-k ₀ ). In the case of a constant transmission frequency according to the state of the art, only the magnitude, but not the sign, of the radial relative velocity can be determined. To determine the sign, state-of-the-art approaches can be used via tracking, ie tracking over several recording cycles, and/or via plausibility checks, e.g. whether the measured value of the radial relative speed corresponds to a stationary object, but this also has disadvantages. These disadvantages can be reduced by the linear frequency modulation superimposed on the phase modulation. According to reference (12): $${\text{e}}_{\text{e}}={\text{e}}_{\text{D}}+{\text{e}}_{\text{r}}=\text{2v}/\mathrm{\lambda }-\text{2r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}},$$

the reception frequency f _e after mixing and also after digitization is composed of the Doppler shift f _D generated by the relative speed v and the frequency shift f _r caused by the linear frequency modulation, where f _r is proportional to the object distance r, which is determined using the phase modulation. The range of the reception frequency f _e corresponding to the relevant relative speed range v _min ,...,v _max thus shifts over the object distance r; this is in 13 for v _min = -80km/h and v _max = +280km/h as well as the design of the linear frequency modulation considered above, i.e. B = 800MHz and T _pm = 13.7µs. As an example, consider the object distance r = 200m; the relevant frequency range extends over f _e = -106.,...,22.3MHz. Since with a real-valued mixer only the magnitude and not the sign of the received frequency can be determined, there is ambiguity for the received frequency in the range f _e = -22.3,...,22.3MHz, which corresponds to the relative speed range v = 156,...,280km/h - for example, no distinction is made between the two relative speeds v = 156km/h and v = 280km/h, or between v = 186km/h and v = 250km/h; for all reception frequencies with measured value |f _e | > 22.3MHz only the negative sign is possible, so that the relative speed can be determined unambiguously. In 13 the ambiguous range of the reception frequency is shown hatched. For the maximum object distance of 249.5m, almost the entire relevant frequency range f _e = -126.2,...,2.9MHz is negative, so that apart from the relative speed range v = 265,...,280km/h, there are no ambiguities.

In order to reduce the range of the receiving frequency with ambiguity, a negative modulation bandwidth can be selected, i.e. with the same amount as above B = -800MHz, ie the transmitting frequency decreases linearly over time. In 14 the conditions for the reception frequency are shown; for object distances r > 73.4m there are no more ambiguities. However, the maximum reception frequency that occurs is now higher in magnitude than with a positive modulation bandwidth (198.1MHz instead of 126.2MHz), so that a higher sampling frequency of at least about f _s = 450MHz is necessary (instead of f _s = 300MHz). By increasing the amount of the modulation bandwidth B, the distance range with ambiguities can be reduced even further; for double the amount and negative sign of the modulation bandwidth, i.e. B = -1600MHz, according to 15 only for object distances r < 36.7m ambiguities arise; however, a further increase in the sampling frequency is then necessary. It should also be mentioned that a higher sampling frequency (and more samples per pixel, whose data acquisition time should remain unchanged) means that the FFT for calculating the two-dimensional correlation can have an unchanged length if - as explained above - the runtime-related part of the frequency shift is eliminated before the FFT by multiplying the corresponding rotation factors and then a corresponding decimation takes place.

With the help of the runtime-dependent frequency shift effect of linear frequency modulation, the ambiguity problem that occurs with a real-valued mixer when determining the relative speed can be reduced to closer object distances. Since tracks are usually already set up in tracking there, the ambiguity can be resolved by assigning the detections generated from a single acquisition cycle to the tracks.

und im Falle der invertierten Modulationsbandbreite -B: $${\text{f}}_{\text{e}\mathrm{,2}}={\text{f}}_{\text{D}}-{\text{f}}_{\text{r}}=\text{2v}/\mathrm{\lambda }+\text{2r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}},$$

wobei die laufzeitabhängige Frequenzverschiebung $${\text{f}}_{\text{r}}=-\text{2r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}}$$

bekannt ist, da die Objektentfernung r ja mit Hilfe der Phasenmodulation bestimmt wird. Die Dopplerverschiebung f_D ergibt sich durch Summation dieser beiden Beziehungen zu $${\text{f}}_{\text{D}}=\left({\text{f}}_{\text{e}\mathrm{,1}}+{\text{f}}_{\text{e}\mathrm{,2}}\right)/2.$$

If the lowest possible sampling frequencies are to be used, only small modulation bandwidths are possible, so that ambiguities in the determination of the relative speed can arise over the entire distance range. This can be resolved by changing the modulation bandwidth over acquisition cycles, in particular by alternating its sign. For the reception frequency, in the case of the modulation bandwidth +B, the following is obtained: $${\text{e}}_{\text{e}\mathrm{,1}}={\text{e}}_{\text{D}}+{\text{e}}_{\text{r}}=\text{2v}/\mathrm{\lambda }-\text{2r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}},$$

and in the case of the inverted modulation bandwidth -B: $${\text{e}}_{\text{e}\mathrm{,2}}={\text{e}}_{\text{D}}-{\text{e}}_{\text{r}}=\text{2v}/\mathrm{\lambda }+\text{2r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}},$$

where the time-dependent frequency shift $${\text{e}}_{\text{r}}=-\text{2r}/\text{c}\cdot \text{B}/{\text{T}}_{\text{pm}}$$

is known, since the object distance r is determined using phase modulation. The Doppler shift f _D is obtained by summing these two relationships to $${\text{e}}_{\text{D}}=\left({\text{e}}_{\text{e}\mathrm{,1}}+{\text{e}}_{\text{e}\mathrm{,2}}\right)/2.$$

und für den Betrag dieser Differenz mit |fD| < f_r: $$\left|\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right|=2\cdot \left|{\text{f}}_{\text{D}}\right|<2\cdot {\text{f}}_{\text{r}}\text{ f}\ddot{\text{u}}\text{r }\left|{\text{f}}_{\text{D}}\right|<{\text{f}}_{\text{r}}.$$

When using a real-valued mixer, however, only the amounts |f _e,1 | and |f _e,2 | of the two receiving frequencies are measured. How the relative speed v can be clearly determined from this will now be derived for a positive time-dependent frequency shift f _r . If the Doppler shift f _D is less than this f _r (i.e. |f _D | < f _r ), then according to equation (47) f _e,1 > 0, i.e. f _e,1 = |f _e,1 |, and f _e,2 < 0, i.e. f _e,2 = -|f _e,2 |, and the difference between the amounts of the receiving frequencies is then given using equation (47): $$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|={\text{e}}_{\text{e}\mathrm{,1}}-\left({\text{e}}_{\text{e}\mathrm{,2}}\right)={\text{e}}_{\text{e}\mathrm{,1}}+{\text{e}}_{\text{e}\mathrm{,2}}=2\cdot {\text{e}}_{\text{D}}\text{e}\ddot{\text{u}}\text{r}\left|{\text{e}}_{\text{D}}\right|<{\text{e}}_{\text{r}},$$

and for the amount of this difference with |fD| < f _r : $$\left|\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right|=2\cdot \left|{\text{e}}_{\text{D}}\right|<2\cdot {\text{e}}_{\text{r}}\text{e}\ddot{\text{u}}\text{r}\left|{\text{e}}_{\text{D}}\right|<{\text{e}}_{\text{r}}.$$

während für Dopplerverschiebung f_D ≤ -f_r beide Empfangsfrequenzen negativ sind, also f_e,1 = -|f_e,1| und f_e,2 = -|f_e,2|, und somit gilt: $$\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|=-{\text{f}}_{\text{e}\mathrm{,1}}-\left(-{\text{f}}_{\text{e}\mathrm{,2}}\right)=-{\text{f}}_{\text{e}\mathrm{,1}}+{\text{f}}_{\text{e}\mathrm{,2}}=-2\cdot {\text{f}}_{\text{r}}{\text{ f}\ddot{\text{u}}\text{r f}}_{\text{D}}\le -{\text{f}}_{\text{r}}.$$

For Doppler shift f _D ≥ f _r both reception frequencies are non-negative, i.e. f _e,1 = |f _e,1 | and f _e,2 = |f _e,2 |, so that using the relation (47) the following applies: $$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|={\text{e}}_{\text{e}\mathrm{,1}}-{\text{e}}_{\text{e}\mathrm{,2}}=2\cdot {\text{e}}_{\text{r}}{\text{e}\ddot{\text{u}}\text{r f}}_{\text{D}}\ge {\text{e}}_{\text{r}},$$

while for Doppler shift f _D ≤ -f _r both reception frequencies are negative, i.e. f _e,1 = -|f _e,1 | and f _e,2 = -|f _e,2 |, and thus: $$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|=-{\text{e}}_{\text{e}\mathrm{,1}}-\left(-{\text{e}}_{\text{e}\mathrm{,2}}\right)=-{\text{e}}_{\text{e}\mathrm{,1}}+{\text{e}}_{\text{e}\mathrm{,2}}=-2\cdot {\text{e}}_{\text{r}}{\text{e}\ddot{\text{u}}\text{r f}}_{\text{D}}\le -{\text{e}}_{\text{r}}.$$

$$\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|=2\cdot {\text{f}}_{\text{r}}\to {\text{f}}_{\text{D}}\ge {\text{f}}_{\text{r}},$$

$$\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|=-2\cdot {\text{f}}_{\text{r}}\to {\text{f}}_{\text{D}}\le -{\text{f}}_{\text{r}}.$$

Thus, the three cases |f _D | < f _r (i.e. -f _r < f _D < f _r ), f _D ≥ f _r and f _D ≤ -f _r can be clearly distinguished from the difference |f _e,1 | - |f _e,2 | of the measured values of the two reception frequencies and thus determined: $$\left|\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right|<2\cdot {\text{e}}_{\text{r}}\to \left|{\text{e}}_{\text{D}}\right|<{\text{e}}_{\text{r}},$$

$$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|=2\cdot {\text{e}}_{\text{r}}\to {\text{e}}_{\text{D}}\ge {\text{e}}_{\text{r}},$$

$$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|=-2\cdot {\text{e}}_{\text{r}}\to {\text{e}}_{\text{D}}\le -{\text{e}}_{\text{r}}.$$

$$\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|=2\cdot {\text{f}}_{\text{r}}\to {\text{f}}_{\text{D}}\ge {\text{f}}_{\text{r}}\to {\text{f}}_{\text{e}\mathrm{,1}}=\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|,{\text{ f}}_{\text{e}\mathrm{,2}}=\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\to {\text{f}}_{\text{D}}=\left(\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right)/\mathrm{2,}$$

$$\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|=-2\cdot {\text{f}}_{\text{r}}\to {\text{f}}_{\text{D}}\le -{\text{f}}_{\text{r}}\to {\text{f}}_{\text{e}\mathrm{,1}}=-\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|,{\text{ f}}_{\text{e}\mathrm{,2}}=-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\to {\text{f}}_{\text{D}}=\left(-\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right)/2.$$

For each of these three cases, the sign of the two reception frequencies is clearly defined (see above), so that the Doppler shift f _D can be determined unambiguously using (48): $$\left|\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right|<2\cdot {\text{e}}_{\text{r}}\to \left|{\text{e}}_{\text{D}}\right|<{\text{e}}_{\text{r}}\to {\text{e}}_{\text{e}\mathrm{,1}}=\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|,{\text{e}}_{\text{e}\mathrm{,2}}=-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\to {\text{e}}_{\text{D}}=\left(\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)/\mathrm{2,}$$

$$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|=2\cdot {\text{e}}_{\text{r}}\to {\text{e}}_{\text{D}}\ge {\text{e}}_{\text{r}}\to {\text{e}}_{\text{e}\mathrm{,1}}=\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|,{\text{e}}_{\text{e}\mathrm{,2}}=\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\to {\text{e}}_{\text{D}}=\left(\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)/\mathrm{2,}$$

$$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|=-2\cdot {\text{e}}_{\text{r}}\to {\text{e}}_{\text{D}}\le -{\text{e}}_{\text{r}}\to {\text{e}}_{\text{e}\mathrm{,1}}=-\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|,{\text{e}}_{\text{e}\mathrm{,2}}=-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\to {\text{e}}_{\text{D}}=\left(-\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)/2.$$

$$\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|=2\cdot \left|{\text{f}}_{\text{r}}\right|\to {\text{f}}_{\text{D}}\le -\left|{\text{f}}_{\text{r}}\right|\to {\text{f}}_{\text{e}\mathrm{,1}}=-\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|,{\text{ f}}_{\text{e}\mathrm{,2}}=-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\to {\text{f}}_{\text{D}}=\left(-\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right)/\mathrm{2,}$$

$$\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|=-2\cdot \left|{\text{f}}_{\text{r}}\right|\to {\text{f}}_{\text{D}}\ge \left|{\text{f}}_{\text{r}}\right|\to {\text{f}}_{\text{e}\mathrm{,1}}=\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|,{\text{ f}}_{\text{e}\mathrm{,2}}=\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\to {\text{f}}_{\text{D}}=\left(\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right)/2.$$

So far, a positive sign of the delay-related frequency shift f _r has been considered (i.e., due to reference (11b), modulation bandwidth B <0); the analogous considerations are valid for a negative sign (i.e., modulation bandwidth B > 0): $$\left|\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right|<2\cdot \left|{\text{e}}_{\text{r}}\right|\to \left|{\text{e}}_{\text{D}}\right|<\left|{\text{e}}_{\text{r}}\right|\to {\text{e}}_{\text{e}\mathrm{,1}}=-\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|,{\text{e}}_{\text{e}\mathrm{,2}}=-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\to {\text{e}}_{\text{D}}=\left(-\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)/\mathrm{2,}$$

$$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|=2\cdot \left|{\text{e}}_{\text{r}}\right|\to {\text{e}}_{\text{D}}\le -\left|{\text{e}}_{\text{r}}\right|\to {\text{e}}_{\text{e}\mathrm{,1}}=-\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|,{\text{e}}_{\text{e}\mathrm{,2}}=-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\to {\text{e}}_{\text{D}}=\left(-\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)/\mathrm{2,}$$

$$\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|=-2\cdot \left|{\text{e}}_{\text{r}}\right|\to {\text{e}}_{\text{D}}\ge \left|{\text{e}}_{\text{r}}\right|\to {\text{e}}_{\text{e}\mathrm{,1}}=\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|,{\text{e}}_{\text{e}\mathrm{,2}}=\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\to {\text{e}}_{\text{D}}=\left(\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)/2.$$

$$\begin{array}{ccc}\text{f}\ddot{\text{u}}\text{r}& \left|\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right|\ge 2\cdot \left|{\text{f}}_{\text{r}}\right|:& \text{v}=-{\text{V}}_{\text{B}}\cdot \mathrm{\lambda }/4\cdot \left(\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right)\end{array},$$

$$\begin{array}{ccc}\text{f}\ddot{\text{u}}\text{r}& \left|\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right|\le -2\cdot \left|{\text{f}}_{\text{r}}\right|:& \text{v}=+{\text{V}}_{\text{B}}\cdot \mathrm{\lambda }/4\cdot \left(\left|{\text{f}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{f}}_{\text{e}\mathrm{,2}}\right|\right)\end{array};$$

dabei ist noch berücksichtigt, dass durch Rechentoleranzen und kleine Änderungen der Relativgeschwindigkeit und/oder Entfernung über die zwei Erfassungszyklen auch Werte von |f_e,1| - |f_e,2| leicht außerhalb der eigentlichen Grenzen ±2·|f_r| möglich sind, weshalb statt auf „=“ auf „≥“ bzw. „≤“ verglichen wird. Die Relativgeschwindigkeit v lässt sich also in eindeutiger Weise dadurch bestimmen, dass zuerst die Differenz |f_e,1|- |f_e,2| der Beträge der beiden Empfangsfrequenzen gebildet und mit ±2·|f_r| verglichen wird, um dann für den sich daraus ergebenden der drei Bereiche die jeweilige Beziehung zur Berechnung von v anzuwenden. Einzig bei f_r = 0, also Entfernung Null, kann das Vorzeichen der Relativgeschwindigkeit v nicht bestimmt werden, denn dann sind der zweite und dritte Fall in obiger Bez. (52) nicht zu unterscheiden und möglich (die Mehrdeutigkeit ist auch dadurch erklärbar, dass die nur vom Betrag her bekannte Empfangsfrequenz dann alleine aus der Dopplerverschiebung besteht). Wegen Toleranzen und Änderungen zwischen zwei Erfassungszyklen kann es auch bei sehr kleinen Werten von f_r noch zu diesem Problem der nicht möglichen Vorzeichenbestimmung der Dopplerverschiebung und damit der Relativgeschwindigkeit kommen); allerdings korrespondieren solche sehr kleinen Werte von f_r zum Bereich unmittelbar vor dem Sensor, wo im Normalfall keine Objekte auftreten und falls doch, deren Vorzeichen der Relativgeschwindigkeit schon aus der Historie bekannt und/oder meistens näherungsweise Null ist. Es sei bemerkt, dass für die Herleitung der Bez. (52) die effektiv aus zwei Erfassungszyklen gemittelte Dopplerverschiebung f_D nach Bez. (48) benutzt wurde; es kann natürlich auch die nur auf einem Erfassungszyklus basierende Dopplerverschiebung f_D nach Bez. (47a) oder (47b) verwendet werden. Des Weiteren kann auch eine in den beiden Erfassungszyklen leicht unterschiedliche entfernungsbedingte Frequenzverschiebung berücksichtigt werden; in Bez. (47a) und (47b) sind dann unterschiedliche Werte f_r,1 und f_r,2 zu verwenden und in Bez. (48) kommt dann noch deren Differenz als kleiner Anteil hinzu.Both relationships can be summarized using the sign V _B = ±1 of the modulation bandwidth B, and the radial relative velocity v = λ/2·f _D then results in a unique way: $$\begin{array}{ccc}\text{e}\ddot{\text{u}}\text{r}& \left|\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right|<2\cdot \left|{\text{e}}_{\text{r}}\right|:& \text{v}=-{\text{V}}_{\text{B}}\cdot \mathrm{\lambda }/4\cdot \left(\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)\end{array},$$

$$\begin{array}{ccc}\text{e}\ddot{\text{u}}\text{r}& \left|\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right|\ge 2\cdot \left|{\text{e}}_{\text{r}}\right|:& \text{v}=-{\text{V}}_{\text{B}}\cdot \mathrm{\lambda }/4\cdot \left(\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)\end{array},$$

$$\begin{array}{ccc}\text{e}\ddot{\text{u}}\text{r}& \left|\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|-\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right|\le -2\cdot \left|{\text{e}}_{\text{r}}\right|:& \text{v}=+{\text{V}}_{\text{B}}\cdot \mathrm{\lambda }/4\cdot \left(\left|{\text{e}}_{\text{e}\mathrm{,1}}\right|+\left|{\text{e}}_{\text{e}\mathrm{,2}}\right|\right)\end{array};$$

It is also taken into account that due to calculation tolerances and small changes in the relative speed and/or distance over the two acquisition cycles, values of |f _e,1 | - |f _e,2 | are also possible slightly outside the actual limits ±2·|f _r |, which is why the comparison is made using “≥” or “≤” instead of “=”. The relative speed v can therefore be determined unambiguously by first calculating the difference |f _e,1 |- |f _e,2 | between the magnitudes of the two reception frequencies and comparing it with ±2·|f _r |, in order to then apply the respective relationship to calculate v for the resulting one of the three ranges. Only when f _r = 0, i.e. distance zero, can the sign of the relative speed v not be determined, because then the second and third cases in relation (52) above cannot be distinguished and are not possible (the ambiguity can also be explained by the fact that the reception frequency, which is only known in terms of its magnitude, then consists solely of the Doppler shift). Due to tolerances and changes between two acquisition cycles, this problem of the impossibility of determining the sign of the Doppler shift and thus the relative speed can still arise even with very small values of f _r ); however, such very small values of f _r correspond to the area immediately in front of the sensor, where no objects normally occur and if they do, their sign of the relative speed is already known from history and/or is usually approximately zero. It should be noted that for the derivation of relation (52), the Doppler shift f _D according to relation (48), effectively averaged over two acquisition cycles, was used; Of course, the Doppler shift f _D based on only one acquisition cycle according to (47a) or (47b) can also be used. Furthermore, a slightly different distance-related frequency shift in the two acquisition cycles can also be taken into account; in (47a) and (47b) different values f _r,1 and f _r,2 are then to be used and in (48) their difference is then added as a small part.

The ambiguity of the relative speed can thus be resolved by comparing the measured reception frequencies of two acquisition cycles. As with normal tracking, an assignment of detections across cycles is necessary. However, the conventional approach to resolving the ambiguity via tracking generally requires tracking over significantly more cycles, since the measured distance progression, i.e. the change in distance measured across cycles, is compared with the change in distance expected from the respective speed hypothesis. However, with a small reception frequency and thus two speed hypotheses that are not far apart, this takes many cycles.

