Abstract:
An embodiment communication system and method for using multiple-input multiple-output use. Kronecker model to determine a symbol coding formulation that achieves ergodic capacity for high signal-to-noise ratios (SNRs). The lower bound on this capacity is achieved by input signals in the form of the product of an isotropically distributed random Grassmannian component and a deterministic component comprising the eigenvectors and the inverse of the eigenvalues of the transmitter covariance matrix.

Description:
[0001]    This application claims the benefit of U.S. Provisional Application No. 61/991,399, filed on May 9, 2014, entitled “The Ergodic High SNR Capacity of the Kronecker Spatially-Correlated Non-Coherent MIMO Channel Within an SNR-Independent Gap,” which application is hereby incorporated herein by reference. 
     
    
     TECHNICAL FIELD 
       [0002]    The present invention relates to a system and method for wireless communication and in particular embodiments relates to a system and method for multiple-input multiple-output communication. 
       BACKGROUND 
       [0003]    Multiple-input multiple-output (MIMO) communication systems provide a spectrally-efficient means for communicating over wireless channels. Such systems can be classified into coherent systems in which the receiver has access to reliable channel state information (CSI) and non-coherent systems in which this information is not available at either the transmitter or the receiver. Coherent systems are more straightforward to design than their non-coherent counterparts. However, the analysis of coherent systems usually does not account for the cost of the resources that have to be expended to acquire reliable CSI. Ignoring this cost in static and slow fading scenarios is generally tolerable. Ignoring it in fast fading scenarios, which arise in wireless systems with high mobility, can be rather misleading. A mobile device that is traveling at 60 miles per hour has a fading coherence of about 3 ms. If the mobile system is operated at 1.9 GHz and has a symbol rate of 30K symbols per second, 3 ms allows for 50-100 symbol periods. If several training symbols per antenna are needed, only a few antennas may be trained during a fading coherence period. 
         [0004]    In order to alleviate the cost of acquiring reliable CSI in fast fading scenarios, it is often believed to be desirable to use non-coherent signaling strategies, that is, strategies that do not require the receiver to have access to CSI. It has been shown that, for spatially-white MIMO channels with independent identically distributed (i.i.d.) block Rayleigh fading coefficients, input matrices that achieve the ergodic capacity can be expressed in the form of an isotropically distributed unitary component and a diagonal component with non-negative entries (Hochwald and Marzetta, “Unitary space-time modulation multiple-antenna communications in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 46, pp. 543-564, March 2000, which is hereby incorporated herein in its entirety by reference). A closed form expression for the asymptotically high signal-to-noise ratio (SNR) ergodic capacity of this type of channel has been determined for the cases where the coherence time of the channel exceeds and is below a certain threshold. When the coherence time of the channel exceeds the threshold, it was shown that, for sufficiently high SNRs, it is optimal, from a capacity perspective, for the diagonal component to be a scaled identity matrix and for the unitary matrix to be isotropically distributed on the so-called Gras smann manifold and to have the number of antennas transmitting less than or equal to half the number of symbol durations spanned by the coherence interval of the channel. Various techniques for designing rate-efficient Grassmannian constellations have been shown (L. Zheng and D. N. C. Tse, “Communication on the Grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel,” IEEE Trans. Inf. Theory, vol. 48, pp. 359-383, February 2002, which is hereby incorporated herein in its entirety by reference). For the case where the SNR is below the threshold, the unitary component was shown to have the same distribution as when above the threshold, but the optimal distribution for the diagonal component was shown to be that of the square root of the eigenvalues of a beta-distributed random matrix. In contrast with the high SNR cases, for low SNRs, it has been shown that the optimal diagonal component has at most one non-zero entry at any given time. 
         [0005]    The proximity of antennas to each other in practical wireless systems renders the realization of spatially-white channels generally difficult, even when the distance between antennas exceeds multiple wavelengths. This proximity induces correlation between the random entries of the channel matrix at the transmitter and receiver sides. A convenient means for characterizing this correlation mathematically is the so-called Kronecker model, whereby the spatially-correlated channel matrix is represented by left and right multiplication of a spatially-white channel matrix with transmitter and receiver covariance matrices, respectively. Fortunately, these matrices are dominated by the locations, geometries and beam patterns of the antennas at the receiver and the transmitter. Being characteristic of the wireless devices on which the antennas are mounted, these matrices can be accurately estimated and made available to the transmitter and the receiver. The impact of these matrices on the proper signaling methodology and the achievable rate can be quite significant. For the case in which the correlation between the channel coefficients is described by the Kronecker model, it has been shown that for spatially white channels, increasing the number of transmit antennas is almost surely beneficial. It has also been shown that the optimal input covariance can be expressed as the product of an isotropically distributed unitary component, a diagonal component with non-negative entries and a deterministic component comprising the eigenvectors of the transmitter covariance matrix. 
       SUMMARY 
       [0006]    An embodiment of the disclosure is a communication system including a plurality of antennas. A precoder is included to combine data to be transmitted with a coding matrix to form transmission symbols. The coding matrix is formed by combining a Grassmannian matrix comprising a distribution on a Gras smannian manifold with a function of a transmitter covariance matrix. A transmitter is connected to the plurality of antennas and is configured to transmit the transmission symbols. 
         [0007]    The coding matrix may be formed according to the formula 
         [0000]    
       
      
       X=Q 
       X 
       DU 
       A 
       † 
      
     
         [0000]    where, Q X  is an isotropically distributed unitary matrix; D is a random diagonal matrix with non-negative entries; and U A  is a matrix containing the eigenvectors of the transmitter covariance matrix, in this case the † indicates that it is the conjugate transpose of that matrix. The random diagonal matrix may be determined according to the formula 
         [0000]    
       
         
           
             D 
             = 
             
               
                 
                   TP 
                   
                     Tr 
                      
                     
                         
                     
                      
                     
                       Λ 
                       A 
                       
                         - 
                         1 
                       
                     
                   
                 
               
                
               
                 Λ 
                 A 
                 
                   
                     - 
                     1 
                   
                   / 
                   2 
                 
               
             
           
         
       
     
         [0000]    where, T is the coherence time, P is the transmit power, Tr is the trace function; and Λ A  is a matrix composed of the eigenvalues of the transmitter covariance matrix. 
         [0008]    Another embodiment is a communication system including a base station with a plurality of antennas configured to transmit to a plurality of mobile receivers. The base station includes a precoder that combines data to be transmitted with a coding matrix to produce transmission symbols. The coding matrix is formed by combining a Grassmannian matrix comprising a distribution on a Gras smannian manifold with a function of a transmitter covariance matrix. A transmitter connected to the plurality of antennas and configured to transmit the transmission symbols. 
         [0009]    The coding matrix may be formed according to the formula 
         [0000]    
       
      
       X=Q 
       X 
       DU 
       A 
       † 
      
     
         [0000]    where, Q X  is an isotropically distributed unitary matrix; D is a random diagonal matrix with non-negative entries; and U A  is a matrix containing the eigenvectors of the transmitter covariance matrix, in this case the † indicates that it is the conjugate transpose of that matrix. The random diagonal matrix may be determined according to the formula 
         [0000]    
       
