Abstract:
A downlink multi-user beamforming scheme for a network of coordinated transmission points where the beamforming weights and power allocation are determined to maximize a jointly-achievable SINR margin under per-transmitter power constraints and the constraint that each data stream is transmitted from a single transmission point.

Description:
[0001]    This application claims the benefit of U.S. Provisional Pat. App. No. 61/094,108, filed on Sep. 4, 2008, which is incorporated herein by this reference. 
     
    
     TECHNICAL FIELD 
       [0002]    The present invention relates to wireless communication systems. More specifically, the present invention relates to beamforming systems and methods. 
       BACKGROUND 
       [0003]    Multi-user multiple-input multiple-output (MIMO) transmission using a Grid-of-Beams (GoB) approach has been shown to be an attractive scheme for the downlink for emerging wireless systems. See e.g., IST-4-027756 WINNER II Deliverable D4.7.3, “Smart antenna based interference mitigation,” June 2007. In the GoB scheme, a grid of beams is created by using closely-spaced antennas at the base stations. Independent data streams are transmitted to terminals in geographical locations served by non-overlapping beams. A hallmark of this scheme is that it requires very little channel state information at the transmitter (CSIT), i.e. beam selection. 
         [0004]    The grid of beams approach relies on fixed beams. Beams can be steered by means of baseband signal processing, providing improved coverage and less interference. The problem of joint adaptive beamforming from a multi-antenna base station to multiple single-antenna terminals has been solved. See e.g., M. Schubert and H. Boche, “Solution of the multi-user beamforming problem with individual SINR constraints,”  IEEE Trans. VT, vol.  53, no. 1, January 2004 (“Schubert”). The beamformers and transmission powers are jointly adjusted to fulfill individual signal-to-interference-plus-noise ratio (SINR) requirements at the terminals. An algorithm is derived that maximizes the jointly-achievable SINR margin (over the SINR requirements) under sum transmission power constraint. A second algorithm is also derived that minimizes the sum transmission power while satisfying the set of SINR requirements. The algorithms require statistical information about the channel models at the transmitter. 
         [0005]    A distributed antenna system (DAS) architecture is being considered for IMT-Advanced systems. DAS differs from a conventional cellular architecture in that DAS cells are connected to a central processing unit (CPU) by means of a fast backhaul. Compared to a conventional cellular network, very high spectral efficiency is possible in a DAS network due to coherently-coordinated transmission from DAS cells in the downlink and joint reception at DAS cells in the uplink. However, coherently-coordinated transmission from DAS cells generally requires a large amount of CSIT which overburdens spectral resources. 
         [0006]    A solution to this problem has been suggested where an approach similar to the multi-user beamforming approach of Schubert has been adapted for a DAS network with multi-antenna transmission points (TP). The scheme only requires statistical information about the channel models at the TPs. 
         [0007]    In the suggested solution, the beamformers and transmission powers are iteratively determined to minimize the sum transmission power while fulfilling the SINR requirements. A problem with this approach is that the feasibility of the solution is not verified beforehand. Due to this, the iterative algorithm may result in an infeasible power allocation and may not converge. This problem has been addressed in Schubert by: first, finding the beamformers and power allocation that maximize the jointly-achievable SINR margin under sum transmission power constraint; second, determining if the SINR requirements are jointly achievable (SINR margin greater than unity); and third, finding the beamformers and power allocation that minimize the sum transmission power while fulfilling the SINR requirements. The solution that minimizes the sum transmission power can be used in a DAS network if the maximum transmission power is less than the corresponding individual power constraint. On the other hand, if the maximum transmission power is higher than the individual power constraint, the solution is considered infeasible. 
         [0008]    A problem with the solution suggested in Schubert is that the sum transmission power constraint, although suitable for single-cell transmission, is not very suitable for multi-cell transmission in a DAS network as each TP antenna is individually power-limited. The maximum jointly-achievable SINR margin obtained under sum power constraint is not meaningful for a DAS network. 
       SUMMARY 
       [0009]    In one aspect, the invention provides a downlink multiuser beamforming scheme for a network of coordinated transmission points where the beamforming weights and power allocation are determined to maximize a jointly-achievable SINR margin under per-transmitter power constraints and the constraint that each data stream is transmitted from a single transmission point. The SINR margin can be used to set appropriate transmission rates that can be achieved jointly in a network of coordinated transmission points. 
         [0010]    More specifically, in one aspect, the invention provides a method performed by a controller for determining a beam forming vector for use in transmitting data from a transmission point to a first mobile device connected to the transmission point. 
         [0011]    In some embodiments, this method includes the following steps: (A) determining a first set of beam forming vectors (e.g., by using equation 20, defined below, and an initial (first) set of virtual dual uplink transmit powers), the first set of beam forming vectors comprising a first beam forming vector for the first mobile device and a first beam forming vector for a second mobile device; (B) determining a first set of downlink transmit powers and a first maximum worst-case signal to interference-plus-noise ratio (SINR) margin using the first set of beam forming vectors, the first set of downlink transmit powers comprising a first downlink transmit power for the first mobile device and a first downlink transmit power for the second mobile device; (C) determining a second set of virtual dual uplink transmit powers using the first maximum worst-case SINR margin (e.g., using the first maximum worst-case SINR margin in solving Equation 19 (defined below)), the second set of virtual dual uplink transmit powers comprising a second virtual dual uplink transmit power for the first mobile device and a second virtual dual uplink transmit power for the second mobile device; (D) determining a second set of beam forming vectors using the second set of virtual dual uplink transmit powers, the second set of beam forming vectors comprising a second beam forming vector for the first mobile device and a second beam forming vector for the second mobile device; (E) determining a second set of downlink transmit powers and a second maximum worst-case SINR margin using the second set of beam forming vectors, the second set of downlink transmit powers comprising a second downlink transmit power for the first mobile device and a second downlink transmit power for the second mobile device; (F) determining whether the maximum worst-case SINR margin has converged using the first maximum worst-case SINR margin and the second maximum worst-case SINR margin; and (G) in response to determining in step (F) that the maximum worst-case SINR margin has converged, then using the second beam forming vector for the first mobile device to transmit the data to the first mobile device. 
