Abstract:
A method of equalizing a received signal compensates for frequency selectivity of the communication channel taking into account channel estimation errors. The method comprises generating channel estimates for the received signal, computing filter weights for an equalizer based on said channel estimates and a covariance of the channel estimation error, and filtering the received signal using the computed filter weights.

Description:
BACKGROUND 
       [0001]    The present invention relates generally to equalization of received signals in a mobile communication system and, more particularly, to an equalizer for a receiver in a single carrier frequency division multiple access (SC-FDMA) system. 
         [0002]    Orthogonal Frequency-Division Multiple Access (OFDMA) is an attractive technique for sharing an available radio resource with multiple users in a high-speed wireless data communication system. Since each subcarrier of an OFDMA signal is simply scaled by a complex-valued scalar after passing through a time dispersive channel, demodulation can be performed for each subcarrier individually, and hence equalization is not needed in the receiver. Moreover, through the use of a cyclic prefix, orthogonality among different subcarriers is preserved even if they are not completely synchronized so long as the relative time delay is limited. This property is particularly desirable for uplink communications because users assigned to different subcarriers are typically only coarsely time aligned. 
         [0003]    A major drawback of OFDMA is the high peak-to-average-power ratio (PAR), or equivalently, the high crest factor (CF) (square root of PAR) of the transmitted waveform, which can cause undesired out-of-band radiation and/or inefficient power amplification in mobile terminals. Because of this limitation, Single-Carrier Frequency Division Multiple Access (SC-FDMA), whose transmitted waveforms have considerably lower peak-to-average-power ratio (PAR) than those of OFDMA, has recently been selected by 3GPP as the standard access method for the uplink (UL) in Evolved UTRA. The low PAR property of SC-FDMA signals enables the mobile terminals to transmit at higher efficiency while reducing undesired out-of-band emissions. 
         [0004]    In an SC-FDMA system, there are two different methods of allocating subcarriers to different users, referred to as the localized or the distributed allocations. The former method allocates contiguous subcarriers to each individual user. This method requires less pilot overhead for channel estimation but provides limited frequency diversity for each user. The second method allocates subcarriers that are evenly distributed over the spectrum assigned to each user. It provides more frequency diversity but generally requires more pilot overhead for channel estimation. Both carrier allocation methods result in transmitted signals that have significantly lower PAR than conventional OFDMA signals. 
         [0005]    Unlike conventional OFDMA systems, where the modulated symbols transmitted over different frequency tones can be demodulated independently of other symbols at the receiver, SC-FDMA requires an equalizer at the receiver to compensate for the frequency selectivity of the channel in order to demodulate the transmitted symbols. Although it is well known that a time-domain maximum likelihood sequence estimation (MLSE) equalizer is optimal in this situation, the complexity of such an equalizer will be exorbitant for the high transmission rates expected in E-UTRA. Consequently, a reduced-complexity, suboptimal frequency-domain equalizer is needed for SC-FDMA. 
       SUMMARY 
       [0006]    A method of equalizing a received signal compensates for frequency selectivity of the communication channel while taking into account channel estimation errors. The method comprises generating channel estimates for the received signal, computing filter weights for an equalizer based on said channel estimates and a covariance of the channel estimation error, and filtering the received signal using the computed filter weights. 
         [0007]    In one exemplary embodiment, the equalizer may comprise a single stage equalization filter that uses error-compensated filter weights to filter the received signal. In another embodiment, the equalizer may comprise a prefilter stage and an equalization stage. The received signal is first filtered in the prefilter to compensate for channel estimation errors and subsequently filtered in the equalization stage to compensate for the frequency selectivity of the channel. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]      FIG. 1  illustrates a receiver for a single carrier FDMA system according to one exemplary embodiment of the present invention. 
           [0009]      FIG. 2  illustrates a first embodiment of an equalizer for the receiver shown in  FIG. 1 . 
           [0010]      FIG. 3  illustrates a second embodiment of an equalizer for the receiver shown in  FIG. 1 . 
           [0011]      FIG. 4  illustrates an exemplary prefilter for the equalizer embodiment shown in  FIG. 3 . 
       
