EP1175668B1 - Device for calculating and generating signals, especially for digital sound synthesis - Google Patents

Abstract

The invention relates to a system for generating signals, especially for the digital sound synthesis, and comprises a computer (35) for calculating coefficients in dependence on the signal to be generated, a coefficient memory (36) for storing the calculated coefficients and an arrangement (38) of a plurality of digital devices (5 to 7) that is linked with the coefficient memory (36). The signal to be generated is output at the output (40) of said digital devices. The system further comprises an excitation device (37) that is linked with the arrangement (38) and that feeds excitation signals to the arrangement (38), a control device (39) for controlling the coefficient memory (36), the excitation device (37) and the arrangement (38) in such a manner that the arrangement (38) is operated with the calculated and stored coefficients and the excitation signals.

Description

The invention relates to a device and a method for signal calculation and generation, in particular for digital sound synthesis, by computer-aided simulation of oscillation processes in acoustic musical instruments or other vibrating structures. Such a simulation is called physical modeling or virtual acoustics.

To classify physical modeling in digital sound synthesis, generally used methods, in particular classic direct synthesis methods, are first discussed.

One of the first widely used methods for digital sound synthesis uses frequency modulation (FM synthesis) and was developed by Chowning in 1973. FM synthesis uses at least two oscillators, with one (modulator) controlling the other (carrier). With this algorithm, complex spectra can be generated, which can also have nonlinearities in the time domain. Quite complex sounds can be produced with several systems of this type connected in parallel, but the authentic reproduction of acoustic musical instruments is not possible.

In the case of sampling, the vibrations of real musical instruments are stored as a sequence of samples and played back on demand. Three different methods are used to save storage space. On the one hand, it is assumed that the oscillation shape changes only slightly after the transient process. As a result, a few samples can be read out in a loop. The minor changes are realized in subsequent filters. The second method is pitch shifting (transposing). In order not to have to record and store all pitches of the real musical instrument, the recorded tone can be transposed. The third method for efficient memory utilization is data reduction (loss of sound quality) and data compression (no loss of sound quality). The advantage of sampling lies in the exact replica of a played sound. This means that the variability of a real instrument cannot be realized.

When synthesizing sounds with wavetables, short samples of musical instruments are faded into one another, making it possible to achieve more complex and variable sounds than with simple sampling.

A sound synthesis method in the frequency domain is an additive synthesis in which sine waves of different frequencies and amplitudes are added in variable mutual phase positions. The difficulty with this sound synthesis lies in the determination of the above. Parameter. These can only be obtained approximately by suitable analysis (short-term Fourier transformation) of the real instrument sound. Noise-like sounds can only be generated with considerable effort with this method.

In contrast to additive synthesis, frequencies of a noise-like source are suppressed by subsequent filtering in subtractive synthesis. This has the advantage that noise-like sounds can be easily generated. Monofrequency tones can only be achieved with a high filter effort.

The formant synthesis assumes that the sound of an acoustic musical instrument has certain frequency ranges, which are emphasized regardless of the pitch currently being played. In the methods based on this, short waveforms are used, which are additively superimposed and faded into one another. All of these short waveforms emphasize the formant frequencies.

The commonality of all these synthetic methods is that they start from the sound of a simulated instrument. In contrast, physical modeling no longer starts from the sound of a musical instrument, but from its constructive structure and the resulting properties. The starting point is partial differential equations, the solution of which determines the waveform and thus the sound of the instrument. The advantage of physical modeling is musical Expressiveness that can be achieved with this approach. This can be done using the following procedures.

In the numerical solution, the partial differential equation is converted into a difference equation with a fixed location and time step size. This can be solved in the computer. The disadvantage of this method is the high numerical effort with a sufficiently small local step size. This method can also be used to solve multi-dimensional model equations, but the numerical effort increases enormously.

Modal synthesis assumes that any complex vibrating structure can be broken down into substructures that can be characterized by their modes (natural vibrations) and damping constants. A coupling (also non-linear) only exists between the same modes, an energy exchange between different modes is not possible. With complex structures, the modes can only be determined experimentally.

In the mass-spring model by Cadoz and others (1983) the vibrating object to be examined is broken down into individual mass points, which are connected with ideal springs and dampers. By discretizing the physically based differential equations, one obtains differential equations that can be implemented in the computer. The disadvantage of this method is the enormous computing capacity even with simple structures.

