Abstract:
This method for carrying out the flexible demodulation of data modulated following an amplitude modulation in quadrature is used in a communications system employing a code division multiple access technique (CDMA) and an optical frequency division multiplexing (OFDM) using non-binary spread codes. The invention includes steps for determining a simplified expression specific to the logarithm of the likelihood ratio.

Description:
PRIORITY CLAIM 
       [0001]    This application is a U.S. nationalization of PCT Application No. PCT/FR2006/002519, filed Nov. 14, 2006, and claims priority to French Patent Application No. 0511613, filed November 16, 2005. 
     
    
     TECHNICAL FIELD  
       [0002]    The present invention relates to a method and a device for soft demodulation (or “soft demapping”) in an OFDM-CDMA system (“Orthogonal Frequency Division Multiplexing—Code Division Multiple Access”). More precisely, the subject of the invention is the definition of several Logarithms of the Likelihood Ratio (LLR) for an OFDM-CDMA system using nonbinary spreading codes. 
       BACKGROUND 
       [0003]    The LLR is used by data decoding algorithms working with a so-called “soft” input, that is to say one that is not reduced to the two “hard” values “0” and “1”. The LLR measures the probability that the bit input to the decoder is a “0” or a “1”. It consists in calculating a soft value for each of the bits inside a complex symbol modulated according to a quadrature amplitude modulation (QAM), doing so independently for the bits of one and the same QAM symbol received. The principle is to demodulate the signal received into one or more soft bit(s) whose sign corresponds to the bit which would be provided by a hard-decision detector and whose absolute value indicates the reliability of the decision of the I/Q demodulation module (or “I/Q demapping module”). 
         [0004]    As shown in  FIG. 1 , in a conventional OFDM-CDMA transmitter with channel coding, a binary information stream d(k) is generated for each user k. Each binary stream is then coded with the aid of a channel coder  10   1 ,  10   2 , . . . , 10   K , K being the total number of users. The types of channel coders that are able to benefit from the present invention are all those whose corresponding decoder at the receiver level uses soft values as input. Thus, the block performing the channel coding may consist for example:
       of a convolutional coder, of a puncturer and of a bit interleaver (the puncturing device makes it possible to obtain the coding rate desired for the application whereas the interleaver avoids error packets on reception), or   of a turbocoder, of a puncturer and of a bit interleaver, or   of a coder using block codes with soft-input decoder, of the LDPC (“Low-Density Parity-Check Code”) type.       
 
         [0008]    The coded binary stream b(k) is then transformed into a series of complex M-QAM symbols a(k), by means of a quadrature amplitude modulator  12   1 ,  12   2 , . . . ,  12   K . The integer M corresponds to the number of complex symbols or points in the constellation associated with the modulation. Throughout what follows, it is assumed that M is a power of 2, i.e. M=2 m  with m a strictly positive integer. One has for example a 2-QAM modulation, also called BPSK, or a 4-QAM modulation, also called QPSK, or a 8-QAM modulation, etc. If m=2u with u a strictly positive integer, one speaks of square QAM constellation (for example 4-QAM, 16-QAM, 64-QAM, etc.) . The 2-QAM modulation (BPSK) using only the in-phase pathway (I pathway), the formulae developed, subsequently in the description, for the quadrature pathway (Q pathway) are to be disregarded for a 2-QAM. 
         [0009]    The symbol a(k)=a 1 (k)+j.a Q (k) corresponds to the complex M-QAM symbol of the user k and {b 1,1  . . . b 1,q  . . . b 1,mI  b Q,1  . . . b Q,q  . . . b Q,mQ } is the corresponding coded binary sequence (“0” or “1”) in the case of a square constellation. The index q corresponds to the q th  bit of the in-phase part of the signal (I pathway) and to the q th  bit of the quadrature part of the signal (Q pathway), and mI and mQ are the number of bits on the I pathway and on the Q pathway, respectively. 
         [0010]    The M-QAM symbols are then spread in a module  14   1 ,  14   2 , . . . ,  14   K  by spreading codes {c k,1 }, 1≦k≦K and 1≦l≦N (N being the number of sub-carriers) specific to each user k, and are then summed in a module  16 . They then undergo a serial-parallel conversion in a module  18 , followed by an inverse fast Fourier transformation (IFFT) in a module  20 . It is assumed that the transmitter separates the OFDM-CDMA symbols with a guard interval (GI) that is long enough to remove the inter-symbol interference (ISI). This guard interval can be a prefix either cyclic or zero, the two techniques allowing simple scalar equalization in the frequency domain. The symbols obtained are sent on a transmission channel. 
         [0011]      FIG. 2  presents the structure of a conventional receiver corresponding to the transmitter illustrated in  FIG. 1 . 
         [0012]    The signal received from the transmission channel is first demodulated with the aid of a module  22  which carries out a synchronization between the clocks of the sender and receiver, removes the guard interval and applies a fast Fourier transformation (FFT) to the signal received. 
         [0013]    Using vector and matrix notation, it is possible to write the signal received, after removing the guard interval and applying the FFT: 
         [0000]      r=H·C√{square root over (Pa+n b )}  (1) 
         [0000]    where n is the noise vector of dimension N×1, N being the number of sub-carriers, n b  containing the samples of complex Gaussian noise iid (independent and identically distributed) of variance σ 2 , H is the diagonal matrix of dimension NXN representing the channel where the diagonal coefficient h(i)=h(z=e j2Πi/N ) (where 
         [0000]    
       
         
           
             
               
                 h 
                  
                 
                   ( 
                   z 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                  
                 
                   
                     g 
                     n 
                   
                   · 
                   
                     z 
                     
                       - 
                       n 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    g n  being the value of the n th  sample of the impulse response of the propagation channel) represents the coefficient of the channel for the i th  sub-carrier, P is a diagonal matrix of dimension K×K containing the powers applied to each of the spreading codes, a and C are respectively the vector of symbols transmitted of dimension N×1 and the matrix of spreading codes of dimension N×K, where the k th  column (1≦k≦K) represents the code {c k,1 } of user k. 
         [0014]    Thereafter, the frequency response of the channel is estimated for each sub-carrier, in a module  24 . These estimates Ĥ are used jointly with the spreading codes so as to carry out a linear equalization of the symbols received, in a module  26 . 
         [0015]    In the case of a mono-user detection, a linear equalizer g k =(g k.1 , . . . , g k,N ) T  followed by a despreading by the code of user k is applied to the vector received r so as to provide an estimate of the transmitted symbol a(k) for user k: 
         [0000]      y k =c k   H g k   H r where c k ={c k,l }, 1&lt;k&lt;K, 1&lt;l&lt;N   (2) 
         [0016]    The purpose of the mono- or multi-user equalizer is to reshape the signal received so that it corresponds as closely as possible to the points of the reference constellation. Nevertheless, in the presence of noise (thermal and multiple access), the points retrieved do not coincide exactly with the initial constellation. This is why, after equalization, a soft I/Q demodulation is carried out, in a module  28 , before decoding the signal in a channel decoding module  30 . 
         [0017]    The I/Q demodulation operation consists in retrieving the binary values from the complex symbols arising from the linear detector. When using a channel decoder with soft inputs, the optimal values to be injected into the channel decoder are soft values, that is to say ones that are not directly hard values “0” and “1”. 
         [0018]    Thus, if a 16-QAM modulation (4 bits) is used, the soft I/Q demodulation (“soft demapping”) will consist in calculating 4 soft values corresponding to the 4 bits of the 16-QAM modulation. The optimal soft values (or metrics) to be injected into the channel decoder correspond to a likelihood ratio logarithm (LLR). 
         [0019]    The LLR associated with bit b I,q  may be written: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           I 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ln 
                      
                     
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             
                               S 
                               
                                 I 
                                 , 
                                 q 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                          
                         
                           Pr 
                            
                           
                             [ 
                             
                               
                                 a 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               = 
                               
                                 λ 
                                  
                                 r 
                               
                             
                             ] 
                           
                         
                       
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             
                               S 
                               
                                 I 
                                 , 
                                 q 
                               
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                          
                         
                           Pr 
                            
                           
                             [ 
                             
                               
                                 a 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               = 
                               
                                 λ 
                                  
                                 r 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the sign Pr denotes probability and:
       b I,q  corresponds to a user k and ought to be written b I,q,k  but for simplicity the index k is omitted in the remainder of the description,   for bit b I,q , the QAM constellation is split into two complementary partitions of complex symbols, respectively containing the          Symbols with a “0” in the position (I,q) and containing the symbols          with a “1” in the position (I,q)a The same procedure is applicable to the bit for the quadrature part b Q,q .   the λ are the reference symbols of the QAM constellation. For example, λ={1+i, 1−i, −1+i, −1−i} for the 4-QAM constellation, well known to the person skilled in the art.       
 