    
    
     DETAILED DESCRIPTION 
       [0012]    Referring now to the drawings,  FIG. 1  illustrates the main functional elements of an SC-FDMA receiver  10  according to one exemplary embodiment. The sampled signals from two or more antennas (not shown) are first converted into the frequency domain by a time-to-frequency converter  12 . The time-to-frequency converter  12  converts the samples into the frequency domain using an N-point Fast Fourier Transform (FFT). After conversion into the frequency domain, a subcarrier mapping function  14  extracts the received signal on the subcarriers of interest from the FFT output, where N C  is the number of subcarriers. The received signal R i  for the i th  antenna may be represented by the signal vector R i =(R i [1], R i [2], . . . , R i [N c ]) T , for i=1, 2, . . . , r. A typical baseband model of the received signal vector R i  for the i th  antenna is given by: 
         [0000]        R   i   =D ( H   i ) S+V   i ,   (1) 
         [0000]    where H i =(H i [1], H i [2], . . . , H i [N c ]) T  denotes a vector of channel coefficients over the desired frequency sub-carriers for the i th  antenna, D(H i ) denotes a diagonal matrix with elements of H i  as the diagonal elements, S=[S[1], S[2], . . . , S[N c ]) T  represents the N C -point FFT of the time-domain, modulated symbols Z=[Z[1], Z[2], . . . , Z[N c ]) T  such that ESS H =I, and V i =(V i [1], V i [2], . . . , V i [N c ]) T  denotes the noise vector at the i th  antenna, which is assumed to be a zero-mean, Gaussian-distributed random vector. The noise vector components are assumed to be uncorrelated across different subcarriers, i.e., EV i [k]V j *[m]=0 if k≠m for any i and j. 
         [0013]    A channel estimator  16  processes the received signals R i  to generate estimates Ĥ i  of the channel coefficients and provides the channel estimates Ĥ i  to a signal combiner  18 . The signal combiner  18  combines the received signals R i  from each antenna i using the channel estimates to produce a combined signal vector R=(R[1], R[2], . . . , R[N c ]) T . The signal combiner  18  may comprise, for example, a maximal ratio combiner (MRC) or Interference Rejection Combiner (IRC). In the case when the noise is uncorrelated across antennas and identically distributed across subcarriers, i.e., E[V i V j   H ]=σ i   2 Iδ(i−j), an MRC can be used. For an MRC, the signal combiner  18  combines the signal vectors R i  from each antenna according to: 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         r 
                       
                        
                       
                         
                           
                             D 
                              
                             
                               ( 
                               
                                 G 
                                 i 
                               
                               ) 
                             
                           
                            
                           
                             R 
                             i 
                           
                         
                         
                           σ 
                           i 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where G i ≡(G i [1], G i [2], . . . , G i [N c ]) T  represent the combining weight for the i th  antenna, and D(G i ) comprises a diagonal matrix with the elements of G i  on its diagonal. The combining weights G i  may be calculated according to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         G 
                         i 
                       
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     = 
                     
                       
                         
                           
                             H 
                             ^ 
                           
                           i 
                           * 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                       
                         
                           H 
                           ^ 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       
                         
                           H 
                           ^ 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                     ≡ 
                     
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             r 
                           
                            
                           
                             
                                
                               
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   i 
                                 
                                  
                                 
                                   [ 
                                   k 
                                   ] 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0014]    In the case when the noise vectors at different antennas are correlated (i.e., the noise is spatially colored), the MRC may be replaced by an IRC in order to achieve the optimal performance, which is well-known to those skilled in the art. For example, with the IRC the signal combiner  18  combines the signal vectors R i  from each antenna according to: 
         [0000]        R =( I   N     c     {circle around (x)} 1     r   T ) D (   G   )Λ   V     −1     R   .   (4) 
       In Equation (4): 
       [0015]          R   =vec([ R   1   , R   2   , . . . R   r ] T ); 
         [0000]      Λ   V     ≡E└  VV     H ┘ where    V   =vec([ V   1   , V   2   , . . . , V   r ] T ); 
         [0000]          G   =([ G   1 [1 ], G   2 [1 ], . . . , G   r [1 ], G   1 [2 ], G   2 [2 ], . . . , G   r [2 ], . . . , G   1   [N   c   ], G   2   [N   c   ], . . . , G   r   [N   c ]] T ) 
         [0016]    where 
         [0000]    
       
         
           
             
               
                 G 
                 i 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   
                     H 
                     ^ 
                   
                   i 
                   * 
                 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
               
                 
                   H 
                   ^ 
                 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
             
           
         
       
     