The approach to resolving the speed ambiguities by varying the modulation bandwidth, e.g. by alternating signs, can not only be applied across two acquisition cycles. Alternatively, a pixel can also be recorded in one acquisition cycle with a different value or sign of the modulation bandwidth B; however, with a given hardware, this would halve the number of pixels, or the number of parallel send-receive paths would have to be doubled. To avoid this, instead of the same pixels, pixels that are close to each other, especially neighboring ones, can be recorded with different values or signs of the modulation bandwidth B; since a real object is normally extended and is thus recorded in several pixels and the relative speed and distance are at least approximately the same, the relation (52) can again be used to clearly determine the relative speed v, where the two reception frequencies belong to two pixels with different signs of the modulation bandwidth B. One approach for different B for pixels that are close to each other, especially neighboring ones, is to that in a system with scanning for different, especially adjacent, scan planes, different B is selected.

Continuous scanning by frequency change

ergibt, wobei „mod_2n“ die symmetrische Modulofunktion zum Modul 2π darstellt, also auf den symmetrischen Bereich -π,...+π abbildet; die Modulobildung berücksichtigt, dass im Allgemeinen zahlreiche Wellenlängen zwischen zwei Koppelstellen liegen, auch um eine möglichst starke Abhängigkeit der Phasendifferenz von der Frequenz zu erzielen, was durch die im 16 angedeutete mäanderförmige Struktur erzielt werden kann (die Mäander können - wie im Bild dargestellt - nach innen gehen oder parallel zur Oberfläche des Chips liegen). Ist die Phasendifferenz Δφ(f) ein ganzzahliges Vielfaches von 2π (also eine ganze Zahl von Wellenlängen liegt jeweils zwischen den Koppelstellen), dann ist die Strahlrichtung senkrecht zum Wellenleiter. Im anderen Fall, also mod_2π(Δφ(f)) ≠ 0, ergibt sich wie in 16 dargestellt eine schräge Strahlrichtung mit Winkel γ₁(f) dadurch definiert, dass die Strahllängendifferenz ΔI zu benachbarten Koppelstellen die Phasendifferenz mod_2π(Δφ(f)) kompensiert; da zur Strahllängendifferenz ΔI die Phasendifferenz 2π·ΔI/λ mit der Freiraumwellenlänge λ = c/f korrespondiert und ΔI = sin(γ₁(f))·λ₀/2 mit λ₀= c/f₀ ist, gilt: $$2\mathrm{\pi }\cdot \text{sin}\left({\mathrm{\gamma }}_1\left(\text{f}\right)\right)\cdot {\mathrm{\lambda }}_0/\left(2\cdot \mathrm{\lambda }\right)=\mathrm{\pi }\cdot \text{sin}\left({\mathrm{\gamma }}_1\left(\text{f}\right)\right)\cdot \text{f}/{\text{f}}_0=-{\underset{\_}{\text{mod}}}_{2\mathrm{\pi }}\left(\mathrm{\Delta \phi }\left(\text{f}\right)\right)$$

und somit $${\mathrm{\gamma }}_1\left(\text{f}\right)=\text{asin}\left(-{\underset{\_}{\text{mod}}}_{2\mathrm{\pi }}\left(\mathrm{\Delta \phi }\left(\text{f}\right)\right)/\mathrm{\pi }\cdot {\text{f}}_0/\text{f}\right),$$

wobei „asin“ die inverse Sinusfunktion darstellt. Ist beispielsweise mod_2π(Δφ) = π/2 und λ = λ₀, dann ergibt sich der Winkel γ₁ = 30°. Der Strahlwinkel gilt sowohl für Senden als auch für Empfangen - dies folgt auch aus dem Grundsatz der Reziprozität.In the following, a possible implementation for continuous scanning in a detection plane is presented. Elements are used whose beam direction (for sending and receiving) depends on the frequency; examples of such elements are dispersive materials, grating structures or waveguides, whereby the focus here is on the latter (but of course the approaches shown can also be transferred analogously to other elements). 16 shows a waveguide 16.2 of length 1 cm realized in a photonic semiconductor chip 16.1 with lateral feed 16.3 with a frequency f and with equidistant coupling points 16.4 at the respective distance of λ ₀ /2 with λ ₀ = 1550nm (free air wavelength at f ₀ = c/λ ₀ = 195THz); the coupling points are in 16 shown as a point, but they can also have a certain extent. The coupling points serve to couple the wave out when transmitting, and to couple it in when receiving. The structure of the waveguide is periodic, ie it takes on the same shape between two neighboring coupling points. Therefore, the phase difference Δφ(f) between two neighboring coupling points, which depends on the frequency f used, is constant, so that the phase of the wave emitted at the k-th coupling point, k = 0,...,K-1 with K = 12900, is $$\begin{array}{ccc}{\mathrm{\phi }}_{\text{k}}\left(\text{e}\right)=\text{k}\cdot \mathrm{\Delta \phi }\left(\text{e}\right)=\text{k}\cdot {\underset{\_}{\text{mod}}}_{2\mathrm{\pi }}\left(\mathrm{\Delta \phi }\left(\text{e}\right)\right)& \text{with}& \text{k}=\mathrm{0,}\dots ,\text{K}-1\end{array}$$

where “mod _2n ” represents the symmetrical modulo function to the modulus 2π, i.e. it maps to the symmetrical range -π,...+π; the modulo formation takes into account that there are generally numerous wavelengths between two coupling points, also in order to achieve the strongest possible dependence of the phase difference on the frequency, which is achieved by the 16 can be achieved (the meanders can - as shown in the picture - go inwards or lie parallel to the surface of the chip). If the phase difference Δφ(f) is an integer multiple of 2π (i.e. a whole number of wavelengths lie between the coupling points), then the beam direction is perpendicular to the waveguide. In the other case, i.e. mod _2π (Δφ(f)) ≠ 0, the result is as in 16 shown an oblique beam direction with angle γ ₁ (f) is defined by the fact that the beam length difference ΔI to neighboring coupling points compensates the phase difference mod _2π (Δφ(f)); since the phase difference 2π·ΔI/λ corresponds to the beam length difference ΔI with the free space wavelength λ = c/f and ΔI = sin(γ ₁ (f))·λ ₀ /2 with λ ₀ = c/f ₀ , the following applies: $$2\mathrm{\pi }\cdot \text{sin}\left({\mathrm{\gamma }}_1\left(\text{e}\right)\right)\cdot {\mathrm{<figure-callout\; id="0"\; label="\lambda "\; filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}_0/\left(2\cdot \mathrm{\lambda }\right)=\mathrm{\pi }\cdot \text{sin}\left({\mathrm{\gamma }}_1\left(\text{e}\right)\right)\cdot \text{e}/{\text{<figure-callout id="0" label="e" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">e</figure-callout>}}_0=-{\underset{\_}{\text{mod}}}_{2\mathrm{\pi }}\left(\mathrm{\Delta \phi }\left(\text{e}\right)\right)$$

and thus $${\mathrm{\gamma }}_1\left(\text{e}\right)=\text{asin}\left(-{\underset{\_}{\text{mod}}}_{2\mathrm{\pi }}\left(\mathrm{\Delta \phi }\left(\text{e}\right)\right)/\mathrm{\pi }\cdot {\text{e}}_0/\text{e}\right),$$

where "asin" represents the inverse sine function. For example, if mod _2π (Δφ) = π/2 and λ = λ ₀ , then the angle γ ₁ = 30° results. The beam angle applies to both sending and receiving - this also follows from the principle of reciprocity.

Continuous spatial scanning is achieved by a continuous frequency change, ie the frequency changes during the data acquisition of a pixel, and this at least approximately linearly. This results in the frequency change superimposed on the phase modulation according to 5 directly as a result of scanning by changing the frequency. As explained above, the required sampling frequency f _s increases with the modulation width B, i.e. the frequency change during a pixel, which de facto limits it - with the pixel duration T _pm = 13.7µs and maximum range of just under 250m considered here, its value |B| ≤ 2GHz (for an f _s ≤ 800MHz). If more than 500 pixels are to be scanned (e.g. for horizontal direction), then the frequency must be changed by about 1THz overall, which corresponds to about 0.5% of the average frequency f ₀ = c/λ ₀ = 195THz (at free air wavelength λ ₀ = 1550nm). For an assumed scan range of -20°,...,+20°, the phase difference Δφ between two neighboring coupling points must change by about -0.34π,...,+0.34π, i.e. by about 0.68π, which requires a high frequency-related sensitivity of the phase difference Δφ for a 0.5% frequency change, which can be realized not only by a meandering waveguide path but also by a waveguide with high dispersion in the frequency range used.

und dass die Phasendifferenz Δφ(f) im betrachteten kleinen Frequenzbereich konstante Steigung hat: $$\begin{array}{cc}\mathrm{\Delta \phi }\left(\text{f}\right)={\text{F}}_{\text{1}}\cdot \left(\text{f}-{\text{f}}_{\text{0}}\right)+\text{L}\cdot 2\mathrm{\pi }& \text{mit L ganzzahlig}\end{array};$$

dabei ist berücksichtigt, dass bei der mittleren Frequenz f₀ die Strahlrichtung Null sein soll (also mod_2π(Δφ(f₀) = 0). Mit Bez. (54) erhält man: $$\mathrm{\pi }\cdot \text{sin}\left({\text{A}}_{\text{1}}\cdot \text{t}\right)\cdot \text{f}/{\text{f}}_{\text{0}}=-{\text{F}}_1\cdot \left(\text{f}-{\text{f}}_{\text{0}}\right)$$

und somit für den benötigten zeitabhängigen Frequenzverlauf f(t): $$\text{f}\left(\text{t}\right)={\text{f}}_{\text{0}}/\left(1+\mathrm{\pi }\cdot \text{sin}\left({\text{A}}_{\text{1}}\cdot \text{t}\right)/\left({\text{f}}_{\text{0}}\cdot {\text{F}}_{\text{1}}\right)\right).$$

First, it is assumed that the beam angle γ ₁ changes linearly over the range -20°,...,+20°, i.e. $${\mathrm{\gamma }}_1\left(\text{t}\right)={\text{A}}_1\cdot \text{t}\text{,}$$

and that the phase difference Δφ(f) has a constant slope in the small frequency range considered: $$\begin{array}{cc}\mathrm{\Delta \phi }\left(\text{e}\right)={\text{F}}_{\text{1}}\cdot \left(\text{e}-{\text{e}}_{\text{0}}\right)+\text{L}\cdot 2\mathrm{\pi }& \text{with L integer}\end{array};$$

It is taken into account that at the average frequency f ₀ the beam direction should be zero (i.e. mod _2π (Δφ(f ₀ ) = 0). Using (54) we get: $$\mathrm{\pi }\cdot \text{sin}\left({\text{A}}_{\text{1}}\cdot \text{t}\right)\cdot \text{e}/{\text{<figure-callout id="0" label="e" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">e</figure-callout>}}_{\text{0}}=-{\text{<figure-callout id="1" label="F" filenames="DE102023201147A1\_0150.png,DE102023201147A1\_0151.png" state="{{state}}">F</figure-callout>}}_1\cdot \left(\text{e}-{\text{e}}_{\text{0}}\right)$$

and thus for the required time-dependent frequency response f(t): $$\text{e}\left(\text{t}\right)={\text{<figure-callout id="0" label="e" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">e</figure-callout>}}_{\text{0}}/\left(1+\mathrm{\pi }\cdot \text{sin}\left({\text{A}}_{\text{1}}\cdot \text{t}\right)/\left({\text{<figure-callout id="0" label="e" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">e</figure-callout>}}_{\text{0}}\cdot {\text{F}}_{\text{1}}\right)\right).$$

For a scan time of 4.5ms (500 partially overlapping pixels at a distance of 9ms) and a given frequency change of 1THz (→ F ₁ = 2.13Ts) 17 the temporal course of the beam angle γ ₁ (t) and the frequency f(t) is shown. It should be noted that the frequency f(t) for transmitting and receiving actually has a small frequency difference (due to the frequency shift of Doppler and transit time), whereby the beam angle difference caused by this is significantly less than the beam width itself.

A temporally linear change in the beam angle γ ₁ (t) according to (56) means a constant angular resolution over the entire detection range with temporally equidistant pixel spacing. However, a different angular resolution can be advantageous, particularly for lidar sensors that look forward, i.e. in the direction of travel: in the direction of travel (γ ₁ = 0) a higher angular resolution is required than outwards (γ ₁ = ±20°; γ ₁ here refers to the horizontal direction, for example), i.e. outwards a higher scanning speed can be achieved by using a faster frequency change; this is shown for example in 18 It should be mentioned that with faster scanning, the area on an object covered by the lidar beam also becomes larger, so that the signal received over the acquisition time of a pixel loses coherence (at least partially different points on the object are acquired at different times); the reduced coherence slightly reduces the sensitivity and thus the range, which is not relevant in the outer area due to its reduced range requirements, and secondly the Doppler accuracy becomes slightly worse (because the magnitude peak in two-dimensional correlation blurs slightly in the Doppler dimension), which is not critical due to the fundamentally very high accuracy of Doppler, i.e. relative speed.

Even at a constant scanning speed according to (56), the frequency response according to (58) is not completely linear in time (especially because of the sine function contained therein). The non-linearity of the temporal response of the frequency becomes even stronger due to a non-constant scanning speed (see also 18 ). Another reason for a temporally non-linear frequency response is a strongly dispersive character of the waveguide (i.e. the linear relationship between the phase difference Δφ(f) and the frequency f according to (57) is no longer valid). Such non-linearities in the temporal frequency response over a detection scan (i.e. 500 pixels in the example above) can lead to a relevant non-linearity of the frequency response even during a single pixel, i.e. to a frequency error f _r,Q (t), in particular with a quadratic form according to (39b). As presented above, such a frequency error can be compensated for by a corresponding correction component in the rotation factors d _n,m ; in the case of a quadratic error, only the register R2 in structure according to 11 according to reference (44). A modulation bandwidth B that changes across pixels (due to a changing slope of the frequency response f(t)) can also be taken into account by adjusting this register value - then the output dimension of the FFT always corresponds to the same relative speeds.

In principle, almost any scanning sequence can be realized via a corresponding frequency change sequence - this is a great advantage of scanning via frequency change. The detection range can also be changed over cycles, especially depending on the traffic situation; for example, a wider detection range may be of interest at low vehicle speeds (e.g. city traffic) than at high vehicle speeds (e.g. motorway). Incorrect adjustment of the sensor (instead of in A deviation in the direction of travel (the sensor looks 2° to the right, for example) can be taken into account simply by adjusting the frequency range used accordingly, i.e. essentially shifting it slightly.

• - Es können keine überlappende Pixel realisiert werden, was bei gegebener Hardware und Zykluszeit zu einer reduzierten Datenaufnahmedauer des einzelnen Pixels führt. Zusätzlich wird die Datenaufnahmezeit noch dadurch verringert, dass das Umschalten der Frequenz eine gewisse Zeit benötigt (insbesondere bis Frequenz auf neuen Wert eingeschwungen ist) und dass danach noch die maximale Laufzeit der Empfangssignale (bei 249.5m maximalere Entfernung etwa 1.67µs) abgewartet werden muss. Bei dem in der bisherigen Auslegung betrachteten zeitlichen Pixelabstand von etwa 9µs bleibt dann etwa nur eine Datenaufnahmezeit von 6µs übrig, was mehr als eine Halbierung im Vergleich zum kontinuierlichen Scannen mit einer Pixeldauer von 13.7µs bedeutet. Hauptnachteil dieser deutlich reduzierten Datenaufnahmezeit pro Pixel ist eine Verringerung der Sensitivität (für obige Verhältnisse etwa um 3.5dB) und damit der Reichweite (etwa um 19%); weniger kritisch ist die reduzierte Auflösung und Genauigkeit der Relativgeschwindigkeitsbestimmung. Noch stärker wird der Verlust, wenn der Pixelabstand z. B. um Faktor 2 geringer gewählt wird, um mit halber Zahl an parallelen Sende-Empfangspfaden auskommen zu können (16 statt 32).
• - Bei einem reellwertigen Mischer lässt sich nicht das Vorzeichen der Relativgeschwindigkeit bestimmen - die oben dargestellten Ansätze zum Lösen der Mehrdeutigkeit durch nicht bekanntes Vorzeichen der Empfangsfrequenz haben ja auf der überlagerten linearen Frequenzmodulation basiert.
• - Wenn durch sehr schnelles Scannen (z. B. nach außen hin) die Strahlbreite kleiner als die Strahlbewegung während der Datenaufnahmepause zwischen zwei Pixeln ist, könnte ein in Scanrichtung sehr schmales Objekt übersehen werden.
• - Ein kontinuierliches Scannen der Frequenz kann einfacher bzw. besser als ein gestuftes realisierbar sein.
• - Ein kontinuierliches Scannen der Frequenz reduziert ein wenig den Speckle-Effekt (also statistisch variierender Empfangspegel von einem diffusen Reflektor), da der Speckle-Effekt frequenzabhängig ist; zusätzlich wird der Speckle-Effekt auch durch räumliches Scannen leicht reduziert, da sich während der Datenaufnahme eines Pixels die beleuchtete Fläche leicht ändert.

Instead of the continuous scanning considered so far, step-by-step scanning could also be used, i.e. the frequency is changed from pixel to pixel, but remains constant within each pixel (in this case, state-of-the-art phase modulation with a constant frequency, i.e. without superimposed frequency modulation, could also be used). However, this step-by-step scanning would have some disadvantages (but it should not be ruled out here as a possible implementation):

• - Overlapping pixels cannot be implemented, which, for a given hardware and cycle time, leads to a reduced data recording time for each individual pixel. The data recording time is also reduced by the fact that switching the frequency takes a certain amount of time (particularly until the frequency has settled at the new value) and that the maximum runtime of the received signals must then be waited for (for a maximum distance of 249.5m, around 1.67µs). With the pixel spacing of around 9µs considered in the previous design, this then leaves a data recording time of around 6µs, which is more than half the time compared to continuous scanning with a pixel duration of 13.7µs. The main disadvantage of this significantly reduced data recording time per pixel is a reduction in sensitivity (for the above conditions by around 3.5dB) and thus the range (by around 19%); less critical is the reduced resolution and accuracy of the relative speed determination. The loss is even greater if the pixel spacing is, for example, around 1.67µs. B. is chosen to be a factor of 2 smaller in order to be able to manage with half the number of parallel transmit/receive paths (16 instead of 32).
• - With a real-valued mixer, the sign of the relative velocity cannot be determined - the approaches presented above to resolve the ambiguity caused by an unknown sign of the receiving frequency were based on superimposed linear frequency modulation.
• - If very fast scanning (e.g. outward) causes the beam width to be smaller than the beam movement during the data acquisition pause between two pixels, an object that is very narrow in the scanning direction could be overlooked.
• - Continuous frequency scanning can be easier or better implemented than step-by-step scanning.
• - Continuous frequency scanning slightly reduces the speckle effect (i.e. statistically varying reception level from a diffuse reflector) because the speckle effect is frequency dependent; in addition, spatial scanning also slightly reduces the speckle effect because the illuminated area changes slightly during data acquisition of a pixel.

In order to implement a frequency change, for example, a changing frequency can either be modulated onto the constant frequency of a laser source (which is difficult here because of the large frequency range required) or a laser source with a directly controllable frequency can be used. Such lasers are typically controlled via a mechanical variable (e.g. using a piezoelectric element) or via an electrical variable (voltage, current). The generation of an electrical control variable can be achieved by digital determination on a computing unit, e.g. a processor, and conversion to the analog range using a digital-analog converter (DAC), possibly followed by low-pass filtering; since the frequency curve and thus the required curve of the control variable is quite low-frequency from a signal theory perspective, a delta-sigma approach can be used to determine the input values of the DAC, which reduces the requirements for the DAC, particularly with regard to its resolution. In addition to direct control, a control loop in the form of a PLL (phase locked loop) can also be used. If a sufficiently large frequency tuning range cannot be achieved with a laser source (on the one hand to address the frequency change of e.g. 1THz required for scanning and/or on the other hand to cover temperature, aging and tolerance effects), several laser sources can be implemented, between which the appropriate source can be selected. Alternatively, the frequency change required for scanning can be greatly reduced by using waveguides with slightly different frequency behavior of the phase difference Δφ(f) adjacent coupling points for the parallel transmit-receive paths - at a frequency f, the waveguides then radiate in different directions and their respective small scan areas adjoin one another (possibly with a certain overlap area); this approach will be explained in more detail later.

Finally, it should be mentioned that the waveguide - as in 16 shown - not only the scanning, but of course also the focusing in its corresponding spatial direction is realized, ie in every cutting plane through the waveguide (more precisely through the line on which its coupling points lie) the wave has a flat wave front.

Scanning in the second spatial direction

In the following, we will explain how focusing and scanning in the second spatial direction (perpendicular to the spatial direction defined by the waveguide) can be realized. As in 19 As shown, focusing can be achieved by means of a lens 19.2; 19 above shows the arrangement from the direction in which the waveguide 19.1 extends. The waveguide is located in the focal plane of the lens, but offset from the focal line (this occurs because several waveguides lying next to each other are considered later), so that the radiation, which forms a plane wavefront after focusing by the lens, is tilted by the beam angle γ _2.1 relative to the optical axis of the lens (in view of 19 above, the symbol γ _2,1 is used for γ _2,1 , which is distinguished by an underline, since only the projected angle is shown there). Since the focusing in the first spatial direction already takes place through the waveguide itself, the lens only has to focus in the second spatial direction, which is perpendicular to the first, so that it has a constant cross-section in the direction of the waveguide; this is in 19 shown below. There you can also see the definition of the two beam angles γ ₁ and γ _2,1 in a three-dimensional context - so they are not angles according to spherical coordinates; in 19 Below, the refraction of the lens on both surfaces is summarized in one place for simplicity.

