# Patent

## Title
Maximum ratio transmission

## Abstract

An arrangement where a transmitter has a plurality of transmitting antennas that concurrently transmit the same symbol, and where the signal delivered to each transmitting antenna is weighted by a factor that is related to the channel transmission coefficients found between the transmitting antenna and receiving antennas. In the case of a plurality of transmit antennas and one receive antenna, where the channel coefficient between the receive antenna and a transmit antenna i is hi, the weighting factor is hi* divided by a normalizing factor, a, which is




  
    
      (
      
        
          ∑
          
            k
            =
            1
          
          K
        
        ⁢
        
          
            
            
              h
              k
            
            
          
          2
        
      
      )
    
    
      1
      /
      2
    
  
  ,




where K is the number of transmitting antennas. When more than one receiving antenna is employed, the weighting factor is





  
    
      1
      a
    
    ⁢
    
      
        (
        gH
        )
      
      H
    
  
  ,




where g=[g1 . . . gL), H is a matrix of channel coefficients, and α is a normalizing factor





  
    
      (
      
        
          ∑
          
            p
            =
            l
          
          L
        
        ⁢
        
          
            ∑
            
              q
              =
              l
            
            L
          
          ⁢
          
            
            
              
                
                  ∑
                  
                    k
                    =
                    l
                  
                
                K
              
              ⁢
              
                
                  h
                  
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    k
                  
                
                ⁢
                
                  h
                  qk
                  *
                
              
            
            
          
        
      
      )
    
    
      1
      /
      2
    
  
  .

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation of U.S. patent application Ser. No. 11/766,853, filed Jun. 22, 2007, now U.S. Pat. No. 7,362,823 which is a continuation of U.S. patent application Ser. No. 10/963,838 filed on Oct. 12, 2004, now U.S. Pat. No. 7,274,752, issued on Sep. 25, 2007, which is a continuation of U.S. patent application Ser. No. 10/177,461 filed on Jun. 19, 2002, now U.S. Pat. No. 6,826,236, issued on Nov. 30, 2004, which is a continuation of U.S. patent application Ser. No. 09/156,066 filed on Sep. 17, 1998, now U.S. Pat. No. 6,459,740, issued on Oct. 1, 2002, each of which is incorporated by reference in their entirety herein.


FIELD OF ART
Aspects described herein relate to a system and method for using transmit diversity in a wireless communications setting.
BACKGROUND OF THE INVENTION
Wireless communications services are provided in different forms. For example, in satellite mobile communications, communications links are provided by satellite to mobile users. In land mobile communications, communications channels are provided by base stations to the mobile users. In PCS, communications are carried out in microcell or picocell environments, including outdoors and indoors. Regardless the forms they are in, wireless telecommunication services are provided through radio links, where information such as voice and data is transmitted via modulated electromagnetic waves. That is, regardless of their forms, all wireless communications services are subjected to vagaries of the propagation environments.
The most adverse propagation effect from which wireless communications systems suffer is the multipath fading. Multipath fading, which is usually caused by the destructive superposition of multipath signals reflected from various types of objects in the propagation environments, creates errors in digital transmission. One of the common methods used by wireless communications engineers to combat multipath fading is the antenna diversity technique, where two or more antennas at the receiver and/or transmitter are so separated in space or polarization that their fading envelopes are de-correlated. If the probability of the signal at one antenna being below a certain level is p (the outage probability), then the probability of the signals from L identical antennas all being below that level is pL. Thus, since p<1, combining the signals from several antennas reduces the outage probability of the system. The essential condition for antenna diversity schemes to be effective is that sufficient de-correlation of the fading envelopes be attained.
A classical combining technique is the maximum-ratio combining (MRC) where the signals from received antenna elements are weighted such that the signal-to-noise ratio (SNR) of their sum is maximized. The MRC technique has been shown to be optimum if diversity branch signals are mutually uncorrelated and follow a Rayleigh distribution. However, the MRC technique has so far been used exclusively for receiving applications. As there are more and more emerging wireless services, more and more applications may require diversity at the transmitter or at both transmitter and receiver to combat severe fading effects. As a result, the interest in transmit diversity has gradually been intensified. Various transmit diversity techniques have been proposed but these transmit diversity techniques were built on objectives other than to maximize the SNR. Consequently, they are sub-optimum in terms of SNR performance.
SUMMARY OF THE INVENTION
Improved performance is achieved with an arrangement where the transmitter has a plurality of transmitting antennas that concurrently transmit the same symbol, and where the signal delivered to each transmitting antenna is weighted by a factor that is related to the channel transmission coefficients found between the transmitting antenna and receiving antenna(s). In the case of a plurality of transmit antennas and one receive antenna, where the channel coefficient between the receive antenna and a transmit antenna i is hi, the weighting factor is hi* divided by a normalizing factor, a, which is




