Method of transmitting a digital signal for a non-orthogonal MS-MARC system, and a corresponding program product and relay device

Abstract

A method for non-orthogonal transmission of a signal intended for a system with N sources, M relays and a single receiver, in which simultaneous transmission over a single spectral resource by the relays is simultaneous with a transmission over a single spectral resource by the sources. The method includes, for each relay: joint iterative detection/decoding of messages transmitted respectively by the sources during first transmission intervals to obtain decoded messages; detecting errors on the decoded messages; interleaving the detected error-free messages, followed by algebraic network coding including a linear combination, in a finite field of an order higher than two, of the interleaved messages to obtain a coded message, the linear combinations being independent, in pairs, between the relays; and channel coding to generate a signal representative of the network coded message and to transmit this signal during the subsequent transmission intervals simultaneously with a transmission by the sources.

Claims

1. A method of non-orthogonal transmission of a signal, for use in a system having N sources, M relays, and a single receiver, wherein non-coordinated simultaneous transmission over a common spectrum resource by each of the M relays takes place simultaneously with simultaneous transmission over the same spectrum resource by the sources, N2, M2, the method comprising for each of the M relays R.sub.j, 1jM: iterative joint multi-user detection and decoding of coded source messages transmitted respectively by the sources during N first transmission intervals to obtain decoded source messages; detecting errors in the decoded source messages; interleaving only the decoded source messages that are detected without error, followed by algebraic network coding including a linear combination in a finite field of order strictly greater than two of the interleaved decoded source messages to obtain a relay message, the linear combinations being independent in pairs between the relays of the system; and shaping comprising channel coding the relay message to obtain the signal representative of the relay message and to transmit this signal during the following (1)N transmission intervals simultaneously with transmission from the sources, where ]0,1[.

2. A method according to claim 1, further including shaping a control signal identifying the decoded source messages involved in the algebraic network coding of the relay message.

3. A method according to claim 1, wherein the channel coding is systematic.

4. A method according to claim 1, wherein the channel coding is binary, and further comprising: binary-to-symbol conversion between the interleaving step and the algebraic coding step; and symbol-to-binary conversion after the algebraic network coding step.

5. A method according to claim 1, wherein the finite field of order strictly greater than two is a Galois field of order four.

6. A method according to claim 1, wherein detecting errors is performed using redundancy information of a cyclic type associated with the coded messages transmitted by the sources.

7. A method according to claim 1, wherein the iterative joint detection and decoding jointly performs multi-user detection and soft input and soft output decoding.

8. A method according to claim 1, wherein the iterative joint detection and decoding implement space-time decoding.

9. A method according to claim 1, wherein the shaping further comprises interleaving and modulation, interleaving being between channel coding and modulation, to obtain the signal representative of the relay message.

10. A method according to claim 1, wherein the algebraic network coding by the M relays is represented by a systematic generator matrix T=[I.sub.NG] with I.sub.N being an identity matrix of dimension N and with G being a matrix of M network coding vectors a.sub.j=[a.sub.1,j a.sub.2,j . . . a.sub.N,j].sup.T, wherein the components of the network coding vectors are determined by independent random drawing in such a manner that maximum diversity of the system is reached.

11. A method of non-orthogonal transmission of a signal, for use in a system having N sources, M relays, and a single receiver, wherein non-coordinated simultaneous transmission over a common spectrum resource by each of the M relays takes place simultaneously with simultaneous transmission over the same spectrum resource by the sources, N2, M2, the method comprising for each of the M relays R.sub.j, 1jM: iterative joint multi-user detection and decoding of coded source messages transmitted respectively by the sources during N first transmission intervals to obtain decoded source messages; detecting errors in the decoded source messages; interleaving only the decoded source messages that are detected without error, followed by algebraic network coding including a linear combination in a finite field of order strictly greater than two of the interleaved decoded source messages to obtain a relay message, the linear combinations being independent in pairs between the relays of the system; and shaping comprising channel coding the relay message to obtain the signal representative of the relay message and to transmit this signal during the following (1)N transmission intervals simultaneously with transmission from the sources, where ]0,1[, wherein the N sources transmit the coded source messages simultaneously over a common spectrum resource, each of which coded source messages comprises a systematic portion and a first redundancy portion during the N first transmission intervals, and the N sources transmit second redundancy portions simultaneously over the same spectrum resource during the following (1)N transmission intervals while the relays are simultaneously transmitting third redundancy portions.

12. A non-transitory data medium including program instructions stored thereon and adapted to implement a method of non-orthogonal transmission of a signal, when said program is loaded in and executed by a relay for a system having N sources, M relays, and a single receiver, wherein non-coordinated simultaneous transmission over a common spectrum resource by each of the M relays takes place simultaneously with simultaneous transmission over the same spectrum resource by the sources, the method comprising for each of the M relays R.sub.j, 1jM, N2, M2: iterative joint multi-user detection and decoding of coded source messages transmitted respectively by the sources during N first transmission intervals to obtain decoded source messages; detecting errors in the decoded source messages; interleaving only the decoded source messages that are detected without error, followed by algebraic network coding including a linear combination in a finite field of order strictly greater than two of the interleaved decoded source messages to obtain a relay message, the linear combinations being independent in pairs between the relays of the system; and shaping comprising channel coding the relay message to obtain the signal representative of the relay message and to transmit this signal during the following (1)N transmission intervals simultaneously with transmission from the sources, where ]0,1[.

13. A relay R.sub.j, j{1, . . . , M} for a non-orthogonal system having N sources, M relays R.sub.j, 1jM, N2, M2, and a single receiver for implementing a method of non-orthogonal transmission of a signal, wherein non-coordinated simultaneous transmission over a common spectrum resource by each of the M relays takes place simultaneously with simultaneous transmission over the same spectrum resource by the sources, the relay R.sub.j comprising: means for iterative joint multi-user detection and decoding coded source messages transmitted respectively by the sources during N first transmission intervals to obtain decoded source messages; means for detecting errors in the decoded source messages; interleaving means for interleaving only those decoded source messages that are detected without error; means for algebraic network coding including a linear combination of the interleaved decoded source messages in a finite field of order strictly greater than two to obtain a relay message, the linear combinations being independent between the relay and each one of the other relays of the system; and means for shaping, comprising channel coding means for channel coding the relay message to obtain the signal representative of the relay message and to transmit this signal during the following (1)N transmission intervals simultaneously with transmission by the sources, with ]0,1[.

14. A system having the N sources, the M relays, and the single non-orthogonal receiver, wherein the relays are relays according to claim 13.

15. A reception method for a receiver of a non-orthogonal system having N sources and M relays, N2, M2, the system being for implementing a method of non-orthogonal transmission of a signal, wherein non-coordinated simultaneous transmission over a common spectrum resource by the M relays takes place simultaneously with simultaneous transmission over the same spectrum resource by the N sources, the reception method comprising in iterative manner a joint channel and network decoding: a first multi-user detection for detecting N coded source messages transmitted simultaneously by the respective N sources to obtain intrinsic soft information about the respective N coded source messages; a second multi-user detection for detecting N source coded messages and M relay messages transmitted simultaneously respectively by the N sources and by the M relays to obtain intrinsic soft information about the respective N coded source messages and about the respective M relays messages which were obtained at the relays by algebraic network coding decoded source messages detected without error by the relays; N soft channel decoding, in parallel, the N coded source messages; and M soft algebraic network decoding in cascade the M relay messages under the control of information received from the relays identifying the decoded source messages involved in the algebraic network coding at the relays; wherein the N soft channel decoding the N coded source messages uses as first a priori information the intrinsic soft information coming from the first multi-user detection and the intrinsic soft information relating to the coded source messages coming from the second multi-user detection in order to provide extrinsic soft information about the decoded source messages involved in the M relay messages decoded in the M soft algebraic network decoding; wherein the M soft algebraic network decoding the M relay messages uses as second a priori information the intrinsic soft information coming from the first and second multi-user detection and provides extrinsic soft information about the decoded relay messages to the second multi-user detection; wherein the N soft channel decoding the N coded source messages and the M soft algebraic network decoding the M relay messages supplying each other mutually with the extrinsic soft information they provide, this mutual supply being under the control of information coming from the M relays identifying the decoded source messages involved in the algebraic network coding at the relays.