         
           
             D 
             = 
             
               
                 
                   TP 
                   
                     Tr 
                      
                     
                         
                     
                      
                     
                       Λ 
                       A 
                       
                         - 
                         1 
                       
                     
                   
                 
               
                
               
                 Λ 
                 A 
                 
                   
                     - 
                     1 
                   
                   / 
                   2 
                 
               
             
           
         
       
     
         [0000]    where, T is the coherence time, P is the transmit power, Tr is the trace function; and Λ A  is the eigenvalues of the transmitter covariance matrix. 
         [0010]    Another embodiment is a communication method including combining data to be transmitted wirelessly using a plurality of antennas with a coding matrix formed by combining a Grassmannian matrix comprising a distribution on a Grassmannian manifold with a transmitter covariance matrix to produce transmission symbols, and then transmitting the transmission symbols. 
         [0011]    The coding matrix may be formed according to the formula 
         [0000]    
       
      
       X=Q 
       X 
       DU 
       A 
       † 
      
     
         [0000]    where, Q X  is an isotropically distributed unitary matrix; D is a random diagonal matrix with non-negative entries; and U A  is a matrix containing the eigenvectors of the transmitter covariance matrix, in this case the † indicates that it is the conjugate transpose of that matrix. The random diagonal matrix may be determined according to the formula 
         [0000]    
       
         
           
             D 
             = 
             
               
                 
                   TP 
                   
                     Tr 
                      
                     
                         
                     
                      
                     
                       Λ 
                       A 
                       
                         - 
                         1 
                       
                     
                   
                 
               
                
               
                 Λ 
                 A 
                 
                   
                     - 
                     1 
                   
                   / 
                   2 
                 
               
             
           
         
       
     
         [0000]    where, T is the coherence time, P is the transmit power, Tr is the trace function; and Λ A  is the eigenvalues of the transmitter covariance matrix. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
         [0012]    For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawing, in which: 
           [0013]      FIG. 1  is a diagram illustrating an example MIMO communications system; 
           [0014]      FIG. 2  is a block diagram of an implementation of a described embodiment; and 
           [0015]      FIG. 3  is a flow chart of an embodiment of the disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0016]    The structure, manufacture and use of the presently preferred embodiments are discussed in detail below. It should be appreciated, however, that the present invention provides many applicable inventive concepts that can be embodied in a wide variety of specific contexts. The specific embodiments discussed are merely illustrative of specific ways to make and use the invention, and do not limit the scope of the invention. 
         [0017]      FIG. 1  is a diagram of a basic MIMO system. Access point (AP)  10  is a base station including multiple antennas. As one example, cell site  10  may be configured as a NodeB in a UMTS cellular communication system. Cell site  10  has M antennas  12 . A particular transmission channel will use more than one of antennas  12  to take advantage of the MIMO transmission system. In the described embodiment, mobile station  14  will also include N antennas  16 . The following description discusses transmissions from cell site  10  to mobile station  14 . However, the principles of this embodiment are equally applicable to transmissions from mobile station  14  to cell site  10 , provided that mobile site  14  includes adequate processing capabilities to implement this embodiment. It should also be noted that the receiver of the MIMO transmission can also be a stationary receiver, and need not be a mobile station. 
         [0018]    In the following discussion, random and deterministic matrices will be denoted by boldface and regular face upper case letters, respectively. The m×m identity matrix will be denoted by I m , and the standard O(•) notation will be used to imply that g 1 (x)=O(g 2 (x)) if g 1 (x)/g 2 (x)→a as x→∞, a is a constant. 
         [0019]    A frequency-flat block Rayleigh fading channel in which the signals emitted from the transmit antennas are correlated and the signals impinging on the receiver antennas are also correlated can be used as a preliminary model. This situation arises in practice when the antenna elements are not sufficiently spaced for their signals to be statistically independent. Denoting the number of transmit and receive antennas by M and N, respectively, the channel matrix can be expressed as H=XA 1/2 H w B 1/2 , where AεC M×M  and BεC N×N  are the Hermitian positive semi-definite transmit and receive covariance matrices, respectively, and H w εC M×N  is a random matrix with zero-mean unit-variance i.i.d. circularly-symmetric complex Gaussian entries. To avoid degenerate scenarios, it can be assumed that both A and B are full rank, and other cases will be alluded to as necessary. 
         [0020]    In the block fading model, the channel assumes a statistically independent realization over each block of T channel uses and remains essentially constant within each block. This model is realistic in perfectly interleaved frequency-hopping systems and is useful in characterizing high mobility scenarios in current Long Term Evolution (LTE) cellular systems in which time-frequency resource blocks are assigned to each user during a coherence interval. In this case, the received signal matrix can be expressed as 
         [0000]        Y=XA   1/2   H   w   B   1/2   +V   (1)
 
         [0000]    where XεC T×M  represented the transmitted signal matrix, and YεC T×N  represents the additive noise matrix; the entries of V are i.i.d. zero mean unit variance circularly-symmetric complex Gaussian random variables. The matrices A and B are normalized so that Tr(A)=Tr(B)=1, and, assuming that the power budget of the transmitter P, there is the following power constraint: 
         [0000]        E{Tr ( XX   † )}≦TP  (2)
 
         [0021]    The matrices A and B can be estimated from the physical properties of the propagation environment and the antenna pattern, and can hence be assumed to be known. The communication scenario that is considered is non-coherent, thereby the matrix H w  in (1) is not known to either the transmitter or the receiver. 
         [0022]    Conditioned on X=X, the matrix Y in (1) is Gaussian distributed with 
         [0000]      vec( Y )=( B   1/2             XA   1/2 )vec( H   w )+vec( V )  (2.5)
 
         [0000]    where vec(•) is the operator that stacks the columns of the matrix argument on top of each other. Now the covariance matrix 
         [0000]        E {vec( Y )vec † ( Y )| X}=B             XAX   †   +I   NT   (3)
 
         [0023]    Hence, conditioned on X=X, the probability density function (PDF) of Y can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           Y 
                           | 
                           X 
                         
                         = 
                         X 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       exp 
                        
                       
                         ( 
                         
                           
                             - 
                             
                               
                                 vec 
                                 † 
                               
                                
                               
                                 ( 
                                 Y 
                                 ) 
                               
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   B 
                                   ⊗ 
                                   
                                     XAX 
                                     † 
                                   
                                 
                                 + 
                                 
                                   I 
                                   NT 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                            
                           
                             vec 
                              
                             
                               ( 
                               Y 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     
                       
                         π 
                         TN 
                       
                        
                       
                         det 
                          
                         
                           ( 
                           
                             
                               B 
                               ⊗ 
                               
                                 XAX 
                                 † 
                               
                             
                             + 
                             
                               I 
                               NT 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using this expression it is straightforward to verify that for any deterministic T×T unitary matrix Φ, p(ΦY|ΦX=ΦX). This along with Lemma 1 in Hochwald et al. (B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation multiple-antenna communications in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 46, pp. 543-564, March 2000, which is hereby incorporated herein in its entirety by reference) can be combined to show that the structure of the matrix X that achieves the ergodic non-coherent capacity of the channel in (1) can be expressed as 
         [0000]        X=Q   X   DU   A   †   (5)
 
