FBMC-MIMO transmission/reception system with ML detection

Abstract

A transmitter of an FBMC-MIMO system in which, for each FBMC-OQAM modulator associated with a transmission antenna, the symbols of a block to be transmitted are grouped in pairs, the input symbols in a pair being combined to output first and second combined symbols, the combined symbols being input to the FBMC-OQAM modulator. The invention also relates to a receiver including an FBMC-OQAM demodulator and an ML detector from the observables obtained, for each reception antenna. The used ML detection allows for not being affected by intrinsic interference and obtaining a high degree of diversity without increasing the number of antennas.

Claims

1. A transmitter of a Filter Bank Multi-Carrier-Multiple Input Multiple Output (FBMC-MIMO) transmission system comprising: a plurality(N.sub.T) of transmission antennas; and a plurality of Filter Bank Multi-Carrier-Offset Quadrature Amplitude Modulation (FBMC-OQAM) modulators equal in number to the plurality of transmission antennas, each FBMC-OQAM modulator comprising an OQAM modulator and a bank of synthesis filters, each FBMC-OQAM modulator transforming a block of input symbols into FBMC symbols to be transmitted on a corresponding transmission antenna, wherein, for each FBMC-OQAM modulator, the input symbols of a block are combined in pairs at the input to linear combination circuits, a linear combination circuit receiving two input symbols thus paired to output a pair including a first combined symbol based on the two input symbols and a second combined symbol based on the two input symbols, the linear combination being obtained by means of combination coefficients that are all not null, the combined symbols thus obtained being interleaved in an interleaver before being input to said FBMC-OQAM modulator.

2. The transmitter of the FBMC-MIMO transmission system according to claim 1, wherein the linear combination is a rotation by angle , the angle being chosen to be not equal to an integer multiple of $\frac{}{2}.$

3. The transmitter of the FBMC-MIMO transmission system according to claim 2, wherein the input symbols are Quadrature Phase Shift Keying (QPSK) symbols and that =0.15.

4. The transmitter of the FBMC-MIMO transmission system according to claim 2, wherein the input symbols are 16-QAM symbols and =0.09.

5. The transmitter of the FBMC-MIMO transmission system according to claim 1, wherein said interleaver interleaves combined symbols from the same pair such that they are carried by FBMC carriers that are not simultaneously affected by the same fading.

6. A receiver of a Filter Bank Multi-Carrier-Multiple Input Multiple Output FBMC-MIMO transmission system comprising a plurality (N.sub.R) of reception antennas and a same plurality of Filter Bank Multi-Carrier-Offset Quadrature Amplitude Modulation (FBMC-OQAM) demodulators, each FBMC-OQAM demodulator comprising an analysis filter bank followed by an OQAM demodulator, each FBMC-OQAM demodulator being associated with a corresponding reception antenna and, starting from a signal received on this antenna, outputting a plurality of complex symbols (y.sub.k,n.sup.l, k=0, . . . , M1) corresponding to different FBMC carriers, M being a number of FBMC carriers, wherein said receiver comprises: a plurality M of multiplexers to group symbols at outputs from FBMC-OQAM demodulators by FBMC carrier, the symbols thus grouped at the output being represented in the form of a complex vector (y.sub.k=(y.sub.k,n.sup.l, . . . , y.sub.k,n.sup.N.sup.R).sup.T) with size N.sub.R, each complex vector being associated with an FBMC carrier, T being a time interval separating consecutive symbol blocks; a deinterleaver to deinterleave the complex vectors thus obtained and to supply them in the form of pairs of complex vectors (y.sub.k,y.sub.k); a plurality M of projection circuits, each projection circuit being associated with the FBMC carrier, k=0, . . . ,M1, and projecting the complex vector associated with this carrier onto the N.sub.T last columns of an orthonormal matrix Q.sub.k with size 2N.sub.R2N.sub.T obtained by QR decomposition of matrix ${\stackrel{.Math.}{H}}_k=\left(\begin{array}{cc}& \\ & \end{array}\right)$ in which H.sub.k is the matrix with size N.sub.RN.sub.T representing a MIMO channel for carrier k, each projection circuit outputting a projected real vector ({tilde over (y)}.sub.k) with size 2N.sub.T, N.sub.T being a number of transmission antennas; a plurality of maximum likelihood detection circuits, each of these circuits receiving a pair of real vectors thus projected and using them to deduce a pair of transmitted symbols ({circumflex over (x)}.sub.k, {circumflex over (x)}.sub.k+1), a most probable having been combined using a linear combination during transmission.