16. A receiver for a non-orthogonal system having N sources and M relays, N2, M2, the system being for implementing a method of non-orthogonal transmission of a signal, wherein non-coordinated simultaneous transmission over a common spectrum resource by the M relays takes place simultaneously with simultaneous transmission over the same spectrum resource by the N sources, and the receiver comprising: first multi-user detection means for multi-user joint detection for detecting N coded source messages coming respectively from the N sources during a first stage of transmission and to generate intrinsic soft information about the N coded source messages; second multi-user detection means for multi-user joint detection for detecting N coded source messages coming respectively from the N sources and M relays messages coming respectively from the M relays during a second stage of transmission and to obtain intrinsic soft information about the respective N coded source messages and about the respective M relays messages which were obtained at the relays by algebraic network coding decoded source messages detected without error by the relays; N soft decoders in parallel for soft decoding the N coded source messages coming respectively from the N sources while using as first a priori information the intrinsic soft information coming from the first multi-user detection means and the intrinsic soft information about the N coded source messages coming from the second multi-user detection means and to provide extrinsic soft information about the decoded source messages involved in M decoded relays messages decoded by M soft algebraic network decoders in cascade; the M soft algebraic network decoders operating in cascade for soft decoding the M relay messages while using as second a priori information the intrinsic soft information coming from the first and second multi-user detection means and provide extrinsic soft information about the decoded relay messages to the second multi-user detection means; wherein the N soft channel decoders and the M soft algebraic network decoders supplying each other mutually with the extrinsic soft information they provide, this mutual supply being under the control of information coming from the M relays identifying the decoded source messages involved in the algebraic network coding at the relays; and decision-taking means for the N soft channel decoders to obtain after some iterations of the mutual supply with the M soft algebraic network decoders an estimate of the source messages.

Description

LIST OF FIGURES

(1) Other characteristics and advantages of the invention appear more clearly on reading the following description of preferred methods given merely as illustrative and non-limiting examples, and with reference to the accompanying drawings, in which:

(2) FIG. 1 shows an example of a prior art MS-MR system with orthogonal links between the various nodes;

(3) FIG. 2 is a diagram showing the topology of MS-MR systems having N sources S.sub.1, . . . , S.sub.N and M relays R.sub.1, . . . , R.sub.N, for a receiver D;

(4) FIG. 3a is a diagram of a non-orthogonal MS-MR system of the invention such that the links are non-orthogonal between the sources and the relays, between the sources and the destination, and between the relays and the destination, the figure showing the first stage of transmission;

(5) FIG. 3b is a diagram of a non-orthogonal MS-MR system of the invention such that the links are non-orthogonal between the sources and the relays, between the sources and the destination, and between the relays and the destination, the figure showing the second stage of transmission;

(6) FIG. 4 is a simplified flow chart of the transmission method of the invention as implemented by a non-orthogonal MS-MR system shown in FIGS. 3a and 3b;

(7) FIG. 5 is a diagram of a first embodiment of an MS-MR system of the invention in which the space-time modulation and coding schemes at the sources and at the relays are defined in the binary field;

(8) FIG. 6 is a diagram of an iterative joint detector and decoder JDDj that jointly detects and decodes messages transmitted simultaneously by the sources and received by the relay;

(9) FIG. 7 is a diagram of a joint detector MAPD-E;

(10) FIG. 8 is a diagram of a decoder DecS.sub.i;

(11) FIG. 9 is a diagram of a second embodiment of an MS-MR system of the invention in which the space-time modulation and coding schemes at the sources and at the relays are defined in the field defined by the algebraic network code;

(12) FIG. 10a is a diagram of a destination receiver of the invention;

(13) FIG. 10b is a diagram of an embodiment of the detector MAPD-EII of the FIG. 10a receiver;

(14) FIG. 11 is a diagram of an embodiment of the SISO decoder Algj when all of the transmissions are performed in the field F.sub.4, i.e. in the second embodiment shown in FIG. 9 in which the channel coding is of the algebraic type;

(15) FIG. 12 is a diagram of an embodiment of the SISO decoder Alg for binary channel code, i.e. as in the first embodiment shown in FIG. 5; and

(16) FIGS. 13 to 15 are diagrams showing the adaptive structure of the receiver as a function of the in-band information transmitted by the relays.

DESCRIPTION OF PARTICULAR IMPLEMENTATIONS

(17) The following notation is used in this application.

(18) All vectors use bold characters. A vector v has its k.sup.th element written [v]k, or v.sub.k.

(19) F.sub.q is the q-element Galois field, is the field of real numbers and is the field of complex numbers.

(20) Matrices/tensors use capital letters in bold. If X is a matrix of dimensions NM belonging to the field E, i.e. XE.sup.NM, then X.sub.k denotes its k.sup.th column (k=1, . . . , M).

(21) Functions use non-italic capital letters. A multi-dimensional function F is applied to input in the form of a matrix A of dimensions mq, where each element a.sub.i,j (for all i=1, . . . , m and j=1, . . . , q) belongs to the set E, and it outputs a matrix B of dimensions np in which each element b.sub.i,j (for all i=1, . . . , n and j=1, . . . , p) belonging to the set G, such that F(A)=B written: F:E.sup.mq.fwdarw.G.sup.np.

(22) The probability density of a complex random variable x with circularly symmetrical Gaussian distribution of mean .sub.x and covariance .sub.x.sup.2 is written: CN(.sub.x,.sub.x.sup.2).

(23) A transmission of the invention is implemented in MS-MR systems having the topology shown in FIG. 2. This topology is such that the sources, nodes S.sub.i, i{1, . . . N} broadcast their coded information sequences for the attention of relays R.sub.j, j{1, . . . , M} and of the destination D; the sources are typically user terminals seeking to reach the same base station with the help of relays. Each relay decodes the signals received from the sources S.sub.1 to S.sub.N and re-encodes the decoded sequences jointly while adding its own redundancy so as to create a network code. At the destination D, the spatially-distributed code sequences comprising the code sequences received directly from the N sources and the code sequences from the M relays are decoded by channel decoding algorithms and by network decoding algorithms.

(24) The invention uses relays with a new non-orthogonal transmission protocol as shown in FIGS. 3a and 3b to improve the spectrum efficiency of transmission while making simple and effective decoding possible at the destination. FIG. 3a shows the first stage of transmission during which only the sources transmit simultaneously to the relays and to the destination. During this stage, the relays are silent. FIG. 3b shows the second stage of transmission during which the sources transmit simultaneously to the relays and to the destination. During this second stage, the relays transmit simultaneously during the same number of transmission intervals to the destination.

(25) Thus, a method 1 of the invention is as shown in FIG. 4.

(26) There is no constraint on the transmission channel; it may suffer from fast fading, with the coefficients of the matrices representing the transmission channel changing on each occasion the channel is used, or it may have slow B-block fading in which the coefficients remain constant over N/B consecutive uses (block) of the channel but change from one block to another, and the channel may be frequency selected, and it may be a MIMO channel. The notations adopted in the description below correspond to a slow 1-block fading channel, i.e. a channel that is quasi-static.

(27) In the description below, the nodes (sources, relays, and destination) are assumed to be perfectly synchronized and the sources are independent (there is no correlation between them).

(28) The non-orthogonal transmission protocol is such that the transmission of information to the destination takes place over N.sub.use uses of the channel arranged in two stages: in a first stage, the sources transmit during N.sub.use=Ch.sub.use uses of the channel, and in a second stage the sources and the relays transmit simultaneously during (1)N.sub.use=Ch.sub.useR uses of the channel.

