Abstract:
A receiver for discrete Fourier transform-spread-orthogonal frequency division multiplexing (DFT-S-OFDM) based systems, including a prefilter for received signal codeword(s); and a log-likelihood ratio LLR module responsive to the prefilter; wherein the prefilter includes a pairing and whitening module that based on channel estimates and data rate enables the LLR module to perform either a Serial-In-Serial-Out (SISO) based log likelihood ratio processing of an output from the paring and whitening module or a two-symbol max-log soft output demodulator (MLSD) based log likelihood ratio processing of an output from the pairing and whitening module.

Description:
[0001]    This application claims the benefit of U.S. Provisional Application No. 61/045,298, entitled “Efficient receiver Algorithms for DFT-Spread Spectrum OFDM Systems”, filed on Apr. 16, 2008, the contents of which is incorporated by reference herein. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    The present invention relates generally to wireless communications, and more particularly, to a receiver for DFT-Spread MIMO-OFDM systems. 
         [0003]    Referring to the diagram in  FIG. 1 , each mobile (or source) transmits its signal using the DFT-S-OFDM technique. The destination or base station receives signals from several mobiles possibly overlapping in time and frequency and has to decode the signal of each mobile. 
         [0004]    Discrete Fourier Transform-spread-Orthogonal Frequency Division Multiple Access (DFT-spread-OFDMA or DFT-S-OFDMA) has emerged as the preferred uplink air interface for the next generation cellular systems such as the 3GPP LTE. The main advantage of this multiple access technique is that it results in considerably lower envelope fluctuations in the signal waveform transmitted by each user and consequently lower peak-to-average-power ratio (PAPR) compared to the classical OFDMA technique. A lower PAPR in turn implies a smaller power back off at the user terminal and hence an improved coverage for the cellular system. Another key technology that will be employed in the upcoming cellular systems is the utilization of antenna arrays at the base station (a.k.a Node-B) and possibly at the user equipment (UE). Multiple antennas when used in point-to-point or multipoint-to-point systems have been shown in theory to result insubstantial capacity improvements, provided that the environment is sufficiently rich in multipath components. However, in practice the capacity improvement obtained by using multiple antennas at the UEs in the uplink can be much smaller due to the fact that multiple antennas will have to be accommodated in the limited space available at the UE, which will result in correlated channel responses that are not conducive to high rate communications. Moreover, installing multiple power amplifiers in each UE is currently deemed impractical based on cost considerations by many vendors. 
         [0005]    A promising scheme, also adopted in 3GPP LTE, which circumvents these two issues is the space-division multiple-access (SDMA) scheme which is sometimes referred to as the virtual multiple-input-multiple-output (MIMO) scheme. In SDMA multiple single-antenna users are scheduled over the same frequency and time resource block in order to boost the system throughput. Since different users are geographically separated, their channel responses seen at the base-station antenna array will be independent and hence capable of supporting high rate communications. Henceforth, the DFT-S-OFDM based uplink employing SDMA will be referred to as the DFT-S-OFDM-SDMA uplink. 
         [0006]    In DFT-S-OFDM systems, which encompass both DFT-S-OFDMA and DFT-S-OFDM-SDMA, as a consequence of the DFT spreading operation at the transmitter, the signal arrives at the base-station with substantial intersymbol interference and the received sufficient statistics can be modeled as the channel output of a large MIMO system. The conventional receiver technique involves tone-by-tone single-tap equalization followed by an inverse DFT operation. While such a simple receiver suffices for the single-user case in the low-rate regime when there is enough receive diversity and where the available frequency diversity can be garnered by the underlying outer code, it results in degraded performance at higher rates as well as with SDMA. 
         [0007]    Unfortunately, unlike classical OFDMA, the large dimension of the equivalent MIMO model in DFT-S-OFDMA does not allow us to leverage the sphere decoder which has an exponential complexity in the problem dimension. Furthermore, the stringent complexity constraints in practical systems also rule out the near-optimal MIMO receivers developed for the narrowband channels. Other promising equalizers for the DFT-S-OFDM systems are the decision feedback equalizers (DFE), in particular the hybrid DFE, where the feedforward filter is realized in the frequency domain and the feedback filter is realized in the time domain, and the iterative block DFE with soft decision feedback that has been proposed by others, where even the cancelation is performed in the frequency domain. However, even the DFE whose iterative process does not include decoding the outer code is substantially more complex and has higher latency especially in the SDMA case, than the conventional receiver. 