  
    a
    =
    
      
        (
        
          
            ∑
            
              k
              =
              1
            
            K
          
          ⁢
          
            
              
              
                h
                k
              
              
            
            2
          
        
        )
      
      
        1
        /
        2
      
    
  
  ,




where K is the number of transmitting antennas. When more than one receiving antenna is employed, the weighting factor is





  
    
      1
      a
    
    ⁢
    
      
        (
        gH
        )
      
      H
    
  
  ,




where g=[g1 . . . gL], H is a matrix of channel coefficients, and a is a normalizing factor





  
    
      (
      
        
          ∑
          
            p
            =
            1
          
          L
        
        ⁢
        
          
            ∑
            
              q
              =
              1
            
            L
          
          ⁢
          
            
            
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                
                K
              
              ⁢
              
                
                  h
                  
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    k
                  
                
                ⁢
                
                  h
                  qk
                  *
                
              
            
            
          
        
      
      )
    
    
      1
      ⁢
      
        /
      
      ⁢
      2
    
  
  .







BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates an arrangement where there is both transmit and receive diversity.
FIG. 2 is a flowchart illustrating a routine performed at the transmitter of FIG. 1.
FIG. 3 is a flowchart illustrating a routine performed at the receiver of FIG. 1.



DETAILED DESCRIPTION
FIG. 1 depicts a system which comprises K antennas for transmission and L antennas for reception. The channel between the transmit antennas and the receive antennas can be modeled by K×L statistically-independent coefficients, as show in FIG. 1. It can conveniently be represented in matrix notation by




  
    
      
        H
        =
        
          
            (
            
              
                
                  
                    h
                    11
                  
                
                
                  ⋯
                
                
                  
                    h
                    
                      1
                      ⁢
                      K
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋰
                
                
                  ⋮
                
              
              
                
                  
                    h
                    
                      L
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
                
                  ⋯
                
                
                  
                    h
                    LK
                  
                
              
            
            )
          
          =
          
            (
            
              
                
                  
                    h
                    1
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    h
                    L
                  
                
              
            
            )
          
        
      
    
    
      
        (
        1
        )
      
    
  




where the entry hpk represents the coefficient for the channel between transmit antenna k and receiver antenna p. It is assumed that the channel coefficients are available to both the transmitter and receiver through some means, such as through a training session that employs pilot signals sent individually through each transmitting antenna (see block 202 of FIG. 2 and block 302 of FIG. 3). Since obtaining these coefficients is well known and does not form a part of this invention additional exposition of the process of obtaining the coefficients is deemed not necessary.

The system model shown in FIG. 1 and also in the routines of FIG. 2 and FIG. 3 is a simple baseband representation. The symbol c to be transmitted is weighted with a transmit weighting vector v to form the transmitted signal vector. The received signal vector, x, is the product of the transmitted signal vector and the channel plus the noise. That is,

X=Hs+n  (2)

where the transmitted signals s is given by

s=[s1 . . . sk]T=c[v1 . . . vk]T,  (3)

the channel is represented by

H=[h1 . . . hk],  (4)

and the noise signal is expressed as

n=[n1 . . . nk]T.  (5)

The received signals are weighted and summed to produce an estimate, ĉ, of the transmitted symbol c.
In accordance with the principles of this invention and as illustrated in block 204 of FIG. 2, the transmit weighting factor, v, is set to




  
    
      
        v
        =
        
          
            
              1
              a
            
            ⁡
            
              [
              
                
                  
                    
                      h
                      1
                    
                  
                  
                    ...
                  