Waveguides are the most widely used method for the physical modeling of musical instruments. This is due to the simple implementation on the one hand and the low computing power required on the other. The waveguide method is based on forward and returning waves on an oscillatory structure, which can be represented by delay lines. The losses and dispersion during the oscillation are concentrated in transfer functions. With this method, vibrations of multidimensional structures can also be realized, but this requires a network of delay lines that communicate with each other by means of multidimensional connections (scattering junctions).

The disadvantages of this synthesis method are the complex filter implementation, especially with small pitch changes, since the delay line can only be changed by whole numerical values. The continuous pitch changes must be implemented in the transfer functions in the feedback branch. As a result, the transfer functions used are no longer physically motivated, but usually have to be determined experimentally. Such a synthesis method is known for example from US-A-5 256 830.

The invention has for its object to provide a signal generating device that allows a relatively simple means to simulate or simulate the oscillations of an oscillatory structure signal generation.

This object is achieved with the features mentioned in claim 1.

Furthermore, the invention creates a method according to claim 8.

Advantageous embodiments of the invention are specified in the subclaims.

The signal generation device according to the invention and the signal generation method according to the invention allow, among other things, digital sound synthesis and are based on an exact model of the physical vibration generation, in particular sound generation. The structure of a system that can be implemented with digital components is then derived from this. The resulting parallel arrangement of digital recursive systems is more than an interconnection of digital oscillators. Depending on the descriptive partial differential equation, the digital recursive systems have degrees 1, 2 or multiples thereof. Each individual digital recursive system thus not only forms the frequency, but also the course of the time envelope of a natural vibration of a fictitious or real sound body in accordance with the underlying physical Model exactly after. The output signal of the parallel arrangement thus corresponds to the overtone spectrum of the physical model in the entire listening area.

It is also advantageous to model and digitally implement the excitation from the initial and boundary values, as well as the excitation function of the partial differential equation of an oscillatory system.

As already mentioned above, the methods with physical modeling differ from the other methods by a possible nuanced way of playing the simulated instruments. This musical phrasing and the simple variability of the instrument and the resulting sound is a great advantage of physical modeling. The method according to the invention is compared below with some other methods to illustrate the advantages achievable according to the invention.

The finite difference method discretizes a partial differential equation of the model and then solves it. This has the disadvantage that the discretization can result in instabilities which are not present in a continuous model and thus in the method presented here. Furthermore, with the finite difference method, all deflections of the location points within the scanning grid must be calculated at all times. This requires a very high computing power, even with simple structures. With the method presented here, however, individual and arbitrary location points can be picked out and their movement can be simulated. This severely limits the computing power required.

In the technique according to the invention, the overall vibration of a system can be synthesized from its partial vibrations. In contrast to modal synthesis, the vibrational forms of very complex models can be solved directly without having to resort to an experimental analysis. In principle, this makes any shapes and boundary conditions of the vibrating structures possible.

The functional transformation method presented here also allows physically exact and separate treatment of excitations, initial conditions and boundary conditions. This is of great importance for understanding the synthetic process and thus for the user. It is also advantageous that digital structures are specified to generate the vibrations.

Compared to the waveguide method, the method presented here has the advantage that the oscillation (e.g. of a string) is e.g. by using the Sturm-Liouville transformation exactly (depending on the exactness of the vibration differential equation). With the waveguide method, however, the vibration can only be calculated approximately. In addition, the listening position or pitch as well as any other physical constant of the musical instrument can be changed in a simple manner in the method presented here. With the waveguide method, however, the transfer functions have to be recalculated.

In addition, in the method presented here, only the physical properties (modulus of elasticity, dimensions, density, etc.) e.g. the string and its boundary conditions (types of attachment at both ends). This promises a direct and easily understandable sound influence in contrast to the prior art.

By using the discrete recursive system to calculate the vibration, real-time capability is given. The method and the device are also characterized by high speed.

