Abstract:
Block-encoded transmissions of a multi-antenna terminal unit are effectively detected in the presence of co-channel interfering transmissions from other multi-antenna terminal units, when the base station has a plurality of antennas, and interference cancellation is combined with maximum likelihood decoding. The signals received in one base station antenna are employed in processing the signals received in a second base station antenna so as to cancel the signals of one terminal unit, while decoding the signals transmitted by the other terminal unit. Zero-forcing and MMSE approaches are presented. In another embodiment of this invention the basic decoding approach is used to obtain an initial estimate for the symbols from each terminal. Assuming that the signals of the first terminal unit has been decoded correctly, the receiver employs this initial decoded signal of the first terminal unit to cancel their contribution to the signals received at the base station antennas while decoding the signals of the second terminal unit. This process is then repeated assuming that the signals of the second terminal unit has been decoded correctly, the receiver employs this initial decoded signal of the second terminal unit to cancel their contribution to the signals received at the base station antennas while decoding the signals of the first terminal unit. The above disclosed techniques are viable for any number K of terminal units concurrently transmitting over a given channel, where each terminal unit is using a space-time block code with N transmit antennas, and a base station has at least K receive antennas.

Description:
REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of application Ser. No. 09/149,163, filed Sep. 4, 1998, now U.S. Pat. No. 6,178,196, which is claims the benefit of U.S. Provisional Application No. 60/061,145, filed Oct. 6, 1997. This application is also related to U.S. application Ser. No. 09/074224, filed May 7, 1998, now U.S. Pat. No. 6,185,258, titled “Transmitter Diversity Technique for Wireless Communications”. 
    
    
     BACKGROUND OF THE INVENTION 
     This invention relates to wireless communication and, more particularly, to techniques for effective wireless communication in the presence of fading, co-channel interference, and other degradations. 
     Rapid growth in mobile computing and other wireless data services is inspiring many proposals for high speed data services in the range of 64-144 kbps for micro cellular wide area and high mobility applications and up to 2 Mbps for indoor applications. Research challenges include the development of efficient coding and modulation, and signal processing techniques to improve the quality and spectral efficiency of wireless communications and better techniques for sharing the limited spectrum among different high capacity users. 
     The physical limitation of the wireless channel presents a fundamental technical challenge for reliable communications. The channel is susceptible to time-varying noise, interference, and multipaths. Power and size limitations of the communications and computing device in a mobile handset constitute another major design consideration. Most personal communications and wireless services portables are meant to be carried in a briefcase and/or pocket and must, therefore, be small and lightweight. This translates to a low power requirement since small batteries must be used. However, many of the signal processing techniques which may be used for reliable communications and efficient spectral utilization demand significant processing power, precluding the use of low power devices. Continuing advances in VLSI and integrated circuit technology for low power applications will provide a partial solution to this problem. Still, placing most of the signal processing burden on fixed locations (base stations) with relatively larger power resources than the mobile units will, likely, continue to be the trend in wireless systems design. 
     Perhaps the single most important parameter in providing reliable communications over wireless channels is diversity. Diversity techniques which may be used include time, frequency, and space diversity 
     Time diversity: channel coding in combination with limited interleaving is used to provide time diversity. However, while channel coding is extremely effective in fast fading environments (high mobility), it offers very little protection under slow fading (low mobility) unless significant interleaving delays can be tolerated. 
     Frequency diversity: the fact that signals transmitted over different frequencies induce different multipath structures and independent fading is. However, when the multipath delay spread is small compared to the symbol period, frequency diversity is not helpful. 
     Space diversity: the receiver/transmitter uses multiple antennas that are separated for reception/transmission and/or differently polarized antennas to create independent fading channels. Currently, multiple antennas at base-stations are used for receive diversity at the base. However, it is difficult to have more than two antennas at the mobile unit due to the size and cost of multiple chains of RF conversion. 
     Previous work on transmit diversity can be classified into three broad categories: schemes using feedback, schemes with feedforward or training information but no feedback, and blind schemes. The third category (blind schemes) relies primarily on multiple transmit antennas combined with channel coding to provide diversity. An example of this approach is disclosed in the aforementioned copending application Ser. No. 09/074224, 1998, titled “Transmitter Diversity Technique for Wireless Communications,” filed May 7. 
     SUMMARY OF THE INVENTION 
     Improved performance is attained in an illustrative arrangement where K synchronized terminal units that transmit on N antennas to a base station having M≧K antennas, by combining interference cancellation (IC) and maximum likelihood (ML) decoding. More specifically, space-time block coding is employed in transmitters that employ N transmit antennas each, and the signals are received in a receiver that employs M receiving antennas. In accordance with the processing disclosed herein, by exploiting the structure of the space-time block code, K-1 interfering transmitting units are cancelled at the receiver, regardless of the number of transmitting antennas, N, when decoding the signals transmitted by a given mobile unit. In another embodiment of the principles of this invention, signals of a first terminal unit are decoded first, and the resulting decoded signals are employed to cancel their contribution to the signals received at the base station antennas while decoding the signals of the remaining K-1 terminal units. The process is repeated among the remaining K-1 terminal units. That is, among the remaining K-1, signals of a first terminal unit is decoded first and the resulting decoded signals are employed to cancel their contribution to the signals received at the base station antennas while decoding the signals of the remaining K-2 terminal units, and so on. This procedure is repeated K times, each time starting with decoding signals of a particular terminal unit. This successive procedure will yield additional performance improvement. 
     Both zero-forcing (ZF) and minimum mean-squared error (MMSE) interference cancellation (IC) and maximum likelihood (ML) techniques are disclosed. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     FIG. 1 depicts an arrangement that, illustratively, includes a receiving base station ( 20 ) and two transmitting terminal units ( 10  and  30 ). 
    
    
     DETAILED DESCRIPTION 
     FIG. 1 illustrates two transmitting units and one receiving unit that comport with the principles disclosed herein. However, this is merely illustrative, and the disclosed method is useful for more than two terminals (K&gt;2). 
     Single Transmitting Unit 
     Transmitting unit  10  may correspond to the transmitting circuitry in a terminal unit, while receiving unit  20  may correspond to the receiving circuitry in a base station. Terminal unit  30  is shown identical to terminal unit  10 . It should be understood, of course, that each terminal unit has a receiving circuit, and the base station has a transmitting circuit. The terminal units are shown to have two antennas each. Receiving unit  20  is also shown to have two receiving antennas. Here, too, it should be kept in mind that, generally, any number, M≧2, of receiving antennas can be had. Particular advantage is realized when M≧K Since the mathematical treatment below is couched in general matrix notations, the expressions are valid for any number K and/or M. 
     Considering terminal unit  10 , the information source provides input symbols to element  13  which develops a block code. The symbols are divided into groups of two symbols each, and at a given symbol period, the two symbols in each group {c 1 , c 2 } are transmitted simultaneously from the two antennas. The signal transmitted from antenna  11  is c 1  and the signal transmitted from antenna  12  is c 2 . In the next symbol period, the signal—c 2 * is transmitted from antenna  11  and the signal c 1 * is transmitted from antenna  12 . The symbols are modulated prior to transmission with constellation mappers  14  and  15 , followed by pulse shapers  16  and  17 , respectively, in a conventional manner. 
     In receiver  20 , signals are received by antennas  21  and  22  and are applied to detector  25 . 
     In the mathematical development of the algorithms disclosed herein, it is assumed that the channel from each of the two transmit antennas remains fixed over two consecutive symbol periods. That is, 
     
       
           h   i ( nT )= h   i (( n+ 1) T ),  i= 1, 2.  (1)  
       