         [0000]    Where Q X  is an isotropically distributed unitary matrix fulfilling the model of a Gras smannian manifold (See Absil et al. Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation. Acta Applicandae Mathematicae, 80(2):199-220, 2004, http://www.cis.upenn.edu/˜cis515/Diffgeom-Gras smann.Absil.pdf, which is hereby incorporated herein in its entirety by reference), D is a random diagonal matrix with non-negative entries and U A  is the matrix containing the eigenvectors of A, i.e. A=U A Λ A U A   † . 
         [0024]    An implementation of equation (5) can be used in an embodiment to provide ergodic capacity (or near ergodic capacity) in a non-coherent transmission system.  FIG. 2  is a block diagram of one such embodiment. The data to be transmitted 20 is provided to a precoder  22 . A code book based on X from equation (5) is generated in code matrix generator  24 . These codes are provided to precoder  22 , which combines the appropriate codes from the code matrix to generate precoded symbols  26 . Specific examples of precoding for several transmission standards are described in Kuo et al., Precoding Techniques for Digital Communication Systems (Springer Science+Business Media 2008), which is hereby incorporated herein in its entirety by reference. Those skilled in the art will appreciate that the precoder can combine matrices using any number of techniques including matrix multiplication, which may be constrained with objectives including the maintenance of orthogonality. These precoded symbols are used by transmitter  28 , which combines them with codes according to the transmission system being used (IS-95 also conventionally referred to as Code Division Multiple Access (CDMA), LTE, or other transmission standard) to prepare symbols for transmission. The transmitter then uses antenna  12  to transmit the symbols wirelessly to mobile unit  16 .  FIG. 3  is a flow diagram of the operation of the embodiment of  FIG. 2 . In step  101 , a covariance matrix is determined for the transmitter. In step  102 , the data to be transmitted is precoded with the covariance matrix and a Grassmannian matrix on a Grassmannian manifold as described herein. In step  104 , the precoded data is transmitted. 
         [0025]    In the current discussion, result of Equation (5) is used as a basis upon which an asymptotically tight expression for the ergodic high SNR non-coherent capacity of the channel model in (1), i.e. the capacity as P goes to infinity, is derived. 
         [0026]    For a given power budget P in (2), let C(P) denote the ergodic non-coherent capacity of the channel in (1). It can then be written: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       C 
                        
                       
                         ( 
                         P 
                         ) 
                       
                     
                     = 
                     
                       p 
                        
                       
                         ( 
                         X 
                         ) 
                       
                     
                   
                   , 
                   
                     
                       E 
                        
                       
                         
                           { 
                           
                             Tr 
                              
                             
                               ( 
                               
                                 XX 
                                 † 
                               
                               ) 
                             
                           
                           } 
                         
                         max 
                       
                     
                     ≤ 
                     
                       
                         TP 
                         
                           1 
                           T 
                         
                       
                        
                       
                         ( 
                         
                           
                             h 
                              
                             
                               ( 
                               Y 
                               ) 
                             
                           
                           - 
                           
                             h 
                              
                             
                               ( 
                               
                                 Y 
                                 | 
                                 X 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    C(P) can be evaluated as P goes to infinity.
 
Evaluating h(Y|X)
 
         [0027]    To evaluate the second term in (6) equation (3) can be used along with the fact that, condition on X, Y, is Gaussian distributed to write: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           h 
                            
                           
                             ( 
                             
                               Y 
                               | 
                               X 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             NT 
                              
                             
                                 
                             
                              
                             log 
                              
                             
                                 
                             
                              
                             π 
                           
                           + 
                           
                             E 
                              
                             
                               { 
                               
                                 log 
                                  
                                 
                                     
                                 
                                  
                                 
                                   det 
                                   ( 
                                   
                                     
                                       B 
                                       ⊗ 
                                       
                                         XAX 
                                         † 
                                       
                                     
                                     + 
                                     
                                       I 
                                       NT 
                                     
                                   
                                   ) 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             NT 
                              
                             
                                 
                             
                              
                             log 
                              
                             
                                 
                             
                              
                             π 
                           
                           + 
                           
                             
                               Σ 
                               
                                 i 
                                 = 
                                 1 
                               
                               M 
                             
                              
                             
                               Σ 
                               
                                 j 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             E 
                              
                             
                               { 
                               
                                 log 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         λ 
                                         
                                           B 
                                           j 
                                         
                                       
                                        
                                       
                                         λ 
                                         
                                           A 
                                           i 
                                         
                                       
                                        
                                       
                                         d 
                                         i 
                                         2 
                                       
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where in writing (7) invokes (5) and the eigen properties of the Kronecker product and used λ A     i    and λ B     j    denote the i-th and j-th eigenvalues of A and B, respectively, and d i  to denote the i-th entry of D. 
         [0028]    As the transmit power goes to infinity, it can be seen that (7) can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     h 
                      
                     
                       ( 
                       
                         Y 
                         | 
                         X 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       NT 
                        
                       
                           
                       
                        
                       log 
                        
                       
                           
                       
                        
                       π 
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                        
                       
                           
                       
                        
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           N 
                         
                          
                         
                             
                         
                          
                         
                           E 
                            
                           
                             { 
                             
                               log 
                                
                               
                                 ( 
                                 
                                   
                                     λ 
                                     
                                       B 
                                       j 
                                     
                                   
                                    
                                   
                                     λ 
                                     
                                       A 
                                       i 
                                     
                                   
                                    
                                   
                                     d 
                                     i 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             } 
                           
                         
                       
                     
                     + 
                     
                       
                         O 
                          
                         
                           ( 
                           
                             1 
                             / 
                             P 
                           
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         8 
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       NT 
                        
                       
                           
                       
                        
                       log 
                        
                       
                           
                       
                        
                       π 
                     
                     + 
                     
                       N 
                        
                       
                           
                       
                        
                       log 
                        
                       
                           
                       
                        
                       det 
                        
                       
                           
                       
                        
                       A 
                     
                     + 
                     
                       M 
                        
                       
                           
                       
                        
                       log 
                        
                       
                           
                       
                        
                       det 
                        
                       
                           
                       
                        
                       B 
                     
                     + 
                     
                       NE 
                        
                       
                         { 
                         
                           log 
                            
                           
                               
                           
                            
                           det 
                            
                           
                               
                           
                            
                           
                             D 
                             2 
                           
                         
                         } 
                       
                     
                     + 
                     
                       O 
                        
                       
                         ( 
                         
                           1 
                           / 
                           P 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    This approximation is valid when D is full rank and its entries scale with P. This condition will be shown to hold for the matrix D to achieve the lower bound on the high SNR ergodic capacity derived below when the number of transmit antennas is less than or equal to the number of receive antennas, i.e., M≦N and the transmit power, P, is sufficiently large. For simplicity, the remainder of this discussion will focus on the case of M=N.
 