         [0008]    Accordingly, there is a need for a receiver at the destination or base station that can receive and decode multiple wireless signals overlapping in time and frequency in a manner that overcomes the limitations of the conventional receiver techniques discussed above. 
       SUMMARY OF THE INVENTION 
       [0009]    In accordance with the invention, there is provided a receiver for discrete fourier transform-spread-orthogonal frequency division multiplexing (DFT-S-OFDM) based systems, including a prefilter for received signal codeword(s); and a log-likelihood ratio LLR module responsive to the prefilter; wherein the prefilter includes a pairing and whitening module that based on channel estimates and data rate enables the LLR module to perform either a Serial-In-Serial-Out (SISO) based log likelihood ratio processing of an output from the pairing and whitening module or a two-symbol max-log soft demodulator (MLSD) based log likelihood ratio processing of an output from the pairing and whitening module. In a preferred embodiment, the prefilter further includes a per-tone equalizer for the received signal codeword(s) and an inverse discrete Fourier transform IDFT module responsive to the equalizer, with the pairing and whitening module being responsive to the IDFT. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0010]    These and other advantages of the invention will be apparent to those of ordinary skill in the art by reference to the following detailed description and the accompanying drawings. 
           [0011]      FIG. 1  is a diagram of an exemplary wireless network, with multiple mobile signal sources  11 - 13  transmitting to a destination base-station  10 , in which the inventive receiver can be employed. 
           [0012]      FIG. 2  is diagram of a two-symbol MLSD DFT-S-OFDM receiver in accordance with the invention. 
           [0013]      FIG. 3  is a diagram of the prefilter, shown in the receiver diagram of  FIG. 2 , demodulating a single signal codeword, in accordance with the invention. 
           [0014]      FIG. 4  is a diagram of the prefilter, shown in the receiver diagram of  FIG. 2 , demodulating multiple signal codewords, in accordance with the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0015]    The invention is directed to a more powerful receiver for DFT-spread OFDM systems that includes an efficient linear pre-filter and a two-symbol max-log soft-output demodulator. The proposed inventive receiver can be applied to both single user per resource block (RB) (DFT-S-OFDMA) and multiple users per RB (DFT-S-OFDM-SDMA) systems and it offers significant performance gains over the conventional method, especially in the high-rate regime, with little attendant increase in computational complexity. 
         [0016]    Referring now to  FIG. 2  there is shown an exemplary two-symbol MLSD DFT-S-OFDM-SDMA Receiver employing the inventive pre-filtering processing. Transmitted data symbols are received at the pre-filter processor  14  and then sent to the two-symbol max-log soft-output demodulator (MLSD)  15  which outputs log Likelihood Ratios (LLR) corresponding to the user equipments  16 . The prefilter  14  structure is depicted in  FIG. 3  demodulating a single user signal codeword and demodulating multiple signal codewords in  FIG. 4 . 
         [0017]    For understanding of the invention and the block diagrams of  FIG. 3  and  FIG. 4 , we present the underlying signal analysis to arrive at the inventive signal receiving. Parenthetical numbers referencing particular signal processes are referred to again when discussing corresponding receiver processes. 
         [0018]    We derive a simple receiver for the DFT-S-OFDM-SDMA uplink. For convenience we consider SDMA with two UEs but the receiver can be extended to more than two UEs as well as the DFT-S-OFDMA uplink with only one UE. 
       Receivers for DFT-S-OFDM Based Systems 1-1.4 
     1 DFT-S-OFDM-SDMA Receivers 
       [0019]    We assume that there are two UEs and for the m-th subcarrier (tone) the n R ×1 channel response vector of the k-th UE is h m   (k) ε           n     R    and the DFT-spread symbol is x m   (k) , k=1, 2. The received signal vector on the m-th tone is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    The noise vector n m  is spatially uncorrelated and satisfies E[n m n m   † ]=I. We define s (k) =[s 1   (k) , s 2   (k) , . . . , s M   (k) ] T  for k=1, 2, where {s m   (k) } are QAM symbols normalized to have unit average energy and let x (k) =[x 1   (k) , x 2   (k) , . . . , x M   (k) ] T =Fs (k) , where F is the M×M DFT matrix. We can now write the received signal over all the M tones in the matrix form as 
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       1.1 Conventional Linear MMSE (LMMSE) Receiver 
       [0020]    The linear MMSE estimate of x m   (k) , k=1, 2, based on y m  in (1) is given by 1   
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    Defining {circumflex over (x)} (k) =[{circumflex over (x)} 1   (k) , . . . , {circumflex over (x)} M   (k) ] T  and applying the inverse DFT on {circumflex over (x)} (k) , we obtain 
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         [0000]    Then, (4) can be simplified as
 