         [0023]    Using Bayes&#39; theorem and making the assumption that the symbols transmitted are distributed equiprobably, the LLR can be expressed as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           I 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ln 
                      
                     
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             
                               S 
                               
                                 I 
                                 , 
                                 q 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                          
                         
                           Pr 
                            
                           
                             [ 
                             
                               
                                 r 
                                  
                                 
                                   a 
                                    
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                               = 
                               λ 
                             
                             ] 
                           
                         
                       
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             
                               S 
                               
                                 I 
                                 , 
                                 q 
                               
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                          
                         
                           Pr 
                            
                           
                             [ 
                             
                               
                                 r 
                                  
                                 
                                   a 
                                    
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                               = 
                               λ 
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0024]    In equation (4), the numerator of the logarithm sums the probabilities for all the symbols having a “1” for the bit in position q and the denominator sums the probabilities for all the symbols having a “0” for the bit in position q. These probabilities are exponentially decreasing functions of the Euclidian distance between the symbols received and the reference symbols λ. 
         [0025]    This distance is weighted by a term which will be detailed later and which forms the subject of the invention. The result is a soft value indicating the degree of confidence with respect to a given bit b I,q , where a positive value indicates a binary “1” while a negative value indicates a binary “0”. 
         [0026]    After the soft demodulation operation is the decoding process which carries out the dual operations of those performed by the coder on transmission. If bit puncturing and interleaving operations are present on transmission, this being the case if convolutional codes or turbocodes are employed, the bit deinterleaving followed by depuncturing operations are therefore carried out after the soft I/Q demodulation. The channel decoding makes it possible finally to retrieve the binary data sent. For example, in the case where a convolutional coder is used on transmission, the channel decoding uses the branch metric calculation, which calls upon the LLR given by equation (4). The branch metric for the state s(i) of the path i at the instant t may be written: 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                     t 
                     
                       s 
                        
                       
                         ( 
                         i 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       M 
                       
                         t 
                         - 
                         1 
                       
                       
                         
                           s 
                           ′ 
                         
                          
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         ( 
                         
                           
                             x 
                             
                               t 
                               , 
                               n 
                             
                             
                               s 
                                
                               
                                 ( 
                                 i 
                                 ) 
                               
                             
                           
                            
                           
                             q 
                             
                               t 
                               , 
                               n 
                             
                           
                            
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 n 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where M t−1   s′ (i) represents the previous branch metric, x t,n   s (i) represents a theoretical transmitted coded bit (according to a trellis diagram) for a coder of rate 1/N, q t,n  represents the puncturing pattern and LLR(b n ) represents the LLR of the coded bit transmitted. 
         [0027]    A calculation of the LLR commonly applied for an MC-CDMA system (“Multi-Carrier—Code Division Multiple Access”) with binary orthogonal spreading codes is known, obtained from equation (4) above. 
         [0028]    The i th  symbol of complex data received for user k after a mono-user detection is expressed in the following form, if it is considered that the data item was transmitted and spread on N sub-carriers: 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                     k 
                   
                   = 
                   
                     
                       
                         a 
                         k 
                       
                        
                       
                         
                           p 
                           k 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                                
                               
                                 c 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                                
                             
                             2 
                           
                            
                           
                             g 
                             
                               k 
                               , 
                               l 
                             
                           
                            
                           
                             h 
                             l 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             i 
                             = 
                             0 
                           
                           
                             i 
                             ≠ 
                             k 
                           
                         
                         
                           K 
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             p 
                             i 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               g 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                              
                             
                               h 
                               l 
                             
                              
                             
                               c 
                               
                                 k 
                                 , 
                                 l 
                               
                               * 
                             
                              
                             
                               c 
                               
                                 i 
                                 , 
                                 l 
                               
                             
                              
                             
                               a 
                               i 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           g 
                           
                             k 
                             , 
                             l 
                           
                         
                          
                         
                           c 
                           
                             k 
                             , 
                             l 
                           
                           * 
                         
                          
                         
                           n 
                           l 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where P k  and P i  are parameters representative of the powers applied respectively to the k th  and to the i th  users, c k,l  is the value of the spreading code for the l th  sub-carrier and the user k, g k,l  is the value of linear equalization coefficient associated with the l th  sub-carrier and with the user k, h l  is the coefficient of the transmission channel for the l th  sub-carrier, n l  is the noise associated with the l th  sub-carrier and the sign * denotes the complex conjugate. 
         [0029]    If Walsh-Hadamard spreading codes are considered, these being real orthogonal codes taking the binary values c k,l =±1, we obtain: 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                     k 
                   
                   = 
                   
                     
                       
                         a 
                         k 
                       
                        
                       
                         
                           p 
                           k 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             g 
                             
                               k 
                               , 
                               l 
                             
                           
                            
                           
                             h 
                             l 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             i 
                             = 
                             0 
                           
                           
                             i 
                             ≠ 
                             k 
                           
                         
                         
                           K 
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             p 
                             i 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               g 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                              
                             
                               h 
                               l 
                             
                              
                             
                               c 
                               
                                 i 
                                 , 
                                 l 
                               
                             
                              
                             
                               c 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                              
                             
                               a 
                               i 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           g 
                           
                             k 
                             , 
                             l 
                           
                         
                          
                         
                           c 
                           
                             k 
                             , 
                             l 
                           
                         
                          
                         
                           n 
                           l 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0030]    A known way of obtaining the exact expression for the LLR of binary spreading codes is now described. 
         [0031]    When a frequency interleaver is applied, the complex coefficients h l  of the channel affecting the data symbols ak may be considered to be independent. Thus, for sufficiently long spreading codes (N≧8), the noise and complex multiple access interference (MAI) terms may be approximated by complex Gaussian additive noise (according to the central limit theorem) of zero mean and with variance: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           σ 
                           MAI 
                           2 
                         
                         ≈ 
                           
                          
                         
                           2 
                            
                           
                             ( 
                             
                               K 
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 E 
                                  
                                 
                                   [ 
                                   
                                     
                                        
                                       
                                         
                                           g 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                         
                                           h 
                                           l 
                                         
                                       
                                        
                                     
                                     2 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 
                                    
                                   
                                     E 
                                      
                                     
                                       [ 
                                       
                                         
                                           g 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                         
                                           h 
                                           l 
                                         
                                       
                                       ] 
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           and 
                            
                           
                               
                           
                            
                           
                             σ 
                             noise 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           N 
                            
                           
                               
                           
                            
                           
                             σ 
                             2 
                           
                            
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                    
                                   
                                     g 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                    
                                 
                                 2 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           σ 
                           MAI 
                           2 
                         
                         ≈ 
                           
                          
                         
                           2 
                            
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     i 
                                     ≠ 
                                     k 
                                   
                                 
                                 
                                   K 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 p 
                                 i 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 E 
                                  
                                 
                                   [ 
                                   
                                     
                                        
                                       
                                         
                                           g 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                         
                                           h 
                                           l 
                                         
                                       
                                        
                                     
                                     2 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 
                                   
                                      
                                     
                                       E 
                                        
                                       
                                         [ 
                                         
                                           
                                             g 
                                             
                                               k 
                                               , 
                                               l 
                                             
                                           
                                            
                                           
                                             h 
                                             l 
                                           
                                         
                                         ] 
                                       
                                     
                                      
                                   
                                   2 
                                 
                                  
                                 
                                     
                                 
                                  
                                 and 
                                  
                                 
                                     
                                 
                                  
                                 
                                   σ 
                                   noise 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           N 
                            
                           
                               
                           
                            
                           
                             σ 
                             2 
                           
                            
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                    
                                   
                                     g 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                    
                                 
                                 2 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0032]    Equation (8) corresponds to the case where all the users have an identical power and equation (9) corresponds to the generalization to the case where the users have different powers. 
         [0033]    The law of large numbers makes it possible to evaluate the mathematical expectations by replacing the expectations by the empirical mean of the terms considered. Consequently, if N≧8, the variances for the noise and MAI terms may be formulated as follows, respectively for identical powers and for different powers: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           σ 
                           MAI 
                           2 
                         
                         ≈ 
                           
                          
                         
                           2 
                            
                           
                             ( 
                             
                               K 
                               - 
                               1 
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   1 
                                   N 
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       l 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                        
                                       
                                         
                                           g 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             ^ 
                                           
                                           l 
                                         
                                       
                                        
                                     
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   1 
                                   
                                     N 
                                     2 
                                   
                                 
                                  
                                 
                                   
                                      
                                     
                                       
                                         ∑ 
                                         
                                           l 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                         
                                           g 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             ^ 
                                           
                                           l 
                                         
                                       
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           and 
                            
                           
                               
                           
                            
                           
                             σ 
                             noise 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           
                             σ 
                             2 
                           
                            
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                  
                                 
                                   g 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           σ 
                           MAI 
                           2 
                         
                         ≈ 
                           
                          
                         
                           2 
                            
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     i 
                                     ≠ 
                                     k 
                                   
                                 
                                 
                                   K 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 p 
                                 i 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   1 
                                   N 
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       l 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                        
                                       
                                         
                                           g 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             ^ 
                                           
                                           l 
                                         
                                       
                                        
                                     
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   1 
                                   
                                     N 
                                     2 
                                   
                                 
                                  
                                 
                                   
                                      
                                     
                                       
                                         ∑ 
                                         
                                           l 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                         
                                           g 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                         
                                           
                                             h 
                                             ^ 
                                           
                                           l 
                                         
                                       
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           and 
                            
                           
                               
                           
                            
                           
                             σ 
                             noise 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           
                             σ 
                             2 
                           
                            
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                  
                                 
                                   g 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ĥ l  is the estimate of h l  originating from the channel estimation module  24 . 
         [0034]    The LLR to be used as input to the channel decoder in the case of a M-QAM modulation may then be written, using equation (4): 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           I 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ln 
                      
                     
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             
                               S 
                               
                                 I 
                                 , 
                                 q 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                          
                         
                           exp 
                           ( 
                           
                             
                               - 
                               
                                 1 
                                 2 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                       y 
                                       k 
                                     
                                     - 
                                     
                                       λ 
                                        
                                       
                                         
                                           p 
                                           k 
                                         
                                       
                                        
                                       
                                         
                                           ∑ 
                                           
                                             l 
                                             = 
                                             0 
                                           
                                           
                                             N 
                                             - 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             g 
                                             
                                               k 
                                               , 
                                               l 
                                             
                                           
                                            