                  
                    
                      h
                      K
                    
                  
                
              
              ]
            
          
          H
        
      
    
    
      
        (
        6
        )
      
    
  




where the superscript H designates the Hermitian operator, and a is a normalization factor given by





  
    
      
        a
        =
        
          
            (
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
                  
                  
                    h
                    k
                  
                  
                
                2
              
            
            )
          
          
            1
            /
            2
          
        
      
    
    
      
        (
        7
        )
      
    
  




is included in the denominator when it is desired to insure that the transmitter outputs the same amount of power regardless of the number of transmitting antennas. Thus, the transmitted signal vector (block 206 of FIG. 2) is





  
    
      
        s
        =
        
          cv
          =
          
            
              
                c
                a
              
              ⁡
              
                [
                
                  
                    
                      
                        h
                        1
                      
                    
                    
                      ...
                    
                    
                      
                        h
                        K
                      
                    
                  
                
                ]
              
            
            H
          
        
      
    
    
      
        (
        8
        )
      
    
  




and the signal received at one antenna is

x=Hs+n=ac+n  (9)

from which the symbol can be estimated with the SNR of





  
    
      
        γ
        =
        
          
            
              a
              2
            
            ⁢
            
              
                σ
                c
                2
              
              
                σ
                n
                2
              
            
          
          =
          
            
              a
              2
            
            ⁢
            
              γ
              0
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




where γ0 denotes the average SNR for the case of a single transmitting antenna (i.e., without diversity). Thus, the gain in the instantaneous SNR is a2 when using multiple transmitting antennas rather than a single transmitting antenna.

The expected value of γ is

 γ=E[a2]γ0=KE∥hk|2|γ0  (11)

and, hence, the SNR with a Kth-order transmitting diversity is exactly the same as that with a Kth-order receiving diversity.

When more than one receiving antenna is employed, the weighting factor, v, is




  
    
      
        v
        =
        
          
            
              1
              a
            
            ⁡
            
              [
              gH
              ]
            
          
          H
        
      
    
    
      
        (
        12
        )
      
    
  




where g=[g1 . . . gL] (see block 204 of FIG. 2). The transmitted signal vector is then expressed as





  
    
      
        s
        =
        
          
            
              c
              a
            
            ⁡
            
              [
              gh
              ]
            
          
          H
        
      
    
    
      
        (
        13
        )
      
    
  




The normalization factor, a, is |gH|, which yields




  
    
      
        a
        =
        
          
            (
            
              
                ∑
                
                  p
                  =
                  1
                
                L
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    1
                  
                  L
                
                ⁢
                
                  
                    g
                    p
                  
                  ⁢
                  
                    g
                    q
                    *
                  
                  ⁢
                  
                    
                      ∑
                      
                        k
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      
                        h
                        
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          k
                        
                      
                      ⁢
                      
                        h
                        qk
                        *
                      
                    
                  
                
              
            
            )
          
          
            1
            /
            2
          
        
      
    
    
      
        (
        14
        )
      
    
  




The received signal vector (block 304 of FIG. 3) is, therefore, given by




  
    
      
        x
        =
        
          
            
              c
              a
            
            ⁢
            
              
                H
                ⁡
                
                  [
                  gH
                  ]
                
              
              H
            
          
          +
          n
        
      
    
    
      
        (
        15
        )
      
    
  




When the receiver's weighting factor, w, is set to be g (see blocks 306 and 308 of FIG. 3), the estimate of the received symbol is given by




  
    
      
        
          c
          _
        
        =
        
          gx
          =
          
            
              
                
                  c
                  d
                
                ⁢
                
                  
                    gH
                    ⁡
                    
                      [
                      gh
                      ]
                    
                  
                  H
                
              
              +
              gn
            
            =
            
              
                a
                ⁢
                
                  
                
                ⁢
                c
              
              +
              gn
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




with the overall SNR given by





  
    