         [0012]    Steps (A)-(G) may be performed periodically while the first mobile terminal is connected to a transmission point controlled by the said controller. 
         [0013]    In some embodiments, the step of determining whether the maximum worst-case SINR margin has converged comprises dividing the second maximum worst-case SINR margin by the first maximum worst-case SINR margin and determining whether the result is less than or equal to a predetermined threshold. 
         [0014]    In some embodiments, the method also includes: (i) determining a third set of virtual dual uplink transmit powers using the second maximum worst-case SINR margin; (ii) determining a third set of beam forming vectors using the third set of uplink transmit powers; (iii) determining a third set of downlink transmit powers and a third maximum worst-case SINR margin using the third set of beam forming vectors; and (iv) determining whether the maximum worst-case SINR margin has, converged using the second maximum worst-case SINR margin and the third maximum worst-case SINR margin, wherein steps (i)-(iv) are performed in response to determining in step (F) that the maximum worst-case SINR margin has not converged. 
         [0015]    In some embodiments, the step of determining the first set of downlink transmit powers and the first maximum worst-case signal SINR margin comprises: (i) assuming that a particular transmission point controlled by the controller transmits at a maximum power allowed under a per-transmitter power constraint (PTPC); (ii) using the assumption and the first set of beam forming vectors to determine a set of downlink transmit powers; (iii) using the determined set of downlink transmit powers to determine whether the PTPC is satisfied; and (iv) setting the first set of downlink transmit powers to the determined set of downlink transmit powers if it is determined that the PTPC is satisfied, otherwise assuming that a different transmission point controlled by the controller transmits at a maximum power allowed under the PTPC and repeating steps (ii)-(iv). In some embodiments, step (ii) comprises solving Equation 12 (defined below). 
         [0016]    In some embodiments, the step of determining the maximum worst-case SINR margin comprises using predetermined target SINR values to solve Equation 3. 
         [0017]    In some embodiments, the method also includes: receiving from the transmission point (1) signal information pertaining to a transmit correlation matrix associated with the first mobile device (e.g., the transmit correlation matrix itself or comprises information from which the transmit correlation matrix can be derived), (2) the noise information, and (3) the interference information, wherein the interference information pertains to a transmit correlation matrix of at least one second mobile device connected to the transmission point, and the step of determining the first set of beam forming vectors comprises using the signal information, noise information and interference information to determine the first set of beam forming vectors. 
         [0018]    In some embodiments, the first and second beam forming vectors are constrained to be zero for all but one transmission point. 
         [0019]    In another aspect, the present invention provides a controller for determining a beam forming vector for use in transmitting data to a first mobile device connected to an transmission point. In some embodiments, the controller includes: a transmitter and receiver for communicating with the transmission point; a data storage system storing software; and a data processing system for executing the software, and the software comprises: (A) computer instructions for determining a first set of beam forming vectors using a first set of virtual dual uplink transmit powers, wherein the first set of beam forming vectors comprises a first beam forming vector for the first mobile device and a first beam forming vector for a second mobile device; (B) computer instructions for determining a first set of downlink transmit powers and a first maximum worst-case signal to interference-plus-noise ratio (SINR) margin using the first set of beam forming vectors, wherein the first set of downlink transmit powers comprises a first downlink transmit power for the first mobile device and a first downlink transmit power for the second mobile device; (C) computer instructions for determining a second set of virtual dual up ink transmit powers using the first maximum worst-case SINR margin, wherein the first set of virtual dual uplink transmit powers comprises a first virtual dual uplink transmit power for the first mobile device and a first virtual dual uplink transmit power for the second mobile device; (D) computer instructions for determining a second set of beam forming vectors using the second set of virtual dual uplink transmit powers, wherein the second set of beam forming vectors comprises a second beam forming vector for the first mobile device and a second beam forming vector for the second mobile device; (E) computer instructions for determining a second set of downlink transmit powers and a second maximum worst-case SINR margin using the second set of beam forming vectors, wherein the second set of downlink transmit powers comprises a second downlink transmit power for the first mobile device and a second downlink transmit power for the second mobile device; and (F) computer instructions for determining whether the maximum worst-case SINR margin has converged using the first maximum worst-case SINR margin and the second maximum worst-case SINR margin. 
         [0020]    The above and other aspects and embodiments are described below with reference to the accompanying drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0021]    The accompanying drawings, which are incorporated herein and form part of the specification, illustrate various embodiments of the present invention and, together with the description, further serve to explain the principles of the invention and to enable a person skilled in the pertinent art to make and use the invention. In the drawings, like reference numbers indicate identical or functionally similar elements. 
           [0022]      FIG. 1  illustrates a system according to an embodiment of the invention. 
           [0023]      FIG. 2  is a flow chart illustrating a process according to some embodiments of the invention 
           [0024]      FIG. 3  is a flow chart illustrating a process according to some embodiments of the invention. 
           [0025]      FIG. 4  is a functional block diagram of a controller according to some embodiments of the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0026]    Referring now to  FIG. 1 ,  FIG. 1  illustrates a wireless communication system  100  with N transmission points (TPs)  104  and K mobile devices  102  (a.k.a., user equipments  102 ). Each TP has M t  transmit antennas  112  and each user equipment (UE) has M r  receive antennas  114 . In some embodiments, however, each TP has a single omni-directional antenna, a single sector antenna or a single antenna array. 
         [0027]    Let the M t ×M r  matrix H i,n  represent the frequency-non-selective fading channel between UE i and TP n. The M r ×NM t  matrix H i =[H i,1  H i,2  . . . H i,N ] represents the channel between UE i and all TPs. The downlink baseband signal model is given by 
         [0000]        y   i   =H   i   x+w   i   Eq. (1) 
         [0000]    where x is an NM t ×1 vector representing the signal transmitted from the transmit antennas of all TPs and w i  is a sample of additive white Gaussian noise with covariance matrix σ i   2 I. The transmitted signal x is given by 
         [0000]    
       