where v i   (k)  contains the residual interference and noise, with variance
 
         [0000]                [| v   i   (k) | 2 ]=α (k) (1−α (k) ).  (7) 
       1.2 New SDMA Receiver 
       [0021]    All operations up-to equation (4) are same as the conventional LMMSE receiver. We thus obtain 
         [0000]    
       
         
           
             
               
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                 ^ 
               
               
                 ( 
                 1 
                 ) 
               
             
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                       ^ 
                     
                     
                       ( 
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                   . 
                 
               
             
           
         
       
     
         [0000]    Let us expand ŝ (1) =[ŝ 1   (1) , . . . , ŝ M   (1) ] T  and ŝ (2) =[ŝ 1   (2) , . . . , ŝ M   (2) ] T . Next, form the pairs ŝ m =[ŝ m   (1) , ŝ m   (2) ] T  for 1≦m≦M. 
         [0022]    We will demodulate each one of the M pairs using a two-symbol max-log demodulator. Before that we need to do a “noise-whitening” operation on each of the M pairs. To do this, we determine 
         [0000]    
       
         
           
             C 
             = 
             
               
                 1 
                 M 
               
                
               
                 
                   ∑ 
                   
                     m 
                     = 
                     1 
                   
                   M 
                 
                  
                 
                   
                     
                       ( 
                       
                         I 
                         + 
                         
                           
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                             m 
                             † 
                           
                            
                           
                             H 
                             m 
                           
                         
                       
                       ) 
                     
                     
                       - 
                       1 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0000]    Note that the terms (I+H m   † H m ) −1 , 1≦m≦M are computed in the LMMSE filter so they need not be re-computed. Next, we compute the 2×2 matrix Qε           2×2  using the Cholesky decomposition 
         [0000]        QQ   † =( I−C ) C   (8) 
         [0000]    and then determine 
         [0000]    
       
         
           
             
               
                 z 
                 m 
               
                
               
                 = 
                 △ 
               
                
               
                 
                   Q 
                   
                     - 
                     1 
                   
                 
                  
                 
                   
                     s 
                     ^ 
                   
                   m 
                 
               
             
             , 
           
         
       
     
         [0000]    1≦m≦M. z m ε           2×1  permits the expansion 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       m 
                     
                     = 
                     
                       
                         
                           
                             
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                                 - 
                                 1 
                               
                             
                              
                             
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                                 - 
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                         m 
                       
                     
                   
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                     ≤ 
                     m 
                     ≤ 
                     M 
                   
                   , 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with Tε           2×2 , s m =[s m   (1) , s m   (2) ] T  and          [{hacek over (n)} m {hacek over (n)} m   † ]=I. The two symbols in s m  can now be jointly demodulated using the two-symbol max-log demodulator on z m  for 1≦m≦M.
 