      
        γ
        =
        
          
            
              
                a
                2
              
              
                gg
                H
              
            
            ⁢
            
              γ
              0
            
          
          =
          
            
              
                a
                2
              
              ⁢
              
                γ
                0
              
            
            
              
                ∑
                
                  p
                  =
                  1
                
                L
              
              ⁢
              
                
                  
                  
                    g
                    p
                  
                  
                
                2
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




From equation (17), it can be observed that the overall SNR is a function of g. Thus, it is possible to maximize the SNR by choosing the appropriate values of g. Since the hqk terms are assumed to be statistically identical, the condition that |g1|=|g2|= . . . =|gL| has to be satisfied for the maximum value of SNR. Without changing the nature of the problem, one can set |gp|=1 for simplicity. Therefore the overall SNR is




  
    
      
        γ
        =
        
          
            
              a
              2
            
            L
          
          ⁢
          
            γ
            0
          
        
      
    
    
      
        (
        18
        )
      
    
  




To maximize γ is equivalent to maximizing a, which is maximized if




  
    
      
        
          
            g
            p
          
          ⁢
          
            g
            q
            *
          
        
        =
        
          
            
              ∑
              
                k
                =
                1
              
              K
            
            ⁢
            
              
                h
                
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  k
                
              
              ⁢
              
                h
                qk
                *
              
            
          
          
            
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
                  h
                  
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    k
                  
                
                ⁢
                
                  h
                  qk
                  *
                
              
            
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




Therefore,




  
    
      
        a
        =
        
          
            (
            
              
                ∑
                
                  p
                  =
                  1
                
                L
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    1
                  
                  L
                
                ⁢
                
                  
                  
                    
                      ∑
                      
                        k
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      
                        h
                        
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          k
                        
                      
                      ⁢
                      
                        h
                        qk
                        *
                      
                    
                  
                  
                
              
            
            )
          
          
            1
            /
            2
          
        
      
    
    
      
        (
        20
        )
      
    
  




which results in the maximum value of γ. It is clear that the gain in SNR is





  
    a
    2
  
  L




when multiple transmitting and receiving antennas are used, as compared to using a single antenna on the transmitting side or the receiving side.

The vector g is determined (block 306 of FIG. 3) by solving the simultaneous equations represented by equation (19). For example, if L=3, equation (19) embodies the following three equations:




  
    
      
        
          
            (
            
              
                g
                1
              
              ⁢
              
                g
                2
                *
              
            
            )
          
          =
          
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
                  h
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    k
                  
                
                ⁢
                
                  h
                  
                    2
                    ⁢
                    k
                  
                  *
                
              
            
            
              
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  K
                
                ⁢
                
                  
                    h
                    
                      1
                      ⁢
                      
                        
                      
                      ⁢
                      k
                    
                  
                  ⁢
                  
                    h
                    
                      3
                      ⁢
                      k
                    
                    *
                  
                
              
              
            
          
        
        ,
        
          
            (
            
              
                g
                1
              
              ⁢
              
                g
                3
                *
              
            
            )
          
          =
          
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
                  h
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    k
                  
                
                ⁢
                
                  h
                  
                    3
                    ⁢
                    k
                  
                  *
                
              
            
            
              
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  K
                
                ⁢
                
                  
                    h
                    
                      1
                      ⁢
                      
                        
                      
                      ⁢
                      k
                    
                  
                  ⁢
                  
                    h
                    
                      3
                      ⁢
                      k
                    
                    *
                  
                
              
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            and
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  g
                  2
                
                ⁢
                
                  g
                  3
                  *
                
              
              )
            
          
          =
          
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
                  h
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    k
                  
                
                ⁢
                
                  h
                  
                    3
                    ⁢
                    k
                  
                  *
                
              
            
            
              
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  K
                
                ⁢
                
                  
                    h
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      k
                    
                  
                  ⁢
                  
                    h
                    
                      3
                      ⁢
                      k
                    
                    *
                  
                
              
              
            
          
        
        ,
      
    
    
      
        (
        21
        )
      