         
           
             
               x 
               _ 
             
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   K 
                 
                  
                 
                   
                     x 
                     _ 
                   
                   i 
                 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   K 
                 
                  
                 
                   
                     
                       p 
                       i 
                     
                   
                    
                   
                     
                       u 
                       _ 
                     
                     i 
                   
                    
                   
                     s 
                     i 
                   
                 
               
             
           
         
       
     
         [0000]    where s i  is a modulation symbol (drawn from a unit-variance symbol alphabet) which is transmitted to UE i using a beamformer or precoder u i  and using power p i . We assume that the signal for each UE is transmitted from a single TP. The beamforming vector is normalized to have unit power 
         [0000]      E[u i   H u i ]=1 for i=1,2, . . . , K. 
         [0000]    The TPs are separated by large distances (much greater than the signal wavelength) therefore their antennas are mutually uncorrelated. The transmit channel covariance matrix for UE i is given by 
         [0000]        R   i   =E[H   i   H   H   i ]=diag( R   i,1   ,R   i,2   , . . . , R   i,N ), 
         [0000]    where R i,n  is the covariance matrix of the channel between TP n and UE i, defined as 
         [0000]      R i,n =E[H i,n   H H i,n ]. 
         [0028]    Assuming that UE i knows its instantaneous channel matrix H i  and employs maximum ratio combining (MRC) at the receiver, its downlink SINR is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                      
                     
                         
                     
                      
                     I 
                      
                     
                         
                     
                      
                     N 
                      
                     
                         
                     
                      
                     
                       
                         R 
                         i 
                         DL 
                       
                       ( 
                       
                         U 
                         , 
                         
                           p 
                           _ 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         p 
                         i 
                       
                        
                       
                         
                           u 
                           _ 
                         
                         i 
                         H 
                       
                        
                       
                         R 
                         i 
                       
                        
                       
                         
                           u 
                           _ 
                         
                         i 
                       
                     
                     
                       
                         
                           ∑ 
                           
                             
                               k 
                               = 
                               1 
                             
                             , 
                             
                               k 
                               ≠ 
                               i 
                             
                           
                           K 
                         
                          
                         
                           
                             p 
                             k 
                           
                            
                           
                             
                               u 
                               _ 
                             
                             k 
                             H 
                           
                            
                           
                             R 
                             i 
                           
                            
                           
                             
                               u 
                               _ 
                             
                             k 
                           
                         
                       
                       + 
                       
                         σ 
                         i 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where U is the matrix of beamformers, given by 
         [0000]      U=[u 1 ,u 2 ,u K ] 
         [0000]    and p is the vector of downlink transmit powers, given by 
         [0000]      p=[p 1 ,p 2 , . . . , p K ]. 
         [0029]    Problem