(29) In a first aspect of the invention, the non-orthogonal transmission method 1 of transmitting a signal is performed by the relays R.sub.j of the system.

(30) A source data block u.sub.iF.sub.q.sup.Ki{1, . . . , N} is a message vector of length K in which the components taken their values in a finite field F.sub.q of integer order q.

(31) The statistically independent sources S.sub.i perform channel coding of the information data u.sub.i while adding a cyclic redundancy check (CRC) to a systematic portion for the purpose of verifying message integrity. The sources S.sub.i having T.sub.s.sub.i transmit antennas use a space-time modulation and coding scheme .sub.i defined on F.sub.q (e.g. the binary field F.sub.2 or the four-element finite field F.sub.4) that associates every block u.sub.i with coded and modulated symbols

(32) $X_i{}_i^{T_{S_i}{N}_{\mathrm{use}}}$
belonging to the complex constellations .sub.i of cardinalities |.sub.i|=2.sup.w.sup.i, w.sub.i. A symbol of F.sub.q has a binary representation containing ln.sub.2(q) bits. If the ratio

(33) $s_i=\frac{w_i}{{\ln }_2\left(q\right)}$
is an integer, it represents the number of symbols of F.sub.q carried by the modulation. The function .sub.i may be described in the following generic manner:

(34) ${}_i:F_q^K.fwdarw.{\mathrm{��}}_i^{T_{S_i}{N}_{\mathrm{use}}}.$
The channel coding and the space-time coding are performed by the coders ENCi.

(35) During the first stage of transmission, the N sources respectively transmit the N symbols

(36) $X_i^{\left(1\right)}{}_i^{T_{S_i}{\mathrm{Ch}}_{\mathrm{use}}}.$
During the second stage of transmission, the N sources transmit respectively the N symbols

(37) $X_i^{\left(2\right)}{}_i^{T_{S_i}{\mathrm{Ch}}_{\mathrm{useR}}}.$

(38) Each relay R.sub.j has N.sub.R.sub.j receive antennas and T.sub.R.sub.j transmit antennas. During the first stage of transmission, each relay receives the disturbed versions of the N source signals transmitted over the source-relay channels. The channel between the sources S.sub.i and a relay R.sub.j is designated by the set of coefficients of the MIMO channel:

(39) $\left\{H_{S_i{R}_j,k,m}{}^{N_{R_j}{T}_{S_i}}\right\}.$
The signal received at the instant k by the relay R.sub.j is expressed in the following form:

(40) $y_{R_j,k}=\underset{i=1}{\overset{N}{.Math.}}\underset{m=0}{\overset{L_m}{.Math.}}{H}_{S_i{R}_j,k,m}{X}_{i,k-m}^{\left(1\right)}+n_{R_j,k}$$k\left\{1,\mathrm{.Math.},C{h}_{\mathrm{use}}\right\},j\left\{1,\mathrm{.Math.},M\right\}$
where

(41) $n_{R_j,k}{}^{N_{R_j}}$
designates the additive noise vector at the relay R.sub.j at the instant k and L.sub.m designates the memory of the channel. The memory effect of the channel is associated with the propagation delay spread. This delay causes transmitted signals to be superposed, thereby giving rise to inter-symbol interference.

(42) The destination has N.sub.D receive antennas. During the first stage of transmission, the destination then receives the disturbed versions of the N source signals transmitted by the source-to-destination channels. Let

(43) $H_{S_{i}D}{}^{N_D{T}_{S_i}}$
be the matrices of fading coefficients associated with the N source-to-destination channels. The signal received by the destination during the first stage of transmission is written:

(44) 0$y_{1D,k}=\underset{i=1}{\overset{N}{.Math.}}\underset{m=0}{\overset{L_m}{.Math.}}{H}_{S_{i}D,k,m}{X}_{i,k-m}^{\left(1\right)}+n_{1D,k},k\left\{1,\mathrm{.Math.},C{h}_{\mathrm{use}}\right\}$
where n.sub.1D,k.sup.N.sup.D designates the additive noise vector during the first stage of transmission at the destination at instant k.

(45) Each relay R.sub.j has a multi-user iterative joint detector and decoder for performing turbo multi-user detection or turbo-MUD that performs joint detection and decoding 2 in iterative manner on the signals received during the first stage in order to obtain the estimated signal vectors from each source .sub.iF.sub.q.sup.K. This joint processing relies on observing the received signal representative of all of the source messages as distorted by the channel and on soft input and output decoding of the channel code from each source. An estimate of the message transmitted by each source is available after a certain number of iterations, which number can be set.

(46) The relay R.sub.j then uses CRC detectors in parallel to detect 3 from the redundant portion of the messages any errors that might have occurred in the previously estimated messages. Let J.sub.j{1, . . . , N} be the set of message indices detected without error by the relay R.sub.j of cardinality .

(47) In parallel, the relay R.sub.j interleaves 4 only the J non-erroneous messages .sub.i,iJ.sub.j with a first type of interleaver .sub.j of size K.

(48) The relay applies 5 a function .sub.R.sub.j to the interleaved messages by means of a network coder in order to obtain a network coded message:
u.sub.R.sub.j=.sub.R.sub.j({.sub.i,iJ.sub.j})F.sub.q.sup.K

(49) The function .sub.R.sub.j linearly combines the interleaved messages in a finite field of dimension strictly greater than two. This linear operation is referred to as network coding. A network code is a linear combination of N source messages received by a relay that can be represented by a vector of dimension N in which each component is non-zero. The network coding associated with the relay R.sub.j is defined by the vector a.sub.j=[a.sub.1,j a.sub.2,j . . . a.sub.N,j].sup.T of dimension N. Assuming that this relay detects only J messages without error of indices contained in the set J.sub.j{1, . . . , N}, the result of the network coding u.sub.R.sub.j is the linear combination u.sub.R.sub.j=.sub.iJ.sub.ja.sub.i,j.sub.i. This amounts to zeroing the coefficients {a.sub.i,j}.sub.i.Math.J.sub.j of the vector a.sub.j having indices that are not included in the set J.sub.j. It is thus possible to define effective network coding for each relay that depends on the source messages that it was not possible to detect without error. In this definition, non-transmission of a relay is an effective network code consisting in a zero vector of dimension N.

(50) In a system having M relays, the order of maximum diversity is M+1. A system having N sources and M relays is said to have maximum diversity if, in the event of outage of any M links from among the NM S-R links and the N+M S-D and R-D links, it is still possible to recover the N messages from the sources (the messages associated with the remaining links are assumed to be detected without error). Let T=[I.sub.NG] be the systematic generator matrix where I.sub.N is the identity matrix of dimension N, and G represents the M vectors of the effective network code of the relays. The vectors a.sub.j=[a.sub.1,j a.sub.2,j . . . a.sub.N,j].sup.T of the network code can be determined by randomly drawing in independent manner the components of the vectors after selecting the order of the field to be strictly greater than two. After the draw, it is necessary to verify that maximum diversity has been reached. This makes it necessary to consider all configurations of Q outages among the MN S-R links, Q{0, 1, . . . , M} and to make sure that the systematic generator matrix T=[I.sub.NG] satisfies the following conditions: after eliminating MQ columns from the matrix T, the remaining N+Q columns have the property of having a rank of not less than N. It should be observed that it suffices to test the case Q=0 (no outages in the MN source destination channels), with the validity of the other conditions being mathematically deduced therefrom.

(51) The channel coded messages from the sources and the corresponding messages interleaved by the relays and then subjected to channel coding by the function .sub.R.sub.j forms a distributed turbo code. The interleaving enables the destination to perform iterative joint channel/network decoding based on the signals received from the sources and the relays. When J=1, the linear operation amounts to identity.

