Patent Publication Number: US-6987819-B2

Title: Method and device for multiple input/multiple output transmit and receive weights for equal-rate data streams

Description:
This application claims the benefit of provisional application 60/259,039 filed on Dec. 29, 2000. 
    
    
     FIELD OF THE INVENTION 
     In general, the invention relates to the field of communication systems. More specifically, the invention relates to strategies for wireless Multiple-Input/Multiple-Output (MIMO) communications and in particular, to establishing transmit and receive weighting matrices for use at the transmitter antenna array and the receiver antenna array. 
     BACKGROUND OF THE INVENTION 
     In a wireless communication system, a major design challenge is to maximize system capacity and performance. One such communication design known in the art is Multiple-Input/Multiple Output (MIMO), and is a means of transmitting multiple data streams on the same time-frequency channel to a single receiver. The MIMO strategy involves deploying multiple antennas on both the transmitter and the receiver. 
     In environments having rich multipath scattering, large increases in capacity can be achieved through the use of MIMO and appropriate transmit and receive signal processing techniques. It is well known that MIMO wireless channels have significantly higher capacities than single-input single-output wireless communication channels, which has motivated the design of wireless communication systems with multiple antennas. Current system algorithms designed to achieve high capacity include spatial multiplexing, space-time coding and adaptive modulation. 
     These MIMO systems include the various BLAST (Bell-labs LAyered Space-Time) type techniques proposed by Lucent as a subset. The BLAST type techniques however, do not use the channel knowledge at the transmitter and thus are sub-optimal. 
     For MIMO systems, the maximum theoretical system capacity can be achieved using transmit and receive weights based on the Singular Value Decomposition (SVD) combined with a water-pouring (optimally assigning power to the individual data streams) strategy for determining the optimal data rates and power distribution. This strategy for maximizing capacity must adaptively control not only the number of independent data streams to be formed but also the choice of modulation and coding to be used on each stream. However, even for full-rank channels (as are known in the art), there are cases where a fixed number of data streams having the same modulation type is desirable, for example to avoid having to make a complex real-time decision on both the number of streams and the modulation/coding type on each stream. Consequently, using the SVD weights for a fixed number of equal-rate data streams is not the best option because fixing the modulation type will not take advantage of the unequal signal-to-noise ratios at the outputs of the receive array. 
     Therefore, it would be desirable to have a method and device for finding transmit and receive weights that are optimized for the use of equal-rate data streams. Further, it would be desirable that these weights have a lower Bit Error Rate (BER) than weights found using the SVD approach. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an overview diagram of a preferred embodiment of a wireless (cellular) communication system in accordance with the invention; 
         FIG. 2  is a block diagram illustrating a preferred embodiment of a receiving device imbedded within the communication system of  FIG. 1  in accordance with the invention; 
         FIG. 3  is a block diagram illustrating details of an antenna combiner imbedded within the receiving device of  FIG. 2  in accordance with the invention; 
         FIG. 3A  is a block diagram illustrating a preferred embodiment of a transmitting device imbedded within the communication system of  FIG. 1  in accordance with the invention; 
         FIG. 4  is a block diagram illustrating one embodiment of a transmit weighting unit imbedded within the transmitting device of  FIG. 3A  in accordance with the invention; 
         FIG. 5  is a flow chart representation of one embodiment of a method for providing Multiple Input Multiple Output (MIMO) receive and transmit weights as performed by the antenna combiner of FIG.  3  and transmitting device of  FIG. 3A , for finding the MMSE linear weights, in accordance with the invention; 
         FIG. 6  is a flow chart representation of an alternative embodiment of a method performed by the antenna combiner of FIG.  3  and transmitting device of  FIG. 3A , for finding the MMSE successive cancellation weights, in accordance with the invention; 
         FIG. 7  is a flow chart representation of an alternative embodiment of a method performed by the antenna combiner of FIG.  3  and transmitting device of  FIG. 3A , for finding the MMSE successive cancellation weights, in accordance with the invention; 
         FIG. 8  is a flow chart representation of an alternative embodiment of a method performed by the antenna combiner of FIG.  3  and transmitting device of  FIG. 3A , for finding the MMSE successive cancellation weights, in accordance with the invention; and 
         FIG. 8A  is a flow chart representation of an embodiment for the method of  FIG. 8  as performed by the antenna combiner of FIG.  3  and transmitting device of FIG.  3 A. 
     
    
    