With physical modeling, acoustic musical instruments can be reproduced much more nuanced and true to the original than with other forms of synthesis such as sampling. The invention can thus be used advantageously, for example, in electronic musical instruments such as keyboards, synthesizers, expanders and computer sound cards with algorithms for physical modeling. In so-called software synthesizers, the algorithms can also be designed according to the invention, which perform the sound calculation directly on the CPU a PC or on special sound cards with digital signal processors (DSPs).

The method presented here thus simulates vibration processes with the aid of a representation using multidimensional models, recursive systems being used for the implementation. It differs from the algorithms already implemented in electronic or digital musical instruments, among others. in the exactness of the result and in the direct input of various vibration excitations. While the z.Z. The usual algorithms for physical modeling, because of their internal structures, the audible vibrations of strings or air columns can only be approximated, the method presented here can reproduce these vibrations exactly. Despite this accuracy, the real-time capability of the presented method is given.

• Fig. 1 zeigt ein schematisches, vereinfachtes Modell einer schwingenden Saite,
• Fig. 2 zeigt eine Ein-Ausgangs-Beschreibung durch Übertragungsfunktionen,
• Fig. 3 zeigt ein digitales System zur Nachbildung des Systems gemäß Fig. 2,
• Fig. 4 zeigt die Struktur eines der für Erregung, Anfangs- oder Randwerte vorgesehenen digitalen Systeme gemäß Fig. 3,
• Fig. 5 zeigt die Struktur eines in Fig. 4 gezeigten digitalen Systems, und
• Fig. 6 zeigt den prinzipiellen Aufbau des Gesamtsystems eines Ausführungsbeispiels zur digitalen Klangsynthese.

The invention is described below using exemplary embodiments with reference to the drawings.

• 1 shows a schematic, simplified model of a vibrating string,
• 2 shows an input-output description by transfer functions,
• 3 shows a digital system for emulating the system according to FIG. 2,
• FIG. 4 shows the structure of one of the digital systems according to FIG. 3 provided for excitation, initial or boundary values,
• Fig. 5 shows the structure of a digital system shown in Fig. 4, and
• 6 shows the basic structure of the overall system of an exemplary embodiment for digital sound synthesis.

The starting point for the explanation below is a physical model in the form of a partial differential equation. It arises from the description of the behavior of strings, air columns or other vibratory structures through the basic equations of acoustics or the elasticity theory. Depending on the level of detail of these physical models, different partial differential equations are obtained, e.g. for air vibrations; Longitudinal waves of a string; or transverse waves of a string with or without consideration of rotation and shear.

Fig. 1 shows a simplified model of a vibrating string. Possible simplifications are, for example, neglecting the thickness versus the length of the string, assuming a completely rigid rest at the ends, neglecting rotation and shear, etc. Under these conditions, transverse vibrations of this string are described by the following differential equations: $$\begin{gathered} {\mathit{\text{c}}}^{\text{2}}\frac{{\text{∂}}^{\text{2}}}{\text{∂}{\mathit{\text{t}}}^{\text{2}}}\mathit{\text{y}}\text{(}\mathit{\text{x, t}}\text{) =}\frac{{\text{∂}}^{\text{4}}}{\text{∂}{\mathit{\text{x}}}^{\text{4}}}\mathit{\text{y}}\text{(}\mathit{\text{x}}\text{.}\mathit{\text{t}}\text{) + Arousal} \\ \text{and initial conditions} \\ \text{and boundary conditions} \end{gathered}$$

The coefficient c contains physical parameters of the string material. The type of assumptions made determines the number and order of the partial derivatives and the coefficients of the differential equation. An excitation term not specified here describes the excitation of the vibration, e.g. through a painted bow. The initial conditions describe the condition of the string at the start of the vibration, e.g. by striking or plucking the string. The boundary conditions indicate how the string attachment affects its vibration behavior.

There are various methods for converting partial differential equations into time- and location-discrete simulation models that can be implemented with a digital computer or with digital technology components. These include finite difference and finite element methods in numerical mathematics, which, however, require the numerical solution of large systems of equations and therefore do not allow simple technical implementation.

In a preferred method, another equivalent model in the form of a multidimensional transfer function is obtained from the description of vibrations by partial differential equations. The mathematical tool for this are suitable functional transformations for the time and location coordinates. They not only convert the partial differential equation into an algebraic equation, but also allow an exact consideration of the initial and boundary conditions. This process is called the functional transformation method. The handling of this mathematical method is outlined in the next section.