                                           
                                             
                                               h 
                                               ^ 
                                             
                                             l 
                                           
                                         
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                               
                                 
                                   σ 
                                   noise 
                                   2 
                                 
                                 + 
                                 
                                   σ 
                                   MAI 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             
                               S 
                               
                                 I 
                                 , 
                                 q 
                               
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                          
                         
                           exp 
                           ( 
                           
                             
                               - 
                               
                                 1 
                                 2 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                       y 
                                       k 
                                     
                                     - 
                                     
                                       λ 
                                        
                                       
                                         
                                           p 
                                           k 
                                         
                                       
                                        
                                       
                                         
                                           ∑ 
                                           
                                             l 
                                             = 
                                             0 
                                           
                                           
                                             N 
                                             - 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             g 
                                             
                                               k 
                                               , 
                                               l 
                                             
                                           
                                            
                                           
                                             
                                               h 
                                               ^ 
                                             
                                             l 
                                           
                                         
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                               
                                 
                                   σ 
                                   noise 
                                   2 
                                 
                                 + 
                                 
                                   σ 
                                   MAI 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0035]    The same relation applies for the bits of the quadrature pathway. 
         [0036]    From equation (12) may be obtained the exact formula for a 4-QAM modulation generalized to the case of different powers: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           I 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         4 
                          
                         
                           
                             p 
                             k 
                           
                         
                          
                         
                            
                           
                             
                               ∑ 
                               
                                 I 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 g 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                             
                           
                            
                         
                       
                       
                         
                           σ 
                           MAI 
                           2 
                         
                         + 
                         
                           σ 
                           noise 
                           2 
                         
                       
                     
                      
                     
                       y 
                       
                         k 
                         , 
                         I 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where y k,I  corresponds to the real part of the complex symbol received after equalization and despreading. The same relation applies for the imaginary part. 
         [0037]    It is also known how to obtain simplified versions of the LLR. 
         [0038]    For 4-QAM, for large spreading factors, the calculation performed in equation (13) may be simplified as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           I 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   ≈ 
                   
                     4 
                      
                     
                       
                         
                           p 
                           k 
                         
                       
                       
                         σ 
                         2 
                       
                     
                      
                     
                       y 
                       
                         k 
                         , 
                         I 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0039]    Indeed, for large spreading factors, the weighting term y k,I  becomes almost constant and therefore has no effect on the soft-input decoding process. 
         [0040]    For small spreading factors, the simplification through equation (14) gives markedly worse results than for large spreading factors, because the central limit theorem is no longer satisfied. An approach giving results similar to those of equation (13) even using an MMSE (“Minimum Mean Square Error”) mono-user equalization consists in weighting the symbol received in the following manner: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           I 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   ≈ 
                   
                     
                       ( 
                       
                         
                           4 
                            
                           
                             
                               
                                 p 
                                 k 
                               
                             
                             
                               σ 
                               2 
                             
                           
                            
                           
                             1 
                             N 
                           
                            
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                         
                          
                         
                             
                         
                         | 
                         
                           
                             h 
                             ^ 
                           
                           l 
                         
                         | 
                       
                       ) 
                     
                      
                     
                       y 
                       
                         k 
                         , 
                         I 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0041]    Useful reference may be made in this regard to the article by Stefan KAISER entitled “Trade-off between Channel Coding and Spreading in Multi-Carrier CDMA Systems” published in Proceedings IEEE Fourth International Symposium on Spread Spectrum Techniques &amp; Applications (ISSSTA&#39;96), Mainz, Germany, pages 1366 to 1370, September 1996. 
         [0042]    In practice, formula (15) is preferable to equation (13) when using a mono-user MMSE equalizer, since the computational complexity is reduced without any significant loss of performance. However, it should be noted that equation (15) gives markedly worse results than equation (13) for MRC (“Maximum Ratio Combining”) and ZF (“Zero Forcing”) equalizers, but not for the EGC (“Equal Gain Combining”) equalizer. 
         [0043]    It is also known how to obtain a simplified expression for the LLR for square modulations of higher order with no CSI (“Channel State Information”). 
         [0044]    In the exact equation (13) or simplified equation (15), the state of the channel is present in the LLR by virtue of the terms using the values h l  or g l . These terms represent the CSI. 
         [0045]    In an article entitled “Simplified Soft-Output Demapper for Binary Interleaved COFDM with Application to HIPERLAN/2” published in IEEE International Conference on Communications ICC 2002, pages 664 to 668, volume 2, F. TOSATO and P. BISAGLIA have proposed a simplification for calculating the LLR for square M-QAM constellations for the COFDM (Coded OFDM) systems only, with no CSI. 
         [0046]    If y k =y k,I +i-y k,Q  denotes the complex QAM symbol received and equalized for user k, the following LLRs are obtained if the CSI are ignored: 
         [0047]    For a 16-QAM with I={b I,1 , b I,2 } and Q={b Q,1 , b Q,2 }: 
         [0048]    for the part I: 
         [0049]    LLR(b I,1 )=y k.I    
         [0050]    LLR(b I,2 )=−|y k,I |+2 for the part Q: 
         [0051]    LLR(b Q,1 )=y k,Q    
         [0052]    LLR(b Q,2 )=−|y k,Q |+2 for a 64-QAM with 
         [0000]      I={b I,1 , b I,2 , b I,3 } and Q={b Q,1 , b Q,2 , b Q,3 }:   (16) for the part I: 
         [0053]    LLR(b I,1 )=y k,I    
         [0054]    LLR(b I,2 )=−|y k,I |+4 
         [0055]    LLR(b I,3 )=−||y k,I |−4|+2 for the part Q: 
         [0056]    LLR(b Q,1 )=y k,Q    
         [0057]    LLR(b Q,2 )=−|y k,Q |+4 
         [0058]    LLR(b Q,3 )=−|y k,Q |−4|+2 
         [0059]    A generic formulation applied to the COFDM for square M-QAM modulations (M&gt;4) is also given in the aforesaid article by F. TOSATO and P. BISAGLIA, namely (the same relations apply for the quadrature part): 
         [0000]      LLR (b I,q )=M I,q    (17) 
         [0000]    where: 
         [0000]      M I,q =y k,I  for q=1 
         [0000]      M I,q =−|M I,q−1 |+m I,q  for q&gt;1   (18) 
         [0000]    where m I,q  corresponds to half the distance between the boundaries of the partition relating to b I,q . For example, for the 16-QAM modulation, M I,2 =2 since the distance between the two boundaries equals 4. 
         [0060]    The procedures presented in the prior art for calculating optimal LLRs or simplified versions of the LLRs have, on the one hand, the drawback of relying on the assumption that the CDMA spreading codes are binary sequences of +1 and −1 and, moreover, these procedures may lead to degradations in the performance of OFDM-CDMA transmission systems using nonbinary codes. 
       SUMMARY 
       [0061]    The aim of the invention is to remedy the drawbacks of the prior art, by optimizing the metric to be injected as input to a channel decoder in an OFDM-CDMA system using nonbinary orthogonal spreading codes. 
         [0062]    The invention finds an application in particular in the field of telecommunications, for communication systems utilizing the OFDM-CDMA technique, such as for example the multi-band OFDM Alliance system. 
         [0063]    With this aim, the present invention proposes a method of soft demodulation of data modulated according to a square quadrature amplitude modulation of order greater than or equal to 4, in a communication system implementing a multi-carrier code division multiple access or OFDM-CDMA technique, using non-binary spreading codes, remarkable in that it comprises steps consisting in determining:
       a simplified expression for the logarithm of the likelihood ratio LLR(b I/q ) for the in-phase pathway for the user k according to the equation:       
 
         [0000]    
       
         
           
             
               LLR 
                
               
                 ( 
                 
                   b 
                   
                     I 
                     , 
                     q 
                   
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     4 
                      
                     
                       
                         
                           p 
                           k 
                         
                       
                       
                         σ 
                         2 
                       
                     
                      
                     
                       1 
                       N 
                     
                      
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   | 
                   
                     c 
                     
                       k 
                       , 
                       l 
                     
                   
                    
                   
                     | 
                     2 
                   
                   | 
                   
                     
                       h 
                       ^ 
                     
                     l 
                   
                   | 
                 
                 ) 
               
               · 
               
                 M 
                 
                   I 
                   , 
                   q 
                 
               
             
           
         
       
     
         [0000]    where
       M I,q =y k,I  for q=1   M I,q =−|M I,q−1 |+m I,q  for q&gt;1
 
where m I,q  corresponds to half the distance between the boundaries of the partition relating to b I,q  and where:
       
 
         [0067]    p k  is a parameter representative of the power applied to the k th  user,
       σ 2 is the variance of the noise,   N is the number of sub-carriers,   l is an index (positive integer) representative of the sub-carrier (1≦l≦N),   c k,l  is the value of the spreading code for the l th  sub-carrier and user k,   ĥ l  is the estimate of the coefficient of the transmission channel for the l th  sub-carrier,   y k,i  corresponds to the real part of the complex symbol after equalization and despreading intended to be demodulated, and   a simplified expression for the logarithm of the likelihood ratio LLR(b Q,q ) for the quadrature pathway for the user k in accordance with the equation:       
 
         [0000]    
       
         
           
             
               LLR 
                
               
                 ( 
                 
                   b 
                   
                     Q 
                     , 
                     q 
                   
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     4 
                      
                     
                       
                         
                           p 
                           k 
                         
                       
                       
                         σ 
                         2 
                       
                     
                      
                     
                       1 
                       N 
                     
                      
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   | 
                   
                     c 
                     
                       k 
                       , 
                       l 
                     
                   
                    
                   
                     | 
                     2 
                   
                   | 
                   
                     
                       h 
                       ^ 
                     
                     l 
                   
                   | 
                 
                 ) 
               
               · 
               
                 M 
                 
                   Q 
                   , 
                   q 
                 
               
             
           
         
       
     
         [0000]    where
       M Q,q =y k,Q  for q=1   M Q,q =−|M Q,q−1 |+m Q,q  for q&gt;1
 
where m Q,q  corresponds to half the distance between the boundaries of the partition relating to b Q,q  and where:
   y k,Q  corresponds to the imaginary part of the complex symbol after equalization and despreading intended to be demodulated.       
 