     
     To ascertain the channel characteristics, the transmitter carries out a calibration session, during which pilot signals or tones are transmitted. The signals received during the calibration session are applied to channel estimator circuits  23  and  24 , which are well known circuits, and the channel characteristics are thus obtained. 
     When only transmitter  10  is considered, the received signals at antenna  21  can be expressed as 
     
       
           r   11   =h   11   c   1   +h   12   c   2   + 72     1   (2) 
       
     
     
       
           r   12   =−h   11   c   2   *+h   12   c   1   *+η   2   (3) 
       
     
     where r 11  and r 12  are the received signals over two consecutive symbol periods, h 11 , denotes the fading channel between transmit antenna  11  and receive antenna  21 , h 12  denotes channel between transmit antenna  12  and receive antenna  21 , and η 1  and η 2  are noise terms, which are assumed to be complex Gaussian random variables with zero mean and power spectral density N 0 /2 per dimension. Defining the vectors r=[r 11 r 12 *] T , c=[c 1 c 2 ] T , and η=[η 1 η 2 *] T , equations (2) and (3) can be rewritten in a matrix form as 
     
       
           R=H·c+η   (4)  
       
     
     where the channel matrix H is defined as              H   =       [           h   11           h   12               h   12   *           -     h   11   *             ]     .             (   5   )                                
     The vector η is a complex Gaussian random vector with zero mean and covariance N 0 ·I. Defining C as the set of all possible symbol-pairs c={c 1 ,c 2 }, and assuming that all symbol pairs are equi-probable, it can be easily shown that the optimum maximum likelihood (ML) decoder selects from C the symbol-pair ĉ that minimizes the expression ∥r−H·ĉ∥ 2 . This can be written as                c   ^     =     arg                     min       c   ^     ∈   C                     r   -     H   ·     c   ^              2     .                 (   6   )                                
     It was shown by S. Alamouti in “Space Block Coding: A simple Transmitter Diversity Scheme for wireless Communications,” submitted to IEEE JSAC, September 1997 that the diversity order of the above space-time block code is equivalent to that of a two branch maximal ratio receive combining (MRRC). Because of the orthogonality of the matrix H, Alamouti also showed that this decoding rule decomposed into two separate decoding rules for c 1  and c 2 . The uncertainty, Δ c , of the decoded symbols ĉ is defined as 
     
       
         Δ c   =∥r−H   1   ·ĉ∥   2 .  (7)  
       
     
     It should be noted that the above analysis addresses only receive antenna  21 . When receiver  20  uses both antennas, i.e., antennas  21  and  22 , two received signal vectors r 1  and r 2  can be defined for antenna  21  and  22 , respectively, as 
     
       
           r   1   =H   1   ·c+η   1   (8)  
       
     
     
       
           r   2   =H   2   ·c+η   2   (9)  
       
     
     where H 1  and H 2  are the channel matrices to receive antennas  21  and  22 , respectively, and η 1  and η 2  are the corresponding noise vectors. That is,                  H   1     =     [           h   11           h   12               h   12   *           -     h   11   *             ]       ,       and                   H   2       =     [           h   21           h   22               h   22   *           -     h   21   *             ]       ,           (9a)                                
     where h 21  denotes the channel between transmit antenna  12  and receive antenna  22 , and h 22  denotes the channel between transmit antenna  11  and receive antenna  22 . In this case, the ML decoding rule is                  c   ^     =     arg                     min       c   ^     =   C            (                r   1     -       H   1     ·     c   ^              2     +              r   2     -       H   2     ·     c   ^              2       )           ,           (   10   )                                
     and the uncertainty of the decoded symbols is defined as 
     
       
           Δ   c   =∥r   1   −H   1   ·ĉ∥+∥r   2   −H   2   ·ĉ∥   2 .  (11)  
       
     
     As before, both the matrices H 1  and H 2  are orthogonal matrices and hence the above decoding rule also decomposes to two separate decoding rules for c 1  and c 2 . Note that the rate of transmission (of information symbols) in the space-time block coding scheme is 1 
     Interference Cancellation and ML Decoding: BASIC CASE 
     FIG. 1, however, shows two terminal units, and the issue that needs to be addressed is the detection performance at the base station receiver when the two terminal units transmit over the same time and frequency channel. 
     In the notation below, g 11  denotes the fading channel between transmit antenna  31  and receive antenna  21 , g 12  denotes the channel between antenna  31  and antenna  21 , g 21  denotes the channel between antenna  32  and antenna  22 , and g 22  denotes the channel between antenna  32  and antenna  22 . Also, {c,c 2 } and {s 1 ,s 2 } denote the two symbols transmitted from terminal units  10  and  30 , respectively. 
     At receiver  20 , the received signals over two consecutive symbol periods at receive antenna  21 , r 11  and r 12 , are 
     
       
           r   11   =h   11   c   1   +h   12   c   2   +g   11   s   1   +g   12   s   2   +η   11   (12) 
       
     
     
       
           r   12   =−h   11   c   2   *+h   12   c   1   *−g   11   s   2   *+g   12   s   1   *+η   12   (13) 
       
     
     Defining r 1 =[r 11 r 12 *] T , c=[c 1  c 2 ] T , s=[s 1  s 2 ] T , and n 1 =[η 11  η 12  *] T  equations (12) and (13) can be rewritten in matrix form as 
     
       
           r   1   =H   1   ·c+G   1   ·s+n   1   (14)  
       
     
     where the channel matrices H 1  and G 1  between the transmitter units  10  and  30  and receive antenna  21  are given by                  H   1     =     [           h   11           h   12               h   12   *           -     h   11   *             ]       ,       and                   G   1       =     [           g   11           g   12               g   12   *           -     g   11   *             ]       ,           (   15   )                                
     The vector n 1  is a complex Gaussian random vector with zero mean and covariance N 0 ·I. Similarly, the received signals over two consecutive symbol periods at receive antenna  22 , r 21  and r 22  are 
     
       
           r   21   =h   21   c   1   +h   22   c   2   +g   21   s   1   +g   22   s   2   +η   21   (16) 
       