     DETAILED DESCRIPTION OF THE PRESENTLY PREFERRED EMBODIMENTS 
       FIG. 1  illustrates a wireless communication system  100  in accordance with one embodiment of the present invention. As shown in  FIG. 1 , a base station  110  provides communication service to a geographic region known as a cell  103 ,  102 . At least one user device  120  and  130  communicate with the base station  110 . In one embodiment of the wireless communication system  100 , at least zero external interference sources  140  share the same spectrum allocated to the base station  110  and subscriber devices  120  and  130 . The external interference sources  140  represent an unwanted source of emissions that interferes with the communication process between the base station  110  and the user devices  120  and  130 . The exact nature and number of the external interference sources  140  will depend on the specific embodiment of the wireless communication system  100 . In the embodiment shown in  FIG. 1 , an external interference source will be another user device  140  (similar in construction and purpose to user device  120 ) that is communicating with another base station  112  in the same frequency spectrum allocated to base station  110  and user devices  120  and  130 . As shown in  FIG. 1 , user devices  120  have a single transmit antenna  101 , while user devices  130  have at least one antenna  101 . One embodiment of the invention provides that the user devices  120  and  130 , as well as the base station  110  may transmit, receive, or both from the at least one antenna  101 . An example of this would be a typical cellular telephone. Additionally, one embodiment of the invention can be implemented as part of a base station  110  as well as part of a user device  120  or  130 . Furthermore, one embodiment provides that user devices as well as base stations may be referred to as transmitting units, receiving units, transmitters, receivers, transceivers, or any like term known in the art, and alternative transmitters and receivers known in the art may be used. 
       FIG. 2 , is a block diagram illustrating one embodiment of a receiving device  200  imbedded within the wireless communication system  100  of  FIG. 1 , in accordance with the present invention. The receiving device  200  includes at least one antenna  101  wherein the outputs of the antennas are each provided to a receiving unit  201 . The outputs of the receiving units  201  are provided to at least one antenna combiner  202 . The signals from the receiving units  201  are also fed into a combiner controller  210 , which may regulate the operation of the at least one antenna combiner  202 . The signals from the receiving units  201  may also be fed into a channel estimation device  208 . A pilot symbol generator  212  generates pilot symbol information that is used by the combiner controller  210  to control the antenna combiner  202 . The pilot symbol information generated by the pilot symbol generator  212  is also used by the channel estimation device  208  to estimate a time-varying frequency response of the transmitting devices of wireless communication system. The output of the antenna combiner  202  is fed into an information decoding unit  206 , which decodes the antenna combiner output  204  and generates data information  213  that was received by the antennas  101 . 
       FIG. 3  is a block diagram illustrating details of the antenna combiner  202  of receiving device  200  of FIG.  2 . In one embodiment, antenna combiner  202  can be coupled to the receiving units  201 , which in turn are coupled to the antennas  101 . The receiving units  201  may include radio frequency pre-amplifiers, filters, and other devices that can be used to convert a radio frequency signal received by the antenna  101 , to a digital stream of baseband equivalent complex symbols. As illustrated in  FIG. 2 , the output (y) of the i th  receiving unit  201  (where i is an integer between 1 and M inclusive, and M is the total number of antenna  101  elements) may be mathematically denoted by y i (k), where k and i are integers, and is provided to the antenna combiner  202 . The antenna combiner  202  can be in the form of at least one complex multipliers  302  that multiply the output of each receiving unit  201  by a weight (w)  304  mathematically denoted as w i (k). A combiner  306  may sum the outputs of the at least one complex multipliers  302 . For one embodiment of the invention, the combiner controller  210  of  FIG. 2  controls the values of the weights  304 . 
       FIG. 3A  is a block diagram illustrating one embodiment of a transmitting device  3 A 00  imbedded within the wireless communication system  100  of  FIG. 1 , in accordance with the present invention. The transmitting device  3 A 00  may include at least one antenna  101  wherein the inputs to the antennas may be provided from at least one transmitting unit  3 A 01 , and at least one transmit combiner  3 A 03 . The inputs to the transmit combiner  3 A 03  may be provided from at least one transmit weighting unit  3 A 05 . At least one Information Bit Stream  3 A 13  may be encoded by an Information Encoding Unit  3 A 07  and converted into a Data Stream  3 A 10 . The Data Stream  3 A 10  is weighted by transmit weighting unit  3 A 05  and the outputs of the transmit weighting unit  3 A 05  may be sent to the transmit combiner  3 A 03 . The Transmit Weighting Unit Controller  3 A 11  may be provided with channel state information  3 A 12 , which may be used by the Transmit Weighting Unit Controller  3 A 10  to control the operation of transmit weighting units  3 A 05 . 
       FIG. 4  is a block diagram illustrating one embodiment of the transmitting weighting unit  3 A 05  of  FIG. 3A , in accordance with the present invention. The symbol (x) on data stream i at time k, x i (k)  410  is weighted  403  by a transmit weight  404 . The transmit weight  404  value may be dependant on which antenna the transmit weight  404  is associated with. In equation form, an antenna m&#39;s signal (v) for data stream i at time k  420  may be expressed as v i,m (k)=V i,m (k)x i (k). The antenna m&#39;s signal at time k for data stream i may be sent to the transmit combiner  3 A 03  for antenna m in order to combine all the data streams before radiating the signal out of antenna m. 
       FIG. 5  is a flow chart representation of a method for providing multiple input multiple output MIMO receive weights as can be performed by the antenna combiner of  FIG. 3 , and the transmitting device  3 A 00 , in accordance with the invention. 
     One embodiment of the invention may transmit and receive N s  data streams, wherein each data stream can have the same modulation type with M T  transmit antennas and M R  receive antennas. The present invention may provide an alternative to the SVD weights that are known in the art. One embodiment of the present invention can minimize the mean square error between the receive array outputs and the transmitted symbols. Using the SVD weights for a fixed number of equal-rate data streams may not be the best option because fixing the modulation type will not take advantage of the unequal signal-to-noise ratios at the outputs of the receive array. This is because the SVD weights diagonalize the channel, which means that there is no cross talk between the data streams. Thus MMSE weights based on the SVD approach will not take advantage of the performance gains possible by trading off the suppression of other streams with gains over noise. Further, if the SVD weights are used while changing each streams&#39; transmitted power so that each data stream is received with equal energy, the optimal data stream will be penalized while an inadequate data stream is built-up, again indicating that this is not the best solution. One embodiment of the present invention computes linear MMSE weights that are not constrained to diagonalize the channel, thereby providing superior performance when all data streams have the same modulation type. 
     In order to compute the linear MMSE weights in accordance with one embodiment of the present invention, a single estimate of the channel gain and phase between each receive antenna and each transmit antenna may be provided to the invention (this is also called the channel). The channel is modeled as stationary and flat-faded. As is known in the art, such channels occur in systems that include, but are not limited to, single carrier systems operating in frequency non-selective channels, OFDM systems in which an embodiment of the invention operates on a set of frequency-domain subcarriers having identical frequency response characteristics, or single carrier system in which an embodiment of the invention operates in the frequency domain on a set of frequency-bins having identical frequency response characteristics, or other similar systems as is known in the art. Also provided is that both the transmitting weighting unit and receiving device may know the channel. 
     In one embodiment, the M R ×M T  channel matrix, H, contains the gain and phase between each transmit and receive element. Also let the M T ×N s  matrix (also known as the transmit weighting matrix), V, denote the transmit weights for the N s  streams (the u th  column of V is denoted v u ), and finally let the N s ×1 vector, x, be the symbols on the N s  streams (at one particular time). Then for this embodiment, a received M R ×1 vector can be given as:
 
 y=HVx+n   (1)
 
where n is a M R ×1 vector of noise with power of σ n   2  each of the received antennas. In one embodiment, in order to keep an average transmit power equal to one, the transmit weighting matrix may be normalized as follows:
 
trace( V   H   V )=1  (2)
 
where trace(A) means the sum of the diagonal elements of the square matrix A.
 