         [0000]    and Ĥ[k] is the square root of the k th  diagonal element of the matrix (I N     c     {circle around (x)} 1     r   T )D(  H ) H  Λ   V     −1 D(  H )(I N     c     {circle around (x)} 1     r ) and 
         [0000]          H = vec([ H   1   , H   2   , . . . , H   r ) T ); 
         [0017]    1 r ≡(1, 1, . . . , 1) T  denotes an all-one column vector of length r; 
         [0018]    I N     c    denotes an N c ×N c  identity matrix; 
         [0019]    vec(·) denotes the vectorization operation of stacking the columns of the argument; and 
         [0020]    {circle around (x)} denotes the Kronecker product. 
         [0021]    The combined signal vector R can then be modeled as: 
         [0000]        R=D ( Ĥ ) S+V,    (5) 
         [0000]    where Ĥ=(Ĥ[1], Ĥ[2], . . . , Ĥ[N c ]) T  denotes a vector of equivalent channel coefficients over the desired frequency sub-carriers after combining, D(Ĥ) denotes a diagonal matrix with elements Ĥ of as the diagonal elements, and V=(V[1], V[2], . . . , V[N c ]) T  denotes a zero-mean Gaussian noise vector with covariance matrix E[VV H ]=I. 
         [0022]    The combined signal vector R is input to a frequency domain equalizer  20 , which compensates the received signal vector R for the frequency selectivity of the uplink channel. A weight calculator  26  receives the channel estimates Ĥ i  for each antenna from the channel estimator  16  and computes filter weights {circumflex over (F)} for the equalizer  20 . The filter weight calculation is performed in a manner that takes into account channel estimation errors. While it is not possible to compute directly the channel estimation error, the covariance of the channel estimation error can be computed and used to refine the filter weight calculation. A frequency-to-time converter  28  converts the equalized signal back into the time domain. The output of the frequency-to-time converter  28  is an estimate {circumflex over (Z)} of the QAM modulated symbols Z. A demodulator  30  and decoder  32  follow the frequency to time converter  28  for demodulating and decoding {circumflex over (Z)} to obtain an estimate of an original information signal I that was transmitted 
         [0023]    In a conventional receiver, filter weights {circumflex over (F)} CONV  for the equalizer can be computed according to: 
         [0000]        {circumflex over (F)}   conv   =D ( Ĥ ) H ( D ( Ĥ ) D ( Ĥ ) H   +I ) −1   =D ( f ),   (6) 
         [0000]    where f=(f[1], f[2], . . . , f[N c ]) T  and 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                      
                     
                       [ 
                       k 
                       ] 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             H 
                             ^ 
                           
                            
                           
                             [ 
                             k 
                             ] 
                           
                         
                         * 
                       
                       
                         
                           
                              
                             
                               
                                 H 
                                 ^ 
                               
                                
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                              
                           
                           2 
                         
                         + 
                         1 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The conventional way of computing the equalizer weights, as described in Equation (6), does not take into the account the channel estimation error. The resulting filter weights {circumflex over (F)} CONV  therefore do not minimize the mean squared error between the transmitted symbols and equalizer output when there are channel estimation errors. 
         [0024]    According to one embodiment of the present invention, channel estimation errors are taken into account to compute the filter weights {circumflex over (F)} for an error-compensated MMSE equalizer. If e=[e[1], e[2], . . . , e[N c ]) T  denotes the channel estimation error such that H=Ĥ+e. The covariance of the channel estimation error e can then be given by: 
         [0000]      Λ e =Eee H .   (8) 
         [0000]    The filter weight {circumflex over (F)} can then be modeled by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           F 
                           ^ 
                         
                         = 
                           
                          
                         
                           
                             
                               arg 
                                
                               
                                   
                               
                                
                               min 
                             
                             F 
                           
                            
                           
                             
                                
                               
                                 S 
                                 - 
                                 FR 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ( 
                             
                               ESR 
                               H 
                             
                             ) 
                           
                            
                           
                             
                               ( 
                               
                                 ERR 
                                 H 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ( 
                             
                               ES 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       D 
                                        
                                       
                                         ( 
                                         
                                           
                                             H 
                                             ^ 
                                           
                                           + 
                                           e 
                                         
                                         ) 
                                       
                                     
                                      
                                     S 
                                   
                                   + 
                                   V 
                                 
                                 ] 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               E 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       D 
                                        
                                       
                                         ( 
                                         
                                           
                                             H 
                                             ^ 
                                           
                                           + 
                                           e 
                                         
                                         ) 
                                       
                                     
                                      
                                     S 
                                   
                                   + 
                                   V 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                             
                            
                           
                             
                               [ 
                               
                                 
                                   D 
                                    
                                   
                                     ( 
                                     
                                       
                                         H 
                                         ^ 
                                       
                                       + 
                                       e 
                                     
                                     ) 
                                   