The representation of the behavior of electronic networks - described by ordinary differential equations - by transfer functions has long been state of the art. Applying the Laplace transform to an ordinary differential equation and the associated initial conditions creates an algebraic equation that contains the initial values as additive terms. Solving the algebraic equation after the Laplace transform of the output signal creates an input-output model in the form of a transfer function. This procedure is applied here to the partial differential equation. First, the transformation for the time variable is discussed.

By using the Laplace transform: $$\mathit{\text{y}}\text{(}\mathit{\text{x, t}}\text{) → L {}\mathit{\text{y}}\text{(}\mathit{\text{x, t}}\text{)}}\mathit{\text{= Y}}\text{(}\mathit{\text{x, s}}\text{)}$$ the initial boundary value problem (1) creates a boundary value problem for the Laplace transform Y (x, s) of the solution y (x, t). $$\begin{gathered} {\mathit{\text{c}}}^{\mathit{\text{2}}}{\mathit{\text{s}}}^{\mathit{\text{2}}}\mathit{\text{Y}}\text{(}\mathit{\text{x, s}}\text{)}\mathit{\text{=}}\frac{{\text{∂}}^{\text{4}}}{\text{∂}{\mathit{\text{x}}}^{\text{4}}}\mathit{\text{Y}}\text{(}\mathit{\text{x, s}}\text{)}\mathit{\text{+}}\text{Arousal + initial values} \\ \text{and boundary conditions} \end{gathered}$$

The second derivative of y (x, t) in (1) becomes a multiplication with the second power of the complex frequency variable s . In addition, the initial conditions from (1) appear as an additive term in (3), which contains the given initial values.

A suitable transformation for the location variable is now carried out, which converts the boundary conditions into an additive term in the same way as the Laplace transformation did with the initial conditions. One such transformation is the Sturm-Liouville transformation: $$\mathit{\text{Y}}\text{(}\mathit{\text{x, s}}\text{) → T {}\mathit{\text{Y}}\text{(}\mathit{\text{x, s}}\text{)} =}\overline{\mathit{\text{Y}}}\text{(}{\mathit{\text{β}}}_{\text{μ}}\text{.}\mathit{\text{s}}\text{).}$$

The local frequency variable β _µ takes discrete values and corresponds to the natural frequencies of the system. The exact definition of the local transformation T depends on the form of the partial differential equation. In the simple case of (1) it is

Here x ₀ and x _{1 are} the coordinates of the start and end point of the string. The transformation core K (β _µ , x) also depends on the partial differential equation (1) and describes the shape of the natural vibrations. In the present case they are sinusoidal, in the case of more complex vibration problems they can take other forms.

The natural frequencies only take discrete values; in the simplest case, they are multiples of the basic vibration. The inverse transformation therefore consists of a sum over the occurring natural vibrations: N _µ is a standardization factor. If the transformation core K (β _µ , x) is a sinoder is a cos function, the back transformation T ^-1 corresponds to a development of Y (x, s) into a Fourier series with the coefficients $\overline{\mathit{\text{Y}}}$ ( β _µ , s ). The forward transformation T then corresponds to the formula for calculating the Fourier coefficients.

The solution of equation (1) is obtained from (6) by inverse Laplace transformation: With $$\overline{\mathit{\text{y}}}\text{(}{\mathit{\text{β}}}_{\mathit{\text{μ}}}\text{.}\mathit{\text{t}}{\text{) = L}}^{\text{-1}}\text{{}\overline{\mathit{\text{Y}}}\text{(}{\mathit{\text{β}}}_{\text{μ}}\mathit{\text{, s}}\text{)}.}$$

However, this form of solution is not yet optimal for practical calculation, since the time profiles of the spectral components $\overline{\mathit{\text{y}}}$ ( β _µ , t ) must be known. To obtain it, an input-output description of equation (1) is required, which is discussed below.