     One embodiment of the invention finds the transmit weighting matrix and a M R ×N s  receive weighting matrix, W, such that an estimate of the transmitted symbols, r, is:
 
 r=W   H   y   (3)
 
(The u th  column of W is denoted w u ) In one embodiment of the invention, the operation of the invention may be optimized for the case where each of the N s  data streams is constrained to have the same modulation type. This embodiment of the invention minimizes the Mean Squared Error (MSE) between the estimated symbol on each stream and the transmitted symbol assuming linear receive and transmit weights. In equation form, this can be expressed as: 
                 min       w   u     ,     v   u         ⁢       ∑     u   =   1       N   s       ⁢           ⁢     E   ⁢              r   u     -     x   u            2           =       min       w   u     ,     v   u         ⁢       ∑     u   =   1       N   s       ⁢           ⁢     E   ⁢                w   u   H     ⁡     (         ∑     l   =   1       N   s       ⁢           ⁢       Hv   l     ⁢     x   l         +   n     )       -     x   u            2                   (   4   )             
 
where r u  is the u th  element of the vector r and x u  is u th  element of the vector x. Using E{x u x l   * }=δ(u−l), (4) becomes: 
               min       w   u     ,     v   u         ⁢     {               ∑     u   =   1       N   s       ⁢         w   u   H     ⁡     (         ∑     l   =   1       N   s       ⁢           ⁢       Hv   l     ⁢     v   l   H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       ⁢     w   u         -                   ∑     u   =   1       N   s       ⁢       w   u   H     ⁢     Hv   u         -       ∑     u   =   1       N   s       ⁢       v   u   H     ⁢     H   H     ⁢     w   u         +     N   s             }             (   5   )             
 
     In addition to the minimization of the objective function in (5), the constraints on the transmit weight vectors (the columns of the matrix V) given in (2) need to be met. One embodiment of the invention meets the constraints by modifying the objective function of (5) as follows: 
               min       w   u     ,     v   u         ⁢     {               ∑     u   =   1       N   s       ⁢         w   u   H     ⁡     (         ∑     l   =   1       N   s       ⁢           ⁢       Hv   l     ⁢     v   l   H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       ⁢     w   u         -                   ∑     u   =   1       N   s       ⁢       w   u   H     ⁢     Hv   u         -       ∑     u   =   1       N   s       ⁢       v   u   H     ⁢     H   H     ⁢     w   u         +     N   s     +               γ   ⁢                ∑     u   =   1       N   s       ⁢       v   u   H     ⁢     v   u         -   1          2             }             (   6   )             
 
where γ is an arbitrary scaling placed on the constraint part of the objective function.
 
     A closed form solution to (6) does not exist for N s &gt;1. However, (6) is in the form of an unconstrained optimization problem so a modified version of an iterative algorithm such as the gradient-based optimization approach is used by one embodiment of the invention to solve for w u  and v u . The gradient of (6) with respect to w u  can be shown to be:
 
∇ w   u =( HVV   H   H   H +σ n   2   I   M     R   ) w   u   −Hv   u   (7)
 
Because (HVV H H H +σ n   2   I   M     R   ) is well conditioned, its inverse can be found. Therefore W can be found at each iteration step as follows (i.e., by setting equation (7) equal to zero):
 
 W =( HVV   H   H   H +σ n   2   I   M     R   ) −1   HV   (8)
 
The gradient of (6) with respect to v u  can be shown to be:
 
∇ v   u =( H   H   WW   H   H+ 2γ(trace( V   H   V )−1) I   M     T   ) v   u   −H   H   w   u   (9)
 
The invention lets G=[∇v 1   . . . ∇v   N     s   ], and then using equation (9), G can be expressed as:
 
 G =( H   H   WW   H   H +2γ(trace( V   H   V )−1) I   M     T   ) V−H   H   W   (10)
 
     The above embodiment of the invention explicitly adds the constraint on the transmit weights in equation (2) to the minimization. Another option for ensuring the constraint is to use a Lagrangian multiplier, λ, as is known in the art. For this method, the gradient matrix in (10) is changed as follows:
 
 G =( H   H   WW   H   H+λI   M     T   ) V−H   H   W   (11)
 
     One addition gradient is needed, the Lagrangian gradient which is:
 
 g   λ =trace( V   H   V −1)  (12)
 
     The previous equations and calculations express a mathematical means for a method of providing linear MIMO transmit and receive weights. 
     The method of providing MIMO receive and transmit weights, shown in the flow chart  500 , may begin by being provided a channel matrix  510 . The channel matrix H, is a M R ×M T  matrix that contains the gain and phase between each transmit antenna (M T  is the total number of transmit antennas) and each receive antenna (M R  is the total number or receive antennas). In one embodiment of the invention, the received vector y, is modeled as y(k)=HVx(k)+n(k), where V is a M T ×N s  matrix of the transmit weight vectors, x(k) is a N s ×1 vector of the transmitted symbols at index k (e.g., time or frequency), and n(k) is a M R ×1 vector of noise at index k. The embodiment of the invention illustrated in  FIG. 5  may require the total transmit power to be limited, therefore the transmit weight vectors are constrained to trace(V H V)=1, where trace(A) means to sum the diagonal elements of the matrix A, and superscript H means the conjugate transpose (also known as the Hermitian) of the matrix. 
     Another embodiment of the invention may find a receive weighting matrix (also referred to as the receive weight vectors) W, that find an estimate, r(k), of x(k) as a function of r(k)=W H y(k). Next for the embodiment  500 , block  520  initializes an iteration number t=0. In one embodiment of the invention, transmit weight vectors can be initialized  530  as 
           V   0     =       1       N   s         ⁡     [           I     N   s                 0       (       M   T     -     N   s       )     ×     N   s               ]         ,       
 
where I N     s    is a N s ×N s  matrix of all zeros except for the diagonal elements which are one, and 0 a×b  is an a×b matrix of all zeros. Utilizing the iteration number t, the receive weight vectors W t , are found  540  as W t =(HV t V t   H H H +σ n   2 I M     R   ) −1 HV t , where V t  are the transmit weight vectors at iteration number t, σ n   2  is the noise power, and I M     R    is a M R ×M R  matrix of all zeros except for the diagonal elements which are one. Again utilizing the iteration number t, the gradient matrix G t , is found  550  as G t =(H H W t W t   H H+2γ(trace(V t   H V t )−1)I M     T   )V t −H H W t , where γ is the constraint weight (in one embodiment, γ=1), and I M     T    is a M T ×M T  matrix of all zeros, except for the diagonal elements that are one. In another embodiment of the invention, the gradient matrix is: G t =(H H W t W t   H H+λ t I M     T   )V t −H H W t , the Lagrangian multiplier at iteration number t can be given as λ t =λ t−1 +αg λ,t−1  and the Lagrangian gradient at iteration number t is: g λ,t =trace(V t   H V t −1).
 