    
  




All of the hpg coefficients are known, so the three equations form a set of three equations and three unknowns, allowing a simple derivation of the g1, g2, and g3 coefficients. The corresponding average SNR is given by




  
    
      
        
          γ
          _
        
        =
        
          
            E
            ⁡
            
              [
              
                a
                2
              
              ]
            
          
          ⁢
          
            
              γ
              0
            
            L
          
        
      
    
    
      
        (
        22
        )
      
    
  




where the value of E[a2] depends on the channel characteristics and, in general is bounded by

LKE[|hk|2]≦E[a2]≦βL2KE[|hk|2]  (23)

## Claims

1. A system comprising:
K transmit antennas;
a transmitter configured to weight a symbol c by a vector of K weighting factors, to thereby generate K weighted versions of the symbol c and configured to transmit respective weighted versions of the symbol c on corresponding antennas of the K transmit antennas,
wherein the vector of K weighting factors is proportional to [gH]H, where g is a vector having L components and the superscript H is the Hermitian operator,
wherein at least one of K and L is greater than one.

2. The system, as recited in claim 1, further comprising:
L receive antennas; and
a receiver configured to receive from the L receive antennas, L received versions of the symbol c transmitted by the K transmit antennas and configured to weight the L received versions of the symbol c by the vector g to generate L weighted, received versions of the symbol c.

3. The system, as recited in claim 2, wherein K is greater than one and L is greater than one.

4. The system, as recited in claim 2, wherein the vector of K weighting factors is normalized by a normalization factor, a, equal to




  
    
      (
      
        
          ∑
          
            p
            =
            1
          
          L
        
        ⁢
        
          
        
        ⁢
        
          
            ∑
            
              q
              =
              1
            
            L
          
          ⁢
          
            
          
          ⁢
          
            
              g
              p
            
            ⁢
            
              g
              q
              *
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
              
              ⁢
              
                
                  h
                  
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    k
                  
                
                ⁢
                
                  h
                  qk
                  *
                
              
            
          
        
      
      )
    
    
      1
      /
      2
    
  
  ,



wherein hpq is a matrix of coefficients for the channel between the K transmit antennas and the L receive antennas.

5. The system as recited in claim 4, wherein the normalization factor, a, is




  
    
      (
      
        
          ∑
          
            p
            =
            1
          
          L
        
        ⁢
        
          
        
        ⁢
        
          
            ∑
            
              q
              =
              1
            
            L
          
          ⁢
          
            
          
          ⁢
          
            
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
              
              ⁢
              
                
                  h
                  
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    k
                  
                
                ⁢
                
                  h
                  qk
                  *
                
              
            
            
          
        
      
      )
    
    
      1
      /
      2
    
  
  .

6. The system as recited in claim 2, wherein the elements of vector g satisfy




  
    
      (
      
        
          g
          p
        
        ⁢
        
          g
          q
          *
        
      
      )
    
    =
    
      
        
          ∑
          
            k
            =
            1
          
          K
        
        ⁢
        
          
            h
            
              p
              ⁢
              
                
              
              ⁢
              k
            
          
          ⁢
          
            h
            qk
            *
          
        
      
      
        
          
        
        ⁢
        
          
          
            
              ∑
              
                k
                =
                1
              
              K
            
            ⁢
            
              
            
            ⁢
            
              
                h
                
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  k
                
              
              ⁢
              
                h
                qk
                *
              
            
          
          
        
      
    
  
  ,
  
    
      where 
      p
    
    =
    1
  
  ,
  2
  ,
  ...
  ⁢
  
    
  
  ,
  
    
  
  ⁢
  
    
      K
      ⁢
      
        
      
      ⁢
      
        and
      
      ⁢
      
        
      
      ⁢
      q
    
    =
    1
  
  ,
  2
  ,
  ...
  ⁢
  
    
  
  ,
  
    
  
  ⁢
  
    L
    .

7. The system as recited in claim 2, wherein the receiver is configured to sum the L weighted, received versions of the symbol c to thereby form an estimated version, ĉ, of the symbol c.