(.) †  denotes the conjugate transpose operator.
 
1.3 New SDMA Receiver with Improved Pairing
 
All operations up-to equation (4) are same as the conventional LMMSE receiver. We thus obtain
 
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         [0000]    Let us expand ŝ (1) =[ŝ 1   (1) , . . . , ŝ M   (1) ] T  and ŝ (2) =[ŝ 1   (2) , . . . , ŝ M   (2) ] T . Suppose we form the pairs ŝ m,q =[ŝ m   (1) , ŝ [m+q]   (2) ] T  for 1≦m≦M and any given q: 0≦q≦M−1 and where [m+q]=(m+q−1)mod(M)+1. Then we determine the matrix X(q) such that 
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                                           / 
                                           M 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     = 
                                     1 
                                   
                                   M 
                                 
                                  
                                 
                                   
                                     h 
                                     k 
                                     
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                        
                                       † 
                                     
                                   
                                    
                                   
                                     R 
                                     k 
                                     
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     h 
                                     k 
                                     
                                       ( 
                                       1 
                                       ) 
                                     
                                   
                                    
                                   
                                     exp 
                                      
                                     
                                       ( 
                                       
                                         j 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             q 
                                              
                                             
                                               ( 
                                               
                                                 k 
                                                 - 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           / 
                                           M 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     = 
                                     1 
                                   
                                   M 
                                 
                                  
                                 
                                   
                                     h 
                                     k 
                                     
                                       
                                         ( 
                                         2 
                                         ) 
                                       
                                        
                                       † 
                                     
                                   
                                    
                                   
                                     R 
                                     k 
                                     
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     h 
                                     k 
                                     
                                       ( 
                                       2 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where R k =I+H k H k   † . Please note that the pairing used in Section 1.2 always uses q=0. Next, we compute the 2×2 matrix Q(q)ε           2×2  using the Cholesky decomposition 
         [0000]        Q ( q ) Q ( q ) \ =( I−X ( q )) X ( q )  (11) 
         [0000]    and then determine 
         [0000]    
       
         
           
             
               
                 z 
                 
                   m 
                   , 
                   q 
                 
               
                
               
                 = 
                 △ 
               
                
               
                 
                   
                     Q 
                      
                     
                       ( 
                       q 
                       ) 
                     
                   
                   
                     - 
                     1 
                   
                 
                  
                 
                   
                     s 
                     ^ 
                   
                   
                     m 
                     , 
                     q 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    1≦m≦M. z m,q ε           2×1  permits the expansion 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       
                         m 
                         , 
                         q 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 Q 
                                  
                                 
                                   ( 
                                   q 
                                   ) 
                                 
                               
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 I 
                                 - 
                                 
                                   X 
                                    
                                   
                                     ( 
                                     q 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           
                              
                             
                               T 
                                
                               
                                 ( 
                                 q 
                                 ) 
                               
                             
                           
                         
                          
                         
                           s 
                           
                             m 
                             , 
                             q 
                           
                         
                       
                       + 
                       
                         
                           n 
                           ⋓ 
                         
                         
                           m 
                           , 
                           q 
                         
                       
                     
                   
                   , 
                   
                     1 
                     ≤ 
                     m 
                     ≤ 
                     M 
                   
                   , 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with T(q)ε           2×2 , s m,q =[s m   (1) , s [m+q]   (2) ] T  and          [{hacek over (n)} m,q {hacek over (n)} m,q   † ]=I. The two symbols in s m,q  can now be jointly demodulated using the two-symbol max-log demodulator on z m,q  for 1≦m≦M. 
         [0023]    To determine the best q (or equivalently the best pair (m, [m+q])) we can use the capacity metric on the model in (12) and determine a suitable {circumflex over (q)} as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           arg 
                            