7. The receiver of the FBMC-MIMO transmission system according to claim 6, wherein a maximum likelihood detection circuit receiving two projected real vectors {tilde over (y)}.sub.k, {tilde over (y)}.sub.k, searches for a most probable transmitted symbols {circumflex over (x)}.sub.k, {circumflex over (x)}.sub.k+1 by means of ${\hat{x}}_{k,k+1}=\underset{x_k,x_{k+1}}{\arg \min }{.Math.{\tilde{y}}_{k,k^{}}^{}-R_{k,k^{}}^{22}\left(A\right)x_{k,k+1}.Math.}^2\mathrm{with}{x}_{k,k+1}=\left(\begin{array}{c}{x}_k\\ x_{k+1}\end{array}\right),{\tilde{y}}_{k,k^{}}^{}=\left(\begin{array}{c}{y}_k^{\%}\\ y_{k^{}}^{\%}\end{array}\right),R_{k,k^{}}^{22}\left(A\right)=\left(\begin{array}{cc}{R}_k^{22}& 0\\ 0& R_{k^{}}^{22}\end{array}\right)$ R.sub.A in which the matrices R.sub.k.sup.22 and R.sub.k.sup.22 were obtained by QR decomposition of matrices H.sub.k and H.sub.k, respectively, and R.sub.A is the matrix defined by: $R_A=\left(\begin{array}{cccccccccc}{a}_{11}& a_{12}& 0& 0& \mathrm{.Math.}& 0& 0& 0& \mathrm{.Math.}& 0\\ 0& 0& 0& \mathrm{.Math.}& 0& a_{11}& a_{12}& 0& \mathrm{.Math.}& 0\\ 0& 0& a_{11}& a_{12}& & 0& 0& 0& & \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& & & & & \mathrm{.Math.}\\ 0& 0& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& 0& a_{11}& a_{12}\\ a_{21}& a_{22}& 0& 0& \mathrm{.Math.}& 0& 0& 0& \mathrm{.Math.}& 0\\ 0& 0& 0& \mathrm{.Math.}& 0& a_{21}& a_{22}& 0& \mathrm{.Math.}& 0\\ 0& 0& a_{21}& a_{22}& & 0& 0& 0& \mathrm{.Math.}& 0\\ \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& & & & & \mathrm{.Math.}\\ 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& 0& a_{21}& a_{22}\end{array}\right)$ in which $A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)$ is the matrix representing the linear combination of symbols x.sub.k, x.sub.k+1, used in transmission.

8. The receiver of the FBMC-MIMO transmission system according to claim 7, wherein matrix A is a rotation matrix to rotate by an angle , chosen to be not equal to an integer multiple of $\frac{}{2}.$

9. The receiver of the FBMC-MIMO transmission system according to claim 6, wherein the maximum likelihood detection circuits have soft outputs.

10. The receiver of the FBMC-MIMO transmission system according to claim 6, wherein each of the maximum likelihood detection circuits uses a sphere decoder.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Other characteristics and advantages of the invention will become clear after reading preferred embodiments of the invention, with reference to the appended figures among which:

(2) FIG. 1 diagrammatically shows a known FBMC-MIMO transmission/reception system according to the state of the art;

(3) FIG. 2 diagrammatically represents a transmission of symbols between the transmitter and the receiver of a system according to FIG. 1;

(4) FIG. 3 diagrammatically represents a transmitter of an FBMC-MIMO system according to one embodiment of the invention;

(5) FIG. 4 diagrammatically represents a receiver of an FBMC-OFDM system according to one embodiment of the invention;

(6) FIG. 5 diagrammatically represents a transmission of symbols between a transmitter according to FIG. 3 and a receiver according to FIG. 4;

(7) FIGS. 6A and 6B illustrate the performances of an FBMC-MIMO according to one example embodiment of the invention in the case in which QPSK symbols are transmitted.