It should be noted that a lens with a constant cross-section in one dimension is easier to manufacture than a lens without this property, in particular a lens for focusing in both spatial directions. Instead of a single lens, a lens system can also be used, whereby the property of a constant cross-section in the waveguide direction is also retained.

For scanning in this second spatial direction, a material that is transparent to the wavelengths used can be used, the dielectric constant of which can be changed by applying a voltage (which generates an electric field in the material) or by passing a current through it (in particular to generate a magnetic field in the material); liquid crystals and ferroelectric materials are examples of materials with such properties. 20 shows above - from the same perspective as 19 above - an exemplary arrangement with the body 20.3 made of such a material, which, like the lens 20.2, has a constant cross-section (with a triangular shape) in the direction of extension of the waveguide 20.1 and thus has a prism-like shape; a side view, ie from a direction rotated by 90°, is shown in 20 shown below - there you can also see the voltage U applied to the two sides of the prism-shaped body 20.3, which changes its dielectric constant either directly or via the current flow through the body generated by it. By changing the applied voltage U, the difference γ _2.2 between the input angle γ _2.1 and the output angle γ ₂ of the prism 20.3 and thus the beam angle γ ₂ itself can be changed, whereby scanning in the second spatial direction, which is orthogonal to the first, can be realized. It should also be mentioned that the prism-shaped body can preferably be larger than required for the actual beam path in order to achieve as homogeneous electrical fields as possible in the relevant area and thus a spatially constant dielectric constant as possible. In principle, one could also consider that the lens and prism form a common body with a constant cross-section in the waveguide direction; However, the change in the dielectric constant must not be constant over the entire cross-section, because otherwise the focusing properties and thus also the focal plane would be changed (a suitable field profile that is not constant over the cross-section would be required). Instead of this prism-shaped body, a flat liquid crystal element with radiation can also be used, particularly in the form of a one-dimensional grid structure, which deflects the beam in the second spatial direction by applying a voltage between the top and bottom.

As an alternative approach for focusing and scanning, a liquid crystal array can be used. In 21 a possible arrangement with a transparent one-dimensional array 21.2 with rod-shaped elements 21.3 is shown; the upper view in 21 shows the array from above, the middle view shows the arrangement and beam path from the direction of the waveguide 21.1, and in the lower view, rotated by 90°, you can see the arrangement and beam path from the side. By applying a voltage to the individual rod-shaped elements (between their top and bottom), the phase difference between the wave entering on one side and the wave exiting on the other side can be changed - the phase difference realized depends on the value of the applied voltage. This can ensure that the phase of the wave on the top has a linear progression, whereby simultaneously focusing (i.e. generation of a plane wave) and a given beam angle γ ₂ can be realized; by changing the applied voltages, the beam angle γ ₂ and thus scanning in this second beam direction can be realized.

Liquid crystals often have the problem that they react slowly to a change in control, i.e. it takes a long time until they have settled into a new control state, which is very critical when using two-dimensional liquid crystal arrays for two-dimensional scanning, since switching has to take place after each pixel. In contrast, the one-dimensional liquid crystal array only has to switch after a complete scan in the first spatial direction, i.e. in the example after about 4.5ms, which is completely unproblematic. In comparison to a liquid crystal array for two-dimensional scanning, there is the additional advantage that much less control voltage is required (since there is only one dimension).

The number of required rod-shaped liquid crystal elements and thus the required control voltages can be reduced by using an additional lens, which essentially takes over the focusing in the second spatial direction and again has a constant cross-section in the direction of the waveguide; because then the liquid crystal array essentially only has to realize the pivoting of the beam direction, which allows a distance that is far above the free space wavelength λ ₀ = 1550 nm, especially with a small pivoting range.

The first approach described above with a lens and a controllable prism has the disadvantage that it requires a high degree of precision in the lens geometry and its distance from the waveguide (otherwise optical blurring occurs, for example due to a shift in the focal plane); the high precision required leads to costs in sensor production and/or increased parts prices due to small mechanical tolerances. In contrast, in the second approach based on a liquid crystal array, position errors can be easily compensated for by appropriately controlling the elements of the liquid crystal array to correct the corresponding phase errors; the same applies to errors in the position and cross-section of any additional lens used.

Instead of the previously considered transparent liquid crystal array according to 21 a reflective liquid crystal array can also be used; for a reflective liquid crystal array, a phase difference between the incoming and outgoing wave can also be effectively achieved by applying a voltage, which can also be seen as a change in the local angle of reflection relative to the angle of incidence. A corresponding arrangement is shown in 22 shown; the upper view shows the arrangement with waveguide 22.1, a polarization-dependent mirror 22.4 (reflects for one polarization and lets through radiation for the polarization perpendicular to it) and a one-dimensional reflective array 22.2 and with rod-shaped elements 22.3 and with 90° polarization rotation from above, the middle view shows the arrangement and beam path from the direction of the waveguide 22.1, and the lower view rotated by 90° shows the arrangement and beam path from the side. The mirror 22.4 for beam deflection enables the photonic chip 22.5, on which the waveguide is located, and the liquid crystal array 22.2 including its control to be located on a circuit board 22.6, which reduces complexity and manufacturing outlay. Of course, an arrangement without a mirror 22.4, i.e. with a direct beam path between the waveguide and the reflective liquid crystal array, could also be realized, for which e.g. B. two circuit boards are required.

One or more additional mirrors can be used to deflect the beam, e.g. if the optical axis of the sensor is to be perpendicular to the circuit board.

All further considerations made above for the arrangement with a transparent liquid crystal array are also valid for the arrangement with a reflective liquid crystal array - also in particular that manufacturing tolerances of the optically relevant components and their arrangement relative to one another can be compensated by appropriate control of the elements of the liquid crystal array.

Until now, the liquid crystal array was considered to be one-dimensional. Due to tolerances (e.g. of the liquid crystal array itself with no constant thickness or no constant optical properties over the extension of the rod-shaped elements), it could be that no absolutely flat wave is generated in the first spatial direction (i.e. the direction defined by the waveguide). To avoid this, as in 23 shown - the rod-shaped elements are divided into several individual elements and phase errors are compensated by controlling these elements accordingly. However, such a two-dimensional liquid crystal array is more complex and requires more control voltages. Compared to a two-dimensional liquid crystal array for two-dimensional scanning, however, significantly fewer elements are required in one dimension (in which rod-shaped elements are divided to compensate for errors).

Instead of a reflective liquid crystal array, a reflective liquid crystal element can also be used, particularly in the form of a one-dimensional grid structure, which deflects the beam in the second spatial direction by applying a voltage. However, the beam must then already be focused beforehand; compared to the arrangement in 22 One then needs either a focusing mirror (approximately parabolic in shape), a reflecting lens (i.e. with a reflective coating on one side) or a combination of mirror and lens, whereby these elements continue to have a constant shape in the direction of the waveguide.

Finally, it should be mentioned that materials whose optical properties can be changed by applying an electrical control variable can be used in arrangements other than those explained above to realize scanning in the second spatial direction.

Waveguide arrangement and scanning pattern

As mentioned several times above, there are parallel transmit-receive paths, e.g. 32, and thus also 32 waveguides; how these can be arranged and designed on the photonic chip will be explained below. In principle, these waveguides can be used to realize additional parallelism when detecting in the first spatial direction (by scanning via waveguides with frequency change) and/or the second spatial direction (by scanning via material with electrically controllable optical properties, e.g. in the form of a liquid crystal array).

• - w bezeichnet die Nummer des jeweiligen Wellenleiters: w =1,...,32,
• - f bezieht sich auf das Scannen mit Hilfe der Wellenleiter durch kontinuierliche Frequenzänderung, also auf das kontinuierliche Scannen in der ersten Raumrichtung und somit für Komponente γ₁ der Strahlrichtung; dieses kontinuierliche Scannen erschließt alle 500 Pixel über diese erste Raumrichtung, wobei f = 1,...,500 als die Nummer der jeweiligen Frequenz definiert ist (genau gesagt der jeweiligen Mittenfrequenz, da sich innerhalb eines Pixels ja die Frequenz kontinuierlich ändert),
• - u bezieht sich auf das schrittweise Scannen mit Hilfe eines Material mit elektrisch steuerbaren optischen Eigenschaften, z. B. in Form eines über Spannungen gesteuerte Flüssigkristall-Arrays, also auf das schrittweise Scannen in der zweiten Raumrichtung und somit für Komponente γ₂ der Strahlrichtung; dabei bezeichnet u die Nummer dieses schrittweisen Scannens (und damit der schrittweise geänderten elektrischen Steuergrößen, z. B. der angelegten Spannungen) - nach einem kontinuierlichen Frequenzscan in erster Raumrichtung, der etwa 4.5ms dauert, wird ein Schritt auf eine neues γ₂ gemacht, d. h. der nächste kontinuierliche Frequenzscan bei einem neuen γ₂ durchgeführt; weil jeder der 32 Wellenleiter schon selber ein unterschiedliches γ₂ realisiert, sind für 320 Pixel in diese zweite Raumrichtung, also für 320 unterschiedliche γ₂, nur 10 Scanschritte u = 1,...,10 nötig, und von Schritt zu Schritt springt die Strahlrichtung um 32 Pixel.

First, we will consider the approach that waveguides are usually used for parallelism, i.e. parallel detection in the second spatial direction. As in 24 As shown, the 32 waveguides 24.2 are then arranged parallel to one another on the photonic chip 24.1 (a possible meandering course is not shown there for reasons of illustration); all waveguides are designed in the same way so that they each realize the same beam angle γ ₁ at the same frequency. Depending on the distance between the waveguides, they can address directly adjacent pixels or pixel lines with respect to the second spatial direction or pixel lines that are further apart and between which other pixel lines lie; at least approximately, the distance between the waveguides is proportional to the distance between the pixel lines they realize (if one ignores larger angles). First, a distance between the waveguides is assumed such that they realize 32 directly adjacent, equidistant pixel lines. 25 shows how the two-dimensional pixel field is then opened up; the nomenclature used “w,f,u” is defined as follows:

• - w denotes the number of the respective waveguide: w =1,...,32,
• - f refers to scanning using the waveguides by continuously changing the frequency, i.e. to continuous scanning in the first spatial direction and thus for component γ ₁ of the beam direction; this continuous scanning covers all 500 pixels via this first spatial direction, where f = 1,...,500 is defined as the number of the respective frequency (more precisely the respective center frequency, since the frequency changes continuously within a pixel),
• - u refers to the step-by-step scanning using a material with electrically controllable optical properties, e.g. in the form of a voltage-controlled liquid crystal array, i.e. to the step-by-step scanning in the second spatial direction and thus for component γ ₂ of the beam direction; where u denotes the number of this step-by-step scanning (and thus the step-by-step changed electrical control variables, e.g. the applied voltages) - after a continuous frequency scan in the first spatial direction, which lasts about 4.5 ms, a step is made to a new γ ₂ , i.e. the next continuous frequency scan is carried out at a new γ ₂ ; because each of the 32 waveguides already realizes a different γ ₂ itself, only 10 scanning steps u = 1,...,10 are necessary for 320 pixels in this second spatial direction, i.e. for 320 different γ ₂ , and the beam direction jumps by 32 pixels from step to step.

The small spacing of the waveguides required for this arrangement can be difficult to implement, especially if the waveguides have a strongly meandering course. Alternatively, the spacing can be increased by a factor of 32 so that the waveguides realize pixel lines at a spacing of 32; this is shown in 26 When scanning in the second spatial direction, only one pixel is jumped per step; this means a strong reduction in the required scanning area in the second spatial direction, i.e. over material with electrically controllable optical properties, which is advantageous for such an approach and can open up simpler or more implementation options. Such an arrangement also allows the approach mentioned above that by choosing different frequency modulation bandwidths B in adjacent scan planes, the sign of the reception frequency in a real-valued mixer is determined, provided an object is seen in at least two adjacent pixels (i.e. from adjacent scan planes, ie pixel lines); over the 10 scan steps u = 1,...,10, an alternating direction of continuous frequency scanning is to be used for the first spatial direction, which realizes an alternating sign of the modulation bandwidth B.

Until now, it was assumed that the waveguides were at least approximately equidistant from each other, which also leads to equidistant spacing of the pixels in the second spatial direction. Often, however, maximum resolution is only required in the central angular range, while it can decrease towards the outside - larger gaps between the pixels are also possible there. This can be achieved by arranging the waveguides in a non-equidistant manner. As a simple case, a small and equal spacing a is assumed for the 16 middle waveguides (such that there are no gaps between the pixel lines generated by them), while the 8 outer waveguides have twice the spacing 2·a between each other and there is a large spacing 146·a between them and the group of 16 middle waveguides. The resulting two-dimensional pixel field is shown in 27 shown; when scanning in the second spatial direction, each step jumps by 16 times the pixel spacing (relative to the pixel spacing in the middle area). In the upper and lower third of the pixel field, the detection is now only half as dense as in the middle.

In contrast to the previous arrangements, the waveguides can also be used for parallelism, i.e. parallel detection in the first spatial direction. For this purpose, the waveguides are designed differently so that they realize different beam angles γ ₁ at the same frequency; the waveguides that are still arranged parallel to each other (as in 24 ) can still have coupling points at the same distance, but then, for example, the length of the meandering waveguide between two coupling points must be designed differently. 28 shows the resulting two-dimensional pixel field for a small and constant distance between the waveguides; the distance is chosen so that the pixel lines realized in the second spatial direction are each one pixel apart. The different design of the 32 waveguides is chosen so that at the same frequency they realize a beam direction component γ ₁ (in the first spatial direction) that differs by 16 pixels; in order to cover the entire first spatial direction (now consisting of 16·32 = 512 pixels), only a continuous frequency scan over 16 pixels is necessary, so that the frequency number only extends over the range f = 1,...,16. The step-by-step scanning in the second spatial direction (with the help of a material with electrically controllable optical properties) must now completely cover this spatial direction, i.e. all 320 pixels, so that 320 steps are necessary. As can be seen in 28 can be seen, the two-dimensional pixel field no longer has a precise rectangular shape, but forms a parallelogram. An approximately rectangular shape with a slight rotation can be achieved by slightly changing the frequency range used for scanning in the first spatial direction for each of the 320 steps for the second spatial direction. The slight rotation of this rectangle can be compensated by arranging the photonic chip accordingly on the circuit board or by rotating the arrangement of the waveguides on the chip itself.

The advantage of this approach is that either the frequency range required for scanning in the first spatial direction is much smaller (by a factor of about 32, which is an advantage for the realization of the laser source), or that with the same frequency range as before, the waveguide requires a much lower frequency-related sensitivity of the phase difference Δφ, which makes its length between two coupling points much shorter, which also reduces its line losses; however, the frequency change per pixel is then much higher, so that continuous scanning could lead to a sampling frequency that is too high, so that a step-by-step frequency change would then be necessary.

A disadvantage of this approach is that the increased number of steps results in a slightly higher overhead for reconfiguring the control variables for scanning (for the material with electrically controllable optical properties and for the frequency modulation of the laser source).

This disadvantage of the increased overhead for reconfiguring the control variables for scanning and also the above-mentioned problem of the excessively high sampling frequency can be eliminated by performing continuous scanning instead of step-by-step scanning in the second spatial direction - the control variable for the material with electrically controllable optical properties is therefore not changed step-by-step, but continuously. During a continuous scan over the entire detection area range in the second spatial direction, the continuous scanning over frequency in the first spatial direction moves only one pixel further; this is in 29 shown. A total of 16 continuous scans in the second spatial direction are therefore required. This approach makes continuous frequency scanning much slower, so that when using a simple waveguide with low frequency-related sensitivity of the phase difference Δφ, the temporal frequency change (i.e. the approximately linear frequency modulation superimposed on the phase modulation) is small enough to manage with a moderate sampling frequency. With this continuous scanning in the second spatial direction, the frequency scanning for the first spatial direction could also be designed step by step, i.e. the frequency could be kept constant during each scan in the second spatial direction and then set to a new value for the next scan in the second spatial direction; this would then use 16 different frequencies. Instead of one laser source whose frequency can be adjusted, 16 laser sources with different constant frequencies could also be used, for example, and switching would then be carried out between these laser sources.

Continuous scanning in a second spatial direction can be carried out at a non-constant scanning speed in order to achieve, for example, a higher angular resolution in the direction of travel for a lidar sensor facing in the direction of travel than towards the edges of the detection area.

Of course, the parallel detection by the 32 waveguides can also be split into both spatial directions in order to combine the resulting advantages (at a reduced level of advantage, of course).

The approach with several different waveguides for different beam direction ranges in the first spatial direction can also be implemented in simpler lidar systems with fewer pixels in such a way that there is only one transmit-receive path that can be switched to several waveguides; this reduces in particular the required frequency tuning range of the laser source.

Alternative approaches for scanning in the second spatial direction

So far, the use of a material with electrically controllable optical properties has been assumed for scanning in the second spatial direction. Of course, other approaches are also conceivable.

In the previous example of continuous scanning in the second spatial direction (i.e. the scan pattern according to 29 ), for example, a mechanical approach could also be used (e.g. with a rotating mirror or prism).

The scan patterns after 25-27 with step-by-step switching in the second spatial direction could be achieved by having not just one waveguide on each of the 32 parallel transmit-receive paths, but for example 10 waveguides between which switching can be carried out; this would give an arrangement with 320 waveguides lying next to one another, of which only 32 are used during a frequency scan for the first spatial direction. It should also be mentioned that with such an approach for switching, cascaded single switches (i.e. with two inputs and one output) are usually used, so that the entire switching matrix then has a power of two of inputs, i.e. 8, which in the above example would mean 256 waveguides and thus 256 pixels in the second spatial direction. If the pixels in the second spatial direction are to be less densely spaced outwards, i.e. have a larger angular distance, then the distance between the adjacent waveguides must be increased towards the outside. If the pixels are less dense towards the outside, then in order to avoid detection gaps it can generally (i.e. not only in this embodiment) be advantageous if the beam width increases outwards in the respective spatial direction, which is often already inherently given by optical focusing errors or can otherwise be realized in this example by positioning the waveguides outside the focal plane.

It should also be mentioned that for all of the above approaches, the two spatial directions that are perpendicular to each other are advantageously placed in the horizontal and vertical detection direction, i.e. azimuth and elevation are assigned, whereby - as already mentioned above - the two beam direction components γ ₁ and γ ₂ do not correspond to the definition of azimuth and elevation in spherical coordinates. In principle, both assignments are possible, ie both spatial directions can be used for the vertical or horizontal detection direction; depending on the different detection range and/or different resolution, there may be an advantageous assignment.

Determination of angular errors

• - Zusammenhang zwischen Strahlwinkel γ₁ des Wellenleiter und der Frequenz,
• - Zusammenhang zwischen Laserfrequenz und ihrer Steuergröße,
• - Zusammenhang zwischen Strahlwinkel γ₂ und der oder den Steuergrößen für das Material mit elektrisch steuerbaren optischen Eigenschaften.

The beam direction with the components γ ₁ and γ ₂ depends on the control variables and the hardware properties, in particular on:

• - Relationship between beam angle γ ₁ of the waveguide and the frequency,
• - Relationship between laser frequency and its control variable,
• - Relationship between beam angle γ ₂ and the control variable(s) for the material with electrically controllable optical properties.

Such relationships can vary from sensor to sensor, over temperature and over aging. In order to correctly record the environment, these relationships must be known precisely, otherwise angle errors will occur (measured and actual angles do not match), in particular distorted and shifted recording images. While the variations between the sensors can be determined initially in production, this is often not possible for temperature dependencies (as otherwise the measurement effort in production would be too high), and aging effects can, in principle, only be determined during the lifetime of a sensor. Therefore, a method is needed that can determine angle errors during operation; based on this, they can be compensated for by changing the control, i.e. changing the choice of control variables.

In addition to angular errors of the sensor due to changed hardware properties, the sensor can also be misaligned in the vehicle (particularly due to mechanical tolerances, vehicle loading and aging effects). Such misalignment has a constant effect across all detection directions of the associated spatial direction, while the hardware-related imaging errors can be angle-dependent.

