WO1998035476A1 - Method for transmitting digital signals by correlated frequencies - Google Patents

Abstract

The invention concerns a method for transmitting a digital signal in the form of blocks, each block comprising at least one bit and corresponding to a symbol to be transmitted, the transmission of each symbol consisting in transmitting at least a frequency constituting a component unequivocally corresponding to this block, the components of different symbols being mutually correlated.

Description

 CORRELATED FREQUENCY DIGITAL SIGNAL TRANSMISSION METHOD

The field of the invention is that of the transmission of digital signals More specifically, the present invention relates to a method of transmission of digital signals where the transmission of each block of signals to be transmitted to the attention of a receiver consists in transmitting to this receiver one or more given frequencies, characteristics of the data included in this block The invention also relates to a transmitter and a receiver of such signals and finds a very particular application to the transmission of signals in non-coherent modulation in channels having a high dispersion

In a transmission system where modulation in M-FSK mode (M- Ary Frequency Shift Keying) is used, the digital data to be transmitted (bits) are grouped in blocks each block comprising a number n of bits, with M = 2 ^π The transmission of these blocks consists in assigning to each block a given frequency chosen from M and in transmitting this frequency to the receiver. For example, for M = 8, each block comprises 3 bits and to each of the blocks capable of being transmitted corresponds one and a single frequency from the set of M frequencies We thus define a unique relationship between the blocks and the frequencies Transmission takes place at a symbolic period Ts, also called symbol time

Figure 1 shows the spectrum of a BFSK signal (n = 1) The amplitude is noted A and the frequency f The central frequencies of the transmitted signals are noted fn and fj Each channel has a certain bandwidth due to the time truncation performed for the transmission of each frequency The difference fi -fn _, is equal to 1 / Ts, that is to say that the frequencies f and fi are orthogonal (absence of correlation) This ensures that the channels do not overlap and that 'there is no interference between these channels (crosstalk) In general, this orthogona ty is also respected between the M frequencies of an M-FSK type modulation

The disadvantage of this known solution is that the spectral efficiency (ratio between the transmitted bit rate and the total bandwidth occupied) is limited More specifically, the maximum bit rate that can be transmitted is limited to a given value depending on the width of total allocated band An increase in bit rate results in a limitation of Ts and therefore necessarily in a greater spacing between the frequencies fn _. and f- | The present invention aims in particular to overcome this drawback

More specifically, one of the objectives of the invention is to provide a method of transmitting digital signals having improved spectral efficiency compared to the aforementioned state of the art

Another objective of the invention is to provide a transmitter and a receiver of digital signals implementing this method

These objectives, as well as others which will appear subsequently, are achieved by a method of transmitting a digital signal in the form of blocks, each of the blocks comprising at least one bit and corresponding to a symbol to be transmitted, the transmission of each of the symbols consisting in transmitting at least one frequency constituting a component unequivocally corresponding to this block, the components of different symbols being correlated with one another There is therefore a non-zero correlation between the components of the symbols transmitted. possible frequencies significantly increase spectral efficiency

Preferably, the method consists in transmitting, for each of the symbols, a number N of components, the components of different symbols being correlated with one another.

These components can be transmitted by time distribution and / or frequency diversity

The invention also relates to a transmitter of a digital signal in the form of blocks, each of the blocks comprising at least one bit and corresponding to a symbol to be transmitted, the transmitter comprising means unequivocally assigning to each of the symbols at least a frequency constituting a component of the symbol, the components assigned to different symbols being correlated with one another

The invention also relates to a receiver of a digital signal transmitted by such a transmitter, characterized in that it comprises a set of suitable filters each centered on one of the components, the samples of the output signals of these filters constituting the components of a vector Z given by

% - (^ ll,. • -, Zl, Q, - • -, Zk, l,..., Z _kι Q,..., Z / y _t ι,.. •, Z ^ .Q)

> _v 's'v'