(52) The message resulting from the network coding is shaped 6, 7 by the relay in order to obtain the signal:

(53) $X_{R_j}{}^{T_{R_j}{\mathrm{Ch}}_{\mathrm{useR}}}.$
Shaping typically comprises an operation of space-time modulation and coding. The shaper signal X.sub.R.sub.j is transmitted from the antennas of the relay. The relay R.sub.j informs the destination D of the J messages involved in the network coding by means of an in-band control signal, e.g. consisting in control bits. Alternatively, the relay may indicate which messages were detected with error.

(54) During the second stage of the non-orthogonal protocol, the sources and the relays transmit simultaneously for the same number of uses Ch.sub.useR of the channel over the same radio resource. The signals transmitted simultaneously by the sources and by the relays interfere at the destination. Let

(55) $H_{R_{j}D}{}^{N_D{T}_{R_j}}$
be the matrices of the coefficients of the M relay-to-destination channels. The signal received at the destination during the second stage is written as follows:
y.sub.2D,k=.sub.j=1.sup.M.sub.m=0.sup.L.sup.mH.sub.R.sub.j.sub.D,k,mX.sub.R.sub.j.sub.,k-m+.sub.i=1.sup.N.sub.m=0.sup.L.sup.mH.sub.S.sub.i.sub.D,k,mX.sub.i,k-m.sup.(2)+n.sub.2D,kk{1, . . . , Ch.sub.useR}
where n.sub.2D,k.sup.N.sup.D is the additive noise during the second stage of transmission at the destination at instant k.

(56) The destination observes a superposition of the symbols transmitted by the sources during the first stage of transmission followed by the symbols transmitted by the sources and by the relays during the second stage. At the end of the second stage of transmission, the destination has signals available to it that come directly from the sources and signals that come simultaneously from the sources and from the relays. The destination can then proceed with joint channel network decoding in order to recover the source messages: u.sub.1 . . . u.sub.N from the received signals.

(57) The space-time modulation and coding schemes at the sources and at the relays are defined either in the binary field or in the field defined by the network coding, i.e. decoding them provides probabilistic information about the representation of the messages from the sources either in the binary field or in the field associated with the network coding.

(58) The description of the invention is given in detail on the basis of particular embodiments of non-orthogonal MS-MR systems having two sources and two relays. Each source has a transmit antenna, each relay has a receive antenna and a transmit antenna, and the destination has a receive antenna.

(59) In a first embodiment of the invention, the modulation and channel coding at the sources and at the relays are defined in the binary field. An embodiment is shown in FIG. 5. The modulation and coding schemes at the sources and at the relays are based on a scheme of the bit interleaved coded modulation (BICM) type that corresponds to concatenating binary channel coding (defined in the binary field) with binary interleaving and with bit-to-symbol Gray coding (modulation). The channel coding is preferably of the systematic type, e.g. the encoders are based on recursive systematic convolutional (RSC) codes, possibly associated with puncturing.

(60) Each source S.sub.i, i=1 or 2, has an encoder ENCci for performing the channel coding on the messages for transmission. A message u.sub.iF.sub.2.sup.K is coded as a vector c.sub.iF.sub.2.sup.n.sup.i where

(61) $\frac{n_i}{K}$
is the coding rate of the encoder associated with the source S.sub.i. The code words c.sub.i are shared between two words c.sub.i.sup.(1)F.sub.2.sup.N.sup.use.sup.s.sup.i and c.sub.i.sup.(2)F.sub.2.sup.(1-)N.sup.use.sup.s.sup.i. They are then interleaved with a binary interleaver .sub.chi. The interleaver is necessary for ensuring independence between the coded values by reducing the correlation between the successive code values. The output from the interleaver corresponds to the sequences V.sub.i.sup.(1)F.sub.2.sup.N.sup.use.sup.s.sup.i and V.sub.i.sup.(2)F.sub.2.sup.(1-)N.sup.use.sup.s.sup.i.

(62) The interleaved words are modulated by a modulator MODi into sequences x.sub.i.sup.(1).sup.N.sup.use and x.sub.i.sup.(2).sup.(1-)N.sup.use with 2.sup.s.sup.i being the order of the modulation; the labeling is performed by the following bijective function: .sub.i:F.sub.2.sup.s.sup.i.fwdarw.X.sub.i. Let v.sub.i,k,l=.sub.i,l.sup.1(x.sub.i,k) be the l.sup.th bit corresponding to the modulated symbol x.sub.i,k (corresponding to the bits of the source i {v.sub.i,k,l: l=1, . . . , s.sub.i}).

(63) Each relay R.sub.j, j=1 or 2, comprises an iterative joint detector and decoder (or Turbo MUD) JDDj with as many outputs as there are sources S.sub.i, an error detector CRC.sub.ji on each output of the joint detector and decoder, an interleaver .sub.ji after each error detector CRC.sub.ji, an algebraic network coder CR.sub.j connected to the outputs of the interleavers .sub.ij, and a shaper module MCSj at the output from the algebraic network coder. In this embodiment, each relay also has as many binary-to-algebraic converters CBAji at the input to the network coding as there are sources together with an algebraic-to-binary converter CABj at the output from the network coding.

(64) Each iterative joint detector and decoder JDDj jointly detects and decodes messages transmitted simultaneously by the sources. It is shown in FIG. 6. It comprises a joint detector MAPD-E, two decoders DecS.sub.1, DecS.sub.2 corresponding to each of the sources, two interleavers .sub.ch1, .sub.ch2 between the output to the decoders and the inputs of the detector, and two deinterleavers .sub.ch1.sup.1, .sub.ch2.sup.1 between the outputs from the detector and the inputs of the decoders. The interleavers are of the same type as those used in the sources. The deinterleavers are the inverses of the interleavers. The relay R.sub.j receives a signal y.sub.R.sub.j.sub.,k resulting from interference between the signals from the two sources S.sub.i:
y.sub.R.sub.j.sub.,k=h.sub.1,R.sub.jx.sub.1,k.sup.(1)+h.sub.2,R.sub.jx.sub.2,k.sup.(1)+n.sub.R.sub.j.sub.,k, k{1, . . . , N.sub.use}
where n.sub.R.sub.j.sub.,k is additive noise with the distribution CN(0,.sub.n.sup.2), and h.sub.i,R.sub.j is the coefficient of the channel between the source S.sub.i and the relay R.sub.j. The joint detector MAPD-E (or multi-user detector (MUD)) based on the maximum a posteriori criterion receives the signal y.sub.R.sub.j.sub.,k and exchanges information with the decoders DecS.sub.1, DecS.sub.2 to extract the information .sub.1, .sub.2 from each source. The exchange is applied to the detected bits and to the decoded bits. On each iteration, the decoders receive as input the intrinsic logarithmic likelihood ratios (LLRs) of the coded bits {L(c.sub.i.sup.1)} and they deliver the extrinsic LLRs {E(c.sub.i.sup.1)} on the coded bits c.sub.i.sup.1. After reinterleaving, the extrinsic LLRs {E(V.sub.i.sup.1)} are used as a priori LLRs at the inputs of the detector. Iterations are stopped when the messages are perfectly decoded or after a certain number of iterations, which number can be set.