     If the embodiment of the invention utilizes an adaptive step size  560 , the step size α, at iteration time t may be found  565  as the argument that minimizes the following objective function, C(V t ,W t ,G t ,γ,α) where 
       C ( V   t   ,W   t   ,G   t , γ,α)= W   t   H   H ( V   t   −αG   t ) H   H   H   W   t   −W   t   H ( V   t   −αG   t )−( V   t   −αG   t ) H   H   H   W   t +γtrace(( V   t   −αG   t ) H ( V   t   −αG   t )) 
     After a step size has been established, the transmit weight vectors can be computed  570  at iteration t+1 as a function of the transmit weight vectors at iteration number t, the step size at iteration number t, and the gradient matrix at iteration number t, using the equation V t+1 =V t −αG t . In another embodiment of the invention, the Lagrangian multiplier may be updated with: λ t+1 =λ t +αg λ,t . After computing the transmit weight vectors  570 , the iteration number is incremented by one  575  as t=t+1. If the iteration number has reached an ‘end’ value  580 , where t end  is an integer designating the maximum number of iterations, then the receive weight vectors are the receive weight vectors at iteration number t, (W=W t )  590  and the transmit weight vectors are the transmit weight vectors at iteration number t, (V=V t ) and this embodiment is completed. If the iteration number has not reached an ‘end’ value,  585  decides if trace(G t−1   H G t−1 ) &lt;ε, and if yes, the receive weight vectors are chosen to be the receive weight vectors at iteration number t, (W=W t ) and the transmit weight vectors are chosen to be the transmit weight vectors at iteration number t, (V=V t )  590  again completing this embodiment. For decision block  585 , trace(G t−1   H G t−1 )&lt;ε means the sum of the diagonal elements of the square matrix (G t−1   H G t−1 ) and where ε is a number indicating how small the gradient matrix at iteration number t should get. In other words, when trace(G t−1   H G t−1 )&lt;ε, the algorithm has almost converged because at the optimal solution Gt t+1 =0. If in decision block  585 , trace(G t−1   H G t−1 )&lt;ε, is false, the process recedes to block  540  and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart  500 . 
     At this point the transmit vectors are, if necessary, renormalized to satisfy the constraint in Equation (2). An alternate embodiment of the invention is a closed form solution to the linear transmit and receive weights that has an equal Mean Squared Error (MSE) on each or the received data streams. At the receiver, an estimate, of the transmitted symbols, r, can be found by using a linear M R ×N s  weight matrix, W, as follows:
 
r=W H y  (13)
 
     An MMSE approach can be used to find W and V as summarized next where a closed form expression for W and V are given. Using the SVD, H can be expressed as follows:
 
H=U H S H Z H   H   (14)
 
where M R ×M R  U H  and M T ×M T  Z H  are unitary matrices and M R ×M T  S H  is a matrix of all zeros except for the upper left r H ×r H  portion which is:
 
[ S   H ] N     s   =diag( s   H,1   , . . . , s   H,r     H   )  (15)
 
where r H  is the rank of H (r H ≦min(M T ,M R )), [A] l  means the first l rows and columns of A, and it is assumed that s H,1 ≧s H,2 ≧. . . ≧s H,r     H   . The transmit weight matrix can also be expressed by a SVD as follows:
 
V=U v S v Z v   H   (16)
 
where M T ×M T  U v  is unitary, N s ×N s  Z v  is unitary, and M T ×N s  S v  may be matrix of all zeros except for the upper N s ×N s  portion which contains the N s  non-zero singular values of the transmit weight matrix as follows:
 
[ S   v ] N     s   =diag( s   v,1   , . . . , s   V,N     s   )  (17)
 
     One way to find the components of (16) may be to set U V =Z H  and the singular values can be found as the solution to: 
               s     V   ,   l     2     =         1     λ       ⁢     σ   n     ⁢     s     H   ,   l       -   1         -       σ   n   2     ⁢     s     H   ,   l       -   2                   (   18   )             
 
where λ is chosen so that the following equation is satisfied (this forces the transmit weights to have unit power): 
                 ∑     l   =   1       N   s       ⁢           ⁢     s     V   ,   l     2       =   1           (   19   )             
 
     The MSE averaged across the data streams may be unchanged regardless of the choice of the right singular vectors of the transmit weight matrix. One option is to use the identity matrix (i.e., Z v =I), however in general this choice gives unequal MSE on each data stream as will be seen. This means that the BER performance can be dominated by the data stream with the highest MSE. Thus if some unitary Z v  can be found that gives equal MSE on each stream, the BER performance will improve while the average MSE will remain unchanged. 
     Using U V =Z H , the received data vector from (1) can be expressed as: 
                   y   =         U   H     ⁢     S   H     ⁢     Z   H   H     ⁢     Z   H     ⁢     S   V     ⁢     Z   V   H     ⁢   x     +   n                 =         U   H     ⁢     S   H     ⁢     S   V     ⁢     Z   V   H     ⁢   x     +   n                   (   20   )             
 
     Let N s ×1 Y N     s   ={U H   H y} N     s    where {a} i  means the first i elements of the vector a, then Y N     s    is: 
                     Y     N   s       =       {         U   H   H     ⁢     U   H     ⁢     S   H     ⁢     S   V     ⁢     Z   V   H     ⁢   x     +       U   H   H     ⁢   n       }       N   s                   =         {       S   H     ⁢     S   V     ⁢     Z   V   H     ⁢   x     }       N   s       +       {       U   H   H     ⁢   n     }       N   s                     =         DZ   V   H     ⁢   x     +     N     N   s                       (   21   )             
 
where D is a real N s ×N s  diagonal matrix equal to diag(s H,1 s V,1 , . . . , s H,N     s   s V,N     s   ) and N N     s    has a covariance matrix equal to σ n   2 I N     s   . (A unitary matrix times a vector of uncorrelated Gaussian random variables does not change its covariance matrix).
 
The MSE on stream l can be shown to be:
 
 MSE   l =1 −Z   V,l   H   {tilde over (D)}Z   V,l   (22)
 
where Z V,l  is the l th  column of Z v , and
 
 {tilde over (D)}=D   2 ( D   2 +σ n   2   I ) −1   (23)
 
     Thus unless all of the diagonal elements of {tilde over (D)} are equal, using Z V =I gives unequal MSEs on each data stream. The MSE averaged across all streams can be shown to be:
 
 {overscore (MSE)}= 1−trace( {tilde over (D)} ) /N   s   (24)
 
     Therefore in order to make all data streams have the same MSE, Z V,     l    must satisfy:
 
 Z   V,l   H   {tilde over (D)}Z   V,l =1 −{overscore (MSE)} =trace( {tilde over (D)} )/ N   s 
 
subject to Z V Z V   H =Z V   H Z V =I N     s     (25)
 
     One type of matrix that solves (25) is the DFT matrix (another choice for when N s  is a power of four is the Hadamard matrix where the Hadamard matrix is known in the art). Thus the linear transmit weights that minimize the average MSE on the data streams and also have the same MSE on each stream are:
 
V=V H S V F N     s     (26)
 
where S V  is given by (17) and (18) and F N     s    is the N s ×N s  DFT matrix normalized such that F N     s     H F N     s   =I. With the transmit weights chosen according to (26), the N s  streams can be recovered without any inter-stream interference as:
 
r=F N     s     H D −1 Y N     s     (27)
 