                           
                               
                           
                            
                           
                             
                               max 
                               
                                 0 
                                 ≤ 
                                 q 
                                 ≤ 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                             
                              
                             
                               det 
                                
                               
                                 ( 
                                 
                                   I 
                                   + 
                                   
                                     
                                       
                                         T 
                                          
                                         
                                           ( 
                                           q 
                                           ) 
                                         
                                       
                                       † 
                                     
                                      
                                     
                                       T 
                                        
                                       
                                         ( 
                                         q 
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         = 
                         
                           arg 
                            
                           
                               
                           
                            
                           
                             
                               max 
                               
                                 0 
                                 ≤ 
                                 q 
                                 ≤ 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                             
                              
                             
                               det 
                                
                               
                                 ( 
                                 
                                   
                                     X 
                                      
                                     
                                       ( 
                                       q 
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           arg 
                            
                           
                               
                           
                            
                           
                             
                               min 
                               
                                 0 
                                 ≤ 
                                 q 
                                 ≤ 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                             
                              
                             
                               
                                 det 
                                  
                                 
                                   ( 
                                   
                                     X 
                                      
                                     
                                       ( 
                                       q 
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, we can equivalently first determine the vector 
         [0000]        r=F[h   1   (1)†   R   1   −1   h   1   (2)   , . . . , h   m   (1)†   R   M   −1   h   M   (2) ] T   (14) 
         [0000]    and compute {circumflex over (q)} as 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     ^ 
                   
                   = 
                   
                     
                       arg 
                        
                       
                           
                       
                        
                       
                         
                           max 
                           
                             1 
                             ≤ 
                             k 
                             ≤ 
                             M 
                           
                         
                          
                         
                           { 
                           
                              
                             
                               r 
                               k 
                             
                              
                           
                           } 
                         
                       
                     
                     - 
                     1. 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    1.4 New OFDMA Receiver with Improved Pairing
 
We only demodulate the symbols of a particular user of interest. Suppose for the m-th subcarrier (tone) the n R ×1 channel response vector of the UE is h m ε           n     R    and the DFT-spread symbol is x m . The received signal vector on the m-th tone is given by
 
         [0000]        y   m   =h   m   x   m   +n   m ,  (16) 
         [0000]    where the noise vector n m  satisfies E[n m n m   † ]=S m . We define R m =h m h m   † +S m  for 1≦m≦M and s=[s 1 , s 2 , . . . , s M ] T , where {s m } are QAM symbols normalized to have unit average energy and let x=[x 1 , x 2 , x M ] T =Fs, where F is the M×M DFT matrix. 
         [0024]    We obtain 
         [0000]      {circumflex over (x)} m =h m   † R m   −1 y m , m=1, . . . , M.  (17) 
         [0000]    Defining {circumflex over (x)}=[{circumflex over (x)} 1 , . . . , {circumflex over (x)} M ] T  and applying the inverse DFT on {circumflex over (x)}, we obtain 
         [0000]    
       
         
           
             
               
                 
                   
                     s 
                     ^ 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               s 
                               ^ 
                             
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             
                               s 
                               ^ 
                             
                             M 
                           
                         
                         ] 
                       
                       T 
                     
                      
                     
                       = 
                       △ 
                     
                      
                     
                       
                         F 
                         † 
                       
                        
                       
                         
                           x 
                           ^ 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0025]    Suppose we form the pair ŝ m,q =[ŝ m , ŝ [m+q] ] T  for any given q: 1≦q≦M−1 and where [m+q]=(m+q−1)mod(M)+1. Please note that the pairing employed in OFDMA before always uses q=1. Then we determine the matrix X(q) such that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                       - 
                       
                         X 
                          
                         
                           ( 
                           q 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         1 
                         M 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   1 
                                 
                                 M 
                               
                                
                               
                                 
                                   h 
                                   k 
                                   † 
                                 
                                  
                                 