DETAILED PRESENTATION OF PARTICULAR EMBODIMENTS

(8) The basic concept of this invention is to group symbols to be transmitted in pairs and to combine the symbols in each pair in the form of two distinct combinations, the combined symbols thus obtained being transmitted on two distinct FBMC carriers. It is thus proposed to introduce an additional degree of diversity by distributing information about a symbol to be transmitted on two independent carriers.

(9) FIG. 3 diagrammatically represents a transmitter of an FBMC-MIMO system according to one embodiment of the invention.

(10) This figure shows the transmitter in relation to an arbitrary antenna l=1, . . . , N.sub.T, all transmitters having the same structure. In particular, the characteristics of synthesis filter banks associated with the different transmission antennas are the same.

(11) The transmitter comprises an FBMC-OQAM modulator 310 composed of an OQAM modulator and a synthesis filter bank that can be made in the time domain (polyphase filter) or in the frequency domain as indicated in the introduction part. This FBMC-OQAM modulator is identical to the modulators 110.sub.1, . . . , 110.sub.N.sub.T represented in FIG. 1.

(12) The transmitter receives real symbols to be transmitted at its input, represented by x.sub.0,n.sup.l, . . . , x.sub.M1,n.sup.l, the x.sub.k,n.sup.l symbol relating to time n (or block n), to carrier k and to antenna l.

(13) Symbols are grouped in pairs and a combination module 305 calculates two distinct linear combinations for each pair of symbols. It should be noted that the symbols of a particular pair are not necessarily contiguous. However, in order to simplify the presentation while remaining perfectly general, we assume in the following that the sequence of symbols in a particular pair is actually contiguous. Thus, two consecutive symbols, x.sub.2,n.sup.l and x.sub.2+1,n.sup.l, even and odd ranks respectively, are combined in the following form:

(14) $\begin{array}{cc}\left(\begin{array}{c}{z}_{k,n}^{}\\ z_{k+1,n}^{}\end{array}\right)=A_k\left(\begin{array}{c}{x}_{k,n}^{}\\ x_{k+1,n}^{}\end{array}\right)& \left(19\right)\end{array}$
in which A.sub.k are non-trivial 22 matrices, in other words all the elements are non-null. In particular, the matrices A.sub.k are distinct from the unit matrix and the null matrix.

(15) The matrices A.sub.k can depend on k but they are preferably chosen to be identical A.sub.k=A. According to one example embodiment, A will be chosen equal to a rotation matrix:

(16) $\begin{array}{cc}A=A_{}=\left(\begin{array}{cc}\cos & \sin \\ -\sin & \cos \end{array}\right)& \left(20\right)\end{array}$
where

(17) $\frac{}{2}$
in which is a relative integer.

(18) The combined symbols are then interleaved by an interleaver, 307, such that two combined signals z.sub.k,n.sup.l and z.sub.k+1,n.sup.l derived from the same pair are transmitted on different carriers. These carriers will advantageously be chosen to be independent, in other words at a sufficient spacing so as to not be affected by fading at the same time. The interleaving function is identical regardless of the transmitter.

(19) In the following, it will be assumed that at the output from the interleaver, the combined symbol z.sub.k,n.sup.l is carried by the carrier k and the combined symbol z.sub.k1,n.sup.l is carried by the carrier k. Thus, after interleaving, z.sub.k,n.sup.l corresponds to z.sub.k+1,n.sup.l.

(20) The combined symbols thus interleaved are input to the FBMC-OQAM modulator to be transmitted on the MIMO channel.

(21) FIG. 4 diagrammatically represents a receiver of an FBMC-OFDM system according to one embodiment of the invention.

(22) This receiver comprises a plurality of FBMC-OQAM demodulators, 410.sub.1, . . . , 410.sub.N.sub.R, each demodulator being associated with one reception antenna l=1, . . . , N.sub.R. Each demodulator comprises an analysis filter bank that can be made in the temporal domain or in the frequency domain, and an OQAM demodulator. These demodulators have the same structure as those shown 140.sub.1, . . . , 140.sub.N.sub.R on FIG. 1. In particular, the characteristics of analysis filter banks associated with the different reception antennas are the same.