If the natural frequencies β _µ and the eigenfunctions K (β _µ , x) are selected correctly, applying the transformation T to the boundary value problem (3) leads to an algebraic equation: $${\mathit{\text{c}}}^{\text{2}}{\mathit{\text{s}}}^{\text{2}}\overline{\mathit{\text{Y}}}{\text{(β}}_{\text{μ}}\mathit{\text{, s}}{\text{) + β}}_{\text{μ}}^{\text{4}}\overline{\mathit{\text{Y}}}\text{(}{\mathit{\text{β}}}_{\text{μ}}\mathit{\text{, s}}\text{) = Excitation + initial values + boundary values.}$$

You can after the transformed $\overline{\mathit{\text{Y}}}$ ( β _µ , s ) of the solution y (x, t) sought. This creates an input-output description with the excitation function, the initial values and the boundary values as inputs and the sought solution as output. The relationships between the inputs and the output are described by transfer functions which contain the constant c from (1), the complex frequency variable s with respect to time t and the discrete frequency variable β _μ with respect to location x . In the simplest case is such a transfer function: $$\overline{\mathit{\text{G}}}\text{(}{\mathit{\text{β}}}_{\mathit{\text{μ}}}\mathit{\text{, s}}\text{) =}\frac{\text{1}}{{\mathit{\text{c}}}^{\text{2}}{\mathit{\text{s}}}^{\text{2}}{\text{+ β}}_{\text{μ}}^{\text{4}}}$$

Depending on the type of excitation, the initial or boundary conditions, the counter can also be a polynomial in s and β _µ .

2 shows this input-output description in the form of a block diagram. Systems 1 to 3 (system 1 ( S _E ) for excitation, system 2 (S _A ) for the initial values, and system 3 ( S _R ) for the boundary values) are each described by transfer functions according to (10). Their output signals are combined via an adder 4 to form the output signal y (x, t).

The mathematical description of a system capable of oscillation represented by natural frequencies and natural oscillations (or eigenvalues and eigenfunctions) is the basis for the construction of a digital system for oscillation synthesis, in particular sound synthesis, which is disclosed here. The design principles are explained in the next section.

The conversion of the mathematical model according to FIG. 2 into a digital system is described below. The purpose of this digital system is to generate sounds based on a physical model. The time course of the vibration amplitude (deflection, sound pressure) should be reproduced at one or more desired location points. This means that the output signal of the digital system corresponds as exactly as possible to the output variable y (x, t) from FIG. 2 at the discrete times t = kT and the discrete location points x = x _n . The discrete times are integer multiples of the sampling interval T, which is to be chosen according to the sampling theorem. The discrete location points can be chosen according to number and location.

The starting point for the construction of the digital system is the representation by transfer functions according to FIG. 2. Each transfer function according to (10) can be used for a fixed value of µ as a description of a location-independent continuous system of the second order. Known transformations exist for the conversion of continuous systems into discrete systems, such as the pulse or step invariant transformation or the bilinear transformation. If such transformations are applied to each of the transfer functions according to (10) and for every value of µ, the time and location continuous system according to FIG. 2 results in a discrete-time system, the reaction of which simulates the output of FIG. 2 for the times t = kT . If the back-transformation according to (6) is additionally evaluated at discrete location points x = x _n , a time- and location-discrete system according to FIG. 3 is created with the same basic structure as in FIG. 2.

3 shows a digital system for emulating the system from FIG. 2, digital subsystems D _E , D _A , D _R (systems 5 to 7) instead of the continuous systems 1 to 3 with the transfer functions S _E , S _A , S _{R are} provided. The output signals of the systems 5 to 7 are also combined here as in FIG. 2 via an adder 8 to form an output signal y (x _n , kT). The structure of the individual subsystems 5 to 7 shown in FIG. 3 ( D _E , D _A , D _R ) is the same in each case and is shown in FIG. 4 for one of the subsystems. At the inputs of input 1 to input m are each components of excitation, initial values and boundary values for the eigenfrequencies of μ. Couplings between the individual recursive systems 9, 11 and 12 also occur in the case of nonlinear models.