     Another embodiment of the invention provides an alternative approach to calculating transmit and receive weights for MIMO equal rate data streamsand is illustrated as FIG.  6 . The embodiment illustrated as flow chart  600 , finds the successive cancellation weights that minimize the MSE between the estimated symbols and the true symbols on each stream. The advantage of successive cancellation weights over linear weights is that some of the receive array&#39;s degrees are freedom are recovered by sequentially subtracting out the estimated contribution from the decoded streams. For example if there are three streams and three receive antennas, when decoding the first stream, two degrees of freedom in the array are needed to null the other two streams. For the second stream, however, only one degree of freedom is needed to null the third stream since the decoded first stream is subtracted out. The extra degree of freedom then can be used in a max-ratio sense to provide an additional gain against noise. Finally, the third stream has all of the degrees of freedom in the array available to provide a gain against noise. 
     In one embodiment, for the successive cancellation method, the estimated symbols are given as: 
                 r   u     =       w   u   H     ⁡     (     y   -       ∑     l   =   1       u   -   1       ⁢           ⁢       Hv   l     ⁢       x   ^     l           )         ,       where   ⁢           ⁢       x   ^     l       =     slice   ⁡     (     r   l     )                 (   28   )             
 
where in one embodiment slice(r l ) means to choose the closest signal constellation point to r l . In another embodiment, slice(r l ) refers to any technique that involves decoding a plurality of received symbols r l  into a bit stream, and then mapping the resulting bit stream to a signal constellation point.
 
The MMSE successive cancellation weights can be found as the solution to the following minimization problem: 
                 min       w   u     ,     v   u         ⁢       ∑     u   =   1       N   s       ⁢           ⁢     E   ⁢              r   u     -     x   u            2           =       min       w   u     ,     v   u         ⁢       ∑     u   =   1       N   s       ⁢           ⁢     E   ⁢                w   u   H     ⁡     (         ∑     l   =   u       N   s       ⁢           ⁢       Hv   l     ⁢     x   l         +   n     )       -     x   u            2                   (   29   )             
 
Note that (29) differs from (4) due to the inner sum being performed from u to N s  instead of 1 to N s  as is done in (4). Using E{x u (n)x l   * (n)}=δ(u−l), (29) becomes: 
               min       w   u     ,     v   u         ⁢     {               ∑     u   =   1       N   s       ⁢         w   u   H     ⁡     (         ∑     l   =   u       N   s       ⁢       Hv   l     ⁢     v   l   H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       ⁢     w   u         -                   ∑     u   =   1       N   s       ⁢       w   u   H     ⁢     Hv   u         -       ∑     u   =   1       N   s       ⁢       v   u   H     ⁢     H   H     ⁢     w   u         +     N   s             }             (   30   )             
 
In addition to the minimization of the objective function in (30), the constraints on the transmit weight vectors given in (2) need to be met. One embodiment of the invention meets those constraints by modifying the objective function of (30) as follows: 
         min       w   u     ,     v   u         ⁢           {               ∑     u   =   1       N   s       ⁢         w   u   H     ⁡     (         ∑     l   =   u       N   s       ⁢       Hv   l     ⁢     v   l   H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       ⁢     w   u         -                   ∑     u   =   1       N   s       ⁢       w   u   H     ⁢     Hv   u         -       ∑     u   =   1       N   s       ⁢       v   u   H     ⁢     H   H     ⁢     w   u         +     N   s     +               γ   ⁢                ∑     u   =   1     N     ⁢       v   u   H     ⁢     v   u         -   1          2             }           (   31   )               
 
where γ is an arbitrary scaling placed on the constraint part of the objective function.
 
     A closed form solution to (31) does not exist for N s &gt;1, therefore a modified version of an iterative algorithm, such as a gradient-based optimization approach, is used to solve for w u  and v u . The gradient of (31) with respect to w u  is: 
               ∇     w   u       =         (         ∑     l   =   u       N   s       ⁢       Hv   l     ⁢     v   l   H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )     ⁢     w   u       -     Hv   u               (   32   )             
 
Equation (32) cannot be put into the same form as (8) because of the sum from u to N s  in the parenthesis. Setting equation (32) equal to zero, w u  can be found at each iteration as: 
               w   u     =         (         ∑     l   =   u       N   s       ⁢       Hv   l     ⁢     v   l   H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       -   1       ⁢     Hv   u               (   33   )             
 
The gradient of (31) with respect to v u  is: 
               ∇     v   u       =         (         ∑     l   =   1     u     ⁢       H   H     ⁢     w   l     ⁢     w   l   H     ⁢   H       +     2   ⁢     γ   ⁡     (       trace   ⁡     (       V   H     ⁢   V     )       -   1     )       ⁢     I     M   T           )     ⁢     v   u       -       H   H     ⁢     w   u                 (   34   )             
 
In another embodiment, the gradient is computed using Lagrangian multipliers as follows: 
               ∇     v   u       =         (         ∑     l   =   1     u     ⁢       H   H     ⁢     w   l     ⁢     w   l   H     ⁢   H       +     λ   ⁢           ⁢     I     M   T           )     ⁢     v   u       -       H   H     ⁢     w   u                 (   35   )             
 
One addition gradient is needed, the Lagrangian gradient which is:
 
  g   λ =trace( V   H   V −1)  (36)
 
The Lagrangian gradient is needed in order to update the Lagrangian multiplier, λ.
 
     The method for calculating the successive cancellation weights is described in  FIG. 6  as a flow chart representation  600  of a preferred embodiment of an alternative method performed by the antenna combiner of FIG.  3  and the transmit device of  FIG. 3A , for finding the MMSE successive cancellation weights, in accordance with the invention.  FIG. 6  begins by being provided  610  a channel matrix H that is a M R ×M T  matrix which contains the gain and phase between each transmit antenna (M T  is the total number of transmit antennas) and each receive antenna (M R  is the total number or receive antennas). The received vector, y, is modeled as y(k)=HVx(k)+n(k), where for one embodiment of the invention, V is a M T ×N s  matrix of the transmit weight vectors, x(k) is a N s ×1 vector of the transmitted symbols at index (e.g., time or frequency), k, and n(k) is a M R ×1 vector of noise at index k. Because the total transmit power has to be limited, in one embodiment of the invention, the transmit weight vectors may be constrained as trace(V H V)=1, where trace (A) means to sum the diagonal elements of the matrix A, and superscript H means the conjugate transpose (also known as the Hermitian) of the matrix. 
     An additional embodiment of the invention finds receive weight vectors which are the columns of the receive weight matrix, W, that can be used to find an estimate, r(k), of x(k) as 
             r   u     ⁢     (   k   )       =       w   u   H     ⁢     (       y   ⁢     (   k   )       -       ∑     l   =   1       u   -   1       ⁢       Hv   l     ⁢         x   ^     l     ⁢     (   k   )             )         ,       
 
where {circumflex over (x)} l (k)=slice(r l (k)) and r u (k) is the u th  element of r(k) slice(r l (k)) may mean to choose the closest signal constellation point to r l (k).
 