(23) Each FBMC-OQAM demodulator, 410, outputs a block of complex symbols y.sub.k,n.sup.l, k=1, . . . , M1 at time n. These symbol blocks are grouped by carrier in the multiplexers 420.sub.1, . . . , 420.sub.M1, each multiplexer 420.sub.k being associated with a carrier k.sup.k and outputting the symbols y.sub.k,n.sup.l, l=1, . . . , N.sub.R, in the form of a complex vector y.sub.k=(y.sub.k,n.sup.1, . . . , y.sub.k,n.sup.N.sup.R).sup.T with size N.sub.R.

(24) The symbol vectors thus obtained are then deinterleaved by a deinterleaver performing the inverse operation (.sup.1) to that done at the transmitters.

(25) In the following, two vectors y.sub.k and y.sub.k corresponding to two carriers k and k+1 before interleaving will be considered.

(26) Vector y.sub.k is projected onto the N.sub.T last column vectors of the matrix Q.sub.k, obtained by QR decomposition of the matrix H.sub.k defined above. The vector thus projected is a real vector, denoted {tilde over (y)}.sub.k, with size N.sub.T.

(27) Similarly, vector y.sub.k is projected onto the N.sub.T last column vectors of the matrix Q.sub.k, obtained by QR decomposition of the matrix H.sub.k. The vector thus projected is a real vector, denoted {tilde over (y)}.sub.k, also with size {tilde over (y)}.sub.k, N.sub.T.

(28) According to (16), we have the following relations:
{tilde over (y)}.sub.k=R.sub.k.sup.22z.sub.k+{tilde over (v)}.sub.k(21-1)
{tilde over (y)}.sub.k=R.sub.k.sup.22z.sub.k+{tilde over (v)}.sub.k(21-2)
that we can formally define more synthetically by:

(29) $\begin{array}{cc}\left(\begin{array}{c}{\tilde{y}}_k\\ {\tilde{y}}_{k^{}}\end{array}\right)=\left(\begin{array}{cc}{R}_k^{22}& 0\\ 0& R_{k^{}}^{22}\end{array}\right)\left(\begin{array}{c}{z}_k\\ z_{k^{}}\end{array}\right)+\left(\begin{array}{c}{\tilde{v}}_k\\ {\tilde{v}}_{k^{}}\end{array}\right)& \left(22\right)\end{array}$

(30) or, allowing for the fact that

(31) $\left(\begin{array}{c}{z}_{k,n}^{}\\ z_{k^{},n}^{}\end{array}\right)=A_{}\left(\begin{array}{c}{x}_{k,n}^{}\\ x_{k+1,n}^{}\end{array}\right)$
for l=1, . . . , N.sub.R:

(32) $\begin{array}{cc}\left(\begin{array}{c}{\tilde{y}}_k\\ {\tilde{y}}_{k^{}}\end{array}\right)=R_{k,k^{}}^{22}\left(\right)\left(\begin{array}{c}{x}_k\\ x_{k+1}\end{array}\right)+\left(\begin{array}{c}{\tilde{v}}_k\\ {\tilde{v}}_{k^{}}\end{array}\right)& \left(23\text{-}1\right)\end{array}$
in which:

(33) $\begin{array}{cc}{R}_{k,k^{}}^{22}\left(\right)=\left(\begin{array}{cc}{R}_k^{22}& 0\\ 0& R_{k^{}}^{22}\end{array}\right)R_{}& \left(23\text{-}2\right)\end{array}$
where R.sub. is the matrix with size 2N.sub.T2N.sub.T defined by:

(34) $\begin{array}{cc}{R}_{}=\left(\begin{array}{cccccccccc}\cos & \sin & 0& 0& \mathrm{.Math.}& 0& 0& 0& \mathrm{.Math.}& 0\\ 0& 0& 0& \mathrm{.Math.}& 0& \cos & \sin & 0& \mathrm{.Math.}& 0\\ 0& 0& \cos & \sin & & 0& 0& 0& & \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& & & & & \mathrm{.Math.}\\ 0& 0& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& 0& \cos & \sin \\ -\sin & \cos & 0& 0& \mathrm{.Math.}& 0& 0& 0& \mathrm{.Math.}& 0\\ 0& 0& 0& \mathrm{.Math.}& 0& -\sin & \cos & 0& \mathrm{.Math.}& 0\\ 0& 0& -\sin & \cos & & 0& 0& 0& \mathrm{.Math.}& 0\\ \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& & & & & \mathrm{.Math.}\\ 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& 0& -\sin & \cos \end{array}\right)& \left(23\text{-}3\right)\end{array}$

(35) Namely for example in the case of 2 transmission antennas:

(36) 0$\begin{array}{cc}{R}_{}=\left(\begin{array}{cccc}\cos & \sin & 0& 0\\ 0& 0& \cos & \sin \\ -\sin & \cos & 0& 0\\ 0& 0& -\sin & \cos \end{array}\right)& \left(23\text{-}4\right)\end{array}$

(37) More generally, when a non-trivial linear combination matrix

(38) $A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)$
is used to combine symbols at the transmitter, the expressions (23-1), (23-2) and (23-3) become:

(39) $\begin{array}{cc}\left(\begin{array}{c}{\tilde{y}}_k\\ {\tilde{y}}_{k^{}}\end{array}\right)=R_{k,k^{}}^{22}\left(A\right)\left(\begin{array}{c}{x}_k\\ x_{k+1}\end{array}\right)+\left(\begin{array}{c}{\tilde{v}}_k\\ {\tilde{v}}_{k^{}}\end{array}\right)& \left(24\text{-}1\right)\\ R_{k,k^{}}^{22}\left(A\right)=\left(\begin{array}{cc}{R}_k^{22}& 0\\ 0& R_{k^{}}^{22}\end{array}\right)R_A& \left(24\text{-}2\right)\end{array}$
where R.sub.A is the matrix with size 2N.sub.T2N.sub.T defined by:

(40) $\begin{array}{cc}{R}_A=\left(\begin{array}{cccccccccc}{a}_{11}& a_{12}& 0& 0& \mathrm{.Math.}& 0& 0& 0& \mathrm{.Math.}& 0\\ 0& 0& 0& \mathrm{.Math.}& 0& a_{11}& a_{12}& 0& \mathrm{.Math.}& 0\\ 0& 0& a_{11}& a_{12}& & 0& 0& 0& & \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& & & & & \mathrm{.Math.}\\ 0& 0& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& 0& a_{11}& a_{12}\\ a_{21}& a_{22}& 0& 0& \mathrm{.Math.}& 0& 0& 0& \mathrm{.Math.}& 0\\ 0& 0& 0& \mathrm{.Math.}& 0& a_{21}& a_{22}& 0& \mathrm{.Math.}& 0\\ 0& 0& a_{21}& a_{22}& & 0& 0& 0& \mathrm{.Math.}& 0\\ \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}& & & & & \mathrm{.Math.}\\ 0& 0& 0& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& 0& a_{21}& a_{22}\end{array}\right)& \left(24\text{-}3\right)\end{array}$

(41) In all cases, the projected real vectors, {tilde over (y)}.sub.k, {tilde over (y)}.sub.k, are input to an ML detector 450.sub.k that estimates the most probable vectors {circumflex over (x)}.sub.k and {circumflex over (x)}.sub.k+1, taking account of these observables, channel matrices H.sub.k and H.sub.k (from which the upper triangular matrices R.sub.k.sup.22 and R.sub.k.sup.22) and the linear combination matrix A are deduced.