FIG. 4 shows the structure of one of the digital systems D _E , D _A , D _R (one of the systems 5 to 7) from FIG. 3 for excitation, initial or boundary values. The structure is basically the same for all systems 5 to 7. With each input 1 to m, a series connection of a system 9, 11 or 13 (system R ₁ , R ₂ or R _m ) and an amplifier or multiplier 10, 12 or 14 is connected, which the output signal of the associated Systems 9, 11 and 13 multiplied by a factor K µ (x _n ) (with µ = 1 to m). The output signals of the multipliers 10, 12, 14 ... are added via an adder 15, which generates the output signal y _l (x _n , kT). The systems R ₁ to R _m arise from the transfer functions (10) by means of the transformations mentioned (pulse, step invariant, bilinear). They are described by difference equations that have the same order as that Have transfer function (10) in the temporal frequency range or a multiple thereof. An advantageous implementation of these systems 10, 12 and 14 is shown in FIG. 5, which illustrates the structure of one of these digital systems. All systems are preferably constructed identically as shown in FIG. 5.

Each system is designed as a recursive digital filter (IIR filter with infinite impulse response), the input signal of which is present at input 20, evaluated via several, here three branches 21, 22 and 23 with different weighting factors b ₀ , b ₁ , b ₂ and at adder 24 , 27, 30 is created. The output signal y _μ (k) emitted by the adder 30 forms the output signal of the entire system 9, 11 or 13 and is fed back via branches 25, 28 to the adders 24 and 27 under evaluation with weakening weighting factors -c ₀ , -c ₁ . The output signal of the adder 24 is applied via a time delay element 26 to a third input of the adder 27, the output signal of which in turn is fed via a time delay element 29 to a second input of the output-side adder 30. The weighting factors b _o , b ₁ , b _{2 are} calculated from the physical quantities of the vibration model. The same applies to the weighting factors -c _o and -c ₁ . The time constants T of the time delay elements 26 and 29 are determined from the sampling frequency.

The output signals y _μ (k) , (with μ = 1, ..., m ) of the adders 30 of the individual systems 9, 11, 13 correspond to the components within the scope of the discrete approximation $\overline{\mathit{\text{y}}}$ ( β _µ , t ) from equation (8), sampled at the times t = kT . These signals describe the time course of the individual natural vibrations. This results from the inverse transformation T ^-1 according to equation (7), the time profile of the entire output signal.

The summation point 15 in FIG. 4 corresponds to the sum in (7).

The factors K _µ (x _n ) of the multipliers 10, 12, 14 (FIG. 4) are formed from the eigenfunctions K (β _µ , x _n ) at discrete location points x _n and the normalization factors N _µ in (7): $${\text{K}}_{\text{μ}}\text{(}{\mathit{\text{x}}}_{\mathit{\text{n}}}\text{) =}\frac{\text{1}}{{\mathit{\text{N}}}_{\text{μ}}}\mathit{\text{K}}{\text{(β}}_{\text{μ}}\text{.}{\mathit{\text{x}}}_{\mathit{\text{n}}}\text{)}$$

Since only the vibration behavior within the listening area is of interest for sound synthesis, the summation only has to record those natural frequencies that are in the listening area, ie, for example, between 16 Hz and 16 kHz (here labeled µ = 1, ..., m ). The output signal y _l (x _n , kT) thus represents a discrete approximation of the audible vibration behavior. I = E, A, R stands for the result based on the excitation, the initial and the boundary values. All three systems 5, 6, 7, which each have the structure according to FIG. 4, together form the output signal of the digital system according to FIG. 3.

The basic structure of the overall system for digital sound synthesis is shown in FIG. 6. The physical model 33 and its parameters 34 are taken from an acoustic model and only serve to define the design parameters of the overall system, but as such do not constitute part of the overall system it doesn't matter whether this acoustic model can be realized with technical means and reasonable effort or not. It is only important that it represents an oscillatory system that is described by known physical laws. The mathematical description of the acoustic model is available as a physical model with its parameters for sound synthesis.

The system shown in Fig. 6 below the dashed dividing line consists of the components: a parallel arrangement 38 of digital recursive systems; an arithmetic unit 35; a coefficient memory 36; an excitation device 37; and a control unit (control device) 39.

The parallel arrangement 38 of digital recursive systems consists of the digital systems 5, 6, 7 (systems D _E , D _A , D _R ) from FIG. 3 with the structure shown in FIGS. 4 and 5.

The individual recursive systems consist of adders, multipliers and Storage elements (time delay elements), as shown in FIG. 5 using an example. The number of storage elements is equal to the number of time derivatives in the underlying partial differential equation or a multiple thereof.