QQQ with Q the number of different values of the components, the receiver comprising calculation means maximizing the following decision variable NOT

en se basant, dans le calcul de la variable de vraisemblance relative à un symbole donné, sur les seules sorties des filtres adaptés centrés sur les fréquences où une des composantes serait émise D'autres caractéristiques et avantages de l'invention apparaîtront à la lecture de la description suivante d'un mode de réalisation préférentiel donné à titre illustratif et non limitatif, et des dessins annexés dans lesquels la figure 1 montre le spectre d'un signal BFSK de l'art connu , la figure 2 montre la répartition temps-fréquence des fréquences transmises dans l'invention, l'exemple étant pris pour Q = 8 , la figure 3 montre la probabilité d'erreur par paire sur le canal de Rayleigh de signaux a composantes corrélées en fonction du rapport signal à bruit, pour N = 4 , la figure 4 montre la probabilité d'erreur sur le canal de Rayleigh de signaux à composantes corrélées en fonction du rapport signal a bruit, pour N = 4 , la figure 5 montre la probabilité d'erreur sur le canal de Rayleigh de signaux à composantes corrélées en fonction du rapport signal a bruit, pour N = 8 , la figure 6 est un schéma synoptique d'un exemple d'un émetteur de signaux numériques mettant en oeuvre le procédé selon l'invention , - la figure 7 est un schéma synoptique d'un exemple de récepteur des signaux numériques transmis par l'émetteur de la figure 6 La figure 1 a ete décrite précédemment en référence à l'état de la technique ^Λ i = ^z ά _k j = 1,. . . ,

based, in the calculation of the likelihood variable relating to a given symbol, on the only outputs of the adapted filters centered on the frequencies where one of the components would be emitted Other characteristics and advantages of the invention will appear on reading the following description of a preferred embodiment given by way of nonlimiting illustration, and of the appended drawings in which FIG. 1 shows the spectrum of a BFSK signal of the prior art, FIG. 2 shows the time-frequency distribution of the frequencies transmitted in the invention, the example being taken for Q = 8, FIG. 3 shows the probability of error per pair on the Rayleigh channel of signals with correlated components as a function of the signal to noise ratio, for N = 4, FIG. 4 shows the probability of error on the Rayleigh channel of signals with correlated components as a function of the signal to noise ratio, for N = 4, FIG. 5 shows the probability of e error on the Rayleigh channel of signals with correlated components as a function of the signal to noise ratio, for N = 8, FIG. 6 is a block diagram of an example of a digital signal transmitter implementing the method according to invention, - Figure 7 is a block diagram of an example of a receiver of digital signals transmitted by the transmitter of Figure 6 Figure 1 has been described previously with reference to the prior art

The present invention is based on the fact that the Q frequencies that can be transmitted by component have a correlation between them, that is to say that the spacing between these frequencies is less than 1 / Ts II therefore exists a non-zero correlation between the components of the symbols transmitted The fact of bringing the transmitted frequencies together makes it possible to significantly increase the spectral efficiency FIG. 2 shows the time-frequency distribution of the frequencies transmitted in the invention, the example being taken for Q = 8 L ′ time axis t is subdivided into elementary time cells of duration Ts and frequency axis f into 8 channels spaced two by two by 1 / 4Ts These frequencies are therefore spaced far less than the distance

1 / Ts guaranteeing orthogonality The total band occupied is equal to W = 2 / Ts, identical to that of a conventional BFSK modulation (Fig 1) We will first consider signals with only one dimension, that is to say that the transmission of a symbol occupies only one time cell Ts In a given cell, at each symbol time Ts, only one frequency is transmitted Providing more than two frequencies in the bandwidth W increases the elementary spectral efficiency of the time-frequency plane thus cut out. We note that we have the possibility of emitting 8 different symbols by associating a symbol with a frequency. The number of bits per symbol is 3. This results in a spectral efficiency of 1.5 bit / s / Hz. By comparing this example to the case of a BFSK which uses only two frequencies in the same elementary band W, it can be seen that the gain in spectral efficiency is by a factor of 3. In other words, 3 times more bits can be transmitted in the same frequency band.