     
     
       
           r   22   =−h   21   c   2   *+h   22   c   1   *−g   21   s   2   *+g   22   s   1   *+η   22   (17) 
       
     
     In a similar fashion, defining r 2 =[r 21 r 22 *] T  and n 2 =[η 21 η 22 *] T  equations (16) and (17) can be rewritten as 
     
       
           r   2   =H   2   ·c+G   2   ·s+η   2   (18) 
       
     
     where the channel matrices H 2  and G 2  between transmitter units  10  and  30  and antenna  22  are given by                  H   2     =     [           h   21           h   22               h   22   *           -     h   21   *             ]       ,       and                   G   2       =     [           g   21           g   22               g   22   *           -     g   21   *             ]               (   19   )                                
     Equations (14) and (18) can be combined to yield the matrix form              r   =       [           r   1               r   2           ]     =         [           H   1           G   1               H   2           G   2           ]                [         c           s         ]     +       [           n   1               n   2           ]     .                 (   20   )                                
     Zero-Forcing IC and ML Decoding Scheme: In this case, a matrix W is chosen such that                W   ·     [           r   1               r   2           ]       =       [             r   ~     1                 r   ~     2           ]     =         [           H   ~         0           0         G   ~           ]     ·     [         c           s         ]       +     [             n   ~     1                 n   ~     2           ]                 (   21   )                                
     We can find an appropriate matrix W by realizing that                    [           H   1           G   1               H   2           G   2           ]                    -   1       =       [           A   1     -   1           0           0         A   2     -   1             ]                [         I           -     G   1            G   2     -   1                     -     H   2            H   1     -   1             I         ]             (   22   )                                
     where 
     
       
           A   1   =H   1   −G   1   G   2   −1   H   2  and  A   2   =G   2   −H   2   H   1   −1   G   1   (23)  
       
     
     Hence, if we select W as              W   =     [         I           -     G   1            G   2     -   1                     -     H   2            H   1     -   1             I         ]             (   24   )                                
     we will have                W   ·     [           r   1               r   2           ]       =       [             r   ~     1                 r   ~     2           ]     =         [           H   ~         0           0         G   ~           ]     ·     [         c           s         ]       +     [             n   ~     1                 n   ~     2           ]                 (   25   )                                
     where                  H   ~     =       H   1     -       G   1          G   2     -   1            H   2                
            G   ~     =         G   2     -       H   2          H   1     -   1              G   1     .     
            n   ~     1           =       n   1     -       G   1          G   2     -   1            n   2                  
              n   ~     2     =       n   2     -       H   2          H   1     -   1            n   1                   (   26   )                                
     From equation (25) it can be easily observed that the modified received signal vector {tilde over (r)} 1  contains signals only from transmitter  10  (i.e. signals from transmitter  30  have been canceled or removed) and, correspondingly, the modified received signal vector {tilde over (r)} 2  contains signals only from transmitter  30  ( i.e. signals from transmitter  10  have been canceled or removed). A number of other attributes can be shown from the above to be true. 
     1) the modified noise vector ñ 1  is a zero mean complex Gaussian random vector with covariance                R   n1     =       N   o     ·     (     1   +       D   g1       D   g2         )     ·   I             (   27   )                                
     where D g1 =|g 11 | 2 +|g 12 | 2  and D g2 =|g 21 | 2 +|g 22 | 2 . Hence, the modified noise vector ñ 1  is also white. 
     2) the modified noise vector ñ 2  is also a zero mean Gaussian random vector with covriance                R   n2     =       N   o     ·     (     1   +       D   h2       D   h1         )     ·   I             (   28   )                                
     where D h1 =|h 11 | 2 +|h 12 | 2  and D h2 =|h 21 | 2 +|h 22 | 2 , and hence it is also white. 
     3) the matrices {tilde over (H)} and {tilde over (G)} the form                  H   ~     =     [             h   ~     1             h   ~     2                 h   ~     2   *           -       h   ~     1   *             ]       ,       and                   G   ~       =       [             g   ~     1             g   ~     2                 g   ~     2   *           -       g   ~     1   *             ]     .               (   29   )                                
     4) Conditioned on G 1  and G 2 , the random variables {tilde over (h)} 1  and {tilde over (h)} 2  are both zero mean complex Gaussian random variables with variance σ g   2 =1+D g1 /D g2 . 
     5) Conditioned on H 1  and H 2 , the random variables {tilde over (g)} 1  and {tilde over (g)} 2  are both zero mean complex Gaussian random variables with variance σ g   2 =1+D h2 /D h1 . 
     6) the modified channel matrices {tilde over (H)} and {tilde over (G)} have a structure similar to that in equation (5), i.e. the modified channel matrices {tilde over (H)} and {tilde over (G)} are orthogonal matrices. 
     Considering the modified received signal vector {tilde over (r)} 1 , which contains signals only from transmitter  10 , i.e., 
     
       
           {tilde over (r)}   1   ={tilde over (H)}·c+ñ   1 ,  (30)  
       
     
     it is noted that this expression resembles the expression in equation (4). Hence, in accordance with the principles disclosed therein, the optimum ML decoder for the symbols transmitted by terminal unit  10  evaluates an equation that is similar to the expression of equation (6), and is given by                c   ^     =         arg                 min         c   ^     ∈   C                         r   ~     1     -       H   ~     ·     c   ^              2     .               (   31   )                                
     The corresponding decoder uncertainty is given by 
     
       
           Δ   c   =∥{tilde over (r)}   1   −{tilde over (H)}·{tilde over (c)}∥   2 .  (32)  
       
     
     Moreover, since the channel Matrix {tilde over (H)} is orthogonal, the ML decoder will also decompose into two separate rules for c 1  and c 2 . 
     In a similar fashion, considering the modified received signal vector {tilde over (r)} 2 , which contains signals only from transmitter  10 , i.e., 
     
       
           {tilde over (r)}   2   ={tilde over (H)}·c+ñ   2 ,  (33)  
       
     
     it is noted that this expression resembles the expression in equation (4). Hence, the optimum ML decoder for the symbols transmitted by terminal unit  30  evaluates an equation that is similar to the expression of equation (6), and is given by                s   ^     =         arg                 min         s   ^     ∈   S                         r   ~     2     -       G   ~     ·     s   ^              2     .               (   34   )                                
     The corresponding decoder uncertainty is given by 
     
       
           Δ   s   =∥{tilde over (r)}   2   −{tilde over (G)}·{tilde over (s)}∥   2 .  (35)  
       