         [0078]    The simplified metric proposed is particularly appropriate in the case of short spreading factors, which gives a sub-optimal decoding but which guarantees low complexity of implementation for a negligible loss of performance. 
         [0079]    When the data are modulated according to a 4-QAM modulation, the simplified expression for the logarithm of the likelihood ratio LLR(b I/q ) for the bits b I,q  of the in-phase pathway for the user k is given by the following equation: 
         [0000]    
       
         
           
             
               LLR 
                
               
                 ( 
                 
                   b 
                   
                     I 
                     , 
                     q 
                   
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     4 
                      
                     
                       
                         
                           p 
                           k 
                         
                       
                       
                         σ 
                         2 
                       
                     
                      
                     
                       1 
                       N 
                     
                      
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   | 
                   
                     c 
                     
                       k 
                       , 
                       l 
                     
                   
                    
                   
                     | 
                     2 
                   
                   | 
                   
                     
                       h 
                       ^ 
                     
                     l 
                   
                   | 
                 
                 ) 
               
                
               
                 y 
                 
                   k 
                   , 
                   I 
                 
               
             
           
         
       
     
         [0000]    where y k,I  corresponds to the real part of the complex symbol received after equalization and despreading, and the simplified expression for the logarithm of the likelihood ratio LLR(b Q,q ) for the bits b Q,q  of the quadrature pathway for the user k is given by the following equation: 
         [0000]    
       
         
           
             
               LLR 
                
               
                 ( 
                 
                   b 
                   
                     Q 
                     , 
                     q 
                   
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     4 
                      
                     
                       
                         
                           p 
                           k 
                         
                       
                       
                         σ 
                         2 
                       
                     
                      
                     
                       1 
                       N 
                     
                      
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   | 
                   
                     c 
                     
                       k 
                       , 
                       l 
                     
                   
                    
                   
                     | 
                     2 
                   
                   | 
                   
                     
                       h 
                       ^ 
                     
                     l 
                   
                   | 
                 
                 ) 
               
                
               
                 y 
                 
                   k 
                   , 
                   Q 
                 
               
             
           
         
       
     
         [0000]    where y k,Q  corresponds to the imaginary part of the complex symbol received after equalization and despreading. 
         [0080]    This also makes it possible to obtain an appreciable improvement in the binary error rate of the system. 
         [0081]    The present invention finds a favored application within the framework of the standard proposed by the MBOA (Multi-Band OFDM Alliance) consortium. 
         [0082]    In a particular embodiment such as this, the logarithm of the likelihood ratio of the bits of the in-phase and quadrature pathways for the bits b m(n) , b m(n)+50 , b m(n)+1 , b m(n)+51  is given by the following equations: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       LLR 
                        
                       
                         ( 
                         
                           b 
                           
                             m 
                              
                             
                               ( 
                               n 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       LLR 
                        
                       
                         ( 
                         
                           b 
                           
                             
                               m 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                             + 
                             50 
                           
                         
                         ) 
                       
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   
                     
                       
                         4 
                          
                         
                           g 
                           n 
                         
                          
                         
                           
                             h 
                             ^ 
                           
                           n 
                         
                       
                       + 
                       
                         
                           g 
                           
                             n 
                             + 
                             50 
                           
                         
                          
                         
                           
                             h 
                             ^ 
                           
                           
                             n 
                             + 
                             50 
                           
                         
                       
                     
                     
                       
                         2 
                          
                         
                           
                             ( 
                             
                               
                                 
                                   g 
                                   n 
                                 
                                  
                                 
                                   
                                     h 
                                     ^ 
                                   
                                   n 
                                 
                               
                               - 
                               
                                 
                                   g 
                                   
                                     n 
                                     + 
                                     50 
                                   
                                 
                                  
                                 
                                   
                                     h 
                                     ^ 
                                   
                                   
                                     n 
                                     + 
                                     50 
                                   
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       + 
                       
                         
                           5 
                         
                          
                         
                           
                             
                               σ 
                               ^ 
                             
                             2 
                           
                            
                           
                             ( 
                             
                               4 
                               | 
                               
                                 g 
                                 n 
                               
                                
                               
                                 | 
                                 2 
                               
                                
                               
                                 + 
                                 
                                   | 
                                   
                                     g 
                                     
                                       n 
                                       + 
                                       50 
                                     
                                   
                                    
                                   
                                     | 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           
                             y 
                             
                               n 
                               , 
                               I 
                             
                           
                         
                       
                       
                         
                           
                             y 
                             
                               n 
                               , 
                               Q 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   
 
                 
                 [ 
                 
                   
                     
                       
                         LLR 
                          
                         
                           ( 
                           
                             b 
                             
                               
                                 m 
                                  
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                               + 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         LLR 
                          
                         
                           ( 
                           
                             b 
                             
                               
                                 m 
                                  
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                               + 
                               51 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
               = 
               
                 
                   
                     
                       
                         g 
                         n 
                       
                        
                       
                         
                           h 
                           ^ 
                         
                         n 
                       
                     
                     + 
                     
                       4 
                        
                       
                         g 
                         
                           n 
                           + 
                           50 
                         
                       
                        
                       
                         
                           h 
                           ^ 
                         
                         
                           n 
                           + 
                           50 
                         
                       
                     
                   
                   
                     
                       2 
                        
                       
                         
                           ( 
                           
                             
                               
                                 g 
                                 n 
                               
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 n 
                               
                             
                             - 
                             
                               
                                 g 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       
                         5 
                       
                        
                       
                         
                           
                             σ 
                             ^ 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             | 
                             
                               g 
                               n 
                             
                              
                             
                               | 
                               2 
                             
                              
                             
                               + 
                               4 
                             
                             | 
                             
                               g 
                               
                                 n 
                                 + 
                                 50 
                               
                             
                              
                             
                               | 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           y 
                           
                             
                               n 
                               + 
                               50 
                             
                             , 
                             I 
                           
                         
                       
                     
                     
                       
                         
                           y 
                           
                             
                               n 
                               + 
                               50 
                             
                             , 
                             Q 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    where:
       m(n)=2n for n=0, 1, 2, . . . , 24 and m(n)=2n+50 for n=25, 26, . . . , 49,   y n,l  and y n+50,l  are the real parts of the complex symbols received after equalization and despreading corresponding respectively to the two complex 4-QAM symbols transmitted,   y n,Q  and y n+50,Q  are the imaginary parts of the complex symbols received after equalization and despreading corresponding respectively to the two complex 4-QAM symbols transmitted,   g n  and g n+50  are the two equalization coefficients employed,   ĥ n  and ĥ n+50  are the estimates of the values of the frequency response of the transmission channel on the two sub-carriers, and   {circumflex over (σ)} 2  is the estimate of the variance of the noise.       
 
         [0089]    The simplified expression for the likelihood ratio of the bits of the in-phase and quadrature pathways for the user k is given by the following equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 
                                   m 
                                    
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 
                                   
                                     m 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   + 
                                   50 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       ( 
                       
                         4 
                         | 
                         
                           
                             h 
                             ^ 
                           
                           n 
                         
                         | 
                         
                           + 
                           
                             | 
                             
                               
                                 h 
                                 ^ 
                               
                               
                                 n 
                                 + 
                                 50 
                               
                             
                             | 
                           
                         
                       
                       ) 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               y 
                               
                                 n 
                                 , 
                                 I 
                               
                             
                           
                         
                         
                           
                             
                               y 
                               
                                 n 
                                 , 
                                 Q 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 
                                   
                                     m 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   + 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 
                                   
                                     m 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   + 
                                   51 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       ( 
                       
                         | 
                         
                           
                             h 
                             ^ 
                           
                           n 
                         
                         | 
                         
                           + 
                           4 
                         
                         | 
                         
                           
                             h 
                             ^ 
                           
                           
                             n 
                             + 
                             50 
                           
                         
                         | 
                       
                       ) 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               y 
                               
                                 
                                   n 
                                   + 
                                   50 
                                 
                                 , 
                                 I 
                               
                             
                           
                         
                         
                           
                             
                               y 
                               
                                 
                                   n 
                                   + 
                                   50 
                                 
                                 , 
                                 Q 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0000]    where: 
         [0090]    m(n)=2n for n=0, 1, 2, . . . , 24 and m(n)=2n+50 for n=25, 26, . . . , 49,
       y n,l  and y n+50,l  are the real parts of the complex symbols received after equalization and despreading corresponding respectively to the two complex 4-QAM symbols transmitted,   y n,Q  and y n+50,Q  are the imaginary parts of the complex symbols received after equalization and despreading corresponding respectively to the two complex 4-QAM symbols transmitted, and   ĥ n  and ĥ n+50  are the estimates of the values of the frequency response of the transmission channel on the two sub-carriers.       
 