     In another embodiment of the invention, the method of  FIG. 6  can initialize an iteration number t, to zero  620 , and the transmit weight vectors  630  may be initialized as 
           V   0     =       1       N   s         ⁡     [           I     N   s                 0       (       M   T     -     N   s       )     ×     N   s               ]         ,       
 
where I N     s    is a N s ×N s  matrix of all zeros except for the diagonal elements which are one, and 0 a×b  is an a×b matrix of all zeros. With the initialization complete, the method of  FIG. 6  computes the receive weight vectors W t  at iteration number t  640 , are found, for u=1 to N s , as 
         w     t   ,   u       =         (         ∑     l   =   u       N   s       ⁢       Hv     t   ,   l       ⁢     v     t   ,   l     H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       -   1       ⁢     Hv     t   ,   u             
 
where w t,u  is the u th  column of W t , v t,u  are the transmit weight vectors at iteration number t, σ n   2  is the noise power, and I M     R    is a M R ×M R  matrix of all zeros except for the diagonal elements which are one.
 
     Flow chart  600  proceeds to compute the gradient vectors at iteration number t  650 . The gradient vectors, g t,u , are found using the equation 
           g     t   ,   u       =         (         ∑     l   =   1     u     ⁢       H   H     ⁢     w     t   ,   l       ⁢     w     t   ,   l     H     ⁢   H       +     2   ⁢     γ   ⁡     (       trace   ⁡     (       V   t   H     ⁢     V   t       )       -   1     )       ⁢     I     M   T           )     ⁢     v     t   ,   u         -       H   H     ⁢     w     t   ,   u             ,       
 
where γ can be the constraint weight (in one embodiment of the invention, γ=1), and I M     T    is a M T ×M T  matrix of all zeros, except for the diagonal elements which are one. In another embodiment of the invention, the gradient vectors may be found according to: 
         g     t   ,   u       =         (         ∑     l   =   1     u     ⁢           ⁢       H   H     ⁢     w     t   ,   l       ⁢     w     t   ,   l     H     ⁢   H       +       λ   t     ⁢     I     M   T           )     ⁢     v     t   ,   u         -       H   H     ⁢       w     t   ,   u       .             
 
It is by computing the gradient vectors, g t,u , instead of the gradient matrix, G t , that distinguishes the successive cancellation weights method of  FIG. 6  from the linear MMSE weights method of FIG.  5 .
 
     If a step size has not been calculated prior to  660 , the step size α, at iteration time t may be found  665  as the argument that minimizes the following objective function, C(V t ,W t ,G t ,γ,α) where 
         C   ⁡     (       V   t     ,     W   t     ,     G   t     ,   γ   ,   α     )       =       {                     ∑     u   =   1       N   s       ⁢           ⁢         w     t   ,   u     H     ⁡     (         ∑     l   =   u       N   s       ⁢           ⁢       H   ⁡     (       v     t   ,   u       -     α   ⁢           ⁢     g     t   ,   u           )       ⁢       (       v     t   ,   u       -     α   ⁢           ⁢     g     t   ,   u           )     H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       ⁢     w     t   ,   u           -                   ∑     u   =   1       N   s       ⁢           ⁢       w     t   ,   u     H     ⁢     H   ⁡     (       v     t   ,   u       -     α   ⁢           ⁢     g     t   ,   u           )           -       ∑     u   =   1       N   s       ⁢         (       v     t   ,   u       -     α   ⁢           ⁢     g     t   ,   u           )     H     ⁢     H   H     ⁢     w     t   ,   u           +     N   s     +                     γ   ⁢                ∑     u   =   1       N   s       ⁢         (       v     t   ,   u       -     α   ⁢           ⁢     g     t   ,   u           )     H     ⁢     (       v     t   ,   u       -     α   ⁢           ⁢     g     t   ,   u           )         -   1          2             }     .         
 
     After a step size has been established, the transmit weight vectors are computed  670  at iteration t+1 as a function of the transmit weight vectors at iteration number t, the step size at iteration number t, and the gradient vectors at iteration number t, using the equation v t+1,u =v t,u −αg t,u , where u=1 to N s . In another embodiment of the invention, the Lagrangian multiplier is updated with: λ t+1 =λ t +αg λ,t . 
     After computing the transmit weight vectors  670 , the iteration number may be incremented by one  675  as t=t+1. If the iteration number has reached an ‘end’ value  680 , where t end  is an integer designating the maximum number of iterations, then the receive weight vectors may be chosen to be the receive weight vectors at iteration number t, (W=W t ) and the transmit weight vectors may be chosen to be the transmit weight vectors at iteration number t, (V=V t )  690  completing this embodiment. If the iteration number has not reached an ‘end’ value,  685  decides if trace(G t−1   H G t−1 )&lt;ε, and if yes, the receive weight vectors are the receive weight vectors at iteration number t, (W=W t ) and the transmit weight vectors are the transmit weight vectors at iteration number t, (V=V t )  690  again completing this embodiment. For decision block  685 , trace(G t−1   H G t−1 )&lt;ε means the sum of the diagonal elements of the square matrix (G t−1   H G t−1 ) and where ε is a number indicating how small the gradient matrix at iteration number t should get. In other words, when trace(G t−1   H G t−1 )&lt;ε, the algorithm has almost converged because at the optimal solution, G t−1 =0. If in decision block  685 , trace(G t−1   H G t−1 )&lt;ε, is false, the process recedes to block  640  and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart  600 . At this point the transmit vectors may be renormalized, if necessary, to satisfy the constraint in Equation (2). 
       FIG. 7  shows a flowchart of another embodiment of the invention  700 . In this embodiment the gradient matrix or vectors may never be calculated, but the transmit and receive linear and successive cancellation weights can be found through means of iteration. Initially,  710  one embodiment of the invention may be provided a channel matrix. The iteration number, t  720 , may next be set to 0 and the transmit and receive vectors can be initialized  730 . In one embodiment of the invention, the transmit and receive vectors can be initialized to: 
         V   0     =       1       N   s         ⁡     [           I     N   s                 0       (       M   T     -     N   s       )     ×     N   s               ]             
For the linear weights, the receive weights are initialized to:
   W   0 =( HV   0   V   0   H   H +σ n   2 I M     R   ) −1   HV   0   
For the successive cancellation weights, the receive weights are initialized to: 
         w     0   ,   u       =         (         ∑     l   =   u       N   s       ⁢           ⁢       Hv     0   ,   l       ⁢     v     0   ,   l     H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       -   1       ⁢     Hv     0   ,   u               
Additionally the objective function at iteration 0, C 0 , is set to an arbitrary large number. (In one embodiment C 0 =infinity.)
 