     
     Moreover, since the channel Matrix {tilde over (G)} is also orthogonal, the ML decoder will also decompose into two separate rules for s 1  and s 2 . 
     The above-disclosed technique can be easily implemented within a detector  25  that comprises a stored program general purpose processor. Specifically, a subroutine (ĉ,Δ)=ZF.DECODE(r 1 , r 2 , H 1 , H 2 , G 1 , G 2 ) can be installed which returns the values ĉ,Δ in response to submitted inputs r 1 , r 2 , H 1 , H 2 , G 1 , and G 2 , as shown below:          (       c   ^     ,     Δ   c       )     =       ZF   .   DECODE                     (       r   1     ,     r   2     ,     H   1     ,     H   2     ,     G   1     ,     G   2       )               {     
                       r   ~     =           r   1     -       G   1          G   2     -   1            r   2              
                     H   ~       =           H   1     -       G   1          G   2     -   1            H   2              
                                  c     ^       =           arg                 min         c   ^     ∈   C                     r   ~     -       H   ~     ·   c            2          
                     Δ   c       =              r   ~     -       H   ~     ·     c   ^              2                  
     }                          
     With such a subroutine, both ŝ and ĉ can be estimated, as follows: 
     
       
         ( ĉ ,Δ)= ZF.DECODE ( r   1   , r   2   , H   1   , H   2   , G   1   , G   2 )  (36)  
       
     
     
       
         ( ŝ ,Δ)= ZF.DECODE ( r   2   , r   1   , G   2   , G   1   , H   2   , H   1 )  (37)  
       
     
     It may be noted that knowledge of the noise power N 0  is not required. Simulation results reveal that the performance of the FIG. 1 system which employs the principles disclosed herein and the ZF.DECODE subroutine is equivalent to that when only one terminal unit exists and the base station uses a single receive antenna. That is, the reception is equivalent to the performance of two branch MRRC diversity receiver. Advantageously, however, the disclosed technique is able to support two co-channel terminal units. 
     The discussion above centered on describing the technique for canceling out the signal of transmitter  10  when detecting the signal of transmitter  30 , and for canceling out the signal of transmitter  30  when detecting the signal of transmitter  10 . Effectively, detector  25  of receiver  20  can either comprise two processors, with one making the subroutine call of equation (31) and the other making the subroutine call of equation (32). Alternatively, the signals can be stored within detector  25  and the subroutine calls of equations  31  and  32  can be made seriatim. 
     Minimum Mean-Squared Error IC and ML Decoding Scheme: The above-disclosed approach for canceling the contribution of an interfering terminal unit is known as the zero-forcing (ZF) as a minimum mean-squared error technique (MMSE). 
     Recalling equation (20), the vector r can also be written as                r   =       H   ·     c   ~       +   n            
            where                   c   ~       =     [                 c   ~     T                   s   ~     T     ]     T     ,                
        r     =     [             r   ~     1   T                   r   ~     2   T     ]     =       [           r   11           r   21   *           r   12           r   22   *           ]     T       ,   and                           (   38   )               H   =     [           H   1           G   1               H   2           G   2           ]             (   39   )                                
     To simplify notations, the vector r is also defined as r [r 1  r 2  r 3  r 4 ] T . 
     When seeking to detect and decode signals {c 1 , c 2 } by minimizing a mean-squared error criterion, the goal is find a linear combination of the received signals such that the mean-squared error in detecting the signals {c 1 , c 2 } is minimized. In general terms, this can be expressed by an error cost finction that is to be minimized, such as the function                J        (     α   ,   β     )       =                  ∑     i   =   1     4            α   i          r   i         -     (         β   1          c   1       +       β   2          c   2         )            2     =              α   ·   r     -     β   ·   c            2               (   40   )                                
     One may note that a minimum is certainly reached when both α and β are equal to 0, but that, of course, is not desired. Therefore, either β 1  or β 2  is set to 1. When β 2  is set to 1, we get the following minimization criterion from equation (40)                  J   1          (       α   1     ,     β   1       )       =                  ∑     i   =   1     5            α     1      i            r     1      i           -     c   2            2     =                  α   ~     1            r   ~     r       -     c   2            2               (   41   )                                
     where {tilde over (α)} 1 =[α 11 , α 12 , α 13 , α 14 , −β 1 ]=[α 1 −β 1 ] and {tilde over (r)} 1 =[r T  c 1 ] T . From this it can be seen that                  r   ~     1     =           [         H         0   T             0       1         ]                [           c   ~               c   1           ]     +     [         n           0         ]       =       R   ·   d     +   η               (   42   )                                
     where 0=[0 0 0 ]. 
     What is needed is to select {tilde over (α)} 1  so that the expected value of the expression in equation (41) is minimized. That is, select {tilde over (α)}× 1  to minimize 
     
       
           E{J   1 ({tilde over (α)} 1 )}= E {( {tilde over (α)}   1 {tilde over (r)} 1   −c   2 ) ( {tilde over (α)}   1 {tilde over (r)} 1   −c   2 )*}.  (43)  
       
     
     Taking the partial derivative with respect to {tilde over (α)} 1  and setting it to zero, what results is                  [         M         h   1               h   1   *         1         ]                [           α   1   *               -     β   1             ]     =     [           h   2             0         ]             (   44   )                                
     where M=HH* +{fraction (I/Γ)}I, Γ is the signal to noise ratio, I is the 4 by 4 identity matrix, h 1  is the first column of H, and h 2  is the second column of H. It follows that 
     
       
         α 1 *=( M−h   1   h   1 *) −1   h   2  and β 1   *=h   1 *( M−h   1   h   1 ) −1   h   2 .  (45)  
       
     
     It can be shown that                    (     M   -       h   1          h   1   *         )       -   1       =       M     -   1       -         M     -   1            h   1          h   1   *          M     -   1           1   -       h   1   *          M     -   1            h   1               ,           (   46   )                                
     which yields                β   1   *     =         h   1   *          M     -   1            h   2         1   -       h   1   *          M     -   1            h   1                   (   47   )                                
     From the structure of the matrix H we can easily verify that h 1  and h 2  are orthogonal. Using this fact and the structure of the matrix M, it can be shown that 
     
       
         β 1   = 0     (48)  
       
     
     
       
         α 1   *=M   −1   h   2   (49)  
       
     
     Hence, the MMSE IC solution given in equations (45) and (46) will minimize the mean-squared error in c 2  without any regard to c 1 . Considering the alternative cost function when β 1  is set to 1, a similar analysis leads to the conclusion that 
     
       
         β 2   = 0     (47)  
       
     
     
       
         α 2   *=M   −1   h   1   (48)  
       