Now we will explain how imaging errors can be determined for sensors looking in the direction of travel. It is known from radar sensors that angular errors (especially due to misalignment) can be determined by comparing the measured radial relative speed of stationary objects with the vehicle's own speed. The radar sensor measures the radial component of the relative movement of an object; a stationary object moves relative to the sensor and in a parallel direction at the vehicle's own speed v _ego , so that for the measured radial component v _m of this movement - as in 30 shown - still has to be multiplied by the cosine of the azimuth angle α of the stationary object: $${\text{v}}_{\text{m}}={\text{v}}_{\text{ego}}\cdot \text{cos}\left(\mathrm{\alpha }\right).$$

However, this relationship only applies at an elevation angle of zero; since radar sensors only have a small detection range at elevation around zero and the above relationship also applies to small elevation angles with a very good approximation, this is usually the only one used for radar sensors. If the radar sensor is misaligned, the azimuth angle α _m measured by the sensor does not correspond to the real azimuth angle α, i.e. the expected measurement speed v _ego ·cos(α _m ) does not correspond to the real measured speed v _ego ·cos(α). The real angle α can basically be determined using the relation (59) and the difference to the measured angle then represents the misalignment. However, it must also be taken into account that radar sensors often only detect objects for small azimuth angle ranges (at least in good quality), the cosine for small angles (i.e. around zero) is very flat and the own speed measured by the vehicle and communicated to the sensor is generally not of good quality. Therefore, a parabolic regression (cosine corresponds to a parabola for small values) is carried out over the measured relative velocities v _m (α _m ) of many stationary objects and the angle of misalignment is then obtained from the parabolic shift.

A coherent lidar system, like a radar system, can directly measure the Doppler, i.e. the radial relative velocity (both systems work coherently and de facto differ only in the frequency range). Thus, the above approach can be transferred to lidar systems. For lidar systems, in addition to misalignment, angle-dependent imaging errors can also be detected by changing hardware properties, and this in the horizontal and vertical spatial direction; instead of relation (59), the relationship is required, which takes into account the beam direction component α in the horizontal spatial direction and the beam direction component β in the vertical spatial direction and in 31 is derived from: $${\text{v}}_{\text{m}}={\text{v}}_{\text{ego}}\cdot \text{sqrt}\left(1-{\text{sin}}^{\text{2}}\left(\mathrm{\alpha }\right)-{\text{sin}}^{\text{2}}\left(\mathrm{\beta }\right)\right).$$

$$\begin{array}{cc}{\text{v}}_{\text{m}}={\text{v}}_{\text{ego}}\cdot \text{cos}\left(\mathrm{\beta }\right)& \text{f}\ddot{\text{u}}\text{r }\mathrm{\alpha }=0\end{array},$$

was auch zur obigen Bez. (59) korrespondiert. Für kleine Betragswerte der Winkel α und β gilt mit sehr guter Näherung: $$\begin{array}{cc}{\text{v}}_{\text{m}}={\text{v}}_{\text{ego}}\cdot \text{cos}\left(\mathrm{\alpha }\right)\cdot \text{cos}\left(\mathrm{\beta }\right)& \text{f}\ddot{\text{u}}\text{r kleine }\left|\mathrm{\alpha }\right|\text{ und }\left|\mathrm{\beta }\right|\end{array}.$$

If one of the two angles is zero, using cos ² (γ) + sin ² (γ) = 1 we get: $$\begin{array}{cc}{\text{v}}_{\text{m}}={\text{v}}_{\text{ego}}\cdot \text{cos}\left(\mathrm{\alpha }\right)& \text{e}\ddot{\text{u}}\text{r}\mathrm{\beta }=0\end{array},$$

$$\begin{array}{cc}{\text{v}}_{\text{m}}={\text{v}}_{\text{ego}}\cdot \text{cos}\left(\mathrm{\beta }\right)& \text{e}\ddot{\text{u}}\text{r}\mathrm{\alpha }=0\end{array},$$

which also corresponds to the above equation (59). For small values of the angles α and β, the following applies with very good approximation: $$\begin{array}{cc}{\text{v}}_{\text{m}}={\text{v}}_{\text{ego}}\cdot \text{cos}\left(\mathrm{\alpha }\right)\cdot \text{cos}\left(\mathrm{\beta }\right)& \text{e}\ddot{\text{u}}\text{r small}\left|\mathrm{\alpha }\right|\text{and}\left|\mathrm{\beta }\right|\end{array}.$$

According to equations (60) and (62c), the measured relative speed depends on both angles α and β of the respective pixel; therefore, if the angular errors in both spatial directions are not known, these cannot be determined from a pixel. However, the angular errors in both spatial directions are independent of each other (they are only determined by hardware properties or control variables for the respective spatial direction); for a pixel field of size 500x320, there are then a total of 820 error sizes (if one assumes the extreme case that the errors of neighboring pixels are independent in the two spatial directions). If there were a stationary object in each pixel, then one would have 500·320 measured values to determine only 820 error sizes, i.e. much more than necessary (820 measured values would be sufficient). In fact, on the one hand, there are fewer pixels with stationary objects, but on the other hand, the errors of neighboring pixels are not independent within a spatial direction. Instead, the course of the angular error can be described with sufficient accuracy using an error curve with a few parameters (e.g. a polynomial, or several sections with a linear or parabolic course) so that only these few unknown parameters need to be determined; for this, there are typically enough measured values per acquisition cycle - and if necessary, the determination can also be extended over several or many cycles, since temperature and aging effects occur comparatively slowly.

Stationary reflections are obtained very reliably from the road surface in the near area; they cover a large or the entire area in the first spatial direction. Because the sensor measures the distance r of the reflections from the road surface and the installation height h _sen is known, the real angle β in the vertical spatial direction is also known, for which 31 applies: $$\text{sin}\left(\mathrm{\beta }\right)={\text{h}}_{\text{sen}}/\text{r}\text{.}$$

Using relation (60) or (62c), in which there is now only one unknown, the course of the angular error for the horizontal spatial direction α can then be determined. For typically given small elevation angles and thus when using relation (62c), after division by cos(β), one can simply average over road reflections of different β (i.e. from different distances) before determining the angular error from the resulting course in the horizontal spatial direction α. If the angular errors in the horizontal spatial direction α and thus the actual value of α for each pixel are known, then the angular error in the vertical spatial direction β can be determined from each pixel (i.e. including those that cannot be assigned to the road surface) (there is then only one unknown in relation (60) or (62c)).

To determine the angular errors using stationary objects, the vehicle's own speed v _ego is required with high accuracy, which is usually not guaranteed by the own speed measured by the vehicle itself and communicated to the sensor. Therefore, the lidar sensor must determine the own speed itself. In the simplest case, this is done by determining the maximum of the measured radial relative speeds, because according to (60), the measured relative speed can assume a maximum of the own speed of the vehicle - this is the case in the direction of travel (i.e. at α = β = 0°). However, this approach assumes that there are stationary reflections in the direction of travel or at least close to it and that the relative speed measurement is very accurate (i.e. in particular not noisy due to a poor signal-to-noise ratio). As mentioned above, the course of the angular errors per spatial direction can be described with sufficient accuracy using an error curve with few parameters (e.g. a polynomial, or several sections with a linear or parabolic course), so that only these few unknown parameters can be deduced from the measured values (i.e. the measured relative velocities of stationary objects); if the velocity is unknown, this is added as a further unknown and is then determined at least implicitly.

If the angle errors are known, they can be corrected in the detection list determined by the sensor, i.e. the real angles can be specified for the detections. In addition, the course of the control variables used for scanning can be adjusted for subsequent acquisition cycles in order to eliminate the angle error (if necessary in an iterative manner) - in particular in order to precisely realize the desired detection range of the two-dimensional pixel field.

Determination of the position of the road surface at greater distances

It was explained above that the road surface can be used to determine angular errors, using reflections of the road surface in the near area.

Reflections of the road surface at a greater distance provide information about the exact position of the road surface at that distance (i.e. in which pixel and thus at which vertical angle the road surface is located, possibly improved to subpixel accuracy by interpolation), which can be helpful for interpreting the recorded data. One example is the exact determination of the height of an object lying on the road; if you know not only the measured reflection points of the object but also the exact position of the road surface at that object, the height determination is more accurate. It should be noted that the position of the road surface is only known a priori in the close range from the known sensor height and because of the assumption of a non-curved road course that is valid over short distances; this no longer applies at greater distances (a curved road course, e.g. in a depression, then has a strong effect) and in addition, even small changes in the vertical sensor alignment (e.g. due to a slight vehicle inclination when braking and accelerating) have a significant effect. However, a lidar sensor generally does not capture the reflections of the road surface at a greater distance sufficiently to be able to detect them; a key reason for this is 32 (Illustration not to scale in the vertical direction for better understanding). This shows a beam with a width of 0.05°, the center of which hits the flat road at a distance of 100m, with the sensor installation height h _sen = 60cm; due to the very small angle of incidence on the road, the beam is projected onto the road over a distance range of around 15m. This means that in the two-dimensional correlation E _m,k the road surface is visible over around 30 discrete distances m (i.e. not just at one m ₀ ), so that the received power is divided up into a corresponding number of values in the two-dimensional correlation, which leads to a very poor signal-to-noise ratio and can therefore prevent detection. This reduction in the received power per distance value m of the two-dimensional correlation can also be explained by the fact that only around 1/30 of the beam width and thus of the transmitted power is effective per distance value.

For better detection of the road surface, one can exploit the fact that it is seen over many pixels in relation to the horizontal spatial direction α, i.e. that there are many pixels where the beam center hits the road surface at the same distance; due to relation (62) these pixels have the same beam angle β, i.e. for the vertical spatial direction. A first approach consists in accumulating the two-dimensional correlation E _m,k non-coherently over these pixels, i.e. in particular by using their magnitude or power values, whereby this is done at least for all values of the discrete distance m where the road can be located. At the distances m where the accumulated value is sufficiently far above the noise level, the road is then located; the noise level in the individual two-dimensional correlation E _m,k is known (for example, it is calculated using the calculation logic according to 8 and 10 determined), and from this one can then calculate the noise level after non-coherent accumulation according to theoretical relationships - or one can estimate it directly using the resulting values of the non-coherent accumulation, in which case a sufficiently large range of m and possibly different values of the discrete frequency k must be evaluated. As explained above, the road is seen in a whole range of discrete distance m, which in the example above extends over about 30 values of m; the center m ₀ of this range represents approximately the actual distance of the road. Instead of doing the non-coherent integration only over values with a discrete running time m, one can also extend it over several neighboring values of m, since the road is also seen over numerous m.

The above evaluation does not have to be done for every discrete frequency, but in principle only for the frequency k ₀ where the road reflections are located; since the vehicle's own speed is known (with the help of the high-quality detection data of the lidar sensor itself), this dis The calculation is usually quite accurate for a specific frequency, so that the calculation only needs to be carried out for one or a few discrete frequencies k. When using a calculation logic according to 8 or 10 the values of the two-dimensional correlation E _m,k for these discrete frequencies are then to be output; if these discrete frequencies can vary, then the realization of the output is very complex; therefore, using the rain logic according to 10 It is advantageous to always have the required discrete frequencies at the same outputs of the calculation logic via a speed-dependent choice of the rotation factors d _n,m .

Instead of a non-coherent integration over values of the two-dimensional correlation E _m,k, the following second approach can also be chosen: The reception sequence e(n) of a pixel is split into two sequences of half the length, with the first sequence being formed from the even-numbered n and the second sequence from the odd-numbered n. The respective two-dimensional correlation is determined for these two sub-sequences and then, for each relevant distance m and frequency k, the value of one correlation is multiplied by the complex conjugate value of the other. There is a phase difference between the two sub-sequences, which results from the reception frequency f _e and the time interval between two samples, i.e. the sampling time T _s ; the product formed from the two correlations has a complex value at the distance m and the frequency k ₀ of the road reflections, the phase of which corresponds to this phase difference, although this statement naturally only refers to the signal component, not to the superimposed noise.

With a value of m, this phase is at least approximately constant across all pixels with road reflections, since the relative speed and thus the reception frequency f _e can be considered constant in the small angular range of the road around 0° (cos(α) in relation (62c) is then approximately =1). Therefore, the product can be integrated coherently across the pixels, i.e. summed up, which results in a greater improvement in the signal-to-noise ratio than with non-coherent integration. If integration is also to be carried out across different m, the distance-dependent frequency component f _r in the reception frequency f _e must be corrected according to relation (12), preferably as part of the rotation factors d _n,m in the calculation logic according to 10 . When using this calculation logic, the two two-dimensional correlations to the subsequences of e(n) with even and odd n are also inherently present - namely at the output of the penultimate stage of the FFT in alternating sequence. The further considerations and evaluations apply as in the first approach with non-coherent integration.

This second approach, which involves forming the product of two correlations and summing them across pixels, can also be generalized. The key point here is that the two correlations at least partially use received signals that come from the same reflection points on the road, i.e. the same areas, so that a defined phase relationship is achieved. For example, the first and second half of the received sequence per pixel could be used to calculate the two correlations, or the correlations of two neighboring pixels, provided they overlap; however, since the received values associated with the two correlations are then further apart in time, one may be sensitive to a changing speed (in which case the phase difference can change across the sequence of pixels).

Determination of the sign of the reception frequency in a real-valued mixer using non-binary phase modulation

In the binary phase modulation considered so far with the two phase values 0° and 180° and using a real-valued mixer, two magnitude peaks occur in the two-dimensional correlation E _m,k at (m ₀ ,+k ₀ ) and (m ₀ ,-k ₀ ), so that only the magnitude of the received frequency can be determined. It was shown above how this ambiguity can be at least partially resolved with the help of a superimposed linear frequency modulation. An alternative approach for determining the sign of the received frequency with a real-valued mixer is presented below.

beschrieben werden, und mit der Bez. (3a) ergibt sich für die Empfangsfolge e(n) eines einzelnen Objekts: $$\text{e}\left(\text{n}\right)=\text{a}\cdot \text{exp}\left(\widehat{\text{j}}\cdot \left[2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{k}}_0-{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right),$$

wobei das negative Vorzeichen von φ_TX(n-m₀) daraus resultiert, dass für den komplexen Mischer - in Konsistenz zu den obigen Herleitungen - angenommen wird, dass er die Phasendifferenzen zwischen empfangenem und gesendetem Signal bildet (im anderen Fall, also wenn Mischer Phasendifferenz zwischen gesendetem und empfangenem Signal bilden würde, wäre Vorzeichen positiv). Bei reellwertigem Mischer erhält man den Realteil davon: $$\text{e}\left(\text{n}\right)=\text{a}/2\cdot \text{exp}\left(+\widehat{\text{j}}\cdot \left[2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{k}}_0-{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right)+\text{a}/2\cdot \text{exp}\left(-\widehat{\text{j}}\cdot \left[2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{k}}_0-{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right).$$

The phase modulation φ _TX (n) should now not consist of the two binary values 0° and 180°, but of a set of J phase values φ _j with j = 0,...,J-1, which can assume general phase values; furthermore, the phase modulation changes irregularly, in particular pseudorandomly, over these J phase values. The phase modulation sequence can then be represented by a complex-valued $$\text{b}\left(\text{n}\right)=\text{ex}\left(\widehat{\text{j}}\cdot {\mathrm{\phi }}_{\text{TX}}\left(\text{n}\right)\right)$$

and with the reference (3a) the reception sequence e(n) of a single object is: $$\text{e}\left(\text{n}\right)=\text{a}\cdot \text{ex}\left(\widehat{\text{j}}\cdot \left[2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{k}}_0-{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right),$$

where the negative sign of φ _TX (nm ₀ ) results from the fact that for the complex mixer - in consistency with the above derivations - it is assumed that it forms the phase differences between the received and transmitted signal (in the other case, i.e. if the mixer would form a phase difference between the transmitted and received signal, the sign would be positive). For a real-valued mixer, the real part of this is obtained: $$\text{e}\left(\text{n}\right)=\text{a}/2\cdot \text{ex}\left(+\widehat{\text{j}}\cdot \left[2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{<figure-callout id="0" label="k" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">k</figure-callout>}}_0-{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right)+\text{a}/2\cdot \text{ex}\left(-\widehat{\text{j}}\cdot \left[2\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{<figure-callout id="0" label="k" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">k</figure-callout>}}_0-{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right).$$

gebildet, wobei b(n) nach Bez. (63) und keine Konjugiert-Komplex-Bildung für b(n-m) benutzt wird, was zu einer Korrelation mit dem ersten Anteil von e(n) in Bez. (65) zu Frequenz +k₀ korrespondiert (eine Benutzung des konjugiert Komplexen von b(n-m) würde zum zweiten Anteil zu Frequenz +k₀ korrespondieren). Mit Bez. (65) erhält man: $$\text{p}\left(\text{n}\right)={\text{p}}_{\text{1}}\left(\text{n}\right)+{\text{p}}_{\text{2}}\left(\text{n}\right)$$

mit $${\text{p}}_{\text{1}}\left(\text{n}\right)=\text{a}/2\cdot \text{exp}\left(\widehat{\text{j}}\cdot \left[{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-\text{m}\right)-{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right)\cdot \text{exp}\left(+\widehat{\text{j}}\cdot \text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{k}}_0\right)\text{ und}$$

$${\text{p}}_{\text{2}}\left(\text{n}\right)=\text{a}/2\cdot \text{exp}\left(-\widehat{\text{j}}\cdot \left[{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-\text{m}\right)+{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right)\cdot \text{exp}\left(+\widehat{\text{j}}\cdot \text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{k}}_0\right).$$

The two-dimensional correlation E _m,k in the form of the relation (8) is calculated as an FFT over the product sequence $$\text{p}\left(\text{n}\right)=\text{e}\left(\text{n}\right)\cdot \text{b}\left(\text{n}-\text{m}\right)=\text{e}\left(\text{n}\right)\cdot \text{ex}\left(\widehat{\text{j}}\cdot {\mathrm{\phi }}_{\text{TX}}\left(\text{n}-\text{m}\right)\right)$$

formed, where b(n) is formed according to Ref. (63) and no complex conjugation is used for b(nm), which corresponds to a correlation with the first part of e(n) in Ref. (65) at frequency +k ₀ (using the complex conjugation of b(nm) would correspond to the second part at frequency +k ₀ ). Using Ref. (65) we get: $$\text{p}\left(\text{n}\right)={\text{p}}_{\text{1}}\left(\text{n}\right)+{\text{p}}_{\text{2}}\left(\text{n}\right)$$

with $${\text{p}}_{\text{1}}\left(\text{n}\right)=\text{a}/2\cdot \text{ex}\left(\widehat{\text{j}}\cdot \left[{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-\text{m}\right)-{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right)\cdot \text{ex}\left(+\widehat{\text{j}}\cdot \text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{k}}_0\right)\text{and}$$

$${\text{p}}_{\text{2}}\left(\text{n}\right)=\text{a}/2\cdot \text{ex}\left(-\widehat{\text{j}}\cdot \left[{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-\text{m}\right)+{\mathrm{\phi }}_{\text{TX}}\left(\text{n}-{\text{m}}_0\right)\right]\right)\cdot \text{ex}\left(+\widehat{\text{j}}\cdot \text{2}\mathrm{\pi }\cdot \text{n}/\text{N}\cdot {\text{k}}_0\right).$$

At the discrete running time, as distance m = m ₀ of the object, the first component p ₁ (n) represents a continuously rotating pointer with the discrete frequency +k ₀ of the object (since φ _TX (nm ₀ )-φ _TX (nm ₀ ) = 0), so that after the FFT, i.e. as a result of the two-dimensional correlation E _m,k, it forms the magnitude peak at the frequency +k ₀ and the discrete running time m ₀ .

The second component p ₂ (n) generates the magnitude peak at frequency -k ₀ and discrete propagation time m ₀ in binary phase modulation (phase values 0° and 180°), because φ _TX (nm ₀ )+φ _TX (nm ₀ ) = 2·φ _TX (nm ₀ ) can only assume the two effectively identical values 0° and 360° (in both cases the complex vector formed by them is =1). If other phase values are selected, however, phase jumps occur in p ₂ (n), since 2·φ _TX (nm ₀ ) does not only assume integer multiples of 360°. As an example, consider the J = 4 phase values φ _j = 0,90°,180°,270° (the modulation sequence b(n) therefore assumes the 4 values 0,+ ĵ,-1,-ĵ). Then exp(-ĵ·2·φ _TX (nm ₀ )) takes on the values +1 and -1, between which a pseudo-random jump is made. Since all 4 phase values are used with the same probability or frequency, the mean value of exp(-ĵ·2·φ _TX (nm ₀ )) is zero, at least with a very good approximation; at the discrete frequency -k ₀ of the object, the contribution of p ₂ (n) to the two-dimensional correlation E _m,k then disappears, so that no peak in magnitude occurs there any more. Because now only one peak in magnitude occurs (at the correct frequency +k ₀ ), the reception frequency is clearly and correctly determined.

realisieren; minimal werden also J = 3 Phasenwerte φ_j = 0, 120°, 240° benötigt. Bei anderer Wahl der Phasenwerte wird die falsche Betragsspitze bei Frequenz -k₀ nicht komplett eliminiert, sondern nur reduziert; im Beispiel der nur J = 2 Phasenwerte φ_j = 0,90° beträgt die Reduktion 3dB. Für die Erkennung der richtigen Betragsspitze wählt man die größere der beiden Betragsspitzen bei ±k₀ aus; abgesehen von einem schlechten Signal-zu-Rauschverhältnis reicht zur korrekten Erkennung schon ein kleiner nominaler Unterschied der Beträge aus. Verallgemeinert kann gesagt werden, dass unter den verwendeten J Phasenwerten φ_j zumindest zwei Phasenwerte sein müssen, die weder gleich- noch gegenphasig sind, um bei einem reellwertigen Mischer das Vorzeichen der Empfangsfrequenz bestimmen zu können.The disappearance of the magnitude peak at the wrong frequency -k ₀ can be explained by each set of J over one revolution, i.e. 360° equally distributed phase values $$\begin{array}{cc}{\mathrm{\phi }}_{\text{j}}=360°/\text{J}\cdot \left(\mathrm{0.1,}\dots ,\text{J}-1\right)& \text{with J}>2\end{array}$$

realize; at a minimum, J = 3 phase values φ _j = 0, 120°, 240° are required. If the phase values are selected differently, the incorrect magnitude peak at frequency -k ₀ is not completely eliminated, but only reduced; in the example of only J = 2 phase values φ _j = 0.90°, the reduction is 3dB. To detect the correct magnitude peak, the larger of the two magnitude peaks at ±k ₀ is selected; apart from a poor signal-to-noise ratio, a small nominal difference in the magnitudes is sufficient for correct detection. In general, it can be said that among the J phase values φ _j used, at least There must be at least two phase values that are neither in phase nor in antiphase in order to be able to determine the sign of the received frequency in a real-valued mixer.