                                   R 
                                   k 
                                   
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   h 
                                   k 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         k 
                                         = 
                                         1 
                                       
                                       M 
                                     
                                      
                                     
                                       
                                         h 
                                         k 
                                         † 
                                       
                                        
                                       
                                         R 
                                         k 
                                         
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                         h 
                                         k 
                                       
                                        
                                       exp 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           q 
                                            
                                           
                                             ( 
                                             
                                               k 
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                         
                                         / 
                                         M 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         k 
                                         = 
                                         1 
                                       
                                       M 
                                     
                                      
                                     
                                       
                                         h 
                                         k 
                                         † 
                                       
                                        
                                       
                                         R 
                                         k 
                                         
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                         h 
                                         k 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                        
                                       exp 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           q 
                                            
                                           
                                             ( 
                                             
                                               k 
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                         
                                         / 
                                         M 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   1 
                                 
                                 M 
                               
                                
                               
                                 
                                   h 
                                   k 
                                   † 
                                 
                                  
                                 
                                   R 
                                   k 
                                   
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   h 
                                   k 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Next, we compute the 2×2 matrix Q(q)ε           2×2  using the Cholesky decomposition 
         [0000]        Q ( q ) Q ( q ) \ =( I−X ( q )) X ( q )  (20) 
         [0000]    and then determine 
         [0000]    
       
         
           
             
               
                 z 
                 
                   m 
                   , 
                   q 
                 
               
                
               
                 = 
                 Δ 
               
                
               
                 
                   
                     Q 
                      
                     
                       ( 
                       q 
                       ) 
                     
                   
                   
                     - 
                     1 
                   
                 
                  
                 
                   
                     s 
                     ^ 
                   
                   
                     m 
                     , 
                     q 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    1≦m≦M. z m,q ε           2×1  permits the expansion 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       
                         m 
                         , 
                         q 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 Q 
                                  
                                 
                                   ( 
                                   q 
                                   ) 
                                 
                               
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 I 
                                 - 
                                 
                                   X 
                                    
                                   
                                     ( 
                                     q 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           
                              
                             
                               T 
                                
                               
                                 ( 
                                 q 
                                 ) 
                               
                             
                           
                         
                          
                         
                           s 
                           
                             m 
                             , 
                             q 
                           
                         
                       
                       + 
                       
                         
                           n 
                           ˘ 
                         
                         
                           m 
                           , 
                           q 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with T(q)ε           2×2 , s m,q =[s m , s [m+q] ] T  and          [{hacek over (n)} m,q {hacek over (n)} m,q   † ]=I. The two symbols in s m,q  can now be jointly demodulated using the two-symbol max-log demodulator on z m,q . 
         [0026]    To determine the best q (or equivalently the best pair (m, [m+q])) we can use the capacity metric on the model in (21) and determine a suitable q as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           arg 
                            
                           
                               
                           
                            
                           
                             
                               max 
                               
                                 1 
                                 ≤ 
                                 q 
                                 ≤ 
                                 
                                   M 
                                   / 
                                   2 
                                 
                               
                             
                              
                             
                               det 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   I 
                                   + 
                                   
                                     
                                       
                                         T 
                                          
                                         
                                           ( 
                                           q 
                                           ) 
                                         
                                       
                                       † 
                                     
                                      
                                     
                                       T 
                                        
                                       
                                         ( 
                                         q 
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         = 
                           
                          
                         
                           arg 
                            
                           
                             
                               max 
                               
                                 1 
                                 ≤ 
                                 q 
                                 ≤ 
                                 
                                   M 
                                   / 
                                   2 
                                 
                               
                             
                              
                             
                               det 
                                
                               
                                 ( 
                                 
                                   
                                     X 
                                      
                                     
                                       ( 
                                       q 
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                            
                           
                             arg 
                              
                             
                               
                                 min 
                                 
                                   1 
                                   ≤ 
                                   q 
                                   ≤ 
                                   
                                     M 
                                     / 
                                     2 
                                   
                                 
                               
                                
                               
                                 
                                   det 
                                    
                                   
                                     ( 
                                     
                                       X 
                                        
                                       
                                         ( 
                                         q 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                                 . 
                               