The increase in spectral efficiency is accompanied by a minimal degradation in performance due to the non-zero correlation between the different frequencies in an elementary cell. However, this degradation is compensated when alphabets with high spectral efficiency are used (multidimensional signals ), as described below

The signals which we consider in the following will always use a frequency and only one at a time per time-frequency box. The symbols emitted all have the same energy A symbol consisting of N components therefore has the following structure

5, = (0, ..., *! ,,, = l, ..., 0, ..., 0, ..., Λ _fc , _ilk = l, ..., 0, ... , 0, ..., β _J yr ₁ i _w = l, ..., 0) QQQ

The receiver which will be described later has a filter adapted for each of the possible frequencies. The outputs of the energy detectors of this receiver constitute the vector Z with ^•

Z = (Zl, l, - ^{• •} , 2l, Q, - -, 2, l, ..., Z / e, Q, ..., -Z_V, 1, • • •, ZJV.Q)

N _v '^, s _v ,'

Q Q Q

Assuming that there is no correlation between the components in the adjacent time-frequency boxes (transmission channel without memory), the joint probability density of the components of the vector Z is written (relation 1):

NOT

The optimal receiver according to the posterior maximum criterion is the one that chooses the symbol S, maximizing this joint probability density. We show easily that maximizing the expression of relation 1 means maximizing the expression

NOT

In other words it suffices to base oneself, in the calculation of the likelihood variable relative to a given symbol S _j , on the only outputs of the adapted filters connected to the frequencies where a logic state 1 would be emitted From this point , we can show that the function of the optimal receiver is to maximize the scalar product {Z, S _j } This maximization consists in calculating the decision variables Λ, given by

NOT

In order to quantify the gain provided by the invention, it is relevant to perform a calculation of probability of error per pair in the event of the emission of a symbol S- _| The probability of a bad detection consists in deciding that the symbol received is S2 ≠ S- _| This probability is none other than the probability P (Λ1 <Λ2), or also (relation 2)

P (S _{1 →} S ₂ ) z _lιfc <z _2tk

In the previous expression the index j ^ is omitted and the indices '1' or '2' refer to the symbols S <\ and S2 respectively

By transforming the previous expression into a probability of a quadratic Hermitian form, we can define the vector p as follows

P = (, l »2 _f l» ^r l, 2) 2,2,. - -, T'l, JV, î ^, 2, Λr)

The expression of relation 2 therefore implies

P (S ₁ → S ₂ ) = P (f = p'Fp '<0)

where f = ρ * Fp * is the hermitian form sought with 1 0 0 -1

F =

1 0 0 - 1

We know that the characteristic function of such a Hermitian form is (relation 3)

Φfti) = det {I - j2ζR * F)

where R = E [p * p ^* ] denotes the correlation matrix of the vector p

The variables r ^ are other than the samples at the outputs of the suitable filters. The symbol S-] being emitted, these variables are given by

where μk designates the existing correlation between the two components of order k of the two symbols If rectangular filters of duration Ts are used, this correlation is expressed as (relation 4)

μk f ^{Tt / 2} (2 _* fι, kt + Φι.ι) _e -j (2τf ₉ , _k t ^. _k ) _dt _ sin (xΔ / _fe r,), JΨl

JT. / 2 7 vrΔAfkι_TT .. with Δfk = f- | k ~ ^ 2 k ' ^has difference between the frequencies of the two components and φk = φ- | k "Φ2 k ^a phase shift taking into account the inconsistency of the oscillators used for each frequency at the receiver We assume below that the correlations are all different

The random variables r-j | <follow Gaussian laws centered on vaπances (relation 5)

For their part, the random variables r2 | <also follow centered Gaussian laws but of vaπaπces (relation 6)

On the other hand, it is important to note that the noise samples b- _j k ^{and D} 2 k are also correlated We have indeed (relation 7) ^ E [b _lιk b _{> k} ] = μ _k σ

It follows from the above that r- | k ^{and r} 2, k ^are correlated such that (relation 8)

By noting that the random variables relating to two distinct components have null correlations, the expressions of relations 5 to 8 make it possible to write the correlation matrix of the vector p like:

où Rk est une matrice 2 x 2 donnée par

where Rk is a 2 x 2 matrix given by

Under these conditions the expression of the characteristic function of relation 3 becomes

A direct calculation of det (l - j2ξRkF _k ) gives

det (I - j2ξR _k 'F _k ) = (1 - j2ξu _k ) (l X j2ξv _k )

with ^•

et

and

Dans les expressions précédentes nous avons défini .