         [0094]    With the same aim as that indicated above, the present invention also proposes a device of soft demodulation of data modulated according to a square quadrature amplitude modulation of order greater than or equal to 4, in a communication system implementing a multi-carrier code division multiple access or OFDM-CDMA technique, using non-binary spreading codes, remarkable in that it comprises means for determining:
       a simplified expression for the logarithm of the likelihood ratio LLR(b I,q ) for the in-phase pathway for the user k according to the equation:       
 
         [0000]    
       
         
           
             
               LLR 
                
               
                 ( 
                 
                   b 
                   
                     I 
                     , 
                     q 
                   
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     4 
                      
                     
                       
                         
                           p 
                           k 
                         
                       
                       
                         σ 
                         2 
                       
                     
                      
                     
                       1 
                       N 
                     
                      
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   | 
                   
                     c 
                     
                       k 
                       , 
                       l 
                     
                   
                    
                   
                     | 
                     2 
                   
                   | 
                   
                     
                       h 
                       ^ 
                     
                     l 
                   
                   | 
                 
                 ) 
               
               · 
               
                 M 
                 
                   I 
                   , 
                   q 
                 
               
             
           
         
       
     
         [0000]    where
       M I,q =y k,I  for q=1   M I,q =−|M I,q−1 |+m I,q  for q&gt;1
 
where m I,q  corresponds to half the distance between the boundaries of the partition relating to b I,q  and where:
   p k  is a parameter representative of the power applied to the k th  user,   σ 2 is the variance of the noise,   N is the number of sub-carriers,   l is an index (positive integer) representative of the sub-carrier (1≦l≦N),   c k,l  is the value of the spreading code for the l th  sub-carrier and the user k,   ĥ l  is the estimate of the coefficient of the transmission channel for the l th  sub-carrier,   y k,I  corresponds to the real part of the complex symbol after equalization and despreading intended to be demodulated, and   a simplified expression for the logarithm of the likelihood ratio LLR(b Q,q ) for the quadrature pathway for the user k in accordance with the equation:       
 
         [0000]    
       
         
           
             
               LLR 
                
               
                 ( 
                 
                   b 
                   
                     Q 
                     , 
                     q 
                   
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       
                         4 
                          
                         
                           
                             p 
                             k 
                           
                         
                       
                       
                         σ 
                         2 
                       
                     
                      
                     
                       1 
                       N 
                     
                      
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   | 
                   
                     c 
                     
                       k 
                       , 
                       l 
                     
                   
                    
                   
                     | 
                     2 
                   
                   | 
                   
                     
                       h 
                       ^ 
                     
                     l 
                   
                   | 
                 
                 ) 
               
               · 
               
                 M 
                 
                   Q 
                   , 
                   q 
                 
               
             
           
         
       
     
         [0000]    where
       M Q,q =y k,Q  for q=1   M Q,q =−|M Q,q−1 |+m Q,q  for q&gt;1
 
where m Q,q  corresponds to half the distance between the boundaries of the partition relating to b Q,q  and where:
   y k,Q  corresponds to the imaginary part of the complex symbol after equalization and despreading intended to be demodulated.       
 
         [0109]    The invention is also directed at a receiver adapted for implementing a method as described above. 
         [0110]    The invention is also directed at a receiver comprising a device such as above. 
         [0111]    The particular characteristics and the advantages of the device and of the receiver being similar to those of the method, they are not repeated here. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS  
         [0112]    Other aspects and advantages of the invention will become apparent on reading the detailed description which follows of particular embodiments, given by way of nonlimiting examples. The description refers to the drawings which accompany it, in which: 
           [0113]      FIG. 1 , already described, represents diagrammatically an OFDM-CDMA transmitter with channel coding of conventional type; 
           [0114]      FIG. 2 , already described, represents diagrammatically a receiver of conventional type corresponding to a transmitter of the type illustrated in  FIG. 1 ; and 
           [0115]      FIG. 3  represents diagrammatically a part of a receiver capable of implementing a method in accordance with the present invention, in a particular embodiment. 
       
    
    
     DETAILED DESCRIPTION  
       [0116]    Throughout what follows, consideration is given to an OFDM-CDMA transmitter using nonbinary orthogonal spreading codes. 
         [0117]    It is recalled that the logarithm of the likelihood ratio (LLR) makes it possible to go from complex symbols (I pathway and Q pathway) originating from an equalizer, such as the equalization/despreading module  32  represented in  FIG. 3 , to real values (one per bit) indicating the reliability of the bit received before entering a channel decoder such as the channel decoder  36 . 
         [0118]    The LLR therefore carries out what is called a soft I/Q demodulation operation  340 , in a soft I/Q demodulation module  34 . 
         [0119]    This demodulation operation intervenes between the channel equalization/despreading and decoding processes. The soft I/Q demodulation process, or calculation of soft decisions on input to the decoder, uses the data arising from the channel estimation, performed by a channel estimation module  38 , and from the equalization performed by the module  32 . 
         [0120]    The output of the soft I/Q demodulation module  34  corresponds to the LLR injected as input to the channel decoder  36 . 
         [0121]    In accordance with the present invention, the same calculations as those presented in the introduction may be conducted as far as equation (6) describing the ith complex data symbol received for the user k after a mono-user detection: 
         [0000]    
       
         
           
             
               y 
               k 
             
             = 
             
               
                 
                   a 
                   k 
                 
                  
                 
                   
                     p 
                     k 
                   
                 
                  
                 
                   ∑ 
                   
                     l 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
               
                
               
                   
               
               | 
               
                 c 
                 
                   k 
                   , 
                   l 
                 
               
                
               
                 | 
                 2 
               
                
               
                 
                   
                     g 
                     
                       k 
                       , 
                       l 
                     
                   
                    
                   
                     h 
                     l 
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       
                         i 
                         = 
                         0 
                       
                        
                       
                         
 
                       
                        
                       
                         i 
                         ≠ 
                         k 
                       
                     
                     
                       K 
                       - 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         p 
                         i 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           g 
                           
                             k 
                             , 
                             l 
                           
                         
                          
                         
                           h 
                           l 
                         
                          
                         
                           c 
                           
                             k 
                             , 
                             l 
                           
                           * 
                         
                          
                         
                           c 
                           
                             i 
                             , 
                             l 
                           
                         
                          
                         
                           a 
                           i 
                         
                       
                     
                   
                 
                 + 
                 
                   
                     ∑ 
                     
                       l 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       g 
                       
                         k 
                         , 
                         l 
                       
                     
                      
                     
                       c 
                       
                         k 
                         , 
                         l 
                       
                       * 
                     
                      
                     
                       n 
                       l 
                     
                   
                 
               
             
           
         
       
     
         [0122]    A nonlimiting example of nonbinary real and orthogonal spreading codes is given by the following code matrix: 
         [0000]    
       
         
           
             
               
                 
                   
                     1 
                     
                       5 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           2 
                         
                         
                           1 
                         
                       
                       
                         
                           1 
                         
                         
                           
                             - 
                             2 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0123]    When a frequency interleaver is applied, the complex coefficients h l  of the channel affecting the data symbols ak may be considered to be independent. Thus, for sufficiently long spreading codes (where the number of sub-carriers N is greater than or equal to 8), the noise and complex multiple access interference (MAI) terms may be approximated by complex Gaussian additive noise (according to the central limit theorem) of zero mean and with variance: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             σ 
                             MAI 
                             2 
                           
                           ≈ 
                           
                             
                               2 
                                
                               
                                 ( 
                                 
                                   K 
                                   - 
                                   1 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     E 
                                      
                                     
                                       [ 
                                       
                                         | 
                                         
                                           
                                             c 
                                             
                                               k 
                                               , 
                                               l 
                                             
                                             * 
                                           
                                            
                                           
                                             c 
                                             
                                               i 
                                               , 
                                               l 
                                             
                                           
                                            
                                           
                                             g 
                                             
                                               k 
                                               , 
                                               l 
                                             
                                           
                                            
                                           
                                             h 
                                             l 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                    
                                   
                                     | 
                                     2 
                                   
                                 
                                 ] 
                               
                             
                             - 
                           
                         
                         | 
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 c 
                                 
                                   k 
                                   , 
                                   l 
                                 
                                 * 
                               
                                
                               
                                 c 
                                 
                                   i 
                                   , 
                                   l 
                                 
                               
                                
                               
                                 g 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                                
                               
                                 h 
                                 l 
                               
                             
                             ] 
                           
                         
                          
                         
                           | 
                           2 
                         
                       
                       ) 
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       σ 
                       noise 
                       2 
                     
                   
                   ≈ 
                   
                       
                   
                    
                   
                     N 
                      
                     
                         
                     
                      
                     
                       σ 
                       2 
                     
                      
                     
                       E 
                        
                       
                         [ 
                         
                           | 
                           
                             
                               c 
                               
                                 k 
                                 , 
                                 l 
                               
                               * 
                             
                              
                             
                               g 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                           
                            
                           
                             | 
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       σ 
                       MAI 
                       2 
                     
                     ≈ 
                     
                       2 
                        
                       
                         ( 
                         
                           
                             ∑ 
                             
                               
                                 i 
                                 = 
                                 0 
                               
                                
                               
                                 
 
                               
                                
                               
                                 i 
                                 ≠ 
                                 k 
                               
                             
                             
                               K 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             p 
                             i 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               E 
                               [ 
                               
                                 | 
                                 
                                   
                                     c 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                     * 
                                   
                                    
                                   
                                     c 
                                     
                                       i 
                                       , 
                                       l 
                                     
                                   
                                    
                                   
                                     g 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                    
                                   
                                     h 
                                     l 
                                   
                                 
                                  
                                 
                                   | 
                                   2 
                                 
                               
                               ] 
                             
                             - 
                           
                           | 
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                   * 
                                 
                                  
                                 