                                    
                                   S 
                                 
                                 + 
                                 V 
                               
                               ] 
                             
                             H 
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               D 
                                
                               
                                 ( 
                                 
                                   H 
                                   ^ 
                                 
                                 ) 
                               
                             
                             H 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                     D 
                                      
                                     
                                       ( 
                                       
                                         H 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       D 
                                        
                                       
                                         ( 
                                         
                                           H 
                                           ^ 
                                         
                                         ) 
                                       
                                     
                                     H 
                                   
                                 
                                 + 
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       
                                         eSS 
                                         H 
                                       
                                        
                                       
                                         e 
                                         H 
                                       
                                     
                                     ] 
                                   
                                 
                                 + 
                                 I 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           D 
                            
                           
                             
                               ( 
                               
                                 H 
                                 ^ 
                               
                               ) 
                             
                             H 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   D 
                                    
                                   
                                     ( 
                                     
                                       H 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     D 
                                      
                                     
                                       ( 
                                       
                                         H 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                   H 
                                 
                               
                               + 
                               
                                 
                                   E 
                                   e 
                                 
                                  
                                 
                                   
                                     { 
                                     
                                       
                                         eE 
                                          
                                         
                                           [ 
                                           
                                             
                                               SS 
                                               H 
                                             
                                              
                                             
                                                
                                               e 
                                               ] 
                                             
                                              
                                             
                                               e 
                                               H 
                                             
                                           
                                           ] 
                                         
                                       
                                       + 
                                       I 
                                     
                                     ) 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           D 
                            
                           
                             
                               ( 
                               
                                 H 
                                 ^ 
                               
                               ) 
                             
                             H 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                     D 
                                      
                                     
                                       ( 
                                       
                                         H 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       D 
                                        
                                       
                                         ( 
                                         
                                           H 
                                           ^ 
                                         
                                         ) 
                                       
                                     
                                     H 
                                   
                                 
                                 + 
                                 
                                   
                                     E 
                                     e 
                                   
                                    
                                   
                                     { 
                                     
                                       
                                         eE 
                                          
                                         
                                           [ 
                                           
                                             SS 
                                             H 
                                           
                                           ] 
                                         
                                       
                                        
                                       
                                         e 
                                         H 
                                       
                                     
                                     ] 
                                   
                                 
                                 + 
                                 I 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                            
                           
                             D 
                              
                             
                               
                                 ( 
                                 
                                   H 
                                   ^ 
                                 
                                 ) 
                               
                               H 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       D 
                                        
                                       
                                         ( 
                                         
                                           H 
                                           ^ 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         D 
                                          
                                         
                                           ( 
                                           
                                             H 
                                             ^ 
                                           
                                           ) 
                                         
                                       
                                       H 
                                     
                                   
                                   + 
                                   I 
                                   + 
                                   
                                     Λ 
                                     e 
                                   
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where it is assumed that the transmitted symbols S are independent of the channel estimation error e. The estimation error covariance matrix Λ e  can often be pre-computed according to the channel estimation method. For example, for maximum-likelihood channel estimator, it can be shown that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           Λ 
                           e 
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 
                                   H 
                                   ^ 
                                 
                                 - 
                                 H 
                               
                               ) 
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   H 
                                   ^ 
                                 
                                 - 
                                 H 
                               
                               ) 
                             
                             H 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               W 
                               N 
                             
                              
                             
                               ( 
                               
                                 J 
                                 , 
                                 I 
                               
                               ) 
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       W 
                                       N 
                                     
                                      
                                     
                                       ( 
                                       
                                         J 
                                         , 
                                         I 
                                       
                                       ) 
                                     
                                   
                                   H 
                                 
                                  
                                 
                                   
                                     D 
                                      
                                     
                                       ( 
                                       P 
                                       ) 
                                     
                                   
                                   H 
                                 
                                  
                                 
                                   D 
                                    
                                   
                                     ( 
                                     P 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     W 
                                     N 
                                   
                                    
                                   
                                     ( 
                                     
                                       J 
                                       , 
                                       I 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 W 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   J 
                                   , 
                                   I 
                                 
                                 ) 
                               