(65) The joint detector MAPD-E is shown in FIG. 7. The multi-user detector implements the following equations:

(66) $\begin{array}{c}\left(v_{i,k,l}^1\right)=\log \left(\frac{P\left(v_{i,k,l}^1=1|y_{R_j}\right)}{P\left(v_{i,k,l}^1=0|y_{R_j}\right)}\right)\\ \log \frac{\underset{a{}_i:{}_{i,l}^{-1}\left(a\right)=1,b{}_j}{.Math.}P\left(Y_{R_j}|x_{i,k}^1=a,x_{j,k}^1=b\right)e^{\left(a\right)+\left(b\right)}}{\underset{a{}_i:{}_{i,l}^{-1}\left(a\right)=0,b{}_j}{.Math.}P\left(Y_{R_j}|x_{i,k}^1=a,x_{j,k}^1=b\right)e^{\left(a\right)+\left(b\right)}}\end{array}$
where (i,j){1,2}.sup.2 such that ij and:

(67) $\{\begin{array}{c}\left(a\right)=\underset{l^{}=1}{\overset{{\log }_2\left(.Math.{}_i.Math.\right)}{.Math.}}{}_{i,l}^{-1}\left(a\right)E\left(v_{i,k,l^{}}^1\right)\\ \left(b\right)=\underset{l^{}=1}{\overset{{\log }_2\left(.Math.{}_j.Math.\right)}{.Math.}}{}_{j,l^{}}^{-1}\left(b\right)E\left(v_{j,k,l^{}}^1\right)\end{array}$
and where E(v.sub.i,k,l.sup.1) is the extrinsic information of the bit v.sub.i,k,l.sup.1 supplied by the decoder corresponding to the source S.sub.i, the l.sup.th bit in the labeling corresponding to the symbol x.sub.i,k.sup.1.

(68) The outputs from the joint detector are then described by: L(V.sub.1.sup.1)=(V.sub.1.sup.1)E(V.sub.1.sup.1) where (V.sub.1.sup.1) is the log a posteriori probability ratio (LAPPR) on the interleaved bits in V.sub.1.sup.1 and E(V.sub.1.sup.1) is the extrinsic information from the decoder associated with the source S.sub.i of these bits.

(69) The decoder DecS.sub.i is shown in FIG. 8. It is a decoder with soft inputs and soft outputs, e.g. of the BCJR type [2]. The equations that characterize soft information exchange at the decoder are given by:
E(c.sub.i.sup.1)=(c.sub.i.sup.1)L(c.sub.i.sup.1)
where (c.sub.i.sup.1) is the LLR corresponding to the coded bits of the code word c.sub.i.sup.1, L(c.sub.i.sup.1) is the a priori information about the bits, and E(c.sub.i.sup.1), which is supplied to the detector, is the extrinsic information about these coded bits.

(70) The error detector CRCji detects the errors so as to select only those messages that have been decoded without error. When there is systematic coding at the sources, the contents of the selected error-free messages correspond to the systematic bits of the code words detected without error.

(71) The interleavers .sub.ji are defined in the binary field, they are identical for a given relay given that the method implements joint network-channel coding. The interleavers .sub.ji interleave the messages that have been detected without error from the source S.sub.i corresponding to the systematic bits of their respective code words, thus making it possible to create a distributed turbo code.

(72) The binary-to-algebraic converters CBAji associate a number n=ln.sub.2(q) of bits (where q is a power of 2 equal to the cardinality of the field, ln.sub.2 is the base 2 logarithm) with an element of the field defined by the network coding. Thus, in the example of a four-element Galois field F.sub.4, the converters rely on the following bijective function:
s:{0,1}{0,1}.fwdarw.F.sub.4
which associates with two bits (b.sub.1,b.sub.2) the symbol q=s(b.sub.1,b.sub.2) of F.sub.4.

(73) This function and its inverse gives the binary representation of an element of F.sub.4. The following notation is used below: b.sub.1=s.sub.1.sup.1(q) and b.sub.2=s.sub.2.sup.1(q) for q=s(b.sub.1,b.sub.2).

(74) The network coder CR.sub.j represented in FIG. 5 by a square with a cross performs the linear combination in the field F.sub.4 of the messages from two sources as detected without error, interleaved, and converted into algebraic form. The network coding vectors of the relays R.sub.1 and R.sub.2 are respectively [1,1].sup.T and [1,2].sup.T. The network coding [n,m].sup.T defined by the vectors [1,1].sup.T and [1,2].sup.T performs the operation n.Math.u.sub.1+m.Math.u.sub.2. The system formed by the independent equations u.sub.2+u.sub.1 and u.sub.1+2.Math.u.sub.2 is invertible in F.sub.4 and thus makes it possible to retrieve the messages u.sub.1 and u.sub.2 in the event of outages on the links S.sub.1 to D and S.sub.2 to D.

(75) The algebraic-to-binary converters transform an element of the finite field into its binary representation, they implement the function that is the inverse of s.

(76) The shaper module MCSj performs a modulation and coding scheme, which may for example be of the bit interleaving coding and modulation (BICM) type, that corresponds to concatenating binary channel coding (defined in the binary field), a binary interleaver, and bit-to-symbol Gray coding (modulator). The binary channel coding at the relays is based for example on recursive systematic codes (RSC). The vector of the coded bits is written:

(77) $c_{R_j}{F}_2^{n_{R_j}}.$

(78) In a particular implementation, channel coding is followed by a puncturing operation , which consists in the recursive systematic coder retaining only bits of determined parity. Under such circumstances, the overall efficiency is equal to

(79) $\frac{n_{R_j}}{K}.$

(80) The coded and interleaved sequences are written:

(81) $V_{R_j}{F}_2^{{\mathrm{Ch}}_{{\mathrm{use}}_R}{S}_{R_j}}.$
The bit-to-symbol coding is described by

(82) ${}_j^R:F_2^{S_{R_j}}.fwdarw.{\mathrm{��}}_j^R,$
where X.sub.j.sup.R designates the constellation of symbols obtained with the order of the modulation being

(83) 0$2^{S_{R_j}}.$

(84) The signal (code word) transmitted by the relay R.sub.j is written:

(85) $x_{R_j}{}^{{\mathrm{Ch}}_{{\mathrm{use}}_R}}.$

(86) When the relays have a plurality of transmit antennas, shaping may for example be of the space-time bit interleaving coding and modulation (ST-BICM) type.

(87) In a second implementation of the invention, the space-time modulation and coding schemes at the sources and at the relays are defined in the field defined by the network coding. A particular implementation is shown in FIG. 9. In this second implementation, the modulation and coding schemes implemented at the sources and at the relays are defined directly by the field defined by the network coding, i.e. on a representation of the messages from the sources in the field. The example shown in FIG. 9 is identical to that of FIG. 5 except that the relays do not have a binary-to-algebraic converter, nor do they have an algebraic-to-binary converter, and except in that the shaper module MCSj shown corresponds to a particular implementation involving puncturing j.

(88) The modulation and coding schemes are based on a scheme of the symbol interleaved coded modulation (SICM) type and corresponds to concatenating channel coding ECcj defined in the field F.sub.4, a symbol interleaver .sub.chRj (not binary) defined on F.sub.4, and a modulator Modj performing Gray bit-to-symbol coding operating on the binary representations of the symbols of F.sub.4. The example is described for the situation when

(89) $s_i=\frac{w_i}{{\ln }_2\left(q\right)}$
is an integer. Thus, a modulated symbol from the source i carries s.sub.i symbols of the field under consideration, i.e. the modulators may be considered as being bijective functions .sub.i:F.sub.4.sup.S.sup.i.fwdarw.X.sub.i.

(90) The detection and the decoding performed by the relay take place in a manner similar to that described above for the example corresponding to the binary implementation shown in FIGS. 5 to 8. The difference comes from the fact that the information exchanged between the detector and the decoders is now in the form of vectors. The linearity of the detection and decoding operations means that it is equivalent to applying on each component of these vectors the same operations as those described for the binary implementation.