Phase values, especially if they are not just in phase opposition, cannot usually be realized with arbitrary precision. As an example, consider the J = 4 equidistant phase values with nominal position φ _j,nom = 0, 90°, 180°, 270°; their real values should be φ _j,real = 15°, 75°, 195°, 255°, i.e. their distances have large errors of ±30°. In this case, phase jumps of ±30° and thus non-constant phase (i.e. phase jitter) occur in the first component p ₁ (n), which leads to a reduction of the magnitude peak at the correct frequency +k ₀ by only 0.3 dB and thus only a small loss of sensitivity (the lost energy migrates into a small noise at other Doppler frequencies k, which is, however, significantly below the noise level in the two-dimensional correlation generated by the phase modulation itself); At the wrong frequency -k ₀ the magnitude peak is no longer completely eliminated, but a small DC component in the phases of p ₂ (n) leads to a magnitude peak that is 11.4 dB below the magnitude peak at the correct frequency +k ₀ , so that it can still be correctly detected. This example shows that even quite large errors in the realization of the phase values are still not critical.

Finally, possible implementations will be discussed. One approach is to switch between different lengths of line; 33 shows the case of J = 3 phase values as an example, ie switching between three different lengths of line is carried out, whereby the different lengths result in a phase difference of 120° or 240°. For the above example of J = 4 equidistant phase values, one can - as in 34 shown - use a combination of a switchable inverter 34.1 and a changeover switch 34.2 between two line sections which have a phase difference of 90°.

Overall system

• - Photonischer Chip, enthält:
  • • in Frequenz verstimmbare Laserquelle (vorzugsweise gibt es nur eine; nur falls Frequenzdurchstimmbereich zu gering ist, werden mehrere benötigt),
  • • Phasenmodulationseinheit bestehend aus schaltbarem Invertierer,
  • • 32 parallele Sende-Empfangspfade mit optischem Verstärker, Zirkulator, reellwertigem Überlagerungsmischer und Photodiode,
  • • 32 Wellenleiter (im Falle der Benutzung einer Schaltmatrix zum Scannen in zweiter Raumrichtung 320 Wellenleiter),
  • • Ausgang mit 32 analogen Empfangssignalen im hohen MHz-Bereich;
• - Digitalchip, enthält:
  • • 32 Analog-Digital-Wandler für Empfangssignale, die vom photonischen Chip ausgegeben werden,
  • • festverdrahte Rechenlogik zur Bestimmung der zweidimensionalen Korrelation mit nachgelagerter Auswertung,
  • • Mikrocontroller und/oder DSP(s) zur weiteren Signalauswertung (insbesondere für Bestimmung einer Detektionsliste) und zur Berechnung der Steuergrößen für Laserfrequenz und Scanner für zweite Raumrichtung,
  • • Steuerausgänge für Laserfrequenz und Scanner für zweite Raumrichtung (können analog oder digital sein);
• - Scanner in eine Raumrichtung (für zweite Raumrichtung), realisiert durch:
  • • Material mit elektrisch steuerbaren optischen Eigenschaften, insbesondere ein Flüssigkristallelement oder -Array, oder
  • • Schaltmatrix für jeden der 32 parallelen Sende-Empfangspfade oder
  • • mechanischer Ansatz, z. B. in Form eines oszillierenden oder rotierenden Spiegels oder Prismas,
  • • wobei für manche der Ansätze noch eine Linseneinheit und/oder Strahlumlenkeinheit benötigt wird.

A lidar system according to the above preferably comprises or consists only of the following three main components:

• - Photonic chip, contains:
  • • frequency-tunable laser source (preferably there is only one; only if the frequency tuning range is too small, several are needed),
  • • Phase modulation unit consisting of switchable inverter,
  • • 32 parallel transmit-receive paths with optical amplifier, circulator, real-valued superheterodyne mixer and photodiode,
  • • 32 waveguides (in case of using a switching matrix for scanning in the second spatial direction 320 waveguides),
  • • Output with 32 analog reception signals in the high MHz range;
• - Digital chip, contains:
  • • 32 analog-digital converters for receiving signals output by the photonic chip,
  • • hard-wired calculation logic for determining the two-dimensional correlation with subsequent evaluation,
  • • Microcontroller and/or DSP(s) for further signal evaluation (in particular for determining a detection list) and for calculating the control variables for laser frequency and scanner for the second spatial direction,
  • • Control outputs for laser frequency and scanner for second spatial direction (can be analog or digital);
• - Scanner in one spatial direction (for second spatial direction), realized by:
  • • Material with electrically controllable optical properties, in particular a liquid crystal element or array, or
  • • Switching matrix for each of the 32 parallel transmit-receive paths or
  • • mechanical approach, e.g. in the form of an oscillating or rotating mirror or prism,
  • • Some of the approaches still require a lens unit and/or beam deflection unit.

In an optimal case, all electronic components are located on one board; for some arrangements, two boards may be necessary. The high semiconductor integration potential of the proposed approach can significantly reduce costs and size.

In order to reduce the amount of hardware required, a halved number of parallel transmit-receive paths can be used by halving the data acquisition time per pixel, which then also halves the size and power consumption of the hard-wired computing logic. A general guideline for this concept is that all components are active (i.e. used) the entire time and that all the generated and radiated power is used for detection (i.e. in particular, power is only radiated in directions from which data is also being received at the same time).

Alternative phase modulation form

So far, phase modulation in the coherent lidar system has been 1 a pseudorandom binary sequence repeating with the period N (i.e. consisting of the two phase values 0° and 180°) according to 2 Instead of a pseudorandom sequence, irregular sequences with a different definition could also be used, for example sequences characterized by low side lobes of the autocorrelates (e.g. a Gold code known from the literature). At least if a high sensitivity is required, irregular sequences always require the complex determination of the two-dimensional correlation E _m,k according to (8) and thus a special calculation logic, in particular according to 8 and 10 If such a calculation logic is not available, but only DSPs (especially with parallel vectorial calculation units), modulation sequences must be used which allow a simpler evaluation (but may also be associated with other disadvantages).

und die zweite Teilfolge b₂(n) der Länge N₂ = N-N₁ ist unregelmäßig, z. B. pseudozufällig: $$\begin{array}{ccc}{\text{b}}_{\text{2}}\left(\text{n}\right)=\pm 1& \text{mit}& \text{n}={\text{N}}_{\text{1}},\dots ,\text{N}-1\end{array}.$$

As an example, we first consider a sequence and an evaluation approach, which are described in the article "Phase-Coded-Based Modulation for Coherent Lidar" by Sebastian Banzhaf and Christian Waldschmidt, published in IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 70, NO. 10, OCTOBER 2021 , are proposed. There, the binary modulation sequence b(n) of length N is composed of two parts: during the first subsequence b ₁ (n) with length N ₁ the phase is constant, e.g. $$\begin{array}{ccc}{\text{b}}_{\text{1}}\left(\text{n}\right)=1& \text{with}& \text{n}=\mathrm{0,}\dots ,{\text{N}}_{\text{1}}-1\end{array},$$

and the second subsequence b ₂ (n) of length N ₂ = NN ₁ is irregular, e.g. pseudorandom: $$\begin{array}{ccc}{\text{b}}_{\text{2}}\left(\text{n}\right)=\pm 1& \text{with}& \text{n}={\text{<figure-callout id="1" label="N" filenames="DE102023201147A1\_0150.png,DE102023201147A1\_0151.png" state="{{state}}">N</figure-callout>}}_{\text{1}},\dots ,\text{N}-1\end{array}.$$

und die Modulationsfolge kann sich für mehrere Erfassungsrichtungen, also Pixel periodisch wiederholen; in 35 ist eine solche Modulationsfolge dargestellt, wobei sich die aus den Teilfolgen zusammengesetzte Folge mit der Periode N=4096 wiederholt.The lengths N ₁ and N ₂ can be equal, so $${\text{N}}_{\text{2}}={\text{N}}_{\text{1}}=\text{N}/\mathrm{2,}$$

and the modulation sequence can be repeated periodically for several detection directions, i.e. pixels; in 35 Such a modulation sequence is shown, where the sequence composed of the subsequences repeats with the period N=4096.

d. h. sie trägt nur die Dopplerfrequenz des jeweiligen Objekts, welche über eine DFT bzw. FFT bestimmt werden kann. Wegen der unbekannten Zeitverschiebungen m_0,i kennt man allerdings nicht die jeweilige genaue zeitliche Lage von e_1,i(n) innerhalb der Empfangsfolge e(n); man kann vereinfachend beispielsweise die Zeitverschiebung Null annehmen, d. h. über die Empfangsteilfolge $$\begin{array}{cc}{\tilde{\text{e}}}_1\left(\text{n}\right)=\text{e}\left(\text{n}\right)& \text{mit n}=\mathrm{0,}\dots ,\text{N}/2-1\end{array}$$

bildet man die FFT $$\begin{array}{cc}{\tilde{\text{E}}}_{\mathrm{1,}\text{k}}={\text{FFT}}_{\text{k}}\left({\tilde{\text{e}}}_1\left(\text{n}\right)\right)={\text{FFT}}_{\text{k}}\left(\text{e}\left(\text{n}\right)\right)& \text{mit n}=\mathrm{0,}\dots ,\text{N}/2-1\end{array}.$$

For the reception subsequence e _1,i (n) generated by an object i for the first, constant modulation subsequence b ₁ (n), the following applies according to (3a): $$\begin{array}{cc}{\text{e}}_{\mathrm{1,}\text{i}}\left(\text{n}\right)={\text{a}}_{\text{i}}\cdot \text{ex}\left(2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\text{k}}_{\mathrm{0,}\text{i}}\right)& \text{with n}={\text{m}}_{\mathrm{0,}\text{i}},\dots ,{\text{m}}_{\mathrm{0,}\text{i}}+\text{N}/2-1\end{array},$$

ie it only carries the Doppler frequency of the respective object, which can be determined via a DFT or FFT. Due to the unknown time shifts m _0,i , however, the exact temporal position of e _1,i (n) within the reception sequence e(n) is not known; for simplicity, one can assume the time shift to be zero, ie via the reception sub-sequence $$\begin{array}{cc}{\tilde{\text{e}}}_1\left(\text{n}\right)=\text{e}\left(\text{n}\right)& \text{with n}=\mathrm{0,}\dots ,\text{N}/2-1\end{array}$$

the FFT is formed $$\begin{array}{cc}{\tilde{\text{E}}}_{\mathrm{1,}\text{k}}={\text{FFT}}_{\text{k}}\left({\tilde{\text{e}}}_1\left(\text{n}\right)\right)={\text{FFT}}_{\text{k}}\left(\text{e}\left(\text{n}\right)\right)& \text{with n}=\mathrm{0,}\dots ,\text{N}/2-1\end{array}.$$

However, the assumption of a time shift of zero means that for other actual time shifts m _0,i , in particular for distant targets, in the first m _0,i -1 values of the received sub-sequence ẽ ₁ (n) there are no values from the constant modulation sub-sequence b ₁ (n), but rather rear values of the modulation sub-sequence b ₂ (n) of the previous period; this reduces the height of the respective magnitude peak of the FFT and thus its distance from the noise, so that the sensitivity is reduced. The magnitude peaks of the FFT Ẽ _1,k of the received sub-sequence ẽ ₁ (n) are checked for a detection threshold. The frequencies k _0,j of the J magnitude peaks above the detection threshold are used for further processing; Typically, these magnitude peaks are seen in two neighboring FFT values (since they are not - as previously considered - at an integer Doppler index k ₀ ), so that their exact position, i.e. a non-integer frequency k _0,j, can be determined by interpolation. These frequencies k _0,j correspond at least approximately to the Doppler frequencies k _0,i of the objects or a subset of them (for objects with very low reflectivity, they may not lead to any magnitude peak above the detection threshold).

The second modulation sub-sequence b ₂ (n) leads in the reception sequence e(n) according to (3a) to components that are time-shifted by m _0,i and multiplied by the respective Doppler frequency k _0,i , i.e. modulated $$\begin{array}{cc}{\text{<figure-callout id="2" label="e" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">e</figure-callout>}}_{\mathrm{2,}\text{i}}\left(\text{n}\right)={\text{a}}_{\text{i}}\cdot {\text{b}}_{\text{2}}\left(\text{n}-{\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\mathrm{0,}\text{i}}\right)\cdot \text{ex}\left(2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\text{<figure-callout id="0" label="k" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">k</figure-callout>}}_{\mathrm{0,}\text{i}}\right)& \text{with n}-{\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\mathrm{0,}\text{i}}=\text{N}/\mathrm{2,}\dots ,\text{N}-1\end{array}.$$

dabei muss der Bereich n = N/2,...,N-1+M-1 der Empfangsfolge e(n) betrachtet werden, in welchem der Empfang der zweiten Modulationsteilfolge b₂(n) liegen kann, wobei M-1 zu der größten anzunehmenden bzw. interessierenden Objektentfernung korrespondiert. Es sei bemerkt, dass der Bereich n = N,...,N-1+M-1 des Empfangssignals nur bei einem kontinuierlichen Scannen benutzt werden kann, allerdings nicht bei einem geschalteten Scannen (dann ist das dortige Signal wegen anderer Erfassungsrichtung nicht kohärent).In order to eliminate the respective modulation by the Doppler frequency, the reception sequence e(n) is rotated back by the frequency k _0,j : $$\begin{array}{cc}{\tilde{\text{e}}}_{\mathrm{2,}\text{j}}\left(\text{n}\right)=\text{e}\left(\text{n}\right)\cdot \text{ex}\left(-2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\text{k}}_{\mathrm{0,}\text{j}}\right)& \text{with n}=\text{N}/\mathrm{2,}\dots ,\text{N}-1+\text{M}-1\text{and j}=\mathrm{1,}\dots \text{J}-1\end{array};$$

The range n = N/2,...,N-1+M-1 of the reception sequence e(n) must be considered, in which the reception of the second modulation sub-sequence b ₂ (n) can take place, where M-1 corresponds to the largest assumed or interesting object distance. It should be noted that the range n = N,...,N-1+M-1 of the reception signal can only be used for continuous scanning, but not for switched scanning (in this case the signal there is not coherent due to a different detection direction).

The sequences ẽ _2,j (n) modified in this way contain shifted modulation subsequences a _i ·b ₂ (nm _0,i ) in the corresponding index j; contributions from objects with other Doppler frequencies k _0,i (i.e. k _0,i ≠ k _0,j ) represent a modulation sequence that is uncorrelated with b ₂ (n), since they are still modulated with the difference frequency k _0,i -k _0,j . Therefore, the sequences ẽ _2,j (n) can now be correlated with the second modulation subsequence b ₂ (n): $$\begin{array}{ccc}{\tilde{\text{E}}}_{\mathrm{2,}\text{j},\text{m}}={\text{sum}}_{\text{n}=\text{N}/\mathrm{2,}\dots ,\text{N}-1}\left({\tilde{\text{e}}}_{\mathrm{2,}\text{j}}\left(\text{n}+\text{m}\right)\cdot {\text{b}}_2\left(\text{n}\right)\right)& \text{with m}=\mathrm{0,}\dots ,\text{M}-1& \text{and j}=\mathrm{1,}\dots \text{J}-1\end{array}.$$

In these one-dimensional correlations Ẽ _2,j,m, magnitude peaks occur at the points m = m _0,i , i.e. at the discrete travel times of objects; in the general case of a non-integer discrete travel time m _0,i, the magnitude peak extends over two neighboring values of m if suitable measures are taken (e.g. smoothing the modulation pulses to an approximately triangular shape), and its non-integer position can be determined by interpolation. The Doppler frequency k 0,i = k _0,j of the object results from the index j of the correlation Ẽ _2,j,m , at which the magnitude peak occurs, and the Doppler frequency k _{0 ,j} associated with this j. In this way, the distances and radial relative velocities of objects in the respective detection direction can be determined from magnitude peaks of the correlations Ẽ _2,j,m that lie above a detection threshold. It should be noted that contributions from objects with different Doppler frequencies k _0,i (i.e. k _0,i ≠ k _0,j ) only lead to noise in the respective correlations due to their uncorrelatedness to b ₂ (n), i.e. to no magnitude peaks above a detection threshold; if reflection signals from objects are approximately equally strong, this noise is significantly below the magnitude peaks of interest and thus does not obscure them - only if reflection signals from objects are very different can objects with a strong reflection signal obscure weak reflection signals with a different Doppler frequency via their noise.

With the modulation sequence according to 35 and the evaluation explained above, in the typical case of an object in a pixel, an FFT is required for the reception sub-sequence ẽ ₁ (n) to the first modulation sub-sequence b ₁ (n) and a correlation between the second modulation sub-sequence b ₂ (n) and the frequency-returned rotated reception sub-sequence ẽ _2,1 (n) to the second modulation sub-sequence. This correlation in the time domain can also be realized by multiplying the FFTs in the frequency domain followed by an inverse FFT (inverse FFT requires the same amount of computing power as the FFT itself); the FFT of the modulation sub-sequence b ₂ (n) can be determined once a priori, so that only the FFT of the reception sub-sequence ẽ _2,1 (n) needs to be determined. Overall, the amount of effort required is three FFTs, while for the two-dimensional correlation E _m,k according to equation (8), a total of M=250 FFTs must be calculated, whereby these FFTs have approximately twice the length if the same duration of a pixel is assumed. This results in a very strong reduction in the required computing power to an order of magnitude that modern DSPs with parallel vector processing units can achieve.

• - Die Sensitivität ist um etwas mehr als 3dB geringer, wenn gleiche Dauer eines Pixels angenommen wird; die halbe Länge der FFT (von der Empfangsteilfolge e₁(n)) wie auch die halbe Länge der zeitlichen Korrelation (von ẽ_2,j(n) zu der Modulationsteilfolge b₂(n)) führt auf 3dB Verlust, und zusätzlich gibt es - wie oben erläutert - den Effekt, dass insbesondere für weit entfernte Ziele die ersten Werte der Empfangsteilfolge ẽ₁(n) nicht von der konstanten Modulationsteilfolge b₁(n) stammen, sondern von hinteren Werten der Modulationsteilfolge b₂(n) der vorhergehenden Periode, so dass diese Werte effektiv nicht zur jeweiligen Betragsspitze der FFT beitragen.
• - Wegen der halben Länge der FFT sind Auflösung und Genauigkeit für die Dopplerbestimmung, d. h. die Bestimmung der radialen Relativgeschwindigkeit um Faktor zwei schlechter.
• - Die zeitliche Korrelation von ẽ_2,j(n) mit b₂(n) zur Bestimmung der Entfernung hat nur die halbe Länge. Damit reduziert sich der Dynamikbereich bei mehr als einem Objekt pro Pixel um 3dB, d. h. ein Objekt mit starkem Reflektionssignal erhöht den Rauschpegel relativ zu dem Pegel eines Objekts mit schwächerem Reflektionssignal um 3dB mehr, so dass die Wahrscheinlichkeit für das Nichtdetektieren eines solchen zweiten Objekts höher ist.
• - Bei kontinuierlich scannenden Systemen können grundsätzlich Pixel überlappen, d. h. manche der Empfangswerte e(n) werden für zwei benachbarte Pixel benutzt, insbesondere um eine längere Datenaufnahmezeit pro Pixel und damit eine bessere Sensitivität zu erzielen. Bei einer Modulationsfolge nach 35 gibt es aber nur die Möglichkeit eines Überlapps von 50%, da man ja in jedem Pixel beide Modulationsteilfolgen benötigt. Kleinere Überlappe, welche häufig bevorzugt werden, sind nicht möglich.