Every natural oscillation (harmonic) of the physical system is realized by a digital recursive system. Their parallel arrangement then reproduces the overtone spectrum. Couplings of these parallel systems occur in non-linear models. The number of recursive systems connected in parallel can preferably be limited to the number of overtones within the listening area without impairing the auditory impression.

The coupled parallel arrangement 38 of digital recursive systems is fundamentally suitable for emulating all oscillation processes which are described by the corresponding partial differential equation (also non-linear). The synthesis of a certain sound requires the determination of the coefficients of the individual digital systems. They are calculated in the arithmetic unit 35 from the parameters of the physical model. These parameters are the physical constants that characterize the vibration process. The calculation rules result from the simulation of the transfer function through a digital implementation. The derivation of the coefficients of the recursive systems from a transfer function ensures that the natural frequencies and the temporal decay behavior of the physical system and the digital implementation correspond exactly.

The coefficient memory 36 receives one or more sets of coefficients from the arithmetic logic unit 35 and loads them into the parallel arrangement 38 of digital recursive systems upon request by the control device 39.

In order to cause the parallel arrangement 38 of digital recursive systems to synthesize an output signal, excitation by one or more input signals is required. These input signals are also simulated according to the physical model and stored in the excitation device. The excitation is derived from the partial differential equation of the oscillatory system and takes into account the initial values (e.g. struck or plucked string), the boundary values (e.g. rope vibrations) and the excitation function (e.g. resonances).

The control device 39 (control unit) takes over the sequence control of the arithmetic unit 35, coefficient memory 36, parallel arrangement 38 and excitation device 37. These possibilities can be used not only to simulate the vibrations of real musical instruments or other sound bodies, but also to synthesize sounds, their natural generation is not possible for technical reasons.

The output signal output at the output 40 of the parallel arrangement 38 represents the desired signal to be generated and can be used or processed in a suitable manner. For example, in order to make generated sound signals audible, a D / A converter can be connected to output 40 and its analog output signal, if necessary. after amplification by an amplifier to be placed on a speaker

In the following, some extensions and variations of the digital system for sound synthesis described fundamentally above and their effects are explained.

• eine andere Struktur eines rekursiven Systems mit gleichem Ein-Ausgangsverhalten, z.B. Regelungsnormalform, Steuerungsnormalform, Zustandsraumstruktur, Leiter (Lattice)-Struktur, Wellendigitalfilter-Struktur;
• eine andere Struktur eines rekursiven Systems, die das Ein-Ausgangsverhalten des Systems aus Fig. 5 approximiert;
• ein nichtrekursives System, das das Ein-Ausgangsverhalten des Systems aus Fig. 5 approximiert.

Instead of the structure of the recursive systems shown in FIG. 5, the following can also be used:

• another structure of a recursive system with the same input-output behavior, for example control normal form, control normal form, state space structure, conductor (lattice) structure, wave digital filter structure;
• another recursive system structure approximating the input-output behavior of the system of FIG. 5;
• a non-recursive system that approximates the input-output behavior of the system of FIG. 5.

• eine andere Struktur eines MIMO-Systems mit gleichem Ein-Ausgangsverhalten,
• eine andere Struktur eines MIMO-Systems, die das Ein-Ausgangsverhalten des Systems aus Fig. 3 und Fig. 4 approximiert.

The possibly coupled parallel arrangement of recursive systems according to FIG. 4 and FIG. 3 shows a special implementation of a system with multiple inputs and multiple outputs (MIMO --- multiple input, multiple output). Instead of this parallel arrangement, the following can also be used:

• another structure of a MIMO system with the same input / output behavior,
• another structure of a MIMO system which approximates the input-output behavior of the system from FIGS. 3 and 4.

A parallel or cascade arrangement of several systems according to FIG. 6 is also possible, the output signal of a system serving as an excitation for the downstream system. In addition to a parallel or cascade connection of several systems for digital sound synthesis according to FIG. 6, combinations thereof are also possible.

3 can emulate the vibration behavior at selected location points x _n . To improve the sound field reconstruction, these location points can also be chosen such that the entire sound field emanating from the vibrating body can be approximated exactly or approximately based on the synthesis results at the points x _n .