In the previous expressions we defined.

le rapport signal sur bruit par symbole La fonction caractéristique de f s'écrit donc maintenant

signal to noise ratio by symbol The characteristic function of f is therefore now written

1

ΦΛJξ) = N

11 (1 - j2ξ _k ) {l + j2ξv _k ) k≈

By decomposing into simple elements of ψf (jξ), we obtain

en considérant que μ, ≠ μ_j pour i ≈ j

considering that μ, ≠ μ _j for i ≈ j

The probability of error is therefore equal

NOT

fei ou encoreP ( _5l → S ₂ ) =

fei or even

PiS, → 5 ₂ )

where r is defined by r = 7

7 + 2.Ν

A simpler expression of this probability of error, as well as an asymptotic expression for large signal-to-noise ratios, are as follows

et (relation 9)

and (relation 9)

The asymptotic expression of the probability of error per pair shows that if j μ _k | 2 _es t _e g _a | to | μ l ² whatever k, the degradation of the signal to noise ratio by symbol γ will be the same whatever the dimension Ν of the symbols

For the particular case Ν = 1, we arrive at the expression of the probability of error of two correlated binary symbols on Rayieigh channel given by

On the other hand, if all the correlations between the components are identical (μk = μ V k), we know how to calculate the probability of error per pair equal to (relation 10)

La figure 3 montre la probabilité d'erreur par paire P(S-_| → S2) sur le canal de

Figure 3 shows the probability of error per pair P (S- _| → S2) on the channel of

Rayleigh of signals with correlated components as a function of the signal to noise ratio γ expressed in dB, for a dimension N = 4

Selected values of correlation | μ | 2 = 0, | μ I 2 = 0.09, 1 μ | 2 = o, 4 and I μ! 2 = 0.8 correspond respectively to frequencies spaced from 1 / Ts (state of the art, reference 30), 1 / 4Ts (reference 31), 1 / 2Ts (reference 32), and 3 / 4Ts (reference 33) La asymptotic performance degradation is in good agreement with the expression of relation 8 which provides for a reduction of the signal to noise ratio by a factor of 1 - | μ | 2

The gain in spectral efficiency obtained using signals with correlated components can be illustrated by the following example

Considering first an alphabet with two signals (BFSK) with a diversity of order N = 4, the two symbols of this alphabet are

S- | = (1, 0, 1, 0, 1, 0, 1, 0)

S ₂ = (0, 1, 0, 1, 0, 1, 0, 1) The performances of this two-symbol alphabet are obtained using the relation 10 with N = 4 and μ = 0 The spectral efficiency is 1 / 8 bit / s / Hz (the spectral expansion factor compared to 1 bit / s / Hz is B _e = 8) This corresponds to the state of the art

The invention for its part also uses a diversity of order N = 4 with signals with correlated components, for Q = 8 By numbering the available frequencies in a time-frequency box from 1 to 8, the M = 8 possible symbols are given below by their frequency numbers used in the various components S- | → 1, 2, 1, 2

5 ₂ → 2, 4, 3, 4

5 ₃ → 3, 1, 5, 6

5 ₄ → 4, 3, 7, 8 S ₅ → 5, 6, 4, 1

5 ₆ → 6, 8, 6, 3

5 ₇ → 7, 5, 8, 7

5 ₈ → 8, 7, 2, 5

For example, the emission of the symbol S- | consists in transmitting frequencies 1, 2, 1 and 2 successively (temporal distribution) or simultaneously (frequency distribution in the total band K ^* W) A combination of temporal and frequency transmission is also possible

The total band K ^* W necessary for the emission of a symbol remains identical to that of the BFSK (NQ / 4Ts = 8 / Ts) but we now have in this band M = 8 symbols and the spectral efficiency goes to 3/8 bit / s / Hz (B _e = 2.66)

By way of comparison, a conventional BFSK modulation occupies a bandwidth of 2 / Ts When the transmitted components are only spaced apart by Ts / 4, the band gain is a factor of 3 for the transmission of the same bit rate It is therefore possible, by occupying the same band as that of the conventional BFSK, to provide a total of 4 sub-bands for the transmission of the components. Transmission can therefore take place in frequency diversity.