     
     In this case, the MMSE IC solution given in equations (45) and (46) will minimize the mean-squared error in c 1  without any regard to c 2 . Therefore, from equations (45)-(48), we can easily see that the MMSE interference canceller for signals from terminal unit  10  will consist of two different sets of weights α 1  and α 2  for c 2  and c 1 , respectively. The weights for decoding signals from terminal  30  can be obtained in a similar fashion, as expected. Thus, the decoding of signals from terminal units  10  and  30  can be performed with a single subroutine MMSE.DECODE in decoder  25  as follows:          (     c   ,     Δ   c       )     =     MMSE   .     DECODE        (       r   1     ,     r   2     ,     H   1     ,     H   2     ,     G   1     ,     G   2     ,   Γ     )                 {     
                         r   ~     =           [           r   1   T           r   1   T           ]     T          
                     H   ~       =         [           H   1           G   1               H   2           G   2           ]          
                   M     =         HH   *     +         1   Γ        I          
                     h   1         =         [           h   11   T           h   21   T           ]     T     =       first                 column                 of                 H        
                     h   2       =         [           h   12   T           h   22   T           ]     T     =       second                 column                 of                 H        
                     α   1   *       =       M     -   1            h   1                       ,       α   2   *     =           M     -   1            h   2            
                     c   1       =         arg                 min           c   ^     1     ∈   C                       α   1   *          r   ~       -       c   ^     1            2           ,       c   2     =           arg                 min           c   ^     1     ∈   C                       α   1   *          r   ~       -       c   ^     2            2          
                     Δ   c       =                  α   1   *          r   ~       -       c   ^     1            2     +                α   1   *          r   ~       -       c   ^     2            2                  
     }                          
     With such a subroutine, both s and c can be estimated, as follows: 
     
       
         ( ĉ ,Δ)= MMSE.DECODE ( r   1   ,r   2   ,H   1   ,H   2   ,G   1   ,G   2 ,Γ)  (49)  
       
     
     
       
         ( ŝ ,Δ)= MMSE.DECODE ( r   2   ,r   1   ,G   1   ,G   2   ,H   1   ,H   2 ,Γ)  (50)  
       
     
     Similar to the zero-forcing case, simulation results reveal that the performance of the disclosed technique MMSE.DECODE is equivalent to that when only one terminal unit exists and the base station uses a single receive antenna which is equivalent to the performance of two branch MRRC diversity. However, this technique is also able to support two co-channel terminal units. In addition, when the SIR (signal-to-interference ratio, which is a ratio between the desired terminal power to the interfering terminal power) increases, the MMSE approach will have a better performance as compared to the ZF case (the ZF performance is almost the same as the performance of the MMSE approach at 0 dB SIR). 
     Two-Step Interference Cancellation: Actually, additional improvement can be realized by employing a two-step interference cancellation approach using either the zero-forcing or the MMSE interference cancellation techniques disclosed above. Below, we will describe this approach based on the MMSE technique. However, as one might expect there a similar approach based on the zero-forcing technique. In this two-step approach, the receiver decodes signals from both terminals using the subroutine MMSE.DECODE disclosed above. Assuming that symbols from the terminal unit  10 , ĉ o , have been decoded correctly, the receiver can, then, perfectly cancel the contribution of the terminal unit  10  in the received signal vectors r 1  and r 2 . The receiver then uses x 1  and x 2 , the received signal vectors after canceling signals from terminal unit  10 , to re-decode symbols from terminal unit  30  ŝ 0  using the optimum ML decoding rule in equation (10). Assuming that the symbols from terminal unit  10  has been decoded correctly, we can easily see that, the performance for terminal unit  30  will be equivalent to that with 2 transmit and 2 receive antennas (which is equivalent to 4 branch MRC diversity). If we let Δ 0 =Δ c     o   +Δ s     o    denotes the overall uncertainty for ĉ o  and ŝ o . The receiver then repeats the above step assuming that symbols from terminal unit  30  ŝ 1  have been decoded correctly using the MMSE.DECODE subroutine. As before, the receiver cancels the contribution of terminal unit  30  in the received signal vectors r 1  and uses y 1  and y 2 , the received signal vectors after cancelling signals from terminal unit 30, to re-decode symbols from terminal unit  10  ĉ 1  using the optimum ML decoding rule in equation (10). As before, assuming that symbols from terminal unit  30 , the performance for terminal unit  10  will be equivalent to that with 2 transmit and 2 receive antennas. Similarly, let Δ 1 =Δ c     1   +Δ s     1    denotes the overall uncertainty for ĉ 1  and ŝ 1 . The receiver then compares the overall uncertainty and chooses the pair (ĉ 0 , ŝ 0 ) if Δ 0 &lt;Δ 1  and (ĉ 1 , ŝ 1 ) otherwise. The two-step interference cancellation approach based on the MMSE technqiue disclosed above is presented below in pseudo-code subroutine II.MMSE.DECODE. As we mentioned earlier, the techniques can be also used with the zero forcing approach. Below, we also present the pseudo code subroutine II.ZF.DECODE for the two-step interference cancellation based on the zero-forcing approach.          (       c   ^     ,     s   ^       )     =       II   .   MMSE   .   DECODE                     (       r   1     ,     r   2     ,     H   1     ,     H   2     ,     G   1     ,     G   2     ,   Γ     )                              (         c   ^     0     ,     Δ     c   ,   o         )     =       MMSE   .   DECODE                     (       r   1     ,     r   2     ,     H   1     ,     H   2     ,     G   1     ,     G   2     ,   Γ     )                                  x   1     =       r   1     -       H   1     ·       c   ^     o           ,       x   2     =       r   2     -       H   2     ·       c   ^     o                                      s   ^     o     =         arg                 min       s   ∈   S            (                x   1     -       G   1     ·   s            2     +              x   2     -       G   2     ·   s            2       )                                Δ     s   ,   o       =                x   1     -       G   1     ·   s            2     +              x   2     -       G   2     ·   s            2                                (         s   ^     1     ,     Δ     s   ,   1         )     =       MMSE   .   DECODE                     (       r   1     ,     r   2     ,     G   1     ,     G   2     ,     H   1     ,     H   2     ,   Γ     )                                  y   1     =       r   1     -       G   1     ·       s   ^     1           ,                    y   ^     2     =       r   2     -       G   2     ·       s   ^     1                                      c   ^     1     =         arg                 min       c   ∈   C            (                y   1     -       H   1     ·   c            2     +              y   2     -       H   2     ·   c            2       )                                Δ     c   ,   1       =                y   1     -       H   1     ·   c            2     +              y   2     -       H   2     ·   c            2                                    If                   (       Δ     c   ,   o       +     Δ     s   ,   o         )       &lt;     (       Δ     c   ,   1       +     Δ     s   ,   1         )       ⇒     (       c   ^     ,     s   ^       )       =     (         c   ^     o     ,       s   ^     o       )                              Else                   (       c   ^     ,     s   ^       )       =     (         c   ^     1     ,       s   ^     1       )                                      (       c   ^     ,     s   ^       )     =       II   .   ZF   .   DECODE                     (       r   1     ,     r   2     ,     H   1     ,     H   2     ,     G   1     ,     G   2       )                              (         c   ^     0     ,     Δ     c   ,   o         )     =       ZF   .   DECODE                     (       r   1     ,     r   2     ,     H   1     ,     H   2     ,     G   1     ,     G   2     ,     )                                  x   1     =       r   1     -       H   1     ·       c   ^     o           ,                  x   2     =       r   2     -       H   2     ·       c   ^     o                                      s   ^     o     =         arg                 min       s   ∈   S            (                x   1     -       G   1     ·   s            2     +              x   2     -       G   2     ·   s            2       )                                Δ     s   ,   o       =                x   1     -       G   1     ·   s            2     +              x   2     -       G   2     ·   s            2                                (         s   ^     1     ,     Δ     s   ,   1         )     =       ZF   .   DECODE                     (       r   1     ,     r   2     ,     G   2     ,     G   1     ,     H   2     ,     H   1     ,     )                                  y   1     =       r   1     -       G   1     ·       s   ^     1           ,                    y   ^     2     =       r   2     -       G   2     ·       s   ^     1                                      c   ^     1     =         arg                 min       c   ∈   C            (                y   1     -       H   1     ·   c            2     +              y   2     -       H   2     ·   c            2       )                                Δ     c   ,   1       =                y   1     -       H   1     ·   c            2     +              y   2     -       H   2     ·   c            2                                    If                   (       Δ     c   ,   o       +     Δ     s   ,   o         )       &lt;     (       Δ     c   ,   1       +     Δ     s   ,   1         )       ⇒     (       c   ^     ,     s   ^       )       =     (         c   ^     o     ,       s   ^     o       )                              Else                   (       c   ^     ,     s   ^       )       =     (         c   ^     1     ,       s   ^     1       )                               
     Interference Cancellation and ML Decoding: GENERAL CASE 
     In the above basic case, we focused on the basic case where we assumed two co-channel terminals (K=2) each uses two transmit antennas (N=2). Both terminals communicate with a base station that is equipped with two transmit antennas (M=2). In this section we will consider the more general case of K≧2 co-channel terminals each is equipped with N≧2 transmitting antennas, both terminals communicate with a base that has receive M≧K antennas. We will develop similar interference cancellation and ML decoding scheme for this case. 
     In a paper submitted to IEEE Transactions on Information Theory, Vahid Tarokh et al. extended the above space-time block coding scheme to the case when more than two antennas are used for transmission (N≧2). There, a technique for constructing space-time block codes (with similar properties to the simple scheme described above) was developed. It was also shown that for real constellations space-time block codes with transmission rate  1  can be constructed. However, for a general complex constellation the rate of transmission for these codes will be less than 1. 
     In general, let us assume that the input information symbols to the transmitter are grouped into groups of Q symbols c 1 , c 2 , . . . , c Q . A space-time block code in this case will map the symbols c 1 , c 2 , . . . , c Q  into an N×L array C whose entries are made ±c 1 , ±c 2 , . . . , ±c Q  and ±c 1 , ±c 2 *, . . . , ±c Q *. At time t, where 1≦t≦L, the t-th column of C is transmitted from the N antennas. In this case, the transmission rate of such code will be Q/L. In general, for a rate Q/L space-ime block code (as constructed by V. Tarokh et al.) designed for N transmit antenna, let r 1 , r 2 , . . . , r L  be the received signals at time t=1, 2, . . . , L. As before, we define the received signal vector as 
     