8. The system, as recited in claim 7, wherein an overall signal-to-noise ratio of the estimated version ĉ is set to a maximum value.

9. The system, as recited in claim 7, wherein an overall signal-to-noise ratio (SNR) of the estimated version ĉ is




  
    γ
    =
    
      
        
          a
          2
        
        L
      
      ⁢
      
        γ
        o
      
    
  
  ,
  
    
      
        where
      
      ⁢
      
        
      
      ⁢
      a
    
    =
    
      
        (
        
          
            ∑
            
              p
              =
              1
            
            L
          
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                q
                =
                1
              
              L
            
            ⁢
            
              
            
            ⁢
            
              
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  K
                
                ⁢
                
                  
                
                ⁢
                
                  
                    h
                    
                      p
                      ⁢
                      
                        
                      
                      ⁢
                      k
                    
                  
                  ⁢
                  
                    h
                    qk
                    *
                  
                
              
              
            
          
        
        )
      
      
        1
        /
        2
      
    
  



and γo is the average SNR for the case of a single transmitting antenna.

10. A system comprising:
K transmit antennas;
L receive antennas;
a transmitter configured to weight a symbol c and configured to transmit respective weighted versions of the symbol c on corresponding antennas of the K transmit antennas; and
a receiver configured to receive from the L receive antennas, L received versions of the symbol c transmitted by the K transmit antennas and configured to weight the L received versions of the symbol and configured to sum the L weighted, received versions of the symbol to thereby form an estimated version, ĉ, of the symbol c,
wherein K is greater than one and L is greater than one and the overall signal-to-noise ratio (SNR) of the estimated version ĉ is





  
    γ
    =
    
      
        
          a
          2
        
        L
      
      ⁢
      
        γ
        o
      
    
  
  ,
  
    
      
        where
      
      ⁢
      
        
      
      ⁢
      a
    
    =
    
      
        (
        
          
            ∑
            
              p
              =
              1
            
            L
          
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                q
                =
                1
              
              L
            
            ⁢
            
              
            
            ⁢
            
              
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  K
                
                ⁢
                
                  
                
                ⁢
                
                  
                    h
                    
                      p
                      ⁢
                      
                        
                      
                      ⁢
                      k
                    
                  
                  ⁢
                  
                    h
                    qk
                    *
                  
                
              
              
            
          
        
        )
      
      
        1
        /
        2
      
    
  
  ,



γo is the average SNR for the case of a single transmitting antenna, and
hpq is the element indexed by p and q of a channel estimate matrix H having K×L elements.

11. The system, as recited in claim 10, wherein the transmitter weights the symbol c by a vector of K weighting factors, the vector of K weighting factors being proportional to [gH]H, where g is a vector having L components and the superscript H is the Hermitian operator.

12. The system, as recited in claim 10, wherein the receiver weights L received versions of the symbol by the vector g the elements of vector g satisfying




  
    
      (
      
        
          g
          p
        
        ⁢
        
          g
          q
          *
        
      
      )
    
    =
    
      
        
          ∑
          
            k
            =
            1
          
          K
        
        ⁢
        
          
            h
            
              p
              ⁢
              
                
              
              ⁢
              k
            
          
          ⁢
          
            h
            qk
            *
          
        
      
      
        
          
        
        ⁢
        
          
          
            
              ∑
              
                k
                =
                1
              
              K
            
            ⁢
            
              
            
            ⁢
            
              
                h
                
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  k
                
              
              ⁢
              
                h
                qk
                *
              
            
          
          
        
      
    
  
  ,
  
    
      where 
      p
    
    =
    1
  
  ,
  2
  ,
  ...
  ⁢
  
    
  
  ,
  
    
  
  ⁢
  
    
      K
      ⁢
      
        
      
      ⁢
      
        and
      
      ⁢
      
        
      
      ⁢
      q
    
    =
    1
  
  ,
  2
  ,
  ...
  ⁢
  
    
  
  ,
  
    
  
  ⁢
  L
  ,
  
    and
  



hpq is an element indexed by p and q of a channel estimate matrix H having K×L elements.