The transmit weight vectors at iteration number t+1 may next be computed  740  as a function of the transmit weight vectors at time t, the receive weight vectors at iteration number t, and the channel matrix as follows (where γ is the constraint weight):
 
For the linear weights:
   V   t+1 =( H   H   W   t   W   t   H   H+ 2γ(trace( V   t   H   V   t )−1) I   M     T   ) −1   H   H   W   t   
For the successive cancellation weights: 
         v       t   +   1     ,   u       =         (         ∑     l   =   1     u     ⁢           ⁢       H   H     ⁢     w     t   ,   l       ⁢     w     t   ,   l     H     ⁢   H       +     2   ⁢     γ   ⁡     (       trace   ⁡     (       V   t   H     ⁢     V   t       )       -   1     )       ⁢     I     M   T           )       -   1       ⁢     H   H     ⁢     w     t   ,   u               
Next, the receive weight vectors at iteration t+1 may be computed as a function of the transmit weight vectors  750  at iteration number t+1, the noise power, and the channel matrix as follows:
 
For the linear weights:
   W   t =( HV   t   V   t   H   H +σ n   2   I   M     R   ) −1   HV   t   
For the successive cancellation weights: 
         w     t   ,   u       =         (         ∑     l   =   u       N   s       ⁢           ⁢       Hv     t   ,   l       ⁢     v     t   ,   l     H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       -   1       ⁢     Hv     t   ,   u               
The objective function at iteration number t+1 may next be computed  760  as a function of the transmit and receive weight vectors at iteration number t+1, the noise power, and the channel matrix as follows:
 
For the linear weights:
   C   t+1   =N   s   +W   t+1   H ( V   t+1 )(V t+1 ) H   H   H   W   t+1   −W   t+1   H   H ( V   t+1 )−( V   t+1 ) H   H   H   W   t+1 +γtrace(( V   t+1 ) H ( V   t +1 )) 
For the successive cancellation weights: 
         C     t   +   1       =     {                     ∑     u   =   1       N   s       ⁢           ⁢         w       t   +   1     ,   u     H     ⁡     (         ∑     l   =   u       N   s       ⁢           ⁢       H   ⁡     (     v       t   +   1     ,   u       )       ⁢       (     v       t   +   1     ,   u       )     H     ⁢     H   H         +       σ   n   2     ⁢     I     M   R           )       ⁢     w       t   +   1     ,   u           -                   ∑     u   =   1       N   s       ⁢           ⁢       w       t   +   1     ,   u     H     ⁢     H   ⁡     (     v       t   +   1     ,   u       )           -       ∑     u   =   1       N   s       ⁢           ⁢         (     v       t   +   1     ,   u       )     H     ⁢     H   H     ⁢     w       t   +   1     ,   u           +     N   s     +                     γ   ⁢                ∑     u   =   1       N   s       ⁢         (     v       t   +   1     ,   u       )     H     ⁢     (     v       t   +   1     ,   u       )         -   1          2             }           
Next,  770  if the objective function at iteration t is less than the objective function at iteration t+1, then the receive weight vectors may be the receive weight vectors at iteration number t, (W=W t ) and the transmit weight vectors may be the transmit weight vectors at iteration number t, (V=V t )  790  completing this embodiment. If the objective function at iteration t is not less than the objective function at iteration t+1, then in  775  the iteration number, t, can be incremented by one. If the iteration number has reached an ‘end’ value  780 , where t end  may be an integer designating the maximum number of iterations, then the receive weight vectors may be chosen to be the receive weight vectors at iteration number t, (W=W t ) and the transmit weight vectors may be chosen to be the transmit weight vectors at iteration number t, (V=V t )  790  completing this embodiment. If the end value is not reached, then the iterative procedure returns back to  740  and performs another iteration. After the iterations are completed, the transmit vectors may need to be renormalized to meet the constraint in Equation (2).
 
     A further embodiment of the invention may provide for the successive cancellation weights to find weights that maximize the theoretical capacity as described next. The capacity of the channel for a given transmit weight matrix, V, can be shown to be: 
             C   =       log   2     ⁡     [     det   ⁡     (     I   +       1     σ   n   2       ⁢     HVV   H     ⁢     H   H         )       ]               (   37   )             
 
     Using (14) and (16), the capacity equation in (37) becomes: 
             C   =       log   2     ⁡     [     det   ⁡     (     I   +       1     σ   n   2       ⁢     U   H     ⁢     S   H     ⁢     Z   H   H     ⁢     U   V     ⁢     S   V     ⁢     Z   V   H     ⁢     Z   V     ⁢     S   V   T     ⁢     U   V   H     ⁢     Z   H     ⁢     S   H   T     ⁢     U   H   H         )       ]               (   38   )             
 
     Using Z V   H Z V =I and making U V =Z H , (38) becomes: 
             C   =       log   2     ⁡     [     det   ⁡     (     I   +       1     σ   n   2       ⁢     U   H     ⁢     S   H     ⁢     S   V     ⁢     S   V   T     ⁢     S   H   T     ⁢     U   H   H         )       ]               (   39   )             
 
     In other words, the capacity is independent of the right singular vectors of the transmit weight matrix. However, even though the capacity is not affected by Z V , different Z V &#39;s will greatly affect the performance of a practical receiver depending on what type of receive algorithm is employed (e.g., successive cancellation or linear weights). Therefore a search can be performed only over Z V  and the resulting weights will not change the theoretical capacity but will improve the receiver performance given the algorithm employed. 
     To maximize the capacity, the r (r≠N s  in general) singular values for the transmit weight vectors are selected according to the water-pouring strategy as known in the art with a total transmit power of one (this enforces the constraint trace(V H V)=1). If the water-pouring strategy says to transmit on more streams than N s , then the singular values of the transmit weight vectors are selected according to the water-pouring strategy using only the largest N s  singular values of H. 
     To summarize, the transmit weights are expressed as:
 
V=Z H S V Z V   H   (40)
 