       
           r=[r   1    r   2    . . . r   L/2    r*   L/2+1    r*   L/2+2    . . . r*   L ] T   (51)  
       
     
     where the L×1 vector r can be written as 
     
       
           r=H·c+η   (52)  
       
     
     and H is the L×Q channel matrix whose entries are from ±h 1 , ±h 2 , . . . , ±h N , ±h* 1 ,±h* 2 , . . . ,±h* N , and it is an orthogonal matrix, c=[c 1  c 2  . . . c Q ] T , and η is an L×1 zero-mean complex Gaussian random vector with covariance N o ·I which models the noise. The ML decoder in this case is similar to that in equation (6), that is                c   ^     =         arg                 min         c   ^     ∈   C                   r   -     H   ·     c   ^              2               (   53   )                                
     and the uncertainty, Δ c , of the decoded symbols ĉ is given by 
     
       
           Δ   c   =∥r−H·{tilde over (c)}∥   2 .  (54)  
       
     
     As before, since the channel matrix H is orthogonal, the decoding rule in (53) decomposes into Q separate decoding rules for c 1 ,c 2 , . . .,c Q . For example, assuming that the terminal unit uses 4 transmit antenna, a rate {fraction (4/8)} (i.e. it is a rate ½) space-time block code is given by          [           c   1               c   2               c   3               c   4           ]     →     [           c   1           -     c   2             -     c   3             -     c   4             c   1   *           -     c   2   *             -     c   3   *             -     c   4   *                 c   2           c   1           c   4           -     c   3             c   2   *           c   1   *           c   4   *           -     c   3   *                 c   3           -     c   4             c   1           c   2           c   3   *           -     c   4   *             c   1   *           c   2   *               c   4           c   3           -     c   2             c   1           c   4   *           c   3   *           -     c   2   *             c   1   *           ]                            
     In this case, at time t=1 c 1 , c 2 , c 3 , c 4  are transmitted from antenna  1  through  4 , respectively. At time t=2, −c 2 , c 1 , −c 4 , c 3  are transmitted from antenna  1  through  4 , respectively, and so on. For this example, let r 1 , r 2 , . . . ,r 8  be the received signals at time t=1, 2, . . . , 8. Define the received signal vector r=[r 1  r 2  r 3  r 4  r* 5  r* 6  r* 7  r* 8 ] T . In this case, we can write the received signal vector r can be written as 
     
       
           r=H·c+η   (55)  
       