Disadvantageous compared to a modulation sequence according to 2 and evaluation via a two-dimensional correlation E _m,k according to (6) or (8) are in particular the following points:

• - The sensitivity is slightly more than 3dB lower if the same duration of a pixel is assumed; half the length of the FFT (from the receive subsequence e ₁ (n)) as well as half the length of the temporal correlation (from ẽ _2,j (n) to the modulation subsequence b ₂ (n)) leads to 3dB loss, and in addition - as explained above - there is the effect that, especially for distant targets, the first values of the receive subsequence ẽ ₁ (n) do not come from the constant modulation subsequence b ₁ (n), but from rear values of the modulation subsequence b ₂ (n) of the previous period, so that these values effectively do not contribute to the respective magnitude peak of the FFT.
• - Because the FFT is half the length, the resolution and accuracy for the Doppler determination, ie the determination of the radial relative velocity, are worse by a factor of two.
• - The temporal correlation of ẽ _2,j (n) with b ₂ (n) for determining the distance is only half the length. This reduces the dynamic range by 3 dB when there is more than one object per pixel, ie an object with a strong reflection signal increases the noise level by 3 dB more relative to the level of an object with a weaker reflection signal, so that the probability of not detecting such a second object is higher.
• - In continuously scanning systems, pixels can generally overlap, ie some of the reception values e(n) are used for two adjacent pixels, in particular to achieve a longer data acquisition time per pixel and thus a better sensitivity. In a modulation sequence according to 35 However, there is only the possibility of an overlap of 50%, since both modulation subsequences are required in each pixel. Smaller overlaps, which are often preferred, are not possible.

$$\begin{array}{cc}{\text{b}}_{\text{2}}\left(\text{n}\right)=\pm 1& \text{mit n}=\mathrm{1,3,5,}\dots ,\text{N}-1\end{array}.$$

These disadvantages of the modulation sequence according to 35 and the associated evaluation explained above, which are state of the art, can be achieved by the new modulation sequence according to 36 can be eliminated to a significant extent without significantly increasing the computing power required for the evaluation. For the modulation sequence b(n) according to 36 the two previous modulation subsequences b ₁ (n) and b ₂ (n) are alternately nested within each other: $$\begin{array}{cc}{\text{b}}_{\text{1}}\left(\text{n}\right)=1& \text{with n}=\mathrm{0,2,4,}\dots ,\text{N}-2\end{array},$$

$$\begin{array}{cc}{\text{b}}_{\text{2}}\left(\text{n}\right)=\pm 1& \text{with n}=\mathrm{1,3,5,}\dots ,\text{N}-1\end{array}.$$

Thus, both modulation subsequences extend over the full modulation duration, taking up only every second grid value.

bei diesem Bereich vom Index n stammen die ersten Werte vom Ende der vorherigen Modulationsperiode (wegen der Laufzeit des jeweiligen Objekts) - im Gegensatz zur vorher betrachteten Modulationsfolge nach 35 kommen diese Werte aber auch von der ersten, konstanten Modulationsfolge, sodass sie auch kohärent zur späteren FFT beitragen. Wegen der zwei möglichen Lagen der Empfangsteilfolgen e_1,i(n) muss die FFT zur Bestimmung der Dopplerfrequenzen der Objekte nun zweimal berechnet werden - einmal über die erste Folge $$\begin{array}{cc}{\tilde{\text{e}}}_{\mathrm{1,1}}\left(\text{n}\right)=\text{e}\left(\text{n}\right)& \text{mit n}=\mathrm{0,2,4,}\dots ,\text{N}-2\end{array},$$

und einmal über die zweite Folge $$\begin{array}{cc}{\tilde{\text{e}}}_{\mathrm{1,2}}\left(\text{n}\right)=\text{e}\left(\text{n}\right)& \text{mit n}=\mathrm{1,3,5,}\dots ,\text{N}-1\end{array}.$$

The reception subsequence e _1,i (n) generated by an object i for the first, constant modulation subsequence b ₁ (n) can be either even or odd values of n, depending on whether the discrete runtime m _0,i of the respective object is even or odd (here it is first assumed that m _0,i is an integer): $$\begin{array}{ccc}{\text{e}}_{\mathrm{1,}\text{i}}\left(\text{n}\right)={\text{a}}_{\text{i}}\cdot \text{ex}\left(2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\text{k}}_{\mathrm{0,}\text{i}}\right)& \text{with n}=\mathrm{0,2,4,}\dots ,\text{N}-2& \text{or n}=\mathrm{1,3,5,}\dots ,\text{N}-1\end{array};$$

in this range of index n, the first values come from the end of the previous modulation period (due to the running time of the respective object) - in contrast to the previously considered modulation sequence after 35 However, these values also come from the first, constant modulation sequence, so that they also contribute coherently to the later FFT. Because of the two possible positions of the reception sub-sequences e _1,i (n), the FFT for determining the Doppler frequencies of the objects must now be calculated twice - once over the first sequence $$\begin{array}{cc}{\tilde{\text{e}}}_{1.1}\left(\text{n}\right)=\text{e}\left(\text{n}\right)& \text{with n}=\mathrm{0,2,4,}\dots ,\text{N}-2\end{array},$$

and once about the second episode $$\begin{array}{cc}{\tilde{\text{e}}}_{1.2}\left(\text{n}\right)=\text{e}\left(\text{n}\right)& \text{with n}=\mathrm{1,3,5,}\dots ,\text{N}-1\end{array}.$$

$$\begin{array}{cc}{\tilde{\text{E}}}_{\mathrm{1,2,}\underset{\_}{\text{k}}}={\text{FFT}}_{\underset{\_}{\text{k}}}\left({\tilde{\text{e}}}_{\mathrm{1,2}}\left(\text{n}\right)\right)={\text{FFT}}_{\underset{\_}{\text{k}}}\left(\text{e}\left(\text{n}\right)\right)& \text{mit n}=\mathrm{1,3,5,}\dots ,\text{N}-1\end{array}$$

beziehen sich auf die Ausgangsdimension, also die diskrete Frequenz k = 0,...,N/2-1, welche mit der bisher betrachteten diskreten Frequenz k wegen der effektiv halben Abtastrate über $$\underset{\_}{\text{k}}={\text{mod}}_{\text{N}/2}\left(\text{k}\right)$$

zusammenhängt. In 37 sind die beiden FFTs nach Bez. (79) betragsmäßig für zwei Objekte gleicher Empfangsamplitude bei (m_0,1,k_0,1) = (300,3846) und (m_0,2, k_0,2) = (101,1000) dargestellt; das erste Objekt bei der geradzahligen diskreten Laufzeit m_0,1 = 300 bildet eine Betragssitze in der ersten FFT Ẽ_1,1,k bei der Dopplerfrequenz k_0,1 = k_0,1-N/2 = 1798, das zweite Objekt bei der ungeradzahligen diskreten Laufzeit m_0,2 = 101 bildet eine Betragssitze in der zweiten FFT Ẽ_1,2,k bei der Dopplerfrequenz k_0,2 = k_0,2 = 1000.The two FFTs $$\begin{array}{cc}{\tilde{\text{E}}}_{\mathrm{1,1,}\underset{\_}{\text{k}}}={\text{FFT}}_{\underset{\_}{\text{k}}}\left({\tilde{\text{e}}}_{1.1}\left(\text{n}\right)\right)={\text{FFT}}_{\underset{\_}{\text{k}}}\left(\text{e}\left(\text{n}\right)\right)& \text{with n}=\mathrm{0,2,4,}\dots ,\text{N}-2\end{array},$$

$$\begin{array}{cc}{\tilde{\text{E}}}_{\mathrm{1,2,}\underset{\_}{\text{k}}}={\text{FFT}}_{\underset{\_}{\text{k}}}\left({\tilde{\text{e}}}_{1.2}\left(\text{n}\right)\right)={\text{FFT}}_{\underset{\_}{\text{k}}}\left(\text{e}\left(\text{n}\right)\right)& \text{with n}=\mathrm{1,3,5,}\dots ,\text{N}-1\end{array}$$

refer to the output dimension, i.e. the discrete frequency k = 0,...,N/2-1, which is related to the previously considered discrete frequency k due to the effectively half sampling rate via $$\underset{\_}{\text{k}}={\text{mod}}_{\text{N}/2}\left(\text{k}\right)$$

related. In 37 the two FFTs according to (79) are shown in terms of magnitude for two objects of the same reception amplitude at (m _0,1 ,k _0,1 ) = (300,3846) and (m _0,2 , k _0,2 ) = (101,1000); the first object with the even-numbered discrete transit time m _0,1 = 300 forms a magnitude seat in the first FFT Ẽ _1,1,k at the Doppler frequency k _0,1 = k _0,1 -N/2 = 1798, the second object with the odd-numbered discrete transit time m _0,2 = 101 forms a magnitude seat in the second FFT Ẽ _1,2,k at the Doppler frequency k _0,2 = k _0,2 = 1000.

eingeführt, wobei m_j = 0 ist, wenn die Betragsspitze in erster FFT Ẽ_1,1,k auftritt und m_j = 1 ist, wenn die Betragsspitze in zweiter FFT Ẽ_1,2,k auftritt.The magnitude peaks of the two FFTs Ẽ _1,1,k and Ẽ _1,2,k are checked for a detection threshold. The frequencies k _0,j of the magnitude peaks J above the detection threshold are used for further processing; typically these magnitude peaks are seen in two adjacent FFT values (since they are not - as in the example above - at an integer Doppler index k _0,j ), so that their exact position, i.e. a non-integer frequency k _0,j, can then be determined by interpolation. Apart from a possibly missing component N/2 (due to the modulo function according to (80)), these frequencies k _0,j correspond at least approximately to the Doppler frequencies k _0,i of the objects or a subset of them (for objects with very low reflectivity, they may not lead to any magnitude peak above the detection threshold). For further processing, it must also be taken into account in which of the two FFTs the magnitude peak at k _0,j was detected, i.e. whether the corresponding discrete runtime is even or odd; for this purpose, the size $${\underset{\_}{\text{m}}}_{\text{i}}\in \left\{0.1\right\}$$

introduced, where m _j = 0 if the magnitude peak occurs in the first FFT Ẽ _1,1,k and m _j = 1 if the magnitude peak occurs in the second FFT Ẽ _1,2,k .

The second modulation subsequence b ₂ (n) according to (76b) leads in the reception sequence e(n) to components that are time-shifted by m _0,i and multiplied by the respective Doppler frequency k _0,i , i.e. modulated $$\begin{array}{cc}{\text{<figure-callout id="2" label="e" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0156.png" state="{{state}}">e</figure-callout>}}_{\mathrm{2,}\text{i}}\left(\text{n}\right)={\text{a}}_{\text{i}}\cdot {\text{b}}_{\text{2}}\left(\text{n}-{\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\mathrm{0,}\text{i}}\right)\cdot \text{ex}\left(2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\text{<figure-callout id="0" label="k" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">k</figure-callout>}}_{\mathrm{0,}\text{i}}\right)& \text{with n}-{\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\mathrm{0,}\text{i}}=\mathrm{1,3,5,}\dots ,\text{N}-1\end{array}.$$

dabei berücksichtigt die in Bez. (81) eingeführte Größe m_j, ob die jeweilige Empfangsteilfolge in einem ungeradzahligen oder geradzahligen Raster liegt (also eine geradzahlige oder ungeradzahlige diskrete Laufzeit hat); und es sei noch bemerkt, dass - wie bei der Bestimmung der FFT - bei diesem Bereich vom Index n die ersten Empfangswerte vom Ende der vorherigen Modulationsperiode stammen (wegen der Laufzeit des jeweiligen Objekts), was für die nachfolgende Korrelation wegen der Periodizität der gesamten Modulationsfolge (hat Periode N) aber nicht die Kohärenz verletzt.To eliminate the respective modulation by the Doppler frequency, the reception sequence e(n) is rotated back by the frequency k _0,j : $$\begin{array}{cc}{\tilde{\text{e}}}_{\mathrm{2,}\text{j}}=\text{e}\left(\text{n}+{\underset{\_}{\text{m}}}_{\text{i}}\right)\cdot \text{ex}\left(-2\mathrm{\pi }\widehat{\text{j}}\cdot \text{n}/\text{N}\cdot {\underset{\_}{\text{k}}}_{\mathrm{0,}\text{j}}\right)& \text{with n}=\mathrm{1,3,5,}\dots ,\text{N}-1\text{and j}=\text{1}\text{,}\dots \text{J}-1\end{array};$$

the quantity m _j introduced in (81) takes into account whether the respective reception subsequence lies in an odd or even grid (i.e. has an even or odd discrete running time); and it should also be noted that - as in the determination of the FFT - in this range of index n the first reception values come from the end of the previous modulation period (due to the running time of the respective object), which does not violate the coherence for the subsequent correlation due to the periodicity of the entire modulation sequence (has period N).

The sequences ẽ _2,j (n) modified in this way contain cyclically shifted modulation subsequences a _i ·b ₂ (mod _N (nm _0,i )) in the corresponding index j. Contributions from objects with other Doppler frequencies k _0,i (i.e. k _0,i ≠ k _0,j ) represent a modulation sequence that is uncorrelated with b ₂ (n), since they are still modulated with the difference frequency k _0,i -k _0,j ; the same applies to contributions with a different m _j , since these are modulated with b ₁ (n). Therefore, the sequences ẽ _2,j (n) can now be cyclically correlated with the second modulation subsequence b ₂ (n): $$\begin{array}{cc}{\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}={\text{sum}}_{\text{n}=\mathrm{1,3,}\dots ,\text{N}-1}\left({\tilde{\text{e}}}_{\mathrm{2,}\text{j}}\left(\text{n}\right)\cdot {\text{b}}_2\left({\text{mod}}_{\text{N}}\left(\text{n}-\underset{\_}{\underset{\_}{\text{m}}}\right)\right)\right)& \text{with}\underset{\_}{\underset{\_}{\text{m}}}=\mathrm{0.2,}\dots ,\text{M}-2\text{and j}=\mathrm{1,}\dots \text{J}-1\end{array}.$$

treten Betragsspitzen an den Stellen ${\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,}\text{j}}={\text{m}}_{\mathrm{0,}\text{i}}-{\underset{\_}{\text{m}}}_{\text{i}}$

auf, also an den diskreten Laufzeiten m_0,i von Objekten minus der in Bez. (83) angewendeten Verschiebung m_j, so dass sich die diskrete Laufzeit jeweils zu ${\text{m}}_{\mathrm{0,}\text{i}}={\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,}\text{j}}+{\underset{\_}{\text{m}}}_{\text{i}}$

ergibt. Für das obige Beispiel mit zwei Objekten sind in 38 die Beträge der beiden Korrelationen ${\tilde{\text{E}}}_{\mathrm{2,1,}\underset{\_}{\underset{\_}{\text{m}}}}$

zu ẽ_2,1(n) mit k_0,1 = 1798 und m₁ = 0 sowie ${\tilde{\text{E}}}_{\mathrm{2,2,}\underset{\_}{\underset{\_}{\text{m}}}}$

zu ẽ_2,2(n) mit k_0,2 = 1000 und m₂ = 1 dargestellt; die Betragsspitzen liegen bei den Werten ${\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,1}}=300$

sowie ${\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,2}}=\mathrm{100,}$

welche zu den diskreten Laufzeiten ${\text{m}}_{\mathrm{0,1}}={\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,1}}+{\underset{\_}{\text{m}}}_1=300$

sowie ${\text{m}}_{\mathrm{0,2}}={\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,2}}+{\underset{\_}{\text{m}}}_2=101$

korrespondieren.In these one-dimensional correlations ${\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}$

peak amounts occur at the points ${\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,}\text{j}}={\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\mathrm{0,}\text{i}}-{\underset{\_}{\text{m}}}_{\text{i}}$

, i.e. the discrete running times m _0,i of objects minus the shift m _j applied in Req. (83), so that the discrete running time is ${\text{<figure-callout id="0" label="m" filenames="DE102023201147A1\_0151.png,DE102023201147A1\_0155.png" state="{{state}}">m</figure-callout>}}_{\mathrm{0,}\text{i}}={\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,}\text{j}}+{\underset{\_}{\text{m}}}_{\text{i}}$

For the above example with two objects, in 38 the amounts of the two correlations ${\tilde{\text{E}}}_{\mathrm{2.1,}\underset{\_}{\underset{\_}{\text{m}}}}$

to ẽ _2,1 (n) with k _0,1 = 1798 and m ₁ = 0 and ${\tilde{\text{E}}}_{\mathrm{2,2,}\underset{\_}{\underset{\_}{\text{m}}}}$

to ẽ _2,2 (n) with k _0,2 = 1000 and m ₂ = 1; the absolute values peak at ${\underset{\_}{\underset{\_}{\text{m}}}}_{0.1}=300$

as well as ${\underset{\_}{\underset{\_}{\text{m}}}}_{0.2}=\mathrm{100,}$

which are at the discrete terms ${\text{m}}_{0.1}={\underset{\_}{\underset{\_}{\text{m}}}}_{0.1}+{\underset{\_}{\text{m}}}_1=300$

as well as ${\text{m}}_{0.2}={\underset{\_}{\underset{\_}{\text{m}}}}_{0.2}+{\underset{\_}{\text{m}}}_2=101$

correspond.

bei welcher die Betragsspitze auftritt, und der zu diesem j gehörigen Dopplerfrequenz k_0,j ergibt sich die Dopplerfrequenz k_0,i des Objekts nur bis auf einen potenziellen Offset von N/2 - siehe Bez. (80), welche berücksichtigt, dass die Dopplerfrequenz aus einer Folge mit halber Abtastrate bestimmt wird, d. h., eine zusätzliche halbe Periode auf volle Abtastrate bezogen kann damit nicht erkannt werden. Allerdings führt eine solche halbe Periode dazu, dass die komplexen Werte bei den jeweiligen Betragsspitzen in FFT Ẽ_1,1,k bzw. Ẽ_1,2,k und Korrelation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j}\text{,}\underset{\_}{\underset{\_}{\text{m}}}},$

zueinander um 180° gedreht sind, während sie im anderen Fall (also keine zusätzliche halbe Periode in Dopplerfrequenz) gleiche Phase haben. Mit diesem Zusammenhang lässt sich die Mehrdeutigkeit zur jeweiligen Dopplerfrequenz k_0,i lösen, indem die beiden komplexen Werte korrigiert um die möglichen Phasenverschiebungen (0° und 180°) aufaddiert werden, also die Summe und Differenz der beiden komplexen Werte gebildet wird - hat die Summe größeren Betrag als die Differenz, dann ist die Dopplerfrequenz k_0,i = k_0,j, im anderen Fall ist die Dopplerfrequenz k_0,i = k_0,j+N/2; im obigen Beispiel ist für das erste Objekt (also für k=k_0,1 und $\underset{\_}{\underset{\_}{\text{m}}}={\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,1}}$

) die Differenz ${\tilde{\text{E}}}_{\mathrm{1,1,}\underset{\_}{\text{k}}}-{\tilde{\text{E}}}_{\mathrm{2,1,}\underset{\_}{\underset{\_}{\text{m}}}}$

größer als die Summe ${\tilde{\text{E}}}_{\mathrm{1,1,}\underset{\_}{\text{k}}}-{\tilde{\text{E}}}_{\mathrm{2,1,}\underset{\_}{\underset{\_}{\text{m}}}}$

(wegen näherungsweiser Gegenphasigkeit der beiden Komponenten), so dass die Dopplerfrequenz k_0,1 = k_0,1+N/2 = 3846 ist, und für das zweite Objekt (also für k=k_0,2 und $\underset{\_}{\underset{\_}{\text{m}}}={\underset{\_}{\underset{\_}{\text{m}}}}_{\mathrm{0,2}}$

) ist die Summe ${\tilde{\text{E}}}_{\mathrm{1,2,}\underset{\_}{\text{k}}}+{\tilde{\text{E}}}_{\mathrm{2,2,}\underset{\_}{\underset{\_}{\text{m}}}}$

größer als die Differenz ${\tilde{\text{E}}}_{\mathrm{1,2,}\underset{\_}{\text{k}}}+{\tilde{\text{E}}}_{\mathrm{2,2,}\underset{\_}{\underset{\_}{\text{m}}}}$

(wegen näherungsweiser Gleichphasigkeit der beiden Komponenten), so dass die Dopplerfrequenz k_0,2 = k_0,2 = 1000 ist. Weil die Dopplerfrequenz nun über die volle Dauer einer Modulationsperiode bestimmt wird, besteht nicht mehr der Nachteil der Modulationsfolge nach 35 (gemäß Stand der Technik), wo Auflösung und Genauigkeit für die Dopplerbestimmung um Faktor zwei reduziert sind, da die FFT nur über halbe Modulationsdauer bestimmt wird.From the index j of the correlation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j}\text{,}\underset{\_}{\underset{\_}{\text{m}}}},$

at which the magnitude peak occurs and the Doppler frequency k _0,j associated with this j, the Doppler frequency k _0,i of the object is only determined up to a potential offset of N/2 - see reference (80), which takes into account that the Doppler frequency is determined from a sequence with half the sampling rate, i.e. an additional half period related to the full sampling rate cannot be detected. However, such a half period leads to the complex values at the respective magnitude peaks in FFT Ẽ _1,1,k or Ẽ _1,2,k and correlation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j}\text{,}\underset{\_}{\underset{\_}{\text{m}}}},$

are rotated by 180° to each other, while in the other case (i.e. no additional half period in Doppler frequency) they have the same phase. With this connection, the ambiguity for the respective Doppler frequency k _0,i can be resolved by adding the two complex values corrected for the possible phase shifts (0° and 180°), i.e. the sum and difference of the two complex values are formed - if the sum has a larger value than the difference, then the Doppler frequency k _0,i = k _0,j , in the other case the Doppler frequency k _0,i = k _0,j +N/2; in the above example, for the first object (i.e. for k=k _0,1 and $\underset{\_}{\underset{\_}{\text{m}}}={\underset{\_}{\underset{\_}{\text{m}}}}_{0.1}$