We note that any pair of symbols has all four different frequencies. We thus obtain the diversity of order 4. On the other hand, by examining the 8 symbols proposed we deduce that the most correlated symbols (S4 and S7 for example) have two components for which the frequencies are adjacent (to

1 / 4Ts), a third component with frequencies separated by 1 / 2Ts and a last component with a spacing of 3/4 Ts The resulting correlations are | μ- _| | 2 =

I μ2 I = 0.8, | μ | 2 = 0.4 and | μ.4 | 2 = 0.09 (we assume that the filters used are rectangular, and therefore that the correlations are given by the relation 4) The performances of the system with correlated components are increased by the bound of the union It indicates that the probability of symbol error is less than M-1 times the greatest probability of error per pair In other words

P _e ≤ (M-1) P (S ₄ → S7) On the other hand, a lower bound is obtained by noting that the probability of error is greater than the probability of error per pair of the two most distant symbols ( that is to say the least correlated) but which are the closest neighbors of each other The two symbols S- | and S2 meet this condition So we have

Pe> P (S- | → S2) The different error probabilities per pair are obtained using the relation P (S- | → S2) previously given Finally, the probability of binary error is approximated by Pj-, = P _e / 2

The performances of the two systems are compared in figure 4 which shows the probability of error on the Rayleigh channel of signals with correlated components according to the signal to noise ratio expressed in dB, for N = 4 The characteristic in solid line corresponds to that of a classic BFSK system (n = 1) for B _e = 8 and those in broken lines at the lower and upper limits mentioned above. We note the very good performance of the correlated signals whose location of the lower and upper limits shows exact performances at least equal to those of the classic system taken as reference Knowing that the symbols with correlated components bring a gain of the spectral efficiency by a factor of 3 we deduce that they are preferable to the classic FSK signals The same remark can be made for a diversity of order 8 (see Fig 5)

The increase in the dimension N makes it easier to absorb more and more correlated components. Indeed, the degradation of the signal to noise ratio is given by the factor

Si N = 1 une corrélation | μ | 2 = 0,95 sur la composante unique du symbole implique une dégradation de -13 dB du rapport signal sur bruit Par contre pour N = 8, la contribution a la dégradation totale d'une composante k ayant la même corrélation I μ I 2 = 0 95 se réduit a -1 ,6 dB, alors qu'elle n'est plus que de -0,4 dB pour N = 32 Quand on sait que cette valeur de la corrélation correspond a des fréquences espacées de 1/8Ts, on comprend bien l'effet constructif des grandes dimensions combinées aux plus grandes densités des fréquences d'émission

If N = 1 a correlation | μ | 2 = 0.95 on the single component of the symbol implies a degradation of -13 dB of the signal to noise ratio On the other hand for N = 8, the contribution to the total degradation of a component k having the same correlation I μ I 2 = 0 95 is reduced to -1.6 dB, whereas it is only -0.4 dB for N = 32 When we know that this value of the correlation corresponds to frequencies spaced 1 / 8Ts apart, we understands the constructive effect of large dimensions combined with higher densities of transmission frequencies

The performances obtained can be further improved by using an optimal diversity alphabet in terms of distance between the components and spectral occupancy. FIG. 6 is a block diagram of an example of a digital signal transmitter implementing the method according to the invention

The transmitter of FIG. 6 comprises a mapping unit 60 ensuring the blocking of a binary train which is applied to it The blocks each contain n bits These blocks are applied to a transformation unit 61 which provides for each of the processed blocks N voltage levels Each voltage level corresponds to a component of a block These voltage levels are then applied to an interleaving unit 62 followed by an oscillator controlled by voltage (VCO) 63 having on its output the interleaved frequencies corresponding to the components of the blocks to be transmitted These frequencies are presented in series to an optional series-parallel converter 64, provided in the case where the transmission had to be carried out in several sub-bands This concept of sub-band is represented in FIG. 2 where two sub-bands SB1 and SB2 are provided. The transmission of the components in sub-bands makes it possible to transmit simultaneously (in the same time interval Ts) several components (frequency diversity)

The different components are then applied to a set of K mixers 65-] at 65 ", with K the number of sub-bands provided The mixers 65-] at 65 ^ ensure the sub-band distribution of the components transmitted The components shifted by frequency are then summed by an adder 66 A mixer 67 receiving a signal from a local oscillator 68 ensures the transposition of the sum signal to a carrier frequency The module signal is then applied to a transmitting antenna 69

The interleaver unit 62 has the function of fighting against selective fading of the transmission channel. Thus, the transmission channel acts independently on the different components.