Asymptotic Bound on h(Y)
 
         [0029]    A high SNR approximation of the first term in (6) is now used. Additionally, because the focus is on the case of M=N, the results in section IIIB of Zheng et al. (L. Zheng and D. N. C. Tse, “Communication on the Grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel,” IEEE Trans. Inf. Theory, vol. 48, pp. 359-383, February 2002, which is hereby incorporated herein in its entirety by reference) are also used. 
         [0030]    As P goes to infinity, the differential entropy, h(Y), is dominated by the differential entropy of X A   1/2 H w   B   1/2 . In particular, using the results in [11] produces 
         [0000]        h ( Y )= h ( XA   1/2   H   w   B   1/2 )+ O (1/ P )  (8.5)
 
         [0000]    Expressing the matrix A in terms of its eigen decomposition, and noting that H w  is isotropically distributed, it can be written that H w           U A   t H w  where “         ” denotes equality in distribution. Using this fact and invoking (5) yields 
         [0000]        h ( XA   1/2   H   w   B   1/2 )= h ( Q   X   DΛ   A   1/2   H   w   B   1/2 )= h (( B   1/2             I   T )vec( Q   X   DΛ   A   1/2   H   w ))  (8.75)
 
         [0031]    Using corollary 9.6.4 from Cover et al. (T. M. Cover and J. A., Thomas,  Elements of Information Theory , New York: Wiley, 1991, which is hereby incorporated herein in its entirety by reference), allows 
         [0000]        h (( B   1/2             I   T )vec( Q   X   DΛ   A   1/2   H   w ))= h ( Q   X   DΛ   A   1/2   H   w )+ T  log det  B   (9)
 
         [0032]    To evaluate h(Q X DΛ A   1/2 H w ) one can follow the methodology in Zheng et al. for the case of M=N to express this entropy in the coordinate system corresponding to the QR-decomposition. (As used herein, the QR decomposition (also called a QR factorization) is a linear algebra operation that is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R.) In particular, the matrix Q X DΛ A   1/2 H w  can be expressed as Q X R, where R is a random upper triangular matrix. Since Q X  is isotropically distributed, the matrix Q X DΛ A   1/2 H w            Q X ΨDΛ A   1/2 H w  where Ψ is an isotropically distributed M×M unitary matrix. Now, the matrix ΨDΛ A   1/2 H w  can be expressed as ΨR. Note that if the matrix DΛ A   1/2  is full rank, then so DΛ A   1/2 H w . It will later be shown that this assumption holds and use the QR decompositions to write 
         [0000]        h ( Q   X   DΛ   A   1/2   H   w )= h ( Q   X )+ h ( R )+ E {log  J   1 }  (10)
 
         [0000]    where J 1  is the Jacobian of the transformation from Cartesian to QR coordinates and is given by Zheng et al., J 1 =Π i=1   M R ii   2(T−i)+1 . Similarly, 
         [0000]        h (Ψ DΛ   A   1/2   H   w )= h (Ψ)+ h ( R )+ E {log  J   2 }  (11)
 
         [0000]    where J=Π i=1   M R ii   2(M−i)+i . 
         [0033]    Substituting for h(R) from (11) into (10) yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           h 
                           ( 
                           
                             
                               Q 
                               X 
                             
                              
                             D 
                              
                             
                                 
                             
                              
                             
                               Λ 
                               A 
                               
                                 1 
                                 2 
                               
                             
                              
                             
                               H 
                               w 
                             
                           
                           ) 
                         
                         = 
                           
                          
                         
                           
                             h 
                             ( 
                             
                               Ψ 
                                
                               
                                   
                               
                                
                               D 
                                
                               
                                   
                               
                                
                               
                                 Λ 
                                 A 
                                 
                                   1 
                                   2 
                                 
                               
                                
                               
                                 H 
                                 w 
                               
                             
                             ) 
                           
                           + 
                           
                             h 
                              
                             
                               ( 
                               
                                 Q 
                                 X 
                               
                               ) 
                             
                           
                           - 
                           
                             h 
                              
                             
                               ( 
                               Ψ 
                               ) 
                             
                           
                           + 
                           
                             E 
                              
                             
                               { 
                               
                                 log 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     J 
                                     1 
                                   
                                   / 
                                   
                                     J 
                                     2 
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             h 
                             ( 
                             
                               Ψ 
                                
                               
                                   
                               
                                
                               D 
                                
                               
                                   
                               
                                
                               
                                 Λ 
                                 A 
                                 
                                   1 
                                   2 
                                 
                               
                                
                               
                                 H 
                                 w 
                               
                             
                             ) 
                           
                           + 
                           
                             log 
                              
                             
                                
                               
                                 
                                    
                                   M 
                                 
                                  
                                 
                                   ( 
                                   
                                     ℂ 
                                     T 
                                   
                                   ) 
                                 
                               
                                
                             
                           
                           + 
                           
                             ( 
                             
                               T 
                               - 
                               M 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           E 
                            
                           
                             { 
                             
                               log 
                                
                               
                                   
                               
                                
                               det 
                                
                               
                                   
                               
                                
                               
                                 H 
                                 w 
                                 † 
                               
                                
                               
                                 D 
                                 2 
                               
                                
                               
                                 Λ 
                                 A 
                               
                                
                               
                                 H 
                                 w 
                               
                             
                             } 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where in writing the second equality the fact that Q X  and Ψ are isotropically distributed is used, whence h(Q X )−h(Ψ) is equal to the logarithm of the volume of the Grassmann manifold, 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     
                       
                          
                         M 
                       
                        
                       
                         ( 
                         
                           ℂ 
                           T 
                         
                         ) 
                       
                     
                      
                   
                   = 
                   
                     
                       
                         ∏ 
                         
                           i 
                           = 
                           
                             t 
                             - 
                             M 
                             + 
                             1 
                           
                         
                         T 
                       
                        
                       
                         2 
                          
                         
                           
                             
                               π 
                               i 
                             
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           ! 
                         
                       
                     
                     
                       
                         ∏ 
                         
                           i 
                           = 
                           1 
                         
                         T 
                       
                        
                       
                         2 
                          
                         
                           
                             
                               π 
                               i 
                             
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           ! 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12.5 
                   ) 
                 
               
             
           
         
       
     
         [0034]    To obtain an upper bound on h(ΨDΛ A   1/2 H w ), it is noted that the covariance 
         [0000]        E {( I   M             ΨDΛ   A   1/2 )vec( H   w )vec † ( H   w )( I   M           Λ A   1/2   DΨhu † )}= I   M             E{ΨD   2 Λ A Ψ † }  (13)
 
         [0000]    To compute the last expectation the following result is used from Hiai et al. (F. Hiai and D. Petz, “Asymptotic freeness almost everywhere for random matrices,” Acta. Sci. Math. (Szeged), vol. 66, pp. 809-834, 2000) and Tulino et al. (A. M. Tulino and S. Verd′u, “Random matrix theory and wireless communications,” Found. Trend. Commun. Inf. Theory, vol. 1, no. 1, pp. 1-182, 2004, which is hereby incorporated herein in its entirety by reference). 
       Lemma 1: 
       [0035]    For any isotropically distributed unitary matrix 
         [0000]    
       