                             
                           
                         
                          
                         
                             
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, we can equivalently first determine the (first M/2+1 rows of the) vector 
         [0000]      r=F[h 1   † R 1   −1 h 1 , . . . , h M   † R M   −1 h M ] T   (23) 
         [0000]    and compute {circumflex over (q)} as 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     ^ 
                   
                   = 
                   
                     
                       arg 
                        
                       
                           
                       
                        
                       
                         
                           max 
                           
                             2 
                             ≤ 
                             k 
                             ≤ 
                             
                               
                                 M 
                                 / 
                                 2 
                               
                               + 
                               1 
                             
                           
                         
                          
                         
                           { 
                           
                              
                             
                               r 
                               k 
                             
                              
                           
                           } 
                         
                       
                     
                     - 
                     1. 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0027]    Referring again to the diagram of  FIG. 3 , for the case of demodulating a single signal codeword, the input to the linear minimum mean square error equalizer LMMSE  100  is a signal vector of the form 
         [0000]        y   m   =h   m   x   m   +n   m   (16). 
         [0000]    The output (17) from the equalizer  100 , according to the form 
         [0000]      {circumflex over (x)} m =h m   † R m   −1 y m , m=1, . . . , M  (17), 
         [0000]    is them handled by the M-point inverse discrete Fourier Transform processing  102  to provide a transformed output of the form 
         [0000]    
       
         
           
             
               
                 
                   
                     s 
                     ^ 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               s 
                               ^ 
                             
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             
                               s 
                               ^ 
                             
                             M 
                           
                         
                         ] 
                       
                       T 
                     
                      
                     
                       = 
                       Δ 
                     
                      
                     
                       
                         F 
                         † 
                       
                        
                       
                         
                           x 
                           ^ 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This output from the IDFT circuit is then handled by a pairing and whitening processing  104  which outputs either the IDFT output according to (18) or 
         [0000]    
       
         
           
             
               z 
               
                 m 
                 , 
                 q 
               
             
             = 
             
               
                 
                   
                     
                       
                         Q 
                          
                         
                           ( 
                           q 
                           ) 
                         
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         I 
                         - 
                         
                           X 
                            
                           
                             ( 
                             q 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   
                      
                     
                       T 
                        
                       
                         ( 
                         q 
                         ) 
                       
                     
                   
                 
                  
                 
                   s 
                   
                     m 
                     , 
                     q 
                   
                 
               
               + 
               
                 
                   
                     n 
                     ˘ 
                   
                   
                     m 
                     , 
                     q 
                   
                 
                  
                 
                     
                 
                  
                 according 
                  
                 
                     
                 
                  
                 to 
                  
                 
                     
                 
                  
                 
                   
                     ( 
                     21 
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
         [0028]    Based on the channel estimates and the data rate, the pairing and whitening module  104  can decide whether or not to process its input signal. In case the pairing and whitening module decides not to process its input signal, then the input to the calculator  106  is of the form (18) and SISO LLR calculator is used in  106 . In case the pairing and whitening module  104  decides to process its input signal, then the output of the pairing and whitening module consists of length-2 vectors of the form (21) and the Two-symbol MLSD function in the calculator  106  is used. An illustrative pairing and whitening procedure is given by the following relationships 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     = 
                     
                       
                         F 
                          
                         
                           [ 
                           
                             
                               
                                 h 
                                 1 
                                 † 
                               
                                
                               
                                 R 
                                 1 
                                 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 h 
                                 1 
                               
                             
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               
                                 h 
                                 M 
                                 † 
                               
                                
                               
                                 R 
                                 M 
                                 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 h 
                                 M 
                               
                             
                           
                           ] 
                         