     Therefore the received data vector can be expressed as:
 
 y=U   H   S   H   Z   H   H   Z   H   S   V   Z   V   H   x+n=U   H   S   H   S   V   Z   V   H   x+n   (41)
 
     Let r×1 Y r ={U H   H y} r  where {a} i  means the first i elements of the vector a, then Y r  is: 
                     Y   r     =       {         U   H   H     ⁢     U   H     ⁢     S   H     ⁢     S   V     ⁢     Z   V   H     ⁢   x     +       U   H   H     ⁢   n       }     r                 =           {       S   H     ⁢     S   V     ⁢     Z   V   H     ⁢   x     }     r     +       {       U   H   H     ⁢   n     }     r       =         DZ   V   H     ⁢   x     +     N   r                       (   42   )             
 
where the diagonalized channel, D, is a real r×r diagonal matrix equal to diag(s H,1 s V,1 , . . . , s H,r s Vr ) and N r  has a covariance matrix equal to σ n   2 I r . Note that s H,r ≠0 because the water-pouring strategy would never dictate sending power on a stream with a singular value of zero.
 
     For the Successive Cancellation MMSE weights, in order to maximize the capacity, r must equal the number of streams that the water-pouring strategy dictates and r must be less than or equal to N s . Assuming this is true (i.e., r≦N s ), the Successive Cancellation MMSE weights that maximize capacity are found by solving: 
                   min     W   ,   T       ⁢       ∑     l   =   1       N   s       ⁢     E   ⁢                w   l   H     ⁡     (       Y   r     -     D   ⁢       ∑     p   =   1       l   -   1       ⁢     t   p         +     x   p       )       -     x   l            2           =       min     W   ,   T       ⁢       ∑     l   =   1       N   s       ⁢     E   ⁢              w   l   H     ⁡     (       D   ⁢       ∑     p   =   1       N   s       ⁢       t   p     ⁢     x   p           +     N   r       )       -     x   l                    ⁢     
     ⁢       subject   ⁢           ⁢     to   :     TT   H         =   I             (   43   )             
 
where the r×N s  right singular matrix, T, is T=[t 1 , . . . , t N     s   ], t 1  through t Ns  are the right singular vectors, T=Z V   H , and M R ×N s  W=[w 1 , . . . w N     s   ].
 
     It can be shown that the receive weight vector for stream l is given by: 
               w   l     =         (       D   ⁢       ∑     p   =   l       N   s       ⁢       t   p     ⁢     t   p   H     ⁢   D         +       σ   n   2     ⁢     I   r         )       -   1       ⁢     Dt   l               (   44   )             
 
     Ignoring the constraint for now, the gradient of the objective function in (44) with respect to t l  (the l th  column of T) can be shown to be: 
               ∇     t   l       =       D   ⁢       ∑     p   =   1     l     ⁢       w   p     ⁢     w   p   H     ⁢     Dt   l           -     Dw   l               (   45   )             
 
     Using (45), T can be found through a gradient search where the constraint of TT H =I is enforced at each step. This procedure is described in a flow chart representation  900  of a preferred embodiment of the invention illustrated in FIG.  8 . 
     The embodiment of the invention illustrated in  FIG. 8  may be provided a channel matrix, H ( 910 ). Next the iteration number, m, can be set to 0 ( 920 ). The right singular matrix at iteration number 0, T 0 , may next be initialized by choosing any arbitrary r×N s  matrix such that T 0 T 0   H =I ( 930 ). Next, the receive weight vectors at iteration number m, w m,l , can be computed as a function of the diagonalized channel, D, and the right singular vectors at iteration number m, t m,l , ( 940 ) as 
         w     m   ,   l       =         (       D   ⁢       ∑     p   =   l       N   s       ⁢       t     m   ,   p       ⁢     t     m   ,   p     H     ⁢   D         +       σ   n   2     ⁢   I       )       -   1       ⁢     Dt     m   ,   l             
 
for l=1, . . . , N s . Then the gradient vectors at iteration number m, g m,l , may be computed as a function of the diagonalized channel, the receive weight vectors at iteration number m, and the right singular vectors at iteration number m ( 950 ) as 
         g     m   ,   l       =       D   ⁢       ∑     p   =   1     l     ⁢       w     m   ,   p       ⁢     w     m   ,   p     H     ⁢     Dt     m   ,   l             -     Dw     m   ,   l             
 
for l=1, . . . , N s . If a step size, α, has not been calculated prior to ( 960 ) then a step size can be calculated that will minimize the objective function in (43) subject to T m T m   H =I ( 965 ). The next step may compute the right singular vectors at iteration number m+1 as a function of the right singular vectors at iteration number m, the step size at iteration number m, and the gradient vectors at iteration number m ( 970 ) as t m,l =t m,l −αg m,l  for l=1, . . . , N s . The iteration number may then be incremented by one ( 975 ) as m=m+1. Then ( 977 ) the right singular matrix at iteration number m, T m , is formed by concatenating the right singular vectors at iteration number m together (T m =[t m,l |, . . . , |t m,N     s   ]) and a Gram-Schmidt orthogonalization (as known in the art) can be performed on the rows of the right singular matrix at iteration number m (this enforces the constraint T m T m   H =I). The flow chart representation  900  continues as an embodiment of the invention in FIG.  8 A.
 
     In  FIG. 8A , if the iteration number has reached an ‘end’ value  980 , where m end  may be an integer designating the maximum number of iterations, then the right singular matrix may be the right singular matrix at iteration number m, (T=T m ) and the receive weight vectors may be chosen to be the receive weight vectors at iteration number m, (w l =w m,l )  990  completing this embodiment. If the iteration number has not reached an ‘end’ value, block  985  can decide if trace(G m   H G m )&lt;ε where G m =[g m,1 |, . . . , |g m,N     s   ], and if yes, the receive weight vectors are chosen to be the receive weight vectors at iteration number m, (w l =w m,l ) and the right singular matrix is chosen to be the right singular matrix at iteration number m, (T=T m )  990  again completing this embodiment. For decision block  985 , trace(G m   H G m ) means the sum of the diagonal elements of the square matrix (G m   H G m )and where ε is a number indicating how small the gradient matrix at iteration number m should get. In other words, when trace(G m   H G m )&lt;ε, the algorithm has almost converged because at the optimal solution, G m =0. If in decision block  985 , trace(G m   H G m )&lt;ε, is false, the process recedes to block  940  of  FIG. 8 , and the successive cancellation weights method continues until all values are satisfied by the techniques described within flow chart  900 . After completion of the technique described within flow chart  900 , the transmit weight vectors may be given as V=Z H S V T. 
     The present invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The described embodiments are to be considered in all respects only as illustrative and not restrictive.