(91) The multi-user detector implements the following equations:

(92) $\begin{array}{c}{}_q\left(v_{i,k,l}^1\right)=\log \left(\frac{P\left(v_{i,k,l}^1=q|y_{R_j}\right)}{P\left(v_{i,k,l}^1=0|y_{R_j}\right)}\right)\\ \log \frac{\underset{a{}_i:{}_{i,l}^{-1}\left(a\right)=q,b{}_j}{.Math.}P\left(Y_{R_j}|x_{i,k}^1=a,x_{j,k}^1=b\right)e^{\left(a\right)+\left(b\right)}}{\underset{a{}_i:{}_{i,l}^{-1}\left(a\right)=0,b{}_j}{.Math.}P\left(Y_{R_j}|x_{i,k}^1=a,x_{j,k}^1=b\right)e^{\left(a\right)+\left(b\right)}}\end{array}$
where (i,j){1,2}.sup.2 such that ij and:

(93) $\{\begin{array}{c}\left(a\right)=\underset{l^{}=1}{\overset{{\log }_4\left(.Math.{}_i.Math.\right)}{.Math.}}{}_{i,l^{}}^{-1}\left(a\right)E\left(v_{i,k,l^{}}^1\right)\\ \left(b\right)=\underset{l^{}=1}{\overset{{\log }_4\left(.Math.{}_j.Math.\right)}{.Math.}}{}_{j,l^{}}^{-1}\left(b\right)E\left(v_{j,k,l^{}}^1\right)\end{array}$

(94) E(v.sub.i,k,l.sup.1) is the extrinsic information supplied by the decoder corresponding to the source S.sub.i, concerning the bit v.sub.i,k,l.sup.1 relative to the value .sub.i,l.sup.1(b) which is the l.sup.th bit in the labeling corresponding to the symbol x.sub.i,k.sup.1. The output from the detector is then written by: L(V.sub.i.sup.1)=(V.sub.i.sup.1)E(V.sub.i.sup.1) where (V.sub.i.sup.1) is the LAPPR of the word V.sub.i.sup.1 and E(V.sub.i.sup.1) is the extrinsic information of this word from the decoder associated with the source S.sub.i. The decoding takes place in a manner similar to that described with reference to FIG. 8.

(95) FIG. 10a shows a destination receiver of the invention. It essentially comprises two stages: a detection stage followed by a decoding stage. The description of the receiver is based on a particular implementation of the transmission method in which the sources transmit their systematic bits and their initial parity bits during the first stage of transmission and additional parity bits during the second stage.

(96) The detection stage performs multi-user iterative detection to detect messages coming directly from the sources during the first stage of the protocol and messages coming from the sources and from the relays during the second stage of the protocol respectively by means of two detectors MAPD-EI and MAPD-EII. The two detectors provide soft information about the detected messages coming respectively from the sources and from the relays. The operation of the detector MAPD-EI is identical to the operation of the detector described with reference to FIG. 7.

(97) During the second stage of the protocol, the destination receives the signals from the sources and also the signals from the relays:
y.sub.D,k.sup.(2)=h.sub.1,Dx.sub.1,k.sup.(2)+h.sub.2,Dx.sub.2,k.sup.(2)+h.sub.R.sub.1.sub.,Dx.sub.R.sub.1.sub.,k+h.sub.R.sub.2.sub.,Dx.sub.R.sub.2.sub.,k+n.sub.D,k.sup.(2)k{1, . . . , (1)N.sub.use}
where n.sub.D.sup.(2) designates additive noise with the distribution CN(0,.sub.n.sup.2), where h.sub.i,D and h.sub.R.sub.j.sub.,D respectively designate the fading coefficients between the source i and the destination D and between the relay R and the destination D. The multi-user detector MAPD-EII shown in FIG. 10b combines as inputs: the extrinsic information E(c.sub.Rj) on the parity bits of the relays coming from the decoders corresponding to the network coding operations at the relays; and the extrinsic information E(c.sub.i.sup.2) on the parity bits of the second stage of transmission from the sources coming from the channel decoders corresponding to the two sources: SISO1 and SISO2.

(98) The equations that generate the operation of the MAPD-E are given below. The calculation of the LLRs of the l.sup.th bit v.sub.Rj,k,l of the symbol received at the instant k and transmitted by the relay R.sub.j is given by:

(99) $\begin{array}{c}{}_q\left(v_{R_j,k,l}\right)=\log \left(\frac{P\left(v_{R_j,k,l}=1|y_{D,k}^2\right)}{P\left(v_{R_j,k,l}=0|y_{D_k}^2\right)}\right)\\ \log \left(\frac{\underset{c{}_{R_j}:{}_{R_j}^{-1}\left(c\right)=1}{.Math.}\underset{a{}_1,b{}_2,d{}_{\stackrel{\_}{R_j}}}{.Math.}P\left(Y_{D,k}^2|x_{1,k}^2=a,x_{2,k}^2=b,x_{R_j,k}=c,x_{\stackrel{\_}{R_j},k}=d\right)e^{\left(a\right)+\left(b\right)+\left(c\right)+\left(d\right)}}{\underset{c{}_{R_j}:{}_{R_j}^{-1}\left(c\right)=0}{.Math.}\underset{a{}_1,b{}_2,d{}_{\stackrel{\_}{R_j}}}{.Math.}P\left(Y_{D,k}^2|x_{1,k}^2=a,x_{2,k}^2=b,x_{R_j,k}=c,x_{\stackrel{\_}{R_j},k}=d\right)e^{\left(a\right)+\left(b\right)+\left(c\right)+\left(d\right)}}\right)\end{array}$
where j{1,2}, where R.sub.j designates the relay other than R.sub.j, where:

(100) $\{\begin{array}{c}\left(a\right)=\underset{l^{}=1}{\overset{{\log }_2\left(.Math.{}_1.Math.\right)}{.Math.}}{}_{1,l^{}}^{-1}\left(a\right)E\left(v_{i,k,l^{}}^2\right)\\ \left(b\right)=\underset{l^{}=1}{\overset{{\log }_2\left(.Math.{}_2.Math.\right)}{.Math.}}{}_{2,l^{}}^{-1}\left(b\right)E\left(v_{2,k,l^{}}^2\right)\\ \left(d\right)=\underset{l^{}=1}{\overset{{\log }_2\left(.Math.{}_{\stackrel{\_}{R_j}}.Math.\right)}{.Math.}}{}_{\stackrel{\_}{R_j}{l}^{}}^{-1}\left(d\right)E\left(v_{\stackrel{\_}{R_j},k,l^{}}\right)\end{array}$
and where E(v.sub.Rj,k,l) is the extrinsic information supplied by the decoder corresponding to the relaying function of the relay R.sub.j of the bit v.sub.Rj,k,l, the l.sup.th bit in the labeling corresponding to the symbol x.sub.Rj,k. These equations are valid only in the presence of the channel interleavers .sub.chU.sub.i where U.sub.i is a source or a relay.

(101) The LLRs corresponding to the second stage of transmission from the sources are given in a manner that is symmetrical relative to the LLRs from the relays. At the output from the detector MAPD, the method calculates the extrinsic information on the parity bits of the second stage of transmission from the sources and also on the parity bits from the relays using the following relationships:
L(V.sub.i.sup.2)=(V.sub.i.sup.2)E(V.sub.i.sup.2) and L(V.sub.R.sub.j)=(V.sub.R.sub.j)E(V.sub.R.sub.j) where i=1,2
designates the sources and where j=1,2 designates the relays.

(102) The decoding stage serves to decode messages respectively from the sources and from the relays on the basis of soft information taken a priori and coming from the detection stage. The decoding is performed by means of as many soft decoders SISO (soft input soft output) as there are sources and as many algebraic decoders SISO algj as there are algebraic network coders, and thus as there are relays. The decoders SISO1, SISO2, and the decoders SISO Alg1, SISO Alg2 are based on the codes implemented respectively in the sources and in the relays.

(103) The soft decoders associated with the sources operate in totally identical manner with one another. Their operation is similar to that of the decoder described with reference to FIG. 8. FIG. 10 shows a demultiplexer between the output of the detector MAPD-EI and each of the soft decoders SISO1, SISO2, and it enables the systematic portion to be distinguished from the parity portion of the code, whereas in FIG. 8 the single input of the decoder implicitly includes both portions. Compared with the decoder of FIG. 8, a difference comes from taking account of the parity information supplied by the algebraic decoders associated with the relays. Furthermore, each decoder SISOi receives the a priori information on the systematic bits and the initial parity bits of the detector MAPD-EI, whereas it receives the a priori information on the second series of parity bits from the second detector MAPD-EII.