) the difference ${\tilde{\text{E}}}_{\mathrm{1,1,}\underset{\_}{\text{k}}}-{\tilde{\text{E}}}_{\mathrm{2.1,}\underset{\_}{\underset{\_}{\text{m}}}}$

greater than the sum ${\tilde{\text{E}}}_{\mathrm{1,1,}\underset{\_}{\text{k}}}-{\tilde{\text{E}}}_{\mathrm{2.1,}\underset{\_}{\underset{\_}{\text{m}}}}$

(due to the approximate antiphase of the two components), so that the Doppler frequency is k _0.1 = k _0.1 +N/2 = 3846, and for the second object (i.e. for k=k _0.2 and $\underset{\_}{\underset{\_}{\text{m}}}={\underset{\_}{\underset{\_}{\text{m}}}}_{0.2}$

) is the sum ${\tilde{\text{E}}}_{\mathrm{1,2,}\underset{\_}{\text{k}}}+{\tilde{\text{E}}}_{\mathrm{2,2,}\underset{\_}{\underset{\_}{\text{m}}}}$

greater than the difference ${\tilde{\text{E}}}_{\mathrm{1,2,}\underset{\_}{\text{k}}}+{\tilde{\text{E}}}_{\mathrm{2,2,}\underset{\_}{\underset{\_}{\text{m}}}}$

(due to the approximate in-phase nature of the two components), so that the Doppler frequency is k _0.2 = k _0.2 = 1000. Because the Doppler frequency is now determined over the full duration of a modulation period, the disadvantage of the modulation sequence according to 35 (according to the state of the art), where resolution and accuracy for the Doppler determination are reduced by a factor of two, since the FFT is only determined over half the modulation duration.

die Entfernungen und radialen Relativgeschwindigkeiten von Objekten in der jeweiligen Erfassungsrichtung bestimmt werden. Es sei bemerkt, dass Beiträge von Objekten mit jeweils anderen Dopplerfrequenzen k_0,i (also k_0,i ≠ k_0,j) und/oder anderem Wert m_j wegen ihrer Unkorreliertheit zu b₂(n) nur zu Rauschen in den jeweiligen Korrelationen führen, also zu keinen Betragsspitzen über einer Detektionsschwelle; bei etwa gleich starken Reflexionssignalen von Objekten liegt dieses Rauschen deutlich unter den interessierenden Betragsspitzen und verdeckt diese damit nicht - nur wenn Reflexionssignale von Objekten stark unterschiedlich sind, können Objekte mit starkem Reflexionssignal über ihr Rauschen schwache Reflexionssignale mit anderer Dopplerfrequenz verdecken.With the relationships and procedures presented above, the peak values of the correlations above a detection threshold can be used to ${\tilde{\text{E}}}_{\mathrm{2,}\text{j}\text{,}\underset{\_}{\underset{\_}{\text{m}}}}$

the distances and radial relative velocities of objects in the respective detection direction are determined. It should be noted that contributions from objects with different Doppler frequencies k _0,i (i.e. k _0,i ≠ k _0,j ) and/or other values m _j only lead to noise in the respective correlations due to their uncorrelatedness to b ₂ (n), i.e. to no magnitude peaks above a detection threshold; if reflection signals from objects are approximately equally strong, this noise is significantly below the magnitude peaks of interest and thus does not obscure them - only if reflection signals from objects are very different can objects with a strong reflection signal obscure weak reflection signals with a different Doppler frequency with their noise.

generiert werden, kann man die oben eingeführten Summen und Differenzen von FFT Ẽ_1,1,k bzw. Ẽ_1,2,k mit k = k_0,j und Korrelation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}$

auf eine Detektionsschwelle prüfen, da derjenige Wert von Summe und Differenz, bei welchem die beiden Komponenten FFT und Korrelation phasenrichtig, d. h. kohärent, aufaddiert werden, ein um 3dB besseres Signal-zu-Rauschverhältnis aufweist als die Korrelation selber; dies ergibt sich daraus, dass bei der Summe bzw. Differenz über die ganze Modulationsfolge b(n) integriert wird, bei der Korrelation aber nur über die halbe. Damit hat die Summe bzw. Differenz auch das gleiche Signal-zu-Rauschverhältnis und damit die gleiche Sensitivität wie die optimale zweidimensionale Korrelation E_m,k nach Bez. (6). Allerdings gilt das nur dann, wenn man in der FFT an der jeweils entsprechenden Stelle, also bei k = k_0,j, schon eine Betragsspitze detektiert hat; da die FFT wegen halber Integrationsdauer ein 3dB schlechteres Signal-zu-Rauschverhältnis aufweist, muss man eine entsprechend reduzierte Detektionsschwelle anwenden. Diese reduzierte Detektionsschwelle hat eine erhöhte Wahrscheinlichkeit, dass fälschlicherweise Rauschspitzen detektiert werden; allerdings werden diese bei der effektiv schärferen Detektionsschwelle von Summe und Differenz von FFT und Korrelation dann wieder verworfen, so dass als einziger Nachteil ein leicht erhöhter Rechenaufwand bleibt (weil Korrelation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}$

öfters berechnet werden muss). Es sei noch bemerkt, dass man Summe und Differenz nicht für jedes $\underset{\_}{\underset{\_}{\text{m}}}$

bestimmen muss, sondern dass man die Korrelation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}$

auch erst mal auf eine etwa 3dB reduzierte Detektionsschwelle prüfen kann und nur an den Stellen, wo diese Detektionsschwelle überschritten wird, Summe und Differenz bildet. Weil der neue Ansatz aus Modulationsfolge nach 36 und oben beschriebener Auswertung die gleiche Sensitivität wie die optimale zweidimensionale Korrelation E_m,k nach Bez. (6) aufweist, ist der Nachteil der Modulationsfolge nach 35 (Stand der Technik) einer 3dB schlechteren Sensitivität behoben. Nun soll noch kurz diskutiert werden, ob der neue Ansatz aus Summe und Differenz von FFT und Korrelation nicht auch bei der Modulationsfolge nach 35 angewendet werden könnte: dies geht nur dann, wenn die Empfangsphase über die ganze Modulationsfolge stabil linear ist (also instantane Frequenz konstant), was insbesondere bei kontinuierlichem räumlichem Scannen wegen Speckle-Effekten nur bedingt gegeben ist.As an alternative to the approach that detections are made from peaks of the correlations above a detection threshold ${\tilde{\text{E}}}_{\mathrm{2,}\text{j}\text{,}\underset{\_}{\underset{\_}{\text{m}}}}$

are generated, the sums and differences of FFT Ẽ _1,1,k and Ẽ _1,2,k introduced above with k = k _0,j and correlation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}$

check for a detection threshold, since the value of sum and difference at which the two components FFT and correlation are added in phase, ie coherently, has a signal-to-noise ratio that is 3dB better than the correlation itself; this results from the fact that the sum or difference is integrated over the entire modulation sequence b(n), but the correlation is only integrated over half of it. The sum or difference therefore also has the same signal-to-noise ratio and thus the same sensitivity as the optimal two-dimensional correlation E _m,k according to (6). However, this only applies if a peak in magnitude has already been detected in the FFT at the corresponding point, i.e. at k = k _0,j ; since the FFT has a 3 dB worse signal-to-noise ratio due to half the integration time, a correspondingly reduced detection threshold must be used. This reduced detection threshold has an increased probability that noise peaks will be detected incorrectly; however, these are then rejected again with the effectively sharper detection threshold of the sum and difference of FFT and correlation, so that the only disadvantage is a slightly increased computational effort (because correlation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}$

It should be noted that the sum and difference cannot be calculated for each $\underset{\_}{\underset{\_}{\text{m}}}$

but that one must determine the correlation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}$

can also first check for a detection threshold reduced by about 3 dB and only form the sum and difference at the points where this detection threshold is exceeded. Because the new approach from modulation sequence according to 36 and the above-described evaluation has the same sensitivity as the optimal two-dimensional correlation E _m,k according to (6), the disadvantage of the modulation sequence according to 35 (state of the art) of a 3dB worse sensitivity. Now we will briefly discuss whether the new approach of sum and difference of FFT and correlation could not also be used for the modulation sequence according to 35 could be applied: this is only possible if the reception phase is stable and linear over the entire modulation sequence (i.e. the instantaneous frequency is constant), which is only partially the case with continuous spatial scanning due to speckle effects.

The approach of sum and difference of FFT and correlation also avoids the disadvantage of the modulation sequence according to 35 (State of the art) a dynamic range reduced by 3dB with more than one object per pixel, because integration is now carried out over the full modulation sequence.

Another advantage of the new modulation sequence according to 36 is that arbitrary overlaps for two neighboring pixels can be realized, since each section of this periodic modulation sequence (period N) has the characteristic property according to (76).

The required computational effort for the new approach with the modulation sequence according to 36 differs from that for the modulation sequence according to 35 (state of the art) by an additional FFT of length N/2 (independent of the number of objects in the respective pixel), whereas the correlations are only half the length, which, however, is not an advantage when implementing the correlation in the frequency domain (i.e. via FFT and inverse FFT). In the typical case of one object in a pixel, two FFTs and one correlation must be determined. In the approach explained above with a reduced detection threshold for the FFTs, additional calculations of the correlation can in principle occur. The required computing effort is therefore still of a magnitude that modern DSPs with parallel vector computing units can achieve, which leads to a cost-effective implementation and does not require any special computing logic.

For the output dimension of the FFT, i.e. the discrete frequency, the non-symmetrical range k = 0,...,N/2-1 was considered above (in this section), as is generally the case - likewise for the discrete frequency after the ambiguities have been resolved, the non-symmetrical range k = 0,...N-1; the actual relative velocities and thus Doppler frequencies can take on either sign, so that the upper range, in particular the upper half of k = 0,...N-1, can be mapped to negative values by subtracting N.

As an alternative to the previously considered new modulation sequence according to 36 one can also use alternating values (i.e. alternating +1 and -1) for the modulation subsequence b ₁ (n) - this is in 39 The only significant difference is that the receive subsequences for this b ₁ (n) have an additional frequency with a period of two (relative to the modulation rate of b ₁ (n)), i.e. the magnitude peaks of the two associated FFTs Ẽ _1,1,k and Ẽ _1,2,k are shifted by half the FFT length N/4, which must be corrected accordingly for further processing.

für beide m_j = 0,1 bestimmt und durch Interpolation der Werte der Betragsspitzen beider Korrelationen das nichtganzzahlige m_0,i ermittelt werden kann (man könnte auch schon durch Interpolation der Werte der Betragsspitzen der beiden FFTs das nichtganzzahlige m_0,i bestimmen oder durch Kombination, also Summe bzw. Differenz der Werte von FFT und Korrelation). Beim Ansatz einer z. B. halb so großen Abtastzeit T_s (im Vergleich zur Modulationsdauer T_m) weist das Ineinanderschachteln der beiden Modulationsteilfolgen bezüglich der Abtastzeit die Periode 4 auf und im Empfangssignal stammen jeweils zwei aufeinanderfolgende Abtastwerte von der ersten Folge b₁(n) und die nächsten zwei Abtastwerte von der zweiten Folge b₂(n); deshalb sind zum einen für die Empfangsteilfolgen e_1,i(n) vier mögliche Lagen zu berücksichtigen, also vier FFTs Ẽ_1,1,k,...,Ẽ_1,4,k zu berechnen, und zum anderen sind dabei die zu b₂(n) korrespondierenden Abtastwerte zu Null zu setzen (weil sie nicht mehr im Raster der Periode 2 liegen und somit einfach weggelassen werden können), so dass die FFT über volle Anzahl der Empfangswerte (im Gegensatz zu halben Anzahl bisher) zu berechnen ist. In zumindest einem Teil dieser FFTs treten dann von einem einzelnen Objekt drei Betragsspitzen auf; die größte an der richtigen Position, also der Dopplerfrequenz des Objekts (Mehrdeutigkeiten gibt es im Gegensatz zu oben keine mehr, was von Vorteil ist), und zwei weitere, um 3dB kleinere Betragsspitzen ein Viertel der FFT-Länge davor und danach (diese sind für die weitere Verarbeitung zu ignorieren). Da in mehreren der vier FFTs Ẽ_1,1,k,...,Ẽ_1,4,k Betragsspitzen auftreten, kann durch Interpolation eine nichtganzzahlige diskrete Laufzeit ermittelt werden (unter Verwendung der Werte der FFTs oder der Korrelationen oder ihrer Kombinationen).So far, this section has considered the case where the discrete transit time m _0,i is an integer and the modulation duration T _m is equal to the sampling repetition time T _s . For a non-integer discrete transit time m _0,i, in the case of an ideal rectangular modulation signal which also retains its ideal shape in the received signal, it could happen that sampling takes place exactly at the edge where no useful information can be obtained. To avoid this, the sampling repetition time T _s of the received sequence can be made smaller, e.g. half as long as the modulation duration T _m and/or the shape of the modulation pulses is adjusted either directly when they are generated or in the receiver to an approximately triangular shape. smoothed (the latter is inherently implemented by a low-pass filter in the receiver). In the approach with smoothing of the modulation signal and the general case of a non-integer discrete delay time m _0,i, magnitude peaks occur in both FFTs Ẽ _1,1,k and Ẽ _1,2,k , so that the correlation ${\tilde{\text{E}}}_{\mathrm{2,}\text{j},\underset{\_}{\underset{\_}{\text{m}}}}$

for both m _j = 0.1 and by interpolating the values of the magnitude peaks of both correlations the non-integer m _0,i can be determined (one could also determine the non-integer m _0,i by interpolating the values of the magnitude peaks of the two FFTs or by combination, i.e. sum or difference of the values of FFT and correlation). If, for example, a sampling time T _s is used that is half as long (compared to the modulation duration T _m ), the nesting of the two modulation subsequences has a period of 4 with respect to the sampling time and in the received signal two consecutive samples come from the first sequence b ₁ (n) and the next two samples from the second sequence b ₂ (n); Therefore, on the one hand, four possible positions must be taken into account for the partial reception sequences e _1,i (n), i.e. four FFTs Ẽ _1,1,k ,...,Ẽ _1,4,k must be calculated, and on the other hand, the samples corresponding to b ₂ (n) must be set to zero (because they are no longer in the grid of period 2 and can therefore simply be omitted), so that the FFT can be calculated over the full number of reception values (in contrast to half the number previously). In at least some of these FFTs, three magnitude peaks from a single object then occur; the largest at the correct position, i.e. the Doppler frequency of the object (unlike above, there are no more ambiguities, which is an advantage), and two further magnitude peaks, 3 dB smaller, a quarter of the FFT length before and after (these are to be ignored for further processing). Since magnitude peaks occur in several of the four FFTs Ẽ _1,1,k ,...,Ẽ _1,4,k , a non-integer discrete running time can be determined by interpolation (using the values of the FFTs or the correlations or their combinations).

In addition to the case considered so far, in which the two modulation subsequences b ₁ (n) and b ₂ (n) are nested alternately, i.e. with period two, longer periods can also be used for nesting and optionally an unequal number of elements of b ₁ (n) and b ₂ (n) per period can be used.

In the lidar system considered so far in this section, 1 the mixer is complex-valued, which represents a significant additional effort (almost twice) compared to a real-valued mixer for the receiving path. When using a real-valued mixer, only the magnitude of the relative speed could be determined, not the sign, since there are two magnitude peaks in the FFTs at +k ₀ and -k _0. To determine the sign, approaches using plausibility checks and/or tracking, i.e. tracking over several recording cycles, would be necessary. Alternatively, as explained above, a complex-valued modulation sequence can be used, e.g. consisting of the 4 values 0, +ĵ, -1, -ĵ (for the 4 equidistant phase values φ _j = 0.90°, 180°, 270°). For the first modulation sub-sequence b ₁ (n), the four values 0, +ĵ, -1, -ĵ are rotated periodically, so that for an object with a relative speed of zero, the peak value is at a quarter of the FFT length - negative relative speeds are below this (and therefore have a lower frequency), positive ones are above this. This first modulation sub-sequence b ₁ (n), which in turn has a period of 4, is also alternately interleaved with the second modulation sub-sequence b ₂ (n), which can also take on these four values 0, +ĵ, -1, -ĵ, e.g. in a pseudo-random sequence. The generation of such a complex-valued sequence requires increased circuit complexity (see e.g. 34 ), but this is only needed once, whereas with a parallel receiver the effort for a complex-valued receiver is required many times over. With a complex-valued modulation sequence, the complex conjugate of the modulation sequence is to be used for the correlation. Alternatively, the complex conjugate of the received sequence can also be used, provided that this is complex-valued.

So far, the case has been considered where no window function is used for the FFT, i.e. no multiplication of the FFT input values with a kind of bell curve; this would only be necessary or useful if two objects with similar relative speed and significantly different reflection strengths can occur at the same distance in a pixel and are to be separated. In particular, if no window function is used at the input of the FFT, the sensitivity at the output of the FFT is reduced (i.e. the detection capability of objects with weak reflectivity and a large distance) if the Doppler index corresponding to the relative speed is not an integer, i.e. the peak value is divided between two neighboring FFT values. This effect can be reduced by choosing the length of the FFT to be longer than that of its input signal, i.e. zeros are appended to the input signal, which is known as zero padding.

Regarding the second modulation subsequence b ₂ (n), it should be noted that it can be formed not only from a pseudorandom alternation between discrete phase values, but also from a code whose autocorrelates have low side lobes - e.g. a Gold code known from the literature. This leads to a higher dynamic range of the correlation, but only if there is no signal nal components (e.g. from objects with a different relative speed) which represent noise in the signal to be correlated.

The modulation sequences presented in this section, consisting of two sub-sequences, which are arranged sequentially according to the state of the art or are nested in a new approach, can also be used according to the invention in combination with a linearly changing frequency according to (9); an example of this is shown in 40 The only effect is that the reception frequency f _e has a propagation time component f _r in addition to the Doppler shift f _D (see reference (12)); all of the above considerations therefore still apply, the only thing to be taken into account is that the propagation time component must be deducted from the reception frequency to determine the relative speed, whereby the propagation time is known from the correlation. The superimposed frequency modulation enables the system approaches discussed above (in particular continuous spatial scanning over frequency) and advantages (in particular determination of the sign of the reception frequency with a real-valued mixer). The approaches presented in the previous sections are also valid for this phase modulation or can be transferred to it.

It should also be noted that for the detection of the road surface at greater distances, the constant or periodic subsequence b ₁ (n) has the advantage that it does not split the received signal over discrete distances, so that the entire beam width is effective and the signal-to-noise ratio is thus much better, which already results in a high probability of detecting the road surface in the individual pixel.

Scanning functionality check

If scanning in one or both spatial directions were not to work, particularly due to a hardware error, there would be a high energy density in some beam directions (because the system is in that direction much more often than in the nominal state), which could result in the permitted limits for eye safety being exceeded. Therefore, scanning must be monitored and as soon as an error occurs, the emission of the lidar sensor must be stopped.

If the sensor no longer scans in one spatial direction, the received signals in this spatial direction are unchanged across all pixels in the same other spatial direction - apart from system noise and moving objects. Therefore, changes in the received signals are checked in both spatial directions. A first approach is to compare the object reflections, i.e. the peak values above a detection threshold; as already indicated above, all moving objects must be excluded, whereby identification is possible via the measured relative speed and the known speed of the object. A second paragraph uses the received signal components from internal reflections and couplings, as well as reflections from a cover, which are approximately at a distance of zero; these are used anyway to compensate for the effects (by correction values c ₁ (n) in hard-wired calculation logic according to 8 and 10 ) - this was explained above. It is used to show that these reception components generally change during scanning.

Concluding remarks

Based on the above application examples, the inventive considerations and designs presented can be easily transferred to general measurements and parameter interpretations, i.e. they can also be applied to other numerical values. For this reason, general parameters are often given in formulas and images.