     
     where η is the 8×1 AWGN noise vector and H is the 8×4 channel matrix given by:              H   =     [           h   1           h   2           h   3           h   4               h   2           -     h   1             h   4           -     h   3                 h   3           -     h   4             -     h   1             h   2               h   4           h   3           -     h   2             -     h   1                 h   1   *           h   2   *           h   3   *           h   4   *               h   2   *           -     h   1   *             h   4   *           -     h   3   *                 h   3   *           -     h   4   *             -     h   1   *             h   2   *               h   4   *           h   3   *           -     h   2   *             -     h   1   *             ]             (   56   )                                
     We can immediately notice that the matrix H is orthogonal, that is H*H=D h ·I where          D   h     =       ∑     i   =   1     4                 h   i          2                              
     and I is a 4×4 identity matrix. 
     Let us now assume a multi-user environment with K co-channel synchronized terminals. Each terminal uses a rate Q/L space-time block code with N transmit antenna (as constructed by V. Tarokh et al). The base station uses M≧Kantennas for reception. We can write the received signal vector at the m-th receive antenna as                  r   m     =         ∑     k   =   1     K            H   km     ·     c   k         +     η   m         ,     m   =   1     ,   2   ,   …              ,   M           (   57   )                                
     where H km  is the L×Q k-th user channel matrix to antenna m, c k =[c k1  c k2  . . . c kQ ] T  is the Q×1 information symbols vector for k-th user, and η m  is the L×1 noise vector. The entries of the k-th user channel matrix H km  are from ±h k,m,1 , ±h k,m,2 , . . . , ±h k,m,N  and ±h* k,m,1 , ±h* k,m,N , where h k,m,n  is the complex channel gain between transmit antenna n of the k-th user and receive antenna m. As we stated earlier, the matrix H km  is orthogonal. 
     Zero-Forcing IC and ML Decoding: Without loss of generality, let us assume that we are interested in suppressing signals from co-channel terminals  2 ,  3 , . . . , K while decoding signals from the first terminal. This can be done in a successive manner as follows. 
     First, let us define r m   (0) =r m . Let us assume that we start by canceling out the contributions of the K-th terminal. We can use the M-th antenna received signal vector r M  to cancel out the contribution of the K-th terminal in the remaining M−1 received signal vectors by forming the modified received signal vectors r m    (1) , m=1, . . . , M−1 as follows: 
     
       
           r   m   (1)   =r   m   (0)   −H   Km   H   +   KM   r   M   M   (0)    m= 1, 2, . . .  M− 1  (58)  
       
     
     where H km   +  is the generalized inverse of the channel matrix H km  and is given by 
     
       
           H   km   + =(H* km   H   km ) −1   H*   km   (59)  
       
     
     We can easily verify that H +   km H km =I, where I is the Q×Q identity matrix. We can easily verify that the modified received signal vectors r m   (1) , m=1, . . . , M−1, do not contain any signal contribution due to the K-th user. Moreover, we can easily verify that r m   (1)  can be written as                  r   m     (   1   )       =         ∑     k   =   1       K   -   1              H   km     (   1   )       ·     c   k         +     η   m     (   1   )           ,     m   =   1     ,   2   ,   …              ,     M   -   1             (   60   )                                
     where H (1)   km  and η (1)   m  are given by 
     
       
           H   (1)   km   =H   (0)   km   −H   (0)   Km ( H   (0)   KM ) +   H   (0)   kM   , m= 1, 2, . . . ,  M− 1  (61)  
       
     
     
       
         η (1)   m =η (0)   m   −H   (0)   km ( H   (0)   KM ) + η (0)   M   , m= 1, 2, . . . ,  M−   (62)  
       
     
     Moreover, it can be shown that for codes constructed by V. Tarokh et al, the modified channel matrix H (1)   km  will have exactly the same structure as that of H km . That is, the entries of the k-th user modified channel matrix H (1)   km  are from ±h (1)   km,1 , ±h (1)   k,m,2 , . . . , ±h (1)   k,m,N  and ±h (1)   k,m,1 , ±h (1)*   k,m,2 , . . . , ±h (1)*   k,m,N , where h (1)   k,m,n  is the modified complex channel gain between transmit antenna n of the k-th user and receive antenna m, m=1, . . . , M−1. Hence, the modified channel matrix H (1)   km  will be orthogonal as well. 
     It can then be observed that the expression for the M−1 modified received signal vector r (1)   m  in equation (60) is the same as that in equation (57) except that we now have one less interfering terminal. In a similar fashion, we can cancel out the contributions of terminal K−1 and obtain M−2 modified received signal vectors r (2)   m , m=1, . . . , M−2 that do not contain any contributions from terminals K-th and K−1. In general, after stagej, where j=1,2, . . . ,K−1 contributions from terminals K,K−1, . . . ,K−j+1 are canceled out and we are left with Mj modified received signal vectors r (j)   m , m=1, . . . , M−j,j=1,2, . . . , K−1 that contain signals due to terminals 1, 2, . . . , K−j only. In this case, we will have                  r   m     (   j   )       =         ∑     k   =   1       K   -   1              H     k   ,   m       (   j   )       ·     c   k         +     η   m     (   j   )           ,     m   =   1     ,   2   ,   …              ,     M   -   j             (   63   )                                
     where H J   km  and η (J)   m  are given by 
     
       
           r   (j)   m   =r   (J−1)   m   −H   (J-1)   K−j,m ( H   (J−1)   K−j,M−J ) +   r   (j−1)   M−j    m= 1, 2 , . . . , M−j   (64)  
       
     
     
       
           H   J)   k,m   =H   (J−1)   k,m   −H   (J−1)   K−j,m ( H   J−1)   K−j,M−j ) +   H   (j−1)   k,M−1 , 1≦ m≦M−j,  1≦ k≦K−j   (65)  
       
     
     
       
         η (J)   m =η (J−1)   m   −H   (J−1)   K−j,m ( H   (J−1)   K−j,M−J ) + η J−1)   M−j   , m= 1, 2, . . . ,  M−j   (66)  
       