(104) The decoders provide soft information about the decoded symbols to the detectors. An iterative exchange of soft information takes place between the decoders associated with the sources and the algebraic decoders associated with the relays, thereby giving rise to iterative joint network channel decoding.

(105) The LLRs of the systematic bits and of the parity bits corresponding to the sequence transmitted by the source i during the first stage are written L.sub.Si and L.sub.pi.sup.1. The LLRs of the parity bits corresponding to the sequence transmitted by the source i during the second stage are written L.sub.pi.sup.2. Let L.sub.SiRj be the LLRs supplied by the relay R.sub.j for the systematic bits from the source S.sub.i. The extrinsic information corresponding to the preceding LLRs is written E.sub.Si, E.sub.pi.sup.1, E.sub.pi.sup.2, E.sub.SiRj.

(106) The decoder SISO Algj is a decoder having soft inputs and outputs that incorporate a logic (algebraic) operation corresponding to the linear transform of the network coding of the relay R.sub.j under consideration.

(107) FIG. 11 is a diagram of an embodiment of the SISO decoder Algj when all of the transmissions are performed in the field F.sub.4, i.e. in the second implementation shown in FIG. 9 where the channel coding is of the algebraic type. In the example shown, the network coding corresponds to the relay R.sub.j=R.sub.2 which performs the linear combination u.sub.R=u.sub.1+2.Math.u.sub.2. The messages from the sources u.sub.1 and u.sub.2 represented in F.sub.4 are detected without error at the relays, and thus u.sub.R=u.sub.1+2.Math.u.sub.2 is included in the parity bits transmitted by the relay R.sub.2. The decoder with soft inputs and outputs SISO Alg2 corresponding to the relay under consideration receives as input a priori information about the parity symbols (because of the puncturing, only the parity bits are transmitted by the relays) from the relay L.sub.pR together with a priori information about the symbols from the two sources u.sub.1 and u.sub.2 (included in the systematic portions of the code words from the sources) which are E.sub.S.sub.1=E.sub.S.sub.1+L.sub.S.sub.1 and E.sub.S.sub.2=E.sub.S.sub.2+L.sub.S.sub.2. This a priori information E.sub.S.sub.1 and E.sub.S.sub.2 about the symbols from the two sources is used to calculate a priori information about the systematic symbols at the input of the channel coding of the relay by using the relationship modeled by the operator 1,2 in FIG. 11:
E.sub.SR,k=.sub.1,2,k(E.sub.s1,E.sub.S2) where kF.sub.4={0,1,2,3}
representing the relationship u.sub.R=u.sub.1+2.Math.u.sub.2. Let E.sub.SR be the set {E.sub.SR,k:k=0, 1, 2, 3, 4}. From the (four component) vector E.sub.SR and L.sub.pR the decoder outputs the extrinsic information for the parity symbols E.sub.pR2 from the relay R.sub.2 under consideration and for the systematic symbols from both sources: E.sub.SiR2 which are associated by the following relationships modeled by the operators 1,2 and 3,3 in FIG. 11:
E.sub.S1R2=(.sub.1,2,k(E.sub.SR,E.sub.S2)).sub.k={0,1,2,3} representing u.sub.1=u.sub.R+2.Math.u.sub.2 and
E.sub.S2R2=(.sub.3,3,k(E.sub.SR,E.sub.S1)).sub.k={0,1,2,3} representing u.sub.2=3.Math.u.sub.R+3.Math.u.sub.1.

(108) FIG. 12 is a diagram of an embodiment of the SISO decoder Algj for binary channel coding, i.e. as in the first embodiment shown in FIG. 5. Operation is on the same principle as that of the SISO decoder Algj shown in FIG. 11 corresponding to algebraic transmission and described above. The difference comes from adding binary-to-algebraic converters CBAs (bit to symbol) at the input of the operators and algebraic-to-binary converters CABs (symbol to bit) at the output from the operators 1,2, 2,1, and 3,3. The binary-to-algebraic converters act on the extrinsic information from the sources in order to retain channel decoding at binary level. The LLRs and the extrinsic information are in the binary domain. Let u=s(u.sub.1, u.sub.2) be an element of F.sub.4, let qF.sub.4, and let

(109) ${}_q\left(u\right)=\log \left(\frac{P\left(u=q\right)}{P\left(u=0\right)}\right)$
be the LLR of u corresponding to the value q. The bit-to-symbol converters CBAs perform the operation that consists in determining the LLR of the symbol u knowing the LLRs of the bits (u.sub.1).sub.i{1,2}:
qF.sub.4, .sub.q(u)=.sub.i{1,2}s.sub.i.sup.1(q)(u.sub.i) where

(110) $\left(u_i\right)=\log \left(\frac{P\left(u_i=1\right)}{P\left(u_i=0\right)}\right)$

(111) At the output from the operators 1,2, 2,1, and 3,3, the symbol-to-bit converters CABs perform the operation that consists in determining the two LLRs associated with each of the bits of the symbol u:

(112) $\left(u_i\right)=\log \left(\frac{\underset{q{F}_4,s_i^{-1}\left(q\right)=1}{.Math.}{e}^{\left({}_q\left(u\right)\right)}}{\underset{q{F}_4,s_i^{-1}\left(q\right)=0}{.Math.}{e}^{\left({}_q\left(u\right)\right)}}\right),i\left\{1,2\right\}$

(113) The SISO decoder Algj shown corresponds to a relay that transmits only the parity bits and that punctures the systematic bits.

(114) The BCJR decoder of FIG. 12, like that of FIG. 11, operates on the two-element finite field like the associated channel coding. Only the binary-to-algebraic and algebraic-to-binary converters and the operators operate in the field F.sub.4.

(115) The basic arrangement of a receiver of the invention is described below for the special case corresponding to an embodiment of relays in which the systematic bits are punctured at the relays.

(116) 1/ The detector MAPD-EI receives as input the multiplexed extrinsic information of the systematic and parity bits respectively from the two sources. Thus, it receives E.sub.1=[E.sub.S.sub.1+.sub.1.sup.1E.sub.S.sub.1.sub.R.sub.1+.sub.2.sup.1E.sub.S.sub.1.sub.R.sub.2, E.sub.p.sub.1.sup.1] and E.sub.2[E.sub.S.sub.2+.sub.1.sup.1E.sub.S.sub.2.sub.R.sub.1+.sub.2.sup.1E.sub.S.sub.2.sub.R.sub.2, E.sub.p2.sup.1]. As output it supplies the LLRs of each of the bit sequences received from the two sources [L.sub.s.sub.1,L.sub.p.sub.1.sup.1] and [L.sub.s.sub.2,L.sub.p.sub.2.sup.1].

(117) 2/ The detector MAPD-EII receives as input the extrinsic information on the parity bits from the two relays E.sub.pR1 and E.sub.pR2 together with the extrinsic information on the parity bits for the second stage of transmission from the two sources E.sub.p1.sup.2 and E.sub.p2.sup.2. It outputs the LLRs of these sequences L.sub.pR1, L.sub.pR2, L.sub.p1.sup.2 and L.sub.p2.sup.2.

(118) 3/ The decoders SISO1 and SISO2 receive as input the LLRs of the parity bits relating to the first stage and to the second stage of transmission from the two sources coming respectively from the detector MAPD-EI and from the detector MAPD-EII, together with the LLRs of the systematic bits from the detector MAPD-EI and the extrinsic information about the respective systematic bits from the algebraic decoders SISO Alg1 and SISO Alg2. This extrinsic information constitutes a priori information about the systematic bits from the two sources. Thus, SISO1 receives L.sub.p.sub.1.sup.1 and L.sub.s.sub.1 from the detector MAPD-EI and L.sub.p.sub.1.sup.2 from the detector MAPD-EII and as a priori information about the systematic bits it receives: .sub.1.sup.1E.sub.S.sub.1.sub.R.sub.1+.sub.2.sup.1E.sub.S.sub.1.sub.R.sub.2 from the algebraic decoders. It calculates the LLRs: .sub.S.sub.1 and .sub.p.sub.1, and from them together with L.sub.s.sub.1, L.sub.p.sub.1.sup.1, and L.sub.p.sub.1.sup.2 it deduces the extrinsic information E.sub.s.sub.1, E.sub.p.sub.1.sup.1, and E.sub.p.sub.1.sup.2 that it delivers as output. SISO2 is symmetrical with SISO1.