         
           
             
               Φ 
               ∈ 
               
                 ℂ 
                 
                   n 
                   × 
                   m 
                 
               
             
             , 
             
               
                 E 
                  
                 
                   { 
                   
                     
                       
                          
                         Φ 
                          
                       
                       ij 
                       * 
                     
                      
                     
                       
                          
                         Φ 
                          
                       
                       kl 
                     
                   
                   } 
                 
               
               = 
               
                 
                   1 
                   n 
                 
                  
                 
                   δ 
                   ik 
                 
                  
                 
                   δ 
                   kl 
                 
               
             
             , 
           
         
       
     
         [0000]    where δ ik  is the Kronecker delta. 
         [0036]    Using this result yields, 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       { 
                       
                         Ψ 
                          
                         
                             
                         
                          
                         
                           D 
                           2 
                         
                          
                         
                           Λ 
                           A 
                         
                          
                         
                           Ψ 
                           
                             
                                 
                             
                             † 
                           
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       1 
                       M 
                     
                      
                     
                       Tr 
                        
                       
                         ( 
                         
                           E 
                            
                           
                             { 
                             
                               D 
                               2 
                             
                             } 
                           
                            
                           
                             Λ 
                             A 
                           
                         
                         ) 
                       
                     
                      
                     
                       I 
                       M 
                     
                   
                 
               
               
                 
                   ( 
                   13.5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Whence an upper bound on h(ΨDΛ A   1/2 H w ) can be readily obtained by applying Theorem 9.6.5 in Cover et al., which yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     h 
                      
                     
                       ( 
                       
                         Ψ 
                          
                         
                             
                         
                          
                         D 
                          
                         
                             
                         
                          
                         
                           Λ 
                           A 
                           
                             1 
                             / 
                             2 
                           
                         
                          
                         
                           H 
                           w 
                         
                       
                       ) 
                     
                   
                   ≤ 
                   
                     log 
                      
                     
                         
                     
                      
                     det 
                      
                     
                         
                     
                      
                     
                       
                         π 
                          
                         
                             
                         
                          
                         e 
                       
                       M 
                     
                      
                     
                       Tr 
                        
                       
                         ( 
                         
                           E 
                            
                           
                             { 
                             
                               D 
                               2 
                             
                             } 
                           
                            
                           
                             Λ 
                             A 
                           
                         
                         ) 
                       
                     
                      
                     
                       I 
                       
                         M 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   = 
                   
                     
                       M 
                       2 
                     
                      
                     log 
                      
                     
                         
                     
                      
                     
                       
                         π 
                          
                         
                             
                         
                          
                         e 
                       
                       M 
                     
                      
                     
                       Tr 
                        
                       
                         ( 
                         
                           E 
                            
                           
                             { 
                             
                               D 
                               2 
                             
                             } 
                           
                            
                           
                             Λ 
                             A 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and equality holds if and only if ΨDΛ A   1/2 H w  is Gaussian and distributed with covariance 
         [0000]    
       
         
           
             
               1 
               M 
             
              
             
               Tr 
                
               
                 ( 
                 
                   E 
                    
                   
                     { 
                     
                       D 
                       2 
                     
                     } 
                   
                    
                   
                     Λ 
                     A 
                   
                 
                 ) 
               
             
              
             
               
                 I 
                 
                   M 
                   2 
                 
               
               . 
             
           
         
       
     
         [0000]    In other words, equality holds if and only if the entries of the matrix are ΨDΛ A   1/2 H w  i.i.d. zero mean Gaussian random variables with variance 
         [0000]    
       
         
           
             
               1 
               M 
             
              
             
               
                 Tr 
                  
                 
                   ( 
                   
                     E 
                      
                     
                       { 
                       
                         D 
                         2 
                       
                       } 
                     
                      
                     
                       Λ 
                       A 
                     
                   
                   ) 
                 
               
               . 
             
           
         
       
     
       Upper and Lower Bounds on Capacity 
     Upper Bound: 
       [0037]    To obtain an upper bound on capacity, (2) is used to bound the right hand side of (15). In particular, let λ A     i   ≧ . . . ≧λ A     M   &gt;0, which yields, 
         [0000]    
       
         
           
             
               
                 
                   
                     h 
                      
                     
                       ( 
                       
                         Ψ 
                          
                         
                             
                         
                          
                         D 
                          
                         
                             
                         
                          
                         
                           Λ 
                           A 
                           
                             1 
                             / 
                             2 
                           
                         
                          
                         
                           H 
                           w 
                         
                       
                       ) 
                     
                   
                   ≤ 
                   
                     
                       M 
                       2 
                     
                      
                     log 
                      
                     
                         
                     
                      
                     
                       
                         π 
                          
                         
                             
                         
                          
                         eT 
                       
                       M 
                     
                      
                     
                       λ 
                       
                         A 
                         1 
                       
                     
                      
                     P 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Notice that this bound is not achievable unless A and E{D 2 } are scaled identities. 
         [0038]    Using (16) with (12) and (9) yields 
         [0000]    
       
         
           
             
               
                 
                   
                     h 
                      
                     
                       ( 
                       Y 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       
                         M 
                         2 
                       
                        
                       log 
                        
                       
                           
                       
                        
                       
                         
                           π 
                            
                           
                               
                           
                            
                           eT 
                         
                         M 
                       
                        
                       
                         λ 
                         
                           A 
                           1 
                         
                       
                        
                       P 
                     
                     + 
                     
                       log 
                        
                       
                           
                       
                        
                       
                          
                         
                           
                              
                             M 
                           
                            
                           
                             ( 
                             
                               ℂ 
                               T 
                             
                             ) 
                           
                         
                          
                       
                     
                     + 
                     
                       
                         ( 
                         
                           T 
                           - 
                           M 
                         
                         ) 
                       
                        
                       E 
                        
                       
                         { 
                         
                           log 
                            
                           
                               
                           
                            
                           det 
                            
                           
                               
                           
                            
                           
                             H 
                             w 
                             † 
                           
                            
                           
                             D 
                             2 
                           
                            
                           
                             Λ 
                             A 
                           
                            
                           
                             H 
                             w 
                           
                         
                         } 
                       
                     
                     + 
                     
                       T 
                        
                       
                           
                       
                        
                       log 
                        
                       
                           
                       
                        
                       det 
                        
                       
                           
                       
                        
                       B 
                     
                   
                 
               
               
                 
                   ( 
                   16.5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Using this bound with (8) and the fact that M=N, yields 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                      
                     
                       ( 
                       P 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       
                         M 
                          
                         
                           ( 
                           
                             
                               M 
                               T 
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                        
                       log 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       e 
                     
                     + 
                     
                       
                         
                           M 
                           2 
                         
                         T 
                       
                        
                       log 
                        
                       
                           
                       
                        
                       
                         TP 
                         M 
                       
                     
                     + 
                     
                       
                         M 
                         T 
                       
                        
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               2 
                             
                           
                           M 
                         
                          
                         
                           log 
                            
                           
                               
                           
                            
                           
                             