                                   c 
                                   
                                     i 
                                     , 
                                     l 
                                   
                                 
                                  
                                 
                                   g 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                                 
                                   h 
                                   l 
                                 
                               
                               ] 
                             
                           
                            
                           
                             | 
                             2 
                           
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       and 
                        
                       
                           
                       
                        
                       
                         σ 
                         noise 
                         2 
                       
                     
                     ≈ 
                     
                       N 
                        
                       
                           
                       
                        
                       
                         σ 
                         2 
                       
                        
                       
                         E 
                          
                         
                           [ 
                           
                             | 
                             
                               
                                 c 
                                 
                                   k 
                                   , 
                                   l 
                                 
                                 * 
                               
                                
                               
                                 g 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                             
                              
                             
                               | 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0124]    Equation (20) corresponds to the case where all the users have an identical power and equation (21) corresponds to the generalization to the case where the users have different powers. 
         [0125]    The law of large numbers makes it possible to evaluate the mathematical expectations by replacing the expectations by the empirical mean of the terms considered. Consequently, if N≧8, the variances for the noise and MAI terms may be formulated as follows, respectively for identical powers (equation (22)) and for different powers (equation (23)): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       σ 
                       MAI 
                       2 
                     
                     ≈ 
                     
                       2 
                        
                       
                         ( 
                         
                           K 
                           - 
                           1 
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                               N 
                             
                              
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                           
                            
                           
                               
                           
                           | 
                           
                             
                               c 
                               
                                 k 
                                 , 
                                 l 
                               
                               * 
                             
                              
                             
                               c 
                               
                                 i 
                                 , 
                                 l 
                               
                             
                              
                             
                               g 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                              
                             
                               
                                 h 
                                 ^ 
                               
                               l 
                             
                           
                            
                           
                             | 
                             2 
                           
                            
                           
                             - 
                             
                               1 
                               
                                 N 
                                 2 
                               
                             
                           
                           | 
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 c 
                                 
                                   k 
                                   , 
                                   l 
                                 
                                 * 
                               
                                
                               
                                 c 
                                 
                                   i 
                                   , 
                                   l 
                                 
                               
                                
                               
                                 g 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                             
                           
                            
                           
                             | 
                             2 
                           
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       and 
                        
                       
                           
                       
                        
                       
                         σ 
                         noise 
                         2 
                       
                     
                     ≈ 
                     
                       
                         σ 
                         2 
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           c 
                           
                             k 
                             , 
                             l 
                           
                           * 
                         
                       
                     
                   
                   | 
                   
                     g 
                     
                       k 
                       , 
                       l 
                     
                   
                    
                   
                     | 
                     2 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       σ 
                       MAI 
                       2 
                     
                     ≈ 
                     
                       2 
                        
                       
                         ( 
                         
                           
                             ∑ 
                             
                               
                                 i 
                                 = 
                                 0 
                               
                                
                               
                                 
 
                               
                                
                               
                                 i 
                                 ≠ 
                                 k 
                               
                             
                             
                               K 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             p 
                             i 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                               N 
                             
                              
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                           
                            
                           
                               
                           
                           | 
                           
                             
                               c 
                               
                                 k 
                                 , 
                                 l 
                               
                               * 
                             
                              
                             
                               c 
                               
                                 i 
                                 , 
                                 l 
                               
                             
                              
                             
                               g 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                              
                             
                               
                                 h 
                                 ^ 
                               
                               l 
                             
                           
                            
                           
                             | 
                             2 
                           
                            
                           
                             - 
                             
                               1 
                               
                                 N 
                                 2 
                               
                             
                           
                           | 
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 c 
                                 
                                   k 
                                   , 
                                   l 
                                 
                                 * 
                               
                                
                               
                                 c 
                                 
                                   i 
                                   , 
                                   l 
                                 
                               
                                
                               
                                 g 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                             
                           
                            
                           
                             | 
                             2 
                           
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       and 
                        
                       
                           
                       
                        
                       
                         σ 
                         noise 
                         2 
                       
                     
                     ≈ 
                     
                       
                         σ 
                         2 
                       
                        
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                     
                   
                    
                   
                       
                   
                   | 
                   
                     
                       c 
                       
                         k 
                         , 
                         l 
                       
                       * 
                     
                      
                     
                       g 
                       
                         k 
                         , 
                         l 
                       
                     
                   
                    
                   
                     | 
                     2 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ĥ is the estimate of h l  originating from the channel estimation module. 
         [0126]    The LLR to be used as input to the channel decoder in the case of a M-QAM modulation may then be written, using equation (4) given in the introduction: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           I 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ln 
                      
                     
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             
                               s 
                               
                                 I 
                                 , 
                                 q 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                          
                         
                             
                         
                          
                         
                           exp 
                            
                           
                             ( 
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                                
                               
                                 
                                   
                                     
                                       y 
                                       k 
                                     
                                     - 
                                     
                                       λ 
                                        
                                       
                                         
                                           p 
                                           k 
                                         
                                       
                                        
                                       
                                         ∑ 
                                         
                                           l 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                   | 
                                   
                                     c 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                    
                                   
                                     | 
                                     2 
                                   
                                    
                                   
                                     
                                       g 
                                       
                                         k 
                                         , 
                                         l 
                                       
                                     
                                      
                                     
                                       
                                         
                                           h 
                                           ^ 
                                         
                                          
                                         
                                             
                                         
                                       
                                       l 
                                     
                                   
                                    
                                   
                                     | 
                                     2 
                                   
                                 
                                 
                                   
                                     σ 
                                     noise 
                                     2 
                                   
                                   + 
                                   
                                     σ 
                                     MAI 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           ∑ 
                           
                             λ 
                             ∈ 
                             
                               s 
                               
                                 I 
                                 , 
                                 q 
                               
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                          
                         
                           exp 
                            
                           
                             ( 
                             
                               
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                                
                               
                                 
                                   | 
                                   
                                     
                                       y 
                                       k 
                                     
                                     - 
                                     
                                       λ 
                                        
                                       
                                         
                                           p 
                                           k 
                                         
                                       
                                        
                                       
                                         ∑ 
                                         
                                           l 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                     
                                   
                                    
                                   
                                       
                                   
                                   | 
                                   
                                     c 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                    
                                   
                                     | 
                                     2 
                                   
                                    
                                   
                                     
                                       g 
                                       
                                         k 
                                         , 
                                         l 
                                       
                                     
                                      
                                     
                                       
                                         h 
                                         ^ 
                                       
                                       l 
                                     
                                   
                                    
                                   
                                     | 
                                     2 
                                   
                                 
                                 
                                   
                                     σ 
                                     noise 
                                     2 
                                   
                                   + 
                                   
                                     σ 
                                     MAI 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where:
       λ is a reference symbol of the constellation associated with the quadrature amplitude modulation,   S I,q   (0)  and S I,q   (1)  are two complementary partitions of complex symbols containing respectively the symbols of the constellation with a “0” in position (I, q) and the symbols of the constellation with a “1” in position (I, q),   y k  corresponds to the complex symbol after equalization and despreading intended to be demodulated,   p k  is a parameter representative of the power applied to the k th  user,   N is the number of sub-carriers,   l is an index (positive integer) representative of the sub-carrier (1≦l≦N),   c k,l  is the value of the spreading code for the l th  sub-carrier and user k,   g k,l  is the value of the linear equalization coefficient associated with the l th  sub-carrier and with the user k,   ĥ l  is the estimate of the coefficient of the transmission channel for the l th  sub-carrier,   σ noise   2  is the variance of the noise, and   σ MAI  is the variance of the complex multiple access interference.       
 
         [0138]    The same relation applies for the quadrature pathway, namely: 
         [0000]    
       
         
           
             
               LLR 
                
               
                 ( 
                 
                   b 
                   
                     Q 
                     , 
                     q 
                   
                 
                 ) 
               
             
             = 
             
               ln 
                
               
                 
                   
                     ∑ 
                     
                       λ 
                       ∈ 
                       
                         S 
                         
                           Q 
                           , 
                           q 
                         
                         
                           ( 
                           1 
                           ) 
                         
                       
                     
                   
                    
                   
                     exp 
                     ( 
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                        
                       
                         
                           
                              
                             
                               
                                 y 
                                 k 
                               
                               - 
                               
                                 λ 
                                  
                                 
                                   
                                     p 
                                     k 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       l 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       
                                          
                                         
                                           c 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                       
                                       2 
                                     
                                      
                                     
                                       g 
                                       
                                         k 
                                         , 
                                         l 
                                       
                                     
                                      
                                     
                                       
                                         h 
                                         ^ 
                                       
                                       l 
                                     
                                   
                                 
                               
                             
                              
                           
                           2 
                         
                         
                           
                             σ 
                             noise 
                             2 
                           
                           + 
                           
                             σ 
                             MAI 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
                 
                   
                     ∑ 
                     
                       λ 
                       ∈ 
                       
                         S 
                         
                           Q 
                           , 
                           q 
                         
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                   
                    
                   
                     exp 
                     ( 
                     
                       
                         - 
                         
                           1 
                           2 
                         
                       
                        
                       
                         
                           
                              
                             
                               
                                 y 
                                 k 
                               
                               - 
                               
                                 λ 
                                  
                                 
                                   
                                     p 
                                     k 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       l 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       
                                          
                                         
                                           c 
                                           
                                             k 
                                             , 
                                             l 
                                           
                                         
                                          
                                       
                                       2 
                                     
                                      
                                     
                                       g 
                                       
                                         k 
                                         , 
                                         l 
                                       
                                     
                                      
                                     
                                       
                                         h 
                                         ^ 
                                       
                                       l 
                                     
                                   
                                 
                               
                             
                              
                           
                           2 
                         
                         
                           
                             σ 
                             noise 
                             2 
                           
                           + 
                           
                             σ 
                             MAI 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where:
       S Q,q   (0)  and S Q,q   (1)  are two complementary partitions of complex symbols containing respectively the symbols of the constellation with a “0” in position (Q, q) and the symbols of the constellation with a “1” in position (Q, q).       
 