(119) 4/ The SISO Alg1 is then activated. As input it receives the extrinsic information about the systematic bits from the sources S.sub.1 and S.sub.2 supplied by the two channel decoders SISO1 and SISO2 together with the extrinsic information calculated at the preceding iteration by the decoder SISO Alg2: .sub.1(E.sub.S.sub.1+.sub.2.sup.1.Math.E.sub.s.sub.1.sub.R.sub.2) and .sub.1(E.sub.S.sub.2+.sub.2.sup.1.Math.E.sub.s.sub.2.sub.R.sub.2). It also receives as input the a priori information about the systematic bits generated by the multi-user detector MAPD-EI: .sub.1.Math.L.sub.S.sub.1 and .sub.1.Math.L.sub.S.sub.2. It also receives as input the LLRs L.sub.pR1 of the parity bits coming from the first relay R.sub.1 supplied by the detector MAPD-EII. The decoder outputs the extrinsic information E.sub.pR.sub.1 about the parity bits from the relay R.sub.1 together with the extrinsic information about the systematic bits from the sources S.sub.1 and S.sub.2:E.sub.s.sub.1.sub.R.sub.1 and E.sub.s.sub.2.sub.R.sub.1. The decoders SISO Alg2 and SISO Alg1 are symmetrical in their operation, but each of them contains a decoder that is specific to their respective algebraic operations. The provision of the extrinsic information E.sub.pR.sub.2 about the parity bits of the relay R.sub.2 and also the extrinsic information E.sub.s.sub.1.sub.R.sub.2 and E.sub.s.sub.2.sub.R.sub.2 about the systematic bits from the sources S.sub.1 and S.sub.2 by the decoder SISO Alg2 terminates the first iteration.

(120) The return to detection step 1/ marks the beginning of the following iteration.

(121) At the end of the iterative process, the method takes from the outputs of the decoders and of the detectors, firstly L.sub.s1+E.sub.s.sub.1+.sub.1.sup.1E.sub.S.sub.1.sub.R.sub.1+.sub.2.sup.1E.sub.S.sub.1.sub.R.sub.20 in order to proceed with taking a decision about .sub.1, and secondly L.sub.s2+E.sub.s.sub.2+.sub.1.sup.1E.sub.S.sub.2.sub.R.sub.1+.sub.2.sup.1E.sub.S.sub.2.sub.R.sub.20 in order to proceed with taking a decision about .sub.2. The sign means that if the output is positive then the decision is that .sub.1 or .sub.2 is one, and if the output is negative then the decision is that .sub.1 or .sub.2 is zero.

(122) The following examples serve to illustrate how the destination receiver uses the information transmitted in-band by the relays. This information specifies which messages have been detected without error by the relays. On the basis of this information, the receiver disconnects or does not disconnect an algebraic decoder of a SISO decoder associated with a source, or it deactivates an algebraic decoder.

(123) In the first example shown in FIG. 13: one of the two relays, the relay R.sub.2 sends only of the source signals, the signal u.sub.2. Under such circumstances, the algebraic SISO decoder SISO alg2 corresponding to the relay R.sub.2 is disconnected from the SISO decoder SISO1 of the non-coded source S.sub.1.

(124) A second situation is shown in FIG. 14: each relay R.sub.1, R.sub.2 sends only one source signal. The SISOi is then disconnected from the decoder SISO Algj where i is the index of the source S.sub.i that is not coded by the relay R.sub.j.

(125) A third situation is shown in FIG. 15: one of the two relays, the relay R.sub.1, does not send any source signal. Under such circumstances, the algebraic SISO decoder SISO alg1 corresponding to the relay R.sub.1 is deactivated.

APPENDIX A

(126) A.1 Field F.sub.4

(127) The primitive polynomial at the origin of the elements of this field is X.sup.2+X+1. Let and .sup.2 be the two roots of this polynomial, and use the notations 0 and 1 respectively for the neutral elements respectively of addition and multiplication for this field. Decimal notation is used for these elements: {0, 1, 2, 3} where 2 and 3 represent and .sup.2, respectively.

(128) TABLE-US-00001 TABLE 1 (+) 0 1 2 3 () 0 1 2 3 0 0 1 2 3 0 0 0 0 0 1 1 0 3 2 1 0 1 2 3 2 2 3 0 1 2 0 2 3 1 3 3 2 1 0 3 0 3 1 2
a) Addition table for F.sub.4 b) Multiplication table for F.sub.4

(129) Let x.sub.1, x.sub.2, and y be three elements of F.sub.4 such that: x.sub.1+2.Math.x.sub.2=y. The transformations that enable x.sub.1 and x.sub.2 to be determined are given by:

(130) 0$\{\begin{array}{c}{x}_1=y+2.Math.x_2\\ x_2=3.Math.y+3.Math.x_1\end{array}$
The notation S.sub.n,m,i is used for the set of pairs (k,l) of F.sub.4 that are solutions to the equation given by n.Math.k+m.Math.l=i. Thus:
S.sub.n,m,i={(k,l)(F.sub.4).sup.2, n.Math.k+m.Math.l=i}
The notation .sub.n,m,i is used for the function defined by:

(131) $4^{}{4}^{}.fwdarw.$$\left(x,y\right).fwdarw.\mathrm{Log}\left(\frac{\underset{\left(k,l\right)S_{n,m,i}}{.Math.}\mathrm{Exp}\left(x_k+y_l\right)}{\underset{\left(k,l\right)S_{n,m,o}}{.Math.}\mathrm{Exp}\left(x_k+y_l\right)}\right)$
A.2 Algebraic Magnitudes
A.2.1. LLR

(132) LLR is the logarithm of the likelihood ratio or probability ratio relating to two symbols.

(133) In the binary version, a bit u may be 1 or 0. Such a ratio is given by a scalar:

(134) $\mathrm{LLR}=\log \left(\frac{P\left(u=1\right)}{P\left(u=0\right)}\right)$
and its sign defines in unique manner the hard decision about the bit u.

(135) In the algebraic version, e.g. in F.sub.4, a symbol u may have four different values. A scalar LLR cannot designate the symbol u in unique manner. We therefore introduce a vector LLR of size four defined as follows:

(136) $q{F}_4,{\mathrm{LLR}}_q\left(u\right)=\log \left(\frac{P\left(u=q\right)}{P\left(u=0\right)}\right)$

(137) The first component is zero since u=0 is considered to be the reference event relative to which the likelihoods of the other events are calculated.

(138) A.2.2. LAPPR

(139) The log a posteriori probability ratio (LAPPR) is a form of LLR that is conditional on side information that is typically provided by a received sequence.

(140) In the binary version, LAPPR is defined by:

(141) $\mathrm{LAPPR}=\log \left(\frac{P\left(u=1|y\right)}{P\left(u=0|y\right)}\right)$

(142) in the algebraic version, e.g. in F.sub.4, use is made of a vector LAPPR of size four defined as follows:

(143) $q{F}_4,{\mathrm{LAPPR}}_q\left(u\right)=\log \left(\frac{P\left(u=q|y\right)}{P\left(u=0|y\right)}\right)$ [1] M. Xiao, M. Skoglund, Design of Network Codes for Multiple Users Multiple Relays Wireless Networks, IEEE International Symposium on Information Theory, 2009. [2] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, Optimal Decoding of Linear Codes for minimizing symbol error rate, IEEE Transactions on Information Theory, vol. IT 20(2), pp. 284-287, March 1974