                               λ 
                               
                                 A 
                                 1 
                               
                             
                             
                               λ 
                               
                                 A 
                                 i 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               M 
                               T 
                             
                           
                         
                         ) 
                       
                        
                       E 
                        
                       
                         { 
                         
                           log 
                            
                           
                               
                           
                            
                           det 
                            
                           
                               
                           
                            
                           
                             D 
                             2 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         T 
                       
                        
                       log 
                        
                       
                          
                         
                           
                              
                             M 
                           
                            
                           
                             ( 
                             
                               ℂ 
                               T 
                             
                             ) 
                           
                         
                          
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             M 
                             T 
                           
                         
                         ) 
                       
                        
                       E 
                        
                       
                         { 
                         
                           log 
                            
                           
                               
                           
                            
                           det 
                            
                           
                               
                           
                            
                           
                             Λ 
                             A 
                           
                            
                           
                             H 
                             w 
                           
                            
                           
                             H 
                             w 
                             † 
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             M 
                             T 
                           
                         
                         ) 
                       
                        
                       log 
                        
                       
                           
                       
                        
                       det 
                        
                       
                           
                       
                        
                       B 
                     
                     + 
                     
                       O 
                        
                       
                         ( 
                         
                           1 
                           / 
                           P 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    When T≧2M, C(P) can be further bounded by using Jensen&#39;s inequality and the fact that log det(•) is a concave function. Doing so, yields E{log detD 2 }≦log det E{D 2 }, with equality if and only if D is deterministic. Subject to the power constraint in (2), the upper bound of this inequality can be further bounded by 
         [0000]    
       
         
           
             M 
              
             
                 
             
              
             log 
              
             
                 
             
              
             
               
                 TP 
                 M 
               
               . 
             
           
         
       
     
         [0039]    Hence, 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                      
                     
                       ( 
                       P 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       
                         M 
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               M 
                               T 
                             
                           
                           ) 
                         
                       
                        
                       log 
                        
                       
                           
                       
                        
                       
                         TP 
                         
                           π 
                            
                           
                               
                           
                            
                           eM 
                         
                       
                     
                     + 
                     
                       
                         M 
                         T 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             2 
                           
                           M 
                         
                          
                         
                           log 
                            
                           
                               
                           
                            
                           
                             
                               λ 
                               
                                 A 
                                 1 
                               
                             
                             
                               λ 
                               
                                 A 
                                 i 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         T 
                       
                        
                       log 
                        
                       
                           
                       
                        
                       
                          
                         
                           
                              
                             M 
                           
                            
                           
                             ( 
                             
                               ℂ 
                               T 
                             
                             ) 
                           
                         
                          
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             M 
                             T 
                           
                         
                         ) 
                       
                        
                       log 
                        
                       
                           
                       
                        
                       AB 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             M 
                             T 
                           
                         
                         ) 
                       
                        
                       E 
                        
                       
                         { 
                         
                           log 
                            
                           
                               
                           
                            
                           det 
                            
                           
                               
                           
                            
                           
                             H 
                             w 
                           
                            
                           
                             H 
                             w 
                             † 
                           
                         
                         } 
                       
                     
                     + 
                     
                       O 
                        
                       
                         ( 
                         
                           1 
                           / 
                           P 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
       Lower Bound: 
       [0040]    To obtain a lower bound on capacity, recalling that, for an arbitrary distribution of D, the right hand side of (14) is achievable if and only if the entries of the matrix ΨDΛ A   1/2 H w  are i.i.d. Gaussian distributed. This condition is satisfied if the matrix D is deterministic and given by 
         [0000]    
       
         
           
             
               D 
               = 
               
                 
                   
                     TP 
                     
                       Tr 
                        
                       
                           
                       
                        
                       
                         Λ 
                         A 
                         
                           - 
                           1 
                         
                       
                     
                   
                 
                  
                 
                   Λ 
                   A 
                   
                     
                       - 
                       1 
                     
                     / 
                     2 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    c.f. (2). Substituting this value of D in (15) yields 
         [0000]    
       
         
           
             
               h 
                
               
                 ( 
                 
                   Ψ 
                    
                   
                       
                   
                    
                   D 
                    
                   
                       
                   
                    
                   
                     Λ 
                     A 
                     
                       1 
                       / 
                       2 
                     
                   
                    
                   
                     H 
                     w 
                   
                 
                 ) 
               
             
             = 
             
               
                 M 
                 2 
               
                
               log 
                
               
                   
               
                
               
                 
                   π 
                    
                   
                       
                   
                    
                   eTP 
                 
                 
                   Tr 
                    
                   
                       
                   
                    
                   
                     Λ 
                     A 
                     
                       - 
                       1 
                     
                   
                 
               
             
           
         
       
     
         [0000]    Thereby, at setting of D, yields 
         [0000]    
       
         
           
             
               h 
                
               
                 ( 
                 Y 
                 ) 
               
             
             = 
             
               
                 
                   M 
                   2 
                 
                  
                 log 
                  
                 
                     
                 
                  
                 π 
                  
                 
                     
                 
                  
                 e 
               
               + 
               
                 log 
                  
                 
                    
                   
                     
                        
                       M 
                     
                      
                     
                       ( 
                       
                         ℂ 
                         T 
                       
                       ) 
                     
                   
                    
                 
               
               + 
               
                 
                   ( 
                   
                     T 
                     - 
                     M 
                   
                   ) 
                 
                  
                 E 
                  
                 
                   { 
                   
                     log 
                      
                     
                         
                     
                      
                     det 
                      
                     
                         
                     
                      
                     
                       H 
                       w 
                     
                      
                     
                       H 
                       w 
                       † 
                     
                   
                   } 
                 
               
               + 
               
                 MT 
                  
                 
                     
                 
                  
                 log 
                  
                 
                     
                 
                  
                 
                   TP 
                   
                     Tr 
                      
                     
                         
                     
                      
                     
                       Λ 
                       A 
                       
                         - 
                         1 
                       
                     
                   
                 
               
               + 
               
                 T 
                  
                 
                     
                 
                  
                 log 
                  
                 
                     
                 
                  
                 det 
                  
                 
                     
                 
                  
                 B 
               
               + 
               
                 O 
                  
                 
                   ( 
                   
                     1 
                     / 
                     P 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             and 
           
         
       
       
         
           
             
                 
             
              
             
               
                 h 
                  
                 
                   ( 
                   
                     Y 
                     | 
                     X 
                   
                   ) 
                 
               
               = 
               
                 
                   MT 
                    
                   
                       
                   
                    
                   log 
                    
                   
                       
                   
                    
                   π 
                 
                  
                 
                     
                 
                 + 
                 
                   M 
                    
                   
                       
                   
                    
                   log 
                    
                   
                       
                   
                    
                   det 
                    
                   
                       
                   
                    
                   B 
                 
                 + 
                 
                   
                     M 
                     2 
                   
                    
                   log 
                    
                   
                       
                   
                    
                   
                     TP 
                     
                       Tr 
                        
                       
                           
                       
                        
                       
                         Λ 
                         A 
                         
                           - 
                           1 
                         
                       
                     