                       
                       T 
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       and 
                        
                       
                           
                       
                        
                       
                         q 
                         ^ 
                       
                     
                     = 
                     
                       
                         arg 
                          
                         
                             
                         
                          
                         
                           
                             max 
                             
                               2 
                               ≤ 
                               k 
                               ≤ 
                               
                                 
                                   M 
                                   / 
                                   2 
                                 
                                 + 
                                 1 
                               
                             
                           
                            
                           
                             { 
                             
                                
                               
                                 r 
                                 k 
                               
                                
                             
                             } 
                           
                         
                       
                       - 
                       1. 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    described in detail above. 
         [0029]    Referring again to the diagram of  FIG. 4 , for the case of demodulating multiple signal codewords, the input to the LMMSE equalizers  200  ( 202 ) is a signal vector of the form 
         [0000]        y   m   =H   m   x   m   +n   m   (1). 
         [0000]    The output of the LMMSE equalizers  200  ( 202 ) and input to inverse discrete Fourier transformers IDFT  204  ( 206 ) is of the form 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 x 
                                 ^ 
                               
                               m 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 ^ 
                               
                               m 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             
                               H 
                               m 
                               † 
                             
                              
                             
                               ( 
                               
                                 I 
                                 + 
                                 
                                   
                                     H 
                                     m 
                                   
                                    
                                   
                                     H 
                                     m 
                                     † 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             - 
                             1 
                           
                         
                          
                         
                           y 
                           m 
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               I 
                               + 
                               
                                 
                                   H 
                                   m 
                                   † 
                                 
                                  
                                 
                                   H 
                                   m 
                                 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                          
                         
                           H 
                           m 
                           † 
                         
                          
                         
                           y 
                           m 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     m 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     M 
                     . 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The output of inverse DFTs  204  ( 206 ) and input to the pairing and whitening module  208  is of the form 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       s 
                       ^ 
                     
                     
                       ( 
                       k 
                       ) 
                     
                   
                    
                   
                     = 
                     Δ 
                   
                    
                   
                     
                       F 
                       † 
                     
                      
                     
                       
                         
                           x 
                           ^ 
                         
                         
                           ( 
                           k 
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The output of the pairing and whitening module  208  is either of the form in (4) or in (12). 
         [0030]    Based on the channel estimates and the data rate, the pairing and whitening module  208  can decide whether or not to process its input signal. In case it decides not to process its input signal, then the input to the calculator  210  is of the form (4) and SISO LLR calculator is used in  210 . In case module  208  decides to process its input, the output of module  208  is produced using a pairing and whitening procedure and consists of length-2 vectors of the form in (12). An illustrative pairing and whitening procedure is given by 
         [0000]    
       
         
           
             
               
                 
                   r 
                   = 
                   
                     
                       F 
                        
                       
                         [ 
                         
                           
                             
                               
                                 h 
                                 1 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                               † 
                             
                              
                             
                               R 
                               1 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               h 
                               1 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             
                               
                                 h 
                                 M 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                               † 
                             
                              
                             
                               R 
                               M 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               h 
                               M 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       and 
                        
                       
                           
                       
                        
                       
                         q 
                         ^ 
                       
                     
                     = 
                     
                       
                         arg 
                          
                         
                             
                         
                          
                         
                           
                             max 
                             
                               1 
                               ≤ 
                               k 
                               ≤ 
                               M 
                             
                           
                            
                           
                             { 
                             
                                
                               
                                 r 
                                 k 
                               
                                
                             
                             } 
                           
                         
                       
                       - 
                       1 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which is described in detail above. 
         [0031]    The present invention has been shown and described in what are considered to be the most practical and preferred embodiments. It is anticipated, however, that departures may be made therefrom and that obvious modifications will be implemented by those skilled in the art. It will be appreciated that those skilled in the art will be able to devise numerous arrangements and variations, which although not explicitly shown or described herein, embody the principles of the invention and are within their spirit and scope.