If the transmission is made by time distribution only, the different components of the symbols to be transmitted are transmitted in different time / frequency boxes and the transmission of a symbol with 4 components (example given above) lasts 4Ts

Of course, the embodiment of this transmitter is given for information only, and many other possibilities exist. FIG. 7 is a block diagram of an example of a receiver of the digital signals transmitted by the transmitter of figure 6

The signal received by an antenna 70 is applied to a mixer 71 receiving a frequency transposition signal from a local oscillator 72 The output signal from the mixer is applied to K sub-band filters 73- _| to 73 "These filters are bandpass filters centered on the center frequencies of the subbands. The filtered signals are then applied to a set of suitable filters 74- _j to 74" * Q, with Q the number of components provided by sub -band The frequencies f- | to fQ correspond to the components and the frequencies F- _| at F ^ at the central frequencies of the sub-bands The output signals of these filters are sampled at the symbol frequency 1 / Ts to provide samples Z _j ι, with i the index corresponding to the sub-band considered and j l ' index corresponding to the component detected These samples are applied to a calculation unit 75 intended to form the vector Z given previously Z - (^ 1,1,..., 2l, Q, •.., -Z / tl, • • -, k, Q,. • •, 2N, 1,..., Zff Q)

^ _v *> _v s _v ,

Q Q Q

It will be noted in this expression of the vector Z that the number of components N of the symbols transmitted is not necessarily equal to K This stems from the fact that it is possible to combine a temporal transmission of the components with a frequency diversity, in the sense of sub- bandaged

The calculation unit 75 preferably operates according to the a priori maximum criterion given above, that is to say that it considers that the block transmitted is that which maximizes the joint probability density of the components of this vector Z

Λ

The calculation unit 75 therefore provides an estimate S equal to the symbol S, which maximizes Λ, gives by

NOT

Λ

The estimate S is then applied to a demapping unit 76 transforming the estimated symbol into bits

The invention therefore makes it possible to use correlated components and therefore a greater density of usable frequencies Consequently, there is greater spectral efficiency

The performances obtained with the correlated signals are good and allow to obtain in the example gives a gain in spectral efficiency by a factor

3 compared to a conventional system occupying the same band and using the same order of diversity The performance in binary error probability remains comparable, if not better, than that of known M-FSK systems

The invention applies in particular to non-coherent modulation The transmission medium used is arbitrary

Claims

1 Method for transmitting a digital signal in the form of blocks, each of said blocks comprising at least one bit and corresponding to a symbol to be transmitted, the transmission of each of said symbols consisting in transmitting at least one frequency constituting a corresponding component of unique way to this block, characterized in that the components of different symbols are correlated with each other 2 Method according to claim 1, characterized in that it consists in transmitting, for each of said symbols, a number N of components, the components of different symbols being correlated with one another 3 The method of claim 2, characterized in that said components are transmitted by time distribution 4 Method according to one of claims 2 and 3, characterized in that said components are transmitted by frequency diversity 5 Transmitter of a digital signal in the form of blocks, each of said blocks comprising at least one bit and corresponding to a symbol to be transmitted, said transmitter comprising means (61, 63) assigning unequivocally to each of said symbols at least a frequency constituting a component of said symbol, characterized in that the components assigned to different symbols are correlated with one another 6 receiver of a digital signal transmitted by a transmitter according to claim 5 characterized in that it comprises a set of suitable filters (74- _] to ⁷⁴ K * Q) each centered on one of said components, the signal samples output of said filters constituting the components of a vector Z given by Z = (Zl.l,. • -, 2l, Q,..., Zfc _f i,..., Z _kι Q,..., ZJV, 1, • • •, Z _N Q) " _V N _v f S QQQ with Q the number of different values of said components, said receiver comprising calculation means (75) maximizing the following decision variable JV

based, in the calculation of the likelihood variable relating to a given symbol, on the only outputs of said adapted filters (74-] to 74 ^ * Q) centered on the frequencies where one of said components would be emitted.