                   
                 
                 + 
                 
                   O 
                    
                   
                     ( 
                     
                       1 
                       / 
                       P 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0041]    Since this setting of D is not necessarily optimal, it can be written 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                      
                     
                       ( 
                       P 
                       ) 
                     
                   
                   ≥ 
                   
                     
                       
                         M 
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               M 
                               T 
                             
                           
                           ) 
                         
                       
                        
                       log 
                        
                       
                           
                       
                        
                       
                         TP 
                         
                           π 
                            
                           
                               
                           
                            
                           e 
                            
                           
                               
                           
                            
                           Tr 
                            
                           
                               
                           
                            
                           
                             Λ 
                             A 
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         T 
                       
                        
                       log 
                        
                       
                          
                         
                           
                              
                             M 
                           
                            
                           
                             ( 
                             
                               ℂ 
                               T 
                             
                             ) 
                           
                         
                          
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             M 
                             T 
                           
                         
                         ) 
                       
                        
                       log 
                        
                       
                           
                       
                        
                       det 
                        
                       
                           
                       
                        
                       B 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             M 
                             T 
                           
                         
                         ) 
                       
                        
                       E 
                        
                       
                         { 
                         
                           log 
                            
                           
                               
                           
                            
                           det 
                            
                           
                               
                           
                            
                           
                             H 
                             w 
                           
                            
                           
                             H 
                             w 
                             † 
                           
                         
                         } 
                       
                     
                     + 
                     
                       O 
                        
                       
                         ( 
                         
                           1 
                           / 
                           P 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
       Bounds on Gap to Capacity 
       [0042]    Comparing (18) to (19), it can be seen that setting 
         [0000]    
       
         
           
             D 
             = 
             
               
                 
                   TP 
                   
                     Tr 
                      
                     
                         
                     
                      
                     
                       Λ 
                       A 
                       
                         - 
                         1 
                       
                     
                   
                 
               
                
               
                 Λ 
                 A 
                 
                   
                     - 
                     1 
                   
                   / 
                   2 
                 
               
             
           
         
       
     
         [0000]    yields a gap to capacity of: 
         [0000]    
       
         
           
             Δ 
             = 
             
               
                 
                   ( 
                   
                     T 
                     - 
                     
                       2 
                        
                       M 
                     
                   
                   ) 
                 
                  
                 log 
                  
                 
                     
                 
                  
                 det 
                  
                 
                     
                 
                  
                 A 
               
               + 
               
                 
                   M 
                   2 
                 
                  
                 log 
                  
                 
                     
                 
                  
                 
                   λ 
                   
                     A 
                     1 
                   
                 
               
               + 
               
                 
                   M 
                    
                   
                     ( 
                     
                       T 
                       - 
                       M 
                     
                     ) 
                   
                 
                  
                 log 
                  
                 
                   
                     
                         
                     
                      
                     
                       TrA 
                       
                         - 
                         1 
                       
                     
                   
                   M 
                 
               
             
           
         
       
     
         [0000]    It should be noted that when 
         [0000]    
       
         
           
             
               A 
               = 
               
                 
                   1 
                   M 
                 
                  
                 
                   I 
                   M 
                 
               
             
             , 
           
         
       
     
         [0000]    Δ=0. Otherwise Δ is strictly greater than zero, and an upper bound on it can be derived as follows. Let κ be the condition number of A; i.e. κ=λ A     i   /λ A     M   . Hence, 
         [0000]    
       
         
           
             
               
                 
                   Δ 
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               M 
                               T 
                             
                           
                         
                         ) 
                       
                        
                       log 
                        
                       
                           
                       
                        
                       
                         Π 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                        
                       
                         
                           λ 
                           
                             A 
                             i 
                           
                         
                         
                           λ 
                           
                             A 
                             1 
                           
                         
                       
                     
                     + 
                     
                       
                         M 
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               M 
                               T 
                             
                           
                           ) 
                         
                       
                        
                       log 
                        
                       
                           
                       
                        
                       
                         Σ 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                        
                       
                         
                           λ 
                           
                             A 
                             1 
                           
                         
                         
                           λ 
                           
                             A 
                             i 
                           
                         
                       
                     
                     - 
                     
                       
                         M 
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               M 
                               T 
                             
                           
                           ) 
                         
                       
                        
                       log 
                        
                       
                           
                       
                        
                       M 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0043]    To obtain a bound on Δ, note that, because λ A     i   ≧ . . . ≧λ A     M   &gt;0, the first term in (20) is non-positive, and the argument of the logarithm in the second term is bounded by M κ . Using these observations, results in 
         [0000]    
       
         
           
             
               
                 
                   Δ 
                   ≤ 
                   
                     
                       M 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             M 
                             T 
                           
                         
                         ) 
                       
                     
                      
                     log 
                      
                     
                         
                     
                      
                     κ 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0044]    Hence, it has been shown that the upper and lower bounds obtained on capacity are within a gap proportional to the logarithm of the condition number of the transmitter covariance matrix, and that that gap does not depend on the SNR or the receiver covariance matrix. 
       Theorem 1 
       [0045]    For the communication scenario described in (1) and (2), when T≧2M and M=N, the ergodic high SNR capacity C(P) satisfies: 
         [0000]    
       
         
           
             
               
                 
                   
                     log 
                      
                     
                         
                     
                      
                     
                       TP 
                       
                         π 
                          
                         
                             
                         
                          
                         eTr 
                          
                         
                             
                         
                          
                         
                           Λ 
                           A 
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                   ≤ 
                   
                     
                       
                         C 
                          
                         
                           ( 
                           P 
                           ) 
                         
                       
                       - 
                       c 
                     
                     
                       M 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             M 
                             / 
                             T 
                           
                         
                         ) 
                       
                     
                   
                   ≤ 
                   
                     log 
                      
                     
                         
                     
                      
                     
                       
                         TP 
                          
                         
                             
                         
                          
                         κ 
                       
                       
                         π 
                          
                         
                             
                         
                          
                         e 
                          
                         
                             
                         
                          
                         Tr 
                          
                         
                             
                         
                          
                         
                           Λ 
                           A 
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22.5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             c 
             = 
             
               
                 
                   1 
                   T 
                 
                  
                 log 
                  
                 
                     
                 
                  
                 
                    
                   
                     
                       G 
                       M 
                     
                      
                     
                       ( 
                       
                         C 
                         T 
                       
                       ) 
                     
                   
                    
                 
               
               + 
               
                 
                   ( 
                   
                     1 
                     - 
                     
                       M 
                       T 
                     
                   
                   ) 
                 
                  
                 log 
                  
                 
                     
                 
                  
                 det 
                  
                 
                     
                 
                  
                 B 
               
               + 
               
                 
                   ( 
                   
                     1 
                     - 
                     
                       M 
                       T 
                     
                   
                   ) 
                 
                  
                 E 
                  
                 
                   { 
                   
                     log 
                      
                     
                         
                     
                      
                     det 
                      
                     
                         
                     
                      
                     
                       H 
                       w 
                     
                      
                     
                       H 
                       w 
                       t 
                     
                   
                   } 
                 
               
               + 
               
                 
                   O 
                    
                   
                     ( 
                     
                       1 
                       / 
                       P 
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
         [0000]    Furthermore, the lower bound on C(P) is achieved by input signal matrices of the from in (5) with 
         [0000]    
       
         
           
             D 
             = 
             
               
                 
                   TP 
                   
                     Tr 
                      
                     
                         
                     
                      
                     
                       Λ 
                       A 
                       
                         - 
                         1 
                       
                     
                   
                 
               
                
               
                 
                   Λ 
                   A 
                   
                     
                       - 
                       1 
                     
                     / 
                     2 
                   
                 
                 . 
               