• - die Ansätze zum Bestimmen und Realisieren der Korrekturwerte, um die Effekte von Verkopplungen und Reflektionen innerhalb des Lidarsystems oder von seiner unmittelbaren Umgebung, insbesondere einer Abdeckung zu kompensieren, können auch ohne überlagerte Frequenzmodulation benutzt werden,
• - die Ansätze mit einer Sende-Empfangs-Einheit mit einem oder mehreren Wellenleitern für Scannen über Frequenz in erste Raumrichtung und einem Scanner für zweite Raumrichtung können auch in Kombination mit anderen Modulationsformen benutzt werden, insbesondere wenn die Frequenz schrittweise geändert wird, oder sogar für nichtkohärente Lidarsysteme,
• - der Ansatz zur Bestimmung von Winkelfehlern (z. B. durch Änderung von Hardwareeigenschaften oder Fehlausrichtung des Sensors) ist auch bei anderen Modulationsformen nutzbar, da er im Wesentlichen nur auf der inhärent gegebenen Dopplermessfähigkeit von kohärenten Lidarsystemen basiert,
• - der Ansatz zur Bestimmung der Lage der Straßenoberfläche in größeren Entfernungen kann auch bei anderen Modulationsformen, also beispielweise bei der reinen linearen Frequenzmodulation (meist aus zwei Frequenzrampen bestehend, deren Steigung entgegengesetztes Vorzeichen haben), benutzt werden; die benutzten Korrelationswerte beruhen dann auf einer entsprechend anderen Korrelationsberechnung (bei rein linearer Frequenzmodulation in Form einer FFT pro Frequenzrampe),
• - der Ansatz zur Vorzeichenbestimmung der Empfangsfrequenz bei reellwertigem Mischer mit Hilfe nichtbinärer Phasenmodulation kann auch ohne überlagerte Frequenzmodulation benutzt werden.

Some of the new approaches presented are not only new in combination with others, but are new in themselves compared to the state of the art; examples include:

• - the approaches for determining and implementing the correction values to compensate for the effects of couplings and reflections within the lidar system or from its immediate environment, in particular a cover, can also be used without superimposed frequency modulation,
• - the approaches with a transmit-receive unit with one or more waveguides for scanning over frequency in the first spatial direction and a scanner for the second spatial direction can also be used in combination with other forms of modulation, in particular when the frequency is changed step by step, or even for non-coherent lidar systems,
• - the approach for determining angular errors (e.g. by changing hardware properties or misalignment of the sensor) can also be used with other forms of modulation, since it is essentially based only on the inherent Doppler measurement capability of coherent lidar systems,
• - the approach for determining the position of the road surface at greater distances can also be used for other forms of modulation, for example for pure linear frequency modulation (usually consisting of two frequency ramps whose slopes have opposite signs); the correlation values used are then based on a correspondingly different correlation calculation (for pure linear frequency modulation in the form of one FFT per frequency ramp),
• - the approach for determining the sign of the reception frequency in a real-valued mixer using non-binary phase modulation can also be used without superimposed frequency modulation.

QUOTES INCLUDED IN THE DESCRIPTION

This list of documents listed by the applicant was generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA accepts no liability for any errors or omissions.

Cited non-patent literature

• "Phase-Coded-Based Modulation for Coherent Lidar" by Sebastian Banzhaf and Christian Waldschmidt, published in IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 70, NO. 10, OCTOBER 2021 [0219]

Claims (46)

Coherently operating lidar system for environmental detection, which emits a phase-modulated signal, in particular an irregular change over discrete phase values, preferably over only two phase values that differ by approximately 180°, characterized in that the frequency to which the phase is modulated changes continuously at least in sections, preferably with at least approximately a linear progression. Lidar system according to Claim 1 , which - receives the signals reflected back from objects, which are delayed compared to the transmitted signal by the distance-dependent propagation time and are shifted in frequency both by the relative speed-dependent Doppler effect and by the distance-dependent propagation time caused by the linear frequency change, and converts them into a low-frequency signal by mixing and digitizes them into a reception sequence, - carries out two-dimensional correlation filtering in digital signal processing means to determine the variable dimensions time shift and frequency shift of signals reflected from objects, and - determines the object distance from the determined time shift and the radial relative speed of the object from the determined frequency shift minus the contribution caused by the time shift. Lidar system according to Claim 2 , in which at least part of the two-dimensional correlation filter is realized by a hard-wired digital circuit designed as a pipeline, wherein several or all output values in one of the two dimensions are determined per clock cycle of the digital circuit and via a sequence of clock cycles in the other dimension. Lidar system according to Claim 3 , in which the hard-wired digital circuit has a front stage in which a signal sequence or its conjugate complex values and a phase modulation sequence or its conjugate complex values are multiplied by a position shifted relative to one another, optionally followed by a decimation of the sequence resulting from this multiplication and/or optionally followed by an expansion with zeros and followed by a fast Fourier transformation implemented in several stages, wherein each individual arithmetic operation is implemented in a dedicated circuit and from cycle to cycle in the first stage the shift between signal and modulation sequence is changed and the result of a Fourier transformation is produced at the output of the rear stage, wherein this result relates in each case to output data of the front stage generated several cycles previously. Lidar system according to one of the Claims 3 or 4 , in which the hard-wired digital circuit is used for several or preferably all pixels due to its high throughput rate, whereby these pixels can be generated in particular by scanning light beams and/or parallel reception paths. Lidar system according to one of the Claims 3 or 4 , in which the hard-wired digital circuit has at least one of the following features: - the components of couplings and reflections contained in the digitized received signal within the lidar system or from its immediate surroundings, in particular a cover, are largely eliminated by adding or subtracting a compensation sequence; - a binary phase modulation, in particular consisting of only two phase values that differ by approximately 180°, is used and the multiplications with the values of the modulation sequence are then implemented by switchable inverters; - the signal multiplications with rotation factors required for the fast Fourier transformation are implemented with a few additions and/or subtractions of shifted signal values, whereby preferably a maximum of one addition or subtraction is used to implement a real-valued multiplication; - the bit length used varies across pipeline stages of the hardwired digital circuit and is preferably only so large that the quantisation noise generated in the digital circuit does not significantly increase the system noise generated in the analogue part of the receiver; - the fast Fourier transform is carried out in the form of a structure with decimation in frequency in order to avoid re-sorting the input data in the form of long lines, as well as having the longest lines of the structure and the non-trivial multiplications in the front stages with their smaller bit length; - truncation is used for quantization and/or purely bit-by-bit inversion is used for inversion and the effects of the resulting mean errors are compensated by adding correction values in a stage of the digital circuit; - the hard-wired digital circuit is extended by one or more further stages for evaluating the result of the two-dimensional correlation, in particular for magnitude or power formation and subsequent summation and/or maximum search. Lidar system according to Claim 6 , in which the proportions of couplings and reflections contained in the digitized received signal within the lidar system or from its immediate surroundings, in particular a cover, are determined by using at least one of the following approaches: - one or more average values are determined via the product of the received sequence and the modulation sequence at the same or similar time, wherein several average values are formed by considering several sections and/or different phase value combinations of the phase modulation sequence; - to form an average over all values of the product of the received sequence and the modulation sequence, the output value of the hard-wired digital circuit, which belongs to the zero frequency of the Fourier transformation and to the zero shift between the signal and modulation sequence in the first stage, is used, preferably by suspending the clocking of the first stage during the calculation of this value; - average values are formed over several acquisition cycles and/or over different, preferably closely spaced pixels. Lidar system according to Claim 7 , in which the components of couplings and reflections contained in the digitized received signal within the lidar system or from its immediate surroundings, in particular a cover, are compensated by a sequence which is formed in the hard-wired digital circuit by assigning, with the correct sign, one or more mean values determined via the product of the received sequence and the modulation sequence. Lidar system according to one of the Claims 2 - 8 , in which - the reception sequence or its conjugate complex values and the phase modulation sequence or its conjugate complex values are multiplied by a position shifted relative to one another, the respective shift corresponding to a running time, - the resulting product sequence is multiplied by a sequence of complex rotation factors with an at least approximately linearly changing phase, the rotation factors for the respective running time at least partially compensating for the frequency contribution of the linear frequency modulation, where appropriate, a frequency offset is additionally implemented, - if appropriate, a decimation of the sampling rate of the sequence is then carried out, - a discrete Fourier transformation follows, preferably implemented via a fast Fourier transformation, and - this is carried out for all running times of interest, the sequence of complex rotation factors changing over the respective time shift. Lidar system according to one of the Claims 2 - 9 , in which the frequency modulation is not exactly linear and linearity errors are taken into account in that in the two-dimensional correlation filtering essentially - the reception sequence or its conjugate complex values and the phase modulation sequence or its conjugate complex values are multiplied by a position shifted relative to one another, wherein the respective shift corresponds to a running time, - the resulting product sequence is multiplied by a sequence of complex rotation factors, wherein the rotation factors for the respective running time at least approximately correct the non-constant frequency contribution of the non-linear frequency modulation, - a discrete Fourier transformation follows, preferably implemented via a fast Fourier transformation, and - this is carried out for all running times of interest, wherein the sequence of complex rotation factors for correcting the linearity errors of the frequency modulation changes over the respective time shift. Lidar system according to one of the Claims 6 - 10 , in which the multiplications with the complex rotation factors for correcting the linearity errors of the frequency modulation and/or for the runtime-dependent frequency shift are also possible in the hard-wired digital circuit. are preferably implemented via individual programmable structures with low precision in the range of a few bits and thus low effort. Lidar system according to one of the above claims, in which a determination of the reception frequency with the correct sign and thus the unambiguous determination of the relative speed of objects when using a real-valued mixer is realized in that, particularly for more distant objects, only a relative speed hypothesis is possible or at least more plausible due to the known time-dependent component of the frequency shift. Lidar system according to one of the above claims, in which a determination of the reception frequency with the correct sign and thus the unambiguous determination of the relative speed of objects when using a real-valued mixer is realized by varying the steepness of the frequency change and collecting and evaluating data for an object at different steepnesses and using the fact that the two frequency shift effects of relative speed and transit time are then in different relations to one another. Lidar system according to Claim 13 , in which the sign, but not the amount of the steepness of the frequency change is varied, so that the two frequency shift effects of relative velocity and transit time add up with different signs and thus the sign of the reception frequency can be determined. Lidar system according to one of the above claims, in which the frequency change is used to change the beam direction, in particular in at least partially continuous manner, in order to be able to record data for several pixels of different directions. Lidar system according to one of the above claims, in which a change in beam direction by frequency change is realized by using a waveguide with several coupling points, preferably in an equidistant grid on a line, for transmission and reception. Lidar system according to Claim 16 , in which the beam direction change by frequency change is used for a first spatial direction and there is a further device for beam direction change for a second spatial direction which is at least approximately perpendicular to the first, wherein preferably one of the two spatial directions is horizontal and the other is then vertical. Lidar system according to one of the Claims 16 or 17 , in which a change in beam direction is realized by irradiating material or reflecting it from material, in which at least one optical material property is changed by a control variable, in particular by applying voltage or passing current, wherein the control variable and thus the change in the optical material property can be locally different. Lidar system according to Claim 17 and 18 , in which a body made of preferably monolithic material is irradiated for the beam direction change for the second spatial direction, which body has a constant, in particular triangular, cross-section in the orientation direction of the waveguide and can optionally be combined with a lens also having a constant cross-section in this dimension for beam bundling for the second spatial direction, wherein the electrical control variable is designed to change the optical property with respect to the orientation direction of the body with a constant body cross-section. Lidar system according to Claim 17 and 18 , in which a flat transparent or reflective liquid crystal element with a one-dimensional lattice structure and control via a voltage is used for beam direction change for a second spatial direction. Lidar system according to Claim 17 and 18 , in which a transparent or reflective liquid crystal array is used for beam direction change for a second spatial direction, wherein this liquid crystal array has only a few elements in the orientation direction of the waveguide, preferably only a single, then rod-shaped element, preferably this liquid crystal array is also used for the sole beam bundling or support of the beam bundling of a lens for the second spatial direction and preferably with the help of the liquid crystal array also manufacturing tolerances of the optically relevant components and their arrangement relative to one another are compensated. Lidar system according to one of the above claims, in which there are several transmit-receive paths operating in parallel, each with a waveguide for beam direction change in the first spatial direction by frequency change, preferably fed by a common laser source with subsequent phase modulation, and these several transmit-receive paths open up different beam directions in parallel in at least one of the two spatial directions. Lidar system according to Claim 22 , in which different beam directions in the second spatial direction are generated both by a plurality of parallel transmitting/receiving paths with preferably similarly radiating and parallel adjacent waveguides and by changing the optical properties of a radiated or reflecting material via an electrical control variable, and the beam directions by a plurality of parallel transmitting/receiving paths for each control of the radiated or reflecting material address either a group of directly adjacent pixels or a group with at least partially further apart pixels, wherein the control of the radiated or reflecting material is successively changed so that the entire beam direction range of interest is covered, optionally with a larger angular distance of the pixels towards the outside via a non-equidistant arrangement of the adjacent waveguides, in particular with spaced-apart groups of waveguides, wherein the distance within a group increases towards the outside, and with then preferably an optical beam width increasing towards the outside in the second spatial direction. Lidar system according to Claim 22 , in which waveguides preferably arranged parallel and close to one another are different in order to open up different beam direction ranges in the first spatial direction by means of the same frequency change, so that only a part of the beam direction range in the first spatial direction is covered by a single waveguide. Lidar system according to Claim 24 , in which the beam direction change in the second spatial direction is realized in a continuous manner with the aid of an electronic or mechanical approach, whereby a slow scanning via frequency change in the first spatial direction is made possible in particular by the fact that during a complete scan in the second spatial direction the scanning via continuous frequency change in the first spatial direction only progresses by approximately one pixel at a time. Lidar system according to one of the above claims, in which there are several transmit-receive paths operating in parallel and preferably fed by a common laser source, each with a switching matrix for sequential switching within a respective group of waveguides, wherein the preferably similar waveguides are each used for beam direction change in the first spatial direction by frequency change and are all arranged next to one another in such a way that they completely open up the second spatial direction, optionally with a larger angular distance of the pixels towards the outside via a non-equidistant arrangement of the adjacent waveguides and with then preferably increasing optical beam width towards the outside in the second spatial direction. Lidar system according to one of the above claims, in which there is a transmission-reception path with a switching matrix for sequential switching between several waveguides, preferably parallel and close to one another, wherein the waveguides are different in order to open up different beam direction ranges in the first spatial direction by means of a frequency change of the same type, so that only a part of the beam direction range in the first spatial direction is covered by a single waveguide and the required frequency tuning range of the laser source is reduced, and the beam direction change in the second spatial direction is realized in a continuous manner with the aid of an electronic or mechanical approach, whereby slow scanning via frequency change in the first spatial direction is made possible in particular by the fact that during a complete scan in the second spatial direction, the scanning via continuous frequency change in the first spatial direction only progresses by about one pixel. Lidar system according to one of the above claims, in which a beam direction change due to frequency change is different speeds, in particular characterized in that it is slower in a central region than in outer regions. Lidar system according to one of the above claims, in which a frequency change for changing the beam direction is generated by one or more frequency-tunable laser sources and, when using several laser sources, each one covers only a part of the frequency range required for changing the beam direction. Lidar system according to one of the above claims, in which a sign-correct determination of the reception frequency and thus the unambiguous determination of the relative speed of objects when using a real-valued mixer is realized in that a frequency change for the beam direction change in the first spatial direction has a different, in particular alternating sign over different beam directions of the second spatial direction and this different sign in different, in particular adjacent beam direction planes in which the same object is detected, leads to the two frequency shift effects of relative speed and transit time adding up with different signs and thus the sign of the reception frequency can be determined. Lidar system according to one of the above claims, in which deviations in the beam directions can occur in particular due to misalignment, due to frequencies that are not precisely known and/or due to the dependence of the optical property on the control variable that is not precisely known, characterized in that these deviations are determined from the measured radial relative velocities of stationary objects in order to later take these deviations into account and/or to correct them. Lidar system according to Claim 31 , in which the road surface is used as stationary objects, preferably with determination of its angle in the vertical direction from the measured distance and the sensor installation height. Lidar system according to one of the above claims, in which the devices for changing the beam direction are used to compensate for a misalignment and/or to adapt the detection range adaptively, in particular depending on the traffic situation. Lidar system according to one of the above claims, which emits a phase-modulated signal, the phase of this signal being generated by switching between discrete values, i.e. from a sequence of phase values, and the times of this phase switching forming a subset of an equidistant time grid, characterized in that - the phase modulation sequence contains two sequentially arranged sequences or two periodically nested sequences, the first sequence having constant or periodic, in particular alternating phase values, while the second sequence changes essentially irregularly between phase values, and - the frequency to which the phase is modulated changes continuously at least in sections, preferably with at least an approximately linear progression. Lidar system according to Claim 34 , which - receives the signals reflected back from objects, which are delayed compared to the transmitted signal by the distance-dependent propagation time and are shifted in frequency both by the relative speed-dependent Doppler effect and by the distance-dependent propagation time caused by the linear frequency change, and converts them into a low-frequency signal by mixing and digitizes them into a reception sequence, - determines the variable dimensions of time shift and frequency shift of signals reflected from objects from this reception sequence in digital signal processing means, whereby the first phase modulation sequence with constant or periodic phase values primarily serves to determine the frequency shift and the second phase modulation sequence with irregular alternation between phase values determines the time shift, and - calculates the object distance from the determined time shift and the radial relative speed of the object from the determined frequency shift minus the contribution caused by the time shift. Lidar system according to Claim 34 or 35 , in which the two phase modulation sequences are alternately nested, ie with period two related to the modulation rate. Lidar system according to one of the above Claims 34 - 36 , in which a phase modulation sequence consisting of two periodically nested sequences is periodically repeated, whereby the cyclical property of the modulation and reception sequence is exploited, particularly during continuous scanning of the laser beam, and detection is realized in different directions. Lidar system according to one of the above Claims 34 - 37 , in which the laser beam scans continuously and overlapping sections of a long, possibly periodic phase modulation sequence, which contains two periodically nested sequences, are used for the successive detection directions. Lidar system according to one of the above Claims 34 - 38 , in which - the phase modulation sequence contains two periodically nested sequences, this nesting having the period N ₁ with respect to the sampling time of the reception sequence, - to determine the frequency shift of objects, N ₁ discrete Fourier transformations are calculated, preferably with the aid of fast Fourier transformations, over values of the reception sequence from the periodic grid of the first sequence, possibly interrupted by zeros or extended with zeros, as well as from N ₁ -1 shifts of this grid, the shift increasing by one grid value in each case, - in each of these N ₁ discrete Fourier transforms, the respective frequencies of magnitude peaks that lie above a detection threshold are determined, - the reception sequence in the periodic and correspondingly shifted grid of the second sequence is turned back in frequency for the respective frequencies and the respective grid shifts, - a correlation is determined between the sequence generated in this way and the second modulation sequence, and - from values of this correlation, in particular of magnitude peaks that lie above a detection threshold, as well as the respective The respective time shift and thus object distance can be determined from the grid shift and the radial relative velocity of the respective object can be determined from the respective frequency minus its contribution caused by the time shift, whereby any ambiguities in the frequency are resolved with the help of the phase relationship between values of the discrete Fourier transform and the correlation. Lidar system according to Claim 39 , in which a first detection threshold is used to determine the magnitude peaks of the discrete Fourier transform, at the magnitude peaks the complex value of the discrete Fourier transform and the respective complex value of the associated correlation at their magnitude peaks are added together several times if necessary while compensating for possible phase shifts, and these sums are checked against a second detection threshold, wherein the first detection threshold is less high above the noise. Lidar system according to one of the above claims, in which the position of the road surface is determined, in particular at greater distances, by summing up the power of a correlation or the complex-valued product between a first correlation and the conjugate complex of a second correlation for better differentiation from system noise at the same vertical angle across several pixels at different, in particular adjacent, horizontal angles and/or within a pixel over different, in particular adjacent distances, wherein in the case of the product of two correlations, their received signals originate at least partially from the same reflection points on the road, in particular by alternating assignment of the values of the received sequence. Lidar system according to one of the above claims, in which a sign-correct determination of the reception frequency and thus the unambiguous determination of the relative speed of objects when using a real-valued mixer is realized in that among the discrete phase values used for the phase modulation in irregular sequence there are at least two that are neither in phase nor out of phase, the correlation between the frequency-shifted reception sequence and the complex-valued modulation sequence or its conjugate complex is calculated for different reception frequencies and it is used to identify the sign of the reception frequency that this correlation has different levels at positive and negative reception frequencies. Lidar system according to Claim 42 , which uses a set of at least three phase values that are nominally evenly distributed over one revolution, but may deviate significantly from these nominal positions. Lidar system according to Claim 42 or 43 , in which the phase values are realized by switching between lines of slightly different lengths or by a combination of switching between lines of slightly different lengths and a switchable inverter. Lidar system according to one of the above claims, in which monitoring of the devices for changing the beam direction, in particular for ensuring eye safety, is realized in that received signals are checked for changes in the respective spatial direction with regard to object reflections or with regard to the proportions of internal reflections and couplings as well as reflections of a cover. Lidar system according to one of the above claims, which consists of the following three main components, the electronic parts of which are preferably located on the same or at most two circuit boards: - photonic chip, in particular containing a frequency-tunable laser source, a phase modulation unit consisting of a switchable inverter, parallel transmission-reception paths and waveguides; - digital chip, in particular containing analog-digital converters, hard-wired computing logic for determining a two-dimensional correlation with downstream evaluation, one or more microcontrollers and/or DSPs and means for generating control signals; - scanner for one spatial direction realized by material with electrically controllable optical properties, in particular a liquid crystal element or a liquid crystal array, or by a switching matrix for each of the parallel transmission-reception paths or by a mechanical approach.

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