     
     This process is repeated until we are left with M−K+1 modified received signal vectors r (K−1)   m , m=1, . . . , M−K+1 that contain only contributions due to the first terminal. In this case we will have 
       r   (K−1)   m   =H   (K−1)   1,m   ·c   1 +η (K−1)   m   , m= 1, 2, . . . ,  M−K+ 1  (67) 
     which contains signals due to the first terminal only. Similarly, the modified channel matrix H (K−1)   1,m , m=1, 2, . . . , M−K+1, will have a similar structure and is also orthogonal. Hence, it is straight forward to see that the ML decoder for signals from the first terminal is given by                  c   ^     1     =         arg                 min           c   ^     1     ∈   C              ∑     m   =   1       M   -   K   +   1                     r   m     (     K   -   1     )       -       H     1   ,   m       (     K   -   1     )       ·       c   ^     1              2                 (   68   )                                
     and the corresponding uncertainty will be given by                Δ   1     =       ∑     m   =   1       M   -   K   +   1                     r   m     (     K   -   1     )       -       H     1   ,   m       (     K   -   1     )       ·       c   ^     1              2               (   69   )                                
     Similarly, since the modified channel matrices H (K−1)   1,m , 1≦m≦M−K+1 are orthogonal, as before, the decoding rule in (68) will decompose into Q separate rules for decoding c 11 , c 12 , . . . , c 1Q . We may observe that the basic case for zero-forcing IC and ML decoding that we discussed in detail earlier is a special case of the above approach.          (         c   ^     1     ,     Δ   1       )     =     G_ZFDECODE                   (         {     r   m     }       1   ≤   m   ≤   M       ,       {     H   km     }         1   ≤   k   ≤   K     ,     1   ≤   m   ≤   M           )               {     
                         r   m     (   0   )       =     r   m       ,       1   ≤   m   ≤   M          
                       H     k   ,   m       (   0   )       =     H     k   ,   m           ,     1   ≤   m   ≤   M     ,       1   ≤   k   ≤     K        
                   for                 j       =       1   →     K   -   1            
                       M   J     =     M   -   j           ,       K   j     =       K   -     j        
                   for                 i       =       1   →     M   j            
                       r   i     (   J   )       =         r   i     (     j   -   1     )       -           H       K   j        i       (     j   -   1     )            (     H       K   j     ,     M   j         (     j   -   1     )       )       +          r     M   J       (     j   -   1     )                
                       H     k   ,   i       (   j   )       =       H     k   ,   i       (     j   -   1     )       -           H       K   j     ,   m       (     j   -   1     )            (     H       K   j     ,     M   j         (     j   -   1     )       )       +          H     k   ,     M   j         (     j   -   1     )                       ,       1   ≤   k   ≤       K   j          
                   end        
                   end        
                       c   ^     1         =           arg                 min           c   ^     1     ∈   C                ∑     m   =   1       M   -   K   +   1                     r   m     (     K   -   1     )       -       H     1   ,   m       (     K   -   1     )       ·       c   ^     1              2            
                     Δ   1         =       ∑     m   =   1       M   -   K   +   1                     r   m     (     K   -   1     )       -       H     1   ,   m       (     K   -   1     )       ·       c   ^     1              2                  
     }                          
     The above-disclosed technique can be easily implemented within a detector  25  that comprises a stored program general purpose processor. Specifically, a subroutine (c,ΔA)=G_ZFDECODE({r m } 1≦m≦M , {H km } 1≦k≦K,1≦m≦M ) can be installed which returns the values c,Δ in response to submitted inputs {r m } 1≦m≦M  and {H km }  1≦k≦K,1≦m≦M , as shown above. 
     Minimum Mean-Squared Error IC and ML Decoding Scheme: The MMSE IC and ML decoding in the general case can be developed in a similar fashion as follows. We recall the received signal vector at the m-th receive antenna in equation (57)                  r   m     =         ∑     k   =   1     K            H   km     ·     c   k         +     η   m         ,     m   =   1     ,   2   ,   …              ,   M           (   70   )                                
     This can be written in a matrix form as in equation (38) 
     
       
           r=H·{tilde over (c)}+n   (71)  
       
     
     where r=[r T   1  r T   2  . . . r T   M ] T  is a ML×1 vector, {tilde over (c)}=[c 1   T  c 2   T  . . . c K   T ] T  is QK×1 a vector, n=[η 1   T  η 2   T  . . . η M   T ] T  is a ML×1 vector, and              H   =     [           H   11           H   21         ⋯         H   K1               H   12           H   22         ⋯         H   K2             ⋮       ⋮       ⋰       ⋮             H     1      M             H     2      M           ⋯         H   KM           ]             (   72   )                                
     is the ML×QK channel matrix. As before, we redefme the vector r as r=[r 1  r 2  . . . r ML ] T . As before, we assume that we are interested in decoding the symbols transmitted from terminal  1  c 11 , c 12 , . . . , c 1Q . As before, when seeking to detect and decode signals c 11 , c 12 , . . . , c 1Q  by minimizing a mean-squared error criterion, the goal is find a linear combination of the received signals such that the mean-squared error in detecting the signals c 11 , c 12 , . . . , c 1Q  is minimized. In general terms, this can be expressed by an error cost function that is to be minimized, such as the function                J        (     α   ,   β     )       =                  ∑     i   =   1     LM                       α   l          r   l         -       ∑     j   =   1     Q                       β   j          c   lj                2     =              α   ·   r     -     β   ·     c   1              2               (   73   )                                
     Similarly, as before we can see that one of the β j , 1≦j≦Q must be set to  1  or else we get an all zero solution for α amd β. Consider the case where we set β j =1. Hence, in this case, the criteria to be minimized is                        J   j          (       α   j     ,     β   j       )       =                ∑     i   =   1       LM   +   Q   -   1                         α   ji          r   ji         -     c     1      j              2                   =                  α   ~     j            r   ~     j       -     c     1      j              2       ,     1   ≤   j   ≤   Q                   (   74   )                                
     where 
     
       
           {tilde over (α)}   j =[α J1 , α J12 . . . , α JLM ,−β 1 , . . . , −β J−1 ,−β J+1 , . . . ,−β Q ]=[α j −β j ]  ( 75 )  
       
     
     
       
           {tilde over (r)}   J   =[r   T   J    c   11    . . . c   1j−1    c   1j+1    . . . c   1Q ] T   (76)  
       
     
     If we follow the same steps as in the basic case, we arrive at the conclusion that 
     
       
         α j *= M   − 1 h   j ,  
       
     
     
       
           1 ≦ j≦   Q    (78) 
       
     
     
       
         α j   *=M   −1   h   j , 1≦ j≦Q   (78)  
       
     
     where h is the j-th column of the channel matrix H, and M=HH*+{fraction ( 1 /Γ)}I, is an ML×ML matrix, Γ is the signal to noise ratio, and I is the ML×ML identity matrix. 
     In this case, as before, the error in decoding the j-th symbol c 1j  will be minimized without any regard to the other symbols. Hence, the MMSE-IC and ML decoder will consist of Q different combiners, one for each symbol. It should be clear now that the MMSI-IC solution for the general case is a straight forward extension to the basic case shown earlier. The MMSEBC solution for the general case can be implemented using the subroutine G 13  MMSE.DECODE shown below.          (       c   ^     ,   Δ     )     =     G_ZDECODE        (         {     r   m     }       1   ≤              m   ≤   M       ,       {     H   km     }         1   ≤   k   ≤   K     ,     1   ≤   m   ≤   M           )               {     
                     r   =           [       r   1   T                     r   2   T                   ⃛                   r   M   T       ]     T          
                   H     =         [           H   11           H   21         ⋯         H   K1               H   12           H   22         ⋯         H   K2             ⋮       ⋮       ⋰       ⋮             H     1      M             H     2      M           ⋯         H   KM           ]          
                   M     =         HH   *     +       1   Γ                   I        
                   for                 j       =       1   →     Q        
                     h   j         =       j        -        th                 column                 of                 H        
                     α   j   *       =         M     -   1            h   j          
                     c   j       =           arg                 min           c   ^     j     ∈   C                                  α   j   *        r     -       c   ^     j            2          
                     Δ   j       =                    α   j   *          r   ~       -     c   j            2          
                   end        
                     c   ^       =           [       c   1                     c     2                     ⃛                   c   Q       ]     T          
                   Δ     =       ∑     j   =   1     Q                     Δ   j                                
     }