         [0140]    From equation (24) can be obtained the exact formula for the LLR for a 4-QAM modulation generalized to the case of different powers: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           l 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         4 
                          
                         
                           
                             p 
                             k 
                           
                         
                          
                         
                            
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     c 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 g 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                             
                           
                            
                         
                       
                       
                         
                           σ 
                           MAI 
                           2 
                         
                         + 
                         
                           σ 
                           noise 
                           2 
                         
                       
                     
                      
                     
                       y 
                       
                         k 
                         , 
                         l 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where y k,I  corresponds to the real part of the complex symbol received after equalization and despreading. The same relation applies for the imaginary part, namely: 
         [0000]    
       
         
           
             
               LLR 
                
               
                 ( 
                 
                   b 
                   
                     Q 
                     , 
                     q 
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   4 
                    
                   
                     
                       p 
                       k 
                     
                   
                    
                   
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                              
                             
                               c 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                              
                           
                           2 
                         
                          
                         
                           g 
                           
                             k 
                             , 
                             l 
                           
                         
                          
                         
                           
                             h 
                             ^ 
                           
                           l 
                         
                       
                     
                      
                   
                 
                 
                   
                     σ 
                     MAI 
                     2 
                   
                   + 
                   
                     σ 
                     noise 
                     2 
                   
                 
               
                
               
                 y 
                 
                   k 
                   , 
                   Q 
                 
               
             
           
         
       
     
         [0000]    where y k,Q  corresponds to the imaginary part of the complex symbol received after equalization and despreading. 
         [0141]    However, the exact formula for the LLR given by equation (24) and even its application to the 4-QAM modulation, given by equation (25), may turn out to be difficult to implement in practice: indeed, the two terms, complex multiple access interference and noise, need to be estimated and, to do this, the receiver must be aware of all the active CDMA codes. Moreover, these equations use the law of large numbers to calculate the mathematical expectations, this possibly no longer being valid when the length of the spreading codes becomes relatively small. 
         [0142]    This is why the present invention also proposes a simplified expression for the LLRs of an OFDM-CDMA system. 
         [0143]    In the case of a 4-QAM modulation, the simplified expression for the LLRs of an OFDM-CDMA system with 4-QAM modulation and with binary codes is weighted to obtain a simplified expression for the LLRs of an OFDM-CDMA system with a 4-QAM modulation and with binary or nonbinary codes: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           l 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         4 
                          
                         
                           
                             p 
                             k 
                           
                         
                       
                       
                         σ 
                         2 
                       
                     
                      
                     
                       1 
                       N 
                     
                      
                     
                       ( 
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                                
                               
                                 c 
                                 
                                   k 
                                   , 
                                   l 
                                 
                               
                                
                             
                             2 
                           
                            
                           
                              
                             
                               
                                 h 
                                 ^ 
                               
                               l 
                             
                              
                           
                         
                       
                       ) 
                     
                      
                     
                       y 
                       
                         k 
                         , 
                         l 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0144]    In practice, the formula (26) is preferable to equation (25) when using a mono-user MMSE equalizer, since the computational complexity is reduced with no significant loss of performance. 
         [0145]    The present invention also proposes a simplified expression for the LLR for square M-QAM constellations, that is to say ones for M=2 2n , where n is a strictly positive integer, for MC-CDMA systems with binary or nonbinary spreading codes. 
         [0146]    If y k =y k,I +i.y k,Q  denotes the complex QAM symbol received and equalized for user k, the following LLRs are obtained: 
         [0147]    For a QAM-16 modulation with I={b I,1 , b I,2 } and Q={b Q,1 , b Q,2 }: 
         [0148]    For the part I: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           l 
                           , 
                           1 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       y 
                       
                         k 
                         , 
                         l 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           l 
                           , 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                           - 
                           
                              
                             
                               y 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                              
                           
                         
                         + 
                         2 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0149]    For the part Q: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           Q 
                           , 
                           1 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       y 
                       
                         k 
                         , 
                         Q 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           Q 
                           , 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                           - 
                           
                              
                             
                               y 
                               
                                 k 
                                 , 
                                 Q 
                               
                             
                              
                           
                         
                         + 
                         2 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0000]      For a 64-QAM modulation with I={b I,1 ,b I,2 ,b I,3 } and Q={b Q,1 ,b Q,2 ,b Q,3 }   (27) 
         [0150]    For the part I: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           l 
                           , 
                           1 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       y 
                       
                         k 
                         , 
                         l 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           l 
                           , 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                           - 
                           
                              
                             
                               y 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                              
                           
                         
                         + 
                         4 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           l 
                           , 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                            
                           
                             
                               - 
                               
                                  
                                 
                                   y 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                             
                             + 
                             4 
                           
                            
                         
                         + 
                         2 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0151]    For the part Q: 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           Q 
                           , 
                           1 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       y 
                       
                         k 
                         , 
                         Q 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           Q 
                           , 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                           - 
                           
                              
                             
                               y 
                               
                                 k 
                                 , 
                                 Q 
                               
                             
                              
                           
                         
                         + 
                         4 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     LLV 
                      
                     
                       ( 
                       
                         b 
                         
                           Q 
                           , 
                           3 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                            
                           
                             
                               - 
                               
                                  
                                 
                                   y 
                                   
                                     k 
                                     , 
                                     Q 
                                   
                                 
                                  
                               
                             
                             + 
                             4 
                           
                            
                         
                         + 
                         2 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0152]    For a square M-QAM modulation for M&gt;4, we obtain (the same relations apply for the quadrature part): 
         [0000]    
       
         
           
             
               
                 
                   
                     LLR 
                      
                     
                       ( 
                       
                         b 
                         
                           l 
                           , 
                           q 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             4 
                              
                             
                               
                                 p 
                                 k 
                               
                             
                           
                           
                             σ 
                             2 
                           
                         
                          
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                  
                                 
                                   c 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                  
                               
                               2 
                             
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 l 
                               
                                
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       M 
                       
                         l 
                         , 
                         q 
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]      M I,q =y k,I  for q=1 
         [0000]      M I,q =−|M I,q−1 |+m I,q  for q&gt;1   (29) 
         [0000]    where m I,q  corresponds to half the distance between the boundaries of the partition relating to b I,q . For example, for the 16-QAM modulation, M I,2 =2, since the distance between the two boundaries equals 4. 
         [0153]    A favored application of the present invention within the framework of the standard proposed by the MBOA (“Multi-Band OFDM Alliance”) consortium is described in what follows. 
         [0154]    The optimal and simplified LLR formulae provided by the present invention apply in fact within the framework of this standard, which uses the OFDM technique and which employs Dual-Carrier Modulation (DCM) for its high bit rate transmission modes. The association of DCM and OFDM is entirely equivalent to an OFDM-CDMA transmitter with nonbinary codes, fully laden, using a 4-QAM modulation and a spreading factor of 2. 
         [0155]    The input bits are first transformed into bipolar symbols, as follows: 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
                   
               
               
                   
                 Input bits b n   
                 Output symbols x n   
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 0 
                 −1 
               
               
                   
                 1 
                 1 
               
               
                   
                   
               
             
          
         
       
     
         [0156]    Next, these bipolar symbols are grouped into 4&#39;s to form 2 complex symbols s, and Snv50, as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             s 
                             n 
                           
                         
                       
                       
                         
                           
                             s 
                             
                               n 
                               + 
                               50 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         1 
                         
                           10 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               2 
                             
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                             
                               
                                 - 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 x 
                                 
                                   m 
                                    
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 jx 
                                 
                                   
                                     m 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   + 
                                   50 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 
                                   
                                     m 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   + 
                                   1 
                                 
                               
                               + 
                               
                                 jx 
                                 
                                   
                                     m 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   + 
                                   51 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where x m(n) , x m(n)+50 , x m(n)+1  and x m(n)+51  are the bipolar symbols corresponding to the bits transmitted b m(n) , b m(n)+50 , b m(n)+1  and b m(n)+51  with: 
         [0000]      m(n)=2n for n=0, 1, 2, . . . , 24 
         [0000]      m(n)=2n+50 for n=25, 26, . . . , 49   (31) 
         [0157]    However, it is possible to view this modulation as 2 symbols a n  and a′ n  originating from a 4-QAM modulation and then used for a CDMA spreading with nonbinary codes of length 2, by rewriting equation (30) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             s 
                             n 
                           
                         
                       
                       
                         
                           
                             s 
                             
                               n 
                               + 
                               50 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         1 
                         
                           5 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               2 
                             
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                             
                               
                                 - 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               a 
                               n 
                             
                           
                         
                         
                           
                             
                               a 
                               n 
                               ′ 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               a 
               n 
             
             = 
             
               
                 1 
                 
                   2 
                 
               
                
               
                 ( 
                 
                   
                     x 
                     
                       m 
                        
                       
                         ( 
                         n 
                         ) 
                       
                     
                   
                   + 
                   
                     jx 
                     
                       
                         m 
                          
                         
                           ( 
                           n 
                           ) 
                         
                       
                       + 
                       50 
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               a 
               n 
               ′ 
             
             = 
             
               
                 1 
                 
                   2 
                 
               
                