Instead of the differential equation (1) for a location coordinate (x), a corresponding differential equation for two or three location coordinates can also be used as a physical model, so that several location dimensions can be simulated. The location functions K µ (x _n ) in FIG. 4 are then also dependent on two or three location dimensions.

The arrangement described above for sound synthesis based on a physical model can also be used for the synthesis of general vibrations, i.e. serve to simulate other physical oscillation processes, if these can be described by partial differential equations. It then represents a digital realization for the simulation of general vibrating bodies, fluids and energy fields.

The arrangement described above can also serve for the simultaneous synthesis of potential and flux size. This is not a scalar differential equation, but to start from a vector differential equation for potential and flux size.

The method presented here can be e.g. Implement in the programming language JAVA for implementation for a vibrating instrument string, which can also take into account the rotational inertia and shear of the string. This has been successfully accomplished by the inventors. In this program, all physical parameters of the real string can be entered, which makes it easy to simulate their vibration behavior.

Depending on the program structure, the non-linearities that occur when a string is excited can also be taken into account and real-time capability can be achieved.

Claims (8)

1. Apparatus for the calculation and the generation signals, particularly for digital sound synthesis with
      a arithmetic unit (35) for the calculation of coefficients depending on a physical model (33) of a vibratory system, where the structure of the physical model (33) is available in form of partial differential equations, and in dependence of the parameters (34) of the physical model (33),
      a coefficient memory (36) for the storage of the calculated coefficients,
      an arrangement (38) of multiple parallel digital recursive systems (9,11,13) reproducing the eigenvibrations of the physical model, connected to the cofficient memory (36), whose output signals result with a weighted summation in the desired signal at the output (40) of the arrangement (38), whereby the recursive systems contain multiple adders (24, 27, 30) among which the time delay elements (26,29), whose delay time constant (T) can be obtained from the sampling frequency, and feedback loops (25, 28), connected to the adders (24,27,30),
      an excitation unit (37) connected with the arrangement (38), applying excitation signals to the arrangement (38) and
      a controller unit (39) for controlling the coefficient memory (36), the excitation unit (37) and the arrangement (38), such that the arrangement (38) is operated with the calculated and stored coefficients and the excitation signals.
2. Apparatus according to claim 1, characterized by the coupling of the digital recursive systems.
3. Apparatus according to claim 1 or 2, characterized by the design of the parallel recursive systems for processing continuous excitation functions, initial values and boundary values of the underlying model in form of PDEs.
4. Apparatus according to one of the previous claims, characterized such that the arithmetic unit (35) determines the coefficients from a transfer function model emulating the vibrational behaviour of a vibrational physical model.
5. Apparatus according to one of the previous claims, characterized by applying physical parameters, representing the physical constants of a vibrational process of a physical model, as inputs to the arithmetic unit (35), and from which the arithmetic unit (35) is calculating the coefficients.
6. Apparatus according to one of the previous claims, characterized by the coefficient memory (36) storing multiplier values for the digital arrangements (9,11,13) as coefficients.
7. Apparatus according to one of the previous claims, characterized by the excitation unit (37) containing at least one signal source for the excitation of the arrangement (38).
8. Method for the calculation and for the generation of signals, particularly for digital sound synthesis with the steps:

   Calculation of coefficients in a arithmetic unit (35) in dependency of a physical model (33) of a vibratory system, whereby the structure of the physical model (33) is given in form of PDEs and in dependency of parameters (34) of the physical model (33),

   storage of the calculated coefficients in a coefficient memory (36),

   emulation of eigenvibrations of the physical model in a arrangement (38) of multiple parallel recursive systems (9,11,13), whose output signals describe the temporal envelopes of the single eigenvibrations, connected to the coefficient memory (36), whereby the weighted output signals are summed up and the signal obtained after the weighted summation, forming the desired signal at the output (40) of the arrangement (38), whereby the recursive systems are arranged with multiple adders (24,27,30) between the time delay units (26,29), whose time delay constant (T) is obtained from the sampling frequency, and which are containing feedback loops (25,28), that are connected to the adders (24,27,30),

   application of excitation signals to the arrangement (38) using an excitation unit (37) connected to the arrangement (38), and

   control of the coefficient memory (36), of the excitation unit (37) and the arrangement (38), such that the arrangement (38) is operated with the calculated and stored coefficients and with the excitation signals.