             
           
         
       
     
       Effect of Transmitter Covariance Condition Number on Achievable Rate 
       [0046]    The rate on the right hand side of (19) is achievable using the strategy outlined in Theorem 1. That theorem shows that this rate is within 
         [0000]    
       
         
           
             
               M 
                
               
                 ( 
                 
                   1 
                   - 
                   
                     M 
                     T 
                   
                 
                 ) 
               
             
              
             log 
              
             
                 
             
              
             κ 
           
         
       
     
         [0000]    of the channel capacity. To investigate the effect of increasing κ on this rate, derive upper and lower bounds on that rate and it will shown that both bounds decrease with ic, indicating that high correlation at the transmitter can be deleterious to the achievable rate if the signaling strategy of Theorem 1 were to be used. 
         [0047]    The transmitter covariance affects the right hand side of (19) through log TrΛ A   −1 . Hence, upper and lower bounds on the achievable rate can be derived by obtaining lower and upper bounds on log TrΛ A   −1 , respectively. 
       Lemma 2 
       [0048]    Any positive definite matrix A, satisfies 
         [0000]    
       
         
           
             
               
                 
                   
                     log 
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         M 
                         - 
                         1 
                         + 
                         κ 
                       
                       ) 
                     
                   
                   ≤ 
                   
                     log 
                      
                     
                         
                     
                      
                     Tr 
                      
                     
                         
                     
                      
                     
                       Λ 
                       A 
                       
                         - 
                         1 
                       
                     
                   
                   ≤ 
                   
                     log 
                      
                     
                       ( 
                       
                         
                           2 
                            
                           
                             ( 
                             
                               M 
                               - 
                               1 
                             
                             ) 
                           
                         
                         + 
                         
                           1 
                           κ 
                         
                         + 
                         
                           
                             
                               ( 
                               
                                 M 
                                 - 
                                 1 
                               
                               ) 
                             
                             2 
                           
                            
                           κ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Proof: To prove the first inequality, write: 
         [0000]    
       
         
           
             
               
                 
                   
                     log 
                      
                     
                         
                     
                      
                     
                       Tr 
                       A 
                       
                         - 
                         1 
                       
                     
                   
                   = 
                     
                    
                   
                     log 
                      
                     
                       ( 
                       
                         
                           1 
                           
                             λ 
                             
                               A 
                               M 
                             
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 λ 
                                 
                                   A 
                                   M 
                                 
                               
                               
                                 λ 
                                 
                                   A 
                                   1 
                                 
                               
                             
                             + 
                             … 
                             + 
                             
                               
                                 λ 
                                 
                                   A 
                                   M 
                                 
                               
                               
                                 λ 
                                 
                                   A 
                                   
                                     M 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                               
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   ≥ 
                     
                    
                   
                     log 
                      
                     
                       ( 
                       
                         κ 
                          
                         
                           ( 
                           
                             
                               1 
                               κ 
                             
                             + 
                             … 
                             + 
                             
                               1 
                               κ 
                             
                              
                             
                                 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     log 
                      
                     
                       ( 
                       
                         M 
                         - 
                         1 
                         + 
                         κ 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0000]    To prove the second inequality write 
         [0000]    
       
         
           
             
               log 
                
               
                   
               
                
               Tr 
                
               
                   
               
                
               
                 Λ 
                 A 
                 
                   - 
                   1 
                 
               
             
             = 
             
               
                 log 
                  
                 
                   ( 
                   
                     
                       1 
                       
                         λ 
                         
                           A 
                           1 
                         
                       
                     
                      
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             λ 
                             
                               A 
                               1 
                             
                           
                           
                             λ 
                             
                               A 
                               2 
                             
                           
                         
                         + 
                         … 
                         + 
                         
                           
                             λ 
                             
                               A 
                               1 
                             
                           
                           
                             λ 
                             
                               A 
                               M 
                             
                           
                         
                         + 
                         1 
                       
                       ) 
                     
                   
                   ) 
                 
               
               ≤ 
               
                 log 
                  
                 
                   ( 
                   
                     
                       
                         
                           λ 
                           
                             A 
                             1 
                           
                         
                         + 
                         … 
                         + 
                         
                           λ 
                           
                             A 
                             M 
                           
                         
                       
                       
                         λ 
                         
                           A 
                           1 
                         
                       
                     
                      
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             ( 
                             
                               M 
                               - 
                               1 
                             
                             ) 
                           
                            
                           κ 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
               ≤ 
               
                 log 
                  
                 
                   ( 
                   
                     
                       2 
                        
                       
                         ( 
                         
                           M 
                           - 
                           1 
                         
                         ) 
                       
                     
                     + 
                     
                       1 
                       κ 
                     
                     + 
                     
                       
                         
                           ( 
                           
                             M 
                             - 
                             1 
                           
                           ) 
                         
                         2 
                       
                        
                       κ 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0049]    This inequality holds with equality when M=2 and with strict inequality when M&gt;2 and the eigenvalues of A are distinct. 
         [0050]    It can be verified that both inequalities are monotonically increasing in κ. Hence, using the result of this lemma in the right hand side of (19), it can be seen that both the lower and upper bounds on the rates achievable by the strategy outlined in Theorem 1 are monotonically decreasing with the condition number of the transmitter covariance matrix, which indicates that the more ill-conditioned this matrix is, the less the rate that can be achieved with this strategy. This result has an intuitive explanation. The signaling strategy in Theorem 1 allocates more power to the weak eigenmodes of the channel. Hence, when the channel matrix is ill-conditioned, most of the power is allocated to that mode, thereby affecting the power available for transmission over the strong eigenmodes of the channel. 
         [0051]    In the above discussion the high-SNR ergodic capacity of spatially-correlated MIMO channels was discussed. Closed form expressions for a lower and an upper bound on this capacity were obtained and it was shown that difference between these bounds is monotonically decreasing in the condition number of the transmitter covariance matrix. In particular, an expression for the high-SNR ergodic capacity of the considered channel that is tight with a constant that depends solely on the signaling dimension and the condition number of the transmitter covariance matrix is obtained. 
         [0052]    The foregoing embodiments are merely intended for describing the technical solutions of the present invention other than limiting the present invention. Although the present invention is described in detail with reference to the foregoing embodiments, persons of ordinary skill in the art should understand that they may still make modifications to the technical solutions described in the foregoing embodiments or make equivalent replacements to some technical features thereof, without departing from the scope of the technical solutions of the embodiments of the present invention.