               
                 ( 
                 
                   
                     x 
                     
                       
                         m 
                          
                         
                           ( 
                           n 
                           ) 
                         
                       
                       + 
                       1 
                     
                   
                   + 
                   
                     jx 
                     
                       
                         m 
                          
                         
                           ( 
                           n 
                           ) 
                         
                       
                       + 
                       51 
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    are the energy-normalized 4-QAM complex symbols, and 
         [0000]    
       
         
           
             
               1 
               
                 5 
               
             
              
             
               [ 
               
                 
                   
                     2 
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     
                       - 
                       2 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    is the CDMA matrix with nonbinary spreading codes with a spreading factor of 2. 
         [0158]    Let ĥ n  and ĥ n+50  be the estimates of the values of the frequency response of the channel on the 2 sub-carriers modulated respectively by s n  and S n+50  and let g n  and g n+50  be respectively the 2 equalization coefficients employed. 
         [0159]    Let y n =y n+I +jy n,Q  and y n+50 =y n+50,I +jy n+50, Q  be the complex symbols received after equalization and despreading corresponding respectively to the 4-QAM complex symbols transmitted a n  and a′ n.    
         [0160]    To obtain the expression for the optimal LLR, it is possible to recalculate equation (6) within the MBOA framework, noting that the powers p k  are all equal (given that the 2 codes are transmitted at the same power) for y n  and y n+50 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       y 
                       n 
                     
                     = 
                     
                       
                         
                           1 
                           5 
                         
                          
                         
                           ( 
                           
                             
                               4 
                                
                               
                                 g 
                                 n 
                               
                                
                               
                                 h 
                                 n 
                               
                             
                             + 
                             
                               
                                 g 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                               
                                 h 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           a 
                           n 
                         
                       
                       + 
                       
                         
                           2 
                           5 
                         
                          
                         
                           ( 
                           
                             
                               
                                 g 
                                 n 
                               
                                
                               
                                 h 
                                 n 
                               
                             
                             - 
                             
                               
                                 g 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                               
                                 h 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           a 
                           n 
                           ′ 
                         
                       
                       + 
                       
                         
                           1 
                           
                             5 
                           
                         
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 g 
                                 n 
                               
                                
                               
                                 n 
                                 n 
                               
                             
                             + 
                             
                               
                                 g 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                               
                                 n 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       n 
                       ′ 
                     
                     = 
                     
                       
                         
                           1 
                           5 
                         
                          
                         
                           ( 
                           
                             
                               
                                 g 
                                 n 
                               
                                
                               
                                 h 
                                 n 
                               
                             
                             + 
                             
                               4 
                                
                               
                                 g 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                               
                                 h 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           a 
                           n 
                           ′ 
                         
                       
                       + 
                       
                         
                           2 
                           5 
                         
                          
                         
                           ( 
                           
                             
                               
                                 g 
                                 n 
                               
                                
                               
                                 h 
                                 n 
                               
                             
                             - 
                             
                               
                                 g 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                               
                                 h 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           a 
                           n 
                         
                       
                       + 
                       
                         
                           1 
                           
                             5 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 g 
                                 n 
                               
                                
                               
                                 n 
                                 n 
                               
                             
                             - 
                             
                               2 
                                
                               
                                 g 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                               
                                 n 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0161]    Within the framework of the MBOA standard, the CDMA spreading factor is very small, hence the law of large numbers is no longer valid for calculating the mathematical expectations in the two MAI terms of equation (33). One then chooses to take the power of these MAI terms directly. Moreover, whatever equalizer is used (MMSE, MRC, EGC or ZF), the product g n ĥ n  is always real and positive. 
         [0162]    In accordance with the present invention, equation (25) yields the following formula for the optimal LLR: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 
                                   m 
                                    
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 
                                   
                                     m 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   + 
                                   50 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               4 
                                
                               
                                 g 
                                 n 
                               
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 n 
                               
                             
                             + 
                             
                               
                                 g 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               g 
                                               n 
                                             
                                              
                                             
                                               
                                                 h 
                                                 ^ 
                                               
                                               n 
                                             
                                           
                                           - 
                                           
                                             
                                               g 
                                               
                                                 n 
                                                 + 
                                                 50 
                                               
                                             
                                              
                                             
                                               
                                                 h 
                                                 ^ 
                                               
                                               
                                                 n 
                                                 + 
                                                 50 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     5 
                                   
                                    
                                   
                                     
                                       
                                         σ 
                                         ^ 
                                       
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           4 
                                            
                                           
                                             
                                                
                                               
                                                 g 
                                                 n 
                                               
                                                
                                             
                                             2 
                                           
                                         
                                         + 
                                         
                                           
                                              
                                             
                                               g 
                                               
                                                 n 
                                                 + 
                                                 50 
                                               
                                             
                                              
                                           
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   y 
                                   
                                     n 
                                     , 
                                     I 
                                   
                                 
                               
                             
                             
                               
                                 
                                   y 
                                   
                                     n 
                                     , 
                                     Q 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         
 
                       
                       [ 
                       
                         
                           
                             
                               LLR 
                                
                               
                                 ( 
                                 
                                   b 
                                   
                                     
                                       m 
                                        
                                       
                                         ( 
                                         n 
                                         ) 
                                       
                                     
                                     + 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               LLR 
                                
                               
                                 ( 
                                 
                                   b 
                                   
                                     
                                       m 
                                        
                                       
                                         ( 
                                         n 
                                         ) 
                                       
                                     
                                     + 
                                     51 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             
                               g 
                               n 
                             
                              
                             
                               
                                 h 
                                 ^ 
                               
                               n 
                             
                           
                           + 
                           
                             4 
                              
                             
                               g 
                               
                                 n 
                                 + 
                                 50 
                               
                             
                              
                             
                               
                                 h 
                                 ^ 
                               
                               
                                 n 
                                 + 
                                 50 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   2 
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             g 
                                             n 
                                           
                                            
                                           
                                             
                                               h 
                                               ^ 
                                             
                                             n 
                                           
                                         
                                         - 
                                         
                                           
                                             g 
                                             
                                               n 
                                               + 
                                               50 
                                             
                                           
                                            
                                           
                                             
                                               h 
                                               ^ 
                                             
                                             
                                               n 
                                               + 
                                               50 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   5 
                                 
                                  
                                 
                                   
                                     
                                       σ 
                                       ^ 
                                     
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                            
                                           
                                             g 
                                             n 
                                           
                                            
                                         
                                         2 
                                       
                                       + 
                                       
                                         4 
                                          
                                         
                                           
                                              
                                             
                                               g 
                                               
                                                 n 
                                                 + 
                                                 50 
                                               
                                             
                                              
                                           
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 y 
                                 
                                   
                                     n 
                                     + 
                                     50 
                                   
                                   , 
                                   I 
                                 
                               
                             
                           
                           
                             
                               
                                 y 
                                 
                                   
                                     n 
                                     + 
                                     50 
                                   
                                   , 
                                   Q 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0163]    To obtain the simplified LLR, use is made of equation (26) in which the term 
         [0000]    
       
         
           
             
               
                 4 
                  
                 
                   
                     p 
                     k 
                   
                 
               
               
                 σ 
                 2 
               
             
              
             
               1 
               N 
             
           
         
       
     
         [0000]    is constant and therefore has no influence on the channel decoding when the channel coder on transmission is of the convolutional type, this being the case for MBO. This therefore yields the following formulae for the simplified LLRs: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 
                                   m 
                                    
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             LLR 
                              
                             
                               ( 
                               
                                 b 
                                 
                                   
                                     m 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   + 
                                   50 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             4 
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 n 
                               
                                
                             
                           
                           + 
                           
                              
                             
                               
                                 h 
                                 ^ 
                               
                               
                                 n 
                                 + 
                                 50 
                               
                             
                              
                           
                         
                         ) 
                       
                       · 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   y 
                                   
                                     n 
                                     , 
                                     I 
                                   
                                 
                               
                             
                             
                               
                                 
                                   y 
                                   
                                     n 
                                     , 
                                     Q 
                                   
                                 
                               
                             
                           
                           ] 
                         
                          
                         
                           
 
                         
                         [ 
                         
                           
                             
                               
                                 LLR 
                                  
                                 
                                   ( 
                                   
                                     b 
                                     
                                       
                                         m 
                                          
                                         
                                           ( 
                                           n 
                                           ) 
                                         
                                       
                                       + 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 LLR 
                                  
                                 
                                   ( 
                                   
                                     b 
                                     
                                       
                                         m 
                                          
                                         
                                           ( 
                                           n 
                                           ) 
                                         
                                       
                                       + 
                                       51 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                              
                             
                               
                                 h 
                                 ^ 
                               
                               n 
                             
                              
                           
                           + 
                           
                             4 
                              
                             
                                
                               
                                 
                                   h 
                                   ^ 
                                 
                                 
                                   n 
                                   + 
                                   50 
                                 
                               
                                
                             
                           
                         
                         ) 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               
                                 y 
                                 
                                   
                                     n 
                                     + 
                                     50 
                                   
                                   , 
                                   I 
                                 
                               
                             
                           
                           
                             
                               
                                 y 
                                 
                                   
                                     n 
                                     + 
                                     50 
                                   
                                   , 
                                   Q 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
         [0164]    Experience shows that a significant decrease in the binary error rate (BER) is noted when the formula (35) is applied to the soft bits at the input of the Viterbi decoder of a high bit rate MBO system (480 Mbps). The results obtained are still better when the impulse response of the channel is long, that is to say when the variations in the channel frequency response (represented by the coefficients h in the formulae for the LLR) are greater and where the LLRs therefore afford the decoder still more information about reliability.