(42) In other words, the vectors {circumflex over (x)}.sub.k and {circumflex over (x)}.sub.k+1 are determined by:

(43) $\begin{array}{cc}{\hat{x}}_{k,k+1}=\underset{x_k,x_{k+1}}{\arg \min }{.Math.{\tilde{y}}_{k,k^{}}-R_{k,k^{}}^{22}\left(A\right)x_{k,k+1}.Math.}^2& \left(25\right)\end{array}$
and more particularly when matrix A is a rotation matrix:

(44) $\begin{array}{cc}{\hat{x}}_{k,k+1}=\underset{x_k,x_{k+1}}{\arg \min }{.Math.{\tilde{y}}_{k,k^{}}-R_{k,k^{}}^{22}\left(\right)x_{k,k+1}.Math.}^2& \left(26\right)\end{array}$
in which

(45) $x_{k,k+1}=\left(\begin{array}{c}{x}_k\\ x_{k+1}\end{array}\right)\mathrm{and}{\tilde{y}}_{k,k^{}}=\left(\begin{array}{c}{\tilde{y}}_k\\ {\tilde{y}}_{k^{}}\end{array}\right).$

(46) The ML detector may be of the type with soft values as described in the paper by M. Caus et al. mentioned above. Alternatively, it may be a Sphere Decoder.

(47) FIG. 5 diagrammatically represents a transmission of symbols between a transmitter and a receiver of an FBMC-OFDM system according to one example embodiment of the invention.

(48) The FBMC-MIMO system considered is also 22, in other words it has 2 transmission antennas and 2 reception antennas. The structures of the transmitters and the receiver are shown in FIGS. 3 and 4 respectively.

(49) At the left of FIG. 5, x.sub.k,n.sup.1 and x.sub.k,n.sup.2 represent the symbols transmitted by antennas 1 and 2 respectively on carrier k, and x.sub.k,n.sup.1 and x.sub.k,n.sup.2 represent the symbols transmitted by the same antennas on carrier k in which k=(k).

(50) Similarly, intrinsic interference generated by the transmultiplexer filter and affecting these same symbols on carrier k is represented by i.sub.k,n.sup.1 and i.sub.k,n.sup.2, and by i.sub.k,n.sup.1 and i.sub.k,n.sup.2

(51) At the receiver end, the observables obtained by projection of vectors

(52) ${\stackrel{.Math.}{y}}_k=\left(\begin{array}{c}\left(y_k\right)\\ \left(y_k\right)\end{array}\right)\mathrm{and}{\stackrel{.Math.}{y}}_{k^{}}=\left(\begin{array}{c}\left(y_{k^{}}\right)\\ \left(y_{k^{}}\right)\end{array}\right)$
onto the N.sub.T last column vectors of Q.sub.k and Q.sub.k, are represented by

(53) ${\tilde{y}}_k=\left(\begin{array}{c}{\tilde{y}}_{k^{}}^1\\ {\tilde{y}}_k^2\end{array}\right)\mathrm{and}{\tilde{y}}_{k^{}}=\left(\begin{array}{c}{\tilde{y}}_{k^{}}^1\\ {\tilde{y}}_{k^{}}^2\end{array}\right)$
respectively.

(54) The fact that two carriers are combined makes it possible to have twice as many observables as in prior art represented in FIG. 2. Everything takes place as if the number of transmission and reception antennas had been virtually doubled.

(55) FIGS. 6A and 6B illustrate the performances of an FBMC-MIMO system according to one example embodiment of the invention, for an EVA (Extended Vehicular A) LTE channel and an ETU (Extended Typical Urban) LTE channel.

(56) The FBMC-MIMO system considered used N.sub.T=2 transmission antennas and N.sub.R=2 reception antennas. The number of carriers was chosen to be equal to 50 out of 1024. A convolutional code with rate was used.

(57) The symbols to be transmitted were QPSK symbols. The ML detector was chosen with soft outputs as in the paper by M. Caus et al. mentioned above.

(58) It can be seen in FIG. 6A that a gain of 1.10 dB on the signal-to-noise ratio can be achieved for a BER (bit error rate) equal to 10.sup.4 if =0.15 is chosen. Performances of the known system according to prior art correspond to =0 (no rotation).

(59) Similarly, it can be seen on FIG. 6B that a gain of 1.5 dB on the signal-to-noise ratio can be reached if we choose =0.15.

(60) More generally, it can be shown that there is an optimum angle for each type of modulation alphabet. Thus, when the symbols to be transmitted belong to a 16-QAM alphabet, the optimum angle =0.09.