                             
                             H 
                           
                           , 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where P=[P[1], P[2], . . . , P[N c ]) T  denotes the vector of pilot symbols over which the channel is estimated, J denotes the index set containing indices of the N c  desired sub-carriers, I denotes the index set containing the indices of the estimated channel tap locations, and W N  (J,I) denotes a sub-matrix of the N-point FFT matrix W N  formed by the rows indexed by the set J and the columns indexed by the set I . When the pilot symbols have a constant magnitude in frequency domain, as it is the case when the pilot symbols are properly designed, Equation (10) reduces to: 
         [0000]      Λ e   =W   N ( J,I )( W   N ( J,I ) H   W   N ( J,I ) 31 1   W   N ( J,I ) H .   (11) 
         [0000]    Furthermore, in the case of distributed sub-carrier allocation where the indices in the set J are uniformly distributed over the N possible indices, Equation (11) reduces further to 
         [0000]      Λ e   =W   N ( J,I ) W   N ( J,I ) H .   (12) 
         [0025]    The filter weights {circumflex over (F)} can be computed according to Equation (9) using one of Equations (10), (11), and (12) to compute the covariance Λ e  of the channel estimation error. In this case, the equalizer  20  may comprise a single MMSE equalization filter  22  where the filter coefficients are given by the filter weights {circumflex over (F)} as shown in  FIG. 2 . 
         [0026]    It may be noted that Equation (9) can be rewritten as follows: 
         [0000]        {circumflex over (F)}=D ( Ĥ ) H ( D ( Ĥ ) D ( Ĥ ) H   +I ) −1 ( D ( Ĥ ) D ( Ĥ ) H   +I )( D ( Ĥ ) D ( Ĥ ) H   +I+Λ   e ) −1 .   (13) 
         [0000]    Note that the first term D(Ĥ) H (D(Ĥ)D(Ĥ) H +I) −1  in Equation (13) is the same as Equation (6). Therefore, Equation (13) can be reduced to: 
         [0000]      {circumflex over (F)}={circumflex over (F)} conv {circumflex over (P)},   (14) 
         [0000]      where 
         [0000]        {circumflex over (P)} ≡( D ( Ĥ)   D ( Ĥ )  H   +I )( D ( Ĥ ) D ( Ĥ ) H   +I+Λ   e ) −1    (15) 
         [0000]    Equation (15) can also be rewritten as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           ^ 
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 
                                   D 
                                    
                                   
                                     ( 
                                     
                                       H 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     D 
                                      
                                     
                                       ( 
                                       
                                         H 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                   H 
                                 
                               
                               + 
                               I 
                               + 
                               
                                 Λ 
                                 e 
                               
                               - 
                               
                                 Λ 
                                 e 
                               
                             
                             ) 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                     D 
                                      
                                     
                                       ( 
                                       
                                         H 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       D 
                                        
                                       
                                         ( 
                                         
                                           H 
                                           ^ 
                                         
                                         ) 
                                       
                                     
                                     H 
                                   
                                 
                                 + 
                                 I 
                                 + 
                                 
                                   Λ 
                                   e 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           I 
                           - 
                           
                             
                               
                                 Λ 
                                 e 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       D 
                                        
                                       
                                         ( 
                                         
                                           H 
                                           ^ 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         D 
                                          
                                         
                                           ( 
                                           
                                             H 
                                             ^ 
                                           
                                           ) 
                                         
                                       
                                       H 
                                     
                                   
                                   + 
                                   I 
                                   + 
                                   
                                     Λ 
                                     e 
                                   
                                 
                                 ) 
                               
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                           
                             I 
                             - 
                             B 
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where B≡Λ e  (D(Ĥ)D(Ĥ) H +I+Λ) −1 . It follows from Equation (16) that an error-compensated MMSE equalizer can be implemented as a pre-filter  24  followed by the conventional MMSE equalizer filter  26 , as shown in  FIG. 3 . The filter weights {circumflex over (P)} for the pre-filter  24  are a function of the estimated channel Ĥ and the error covariance matrix Λ e . The filter weights {circumflex over (F)} conv  for the conventional MMSE equalizer are a function of the channel estimates Ĥ. 
         [0027]      FIG. 4  illustrates one exemplary embodiment of the pre-filter  24 . In this embodiment, the received signal is passed through a compensation filter  34  whose coefficients are given by B to obtain an estimate of the signal contribution due to the channel estimate error. Subtractor  36  subtracts the output of the compensation filter  28  from the received signal R before passing through the conventional MMSE equalizer  22 . 
         [0028]    Those skilled in the art will appreciate that the inventive receiver  10  can be implemented with a digital signal processor by executing code stored in a memory. The received signals on each antenna may be downconverted to baseband, sampled, and digitized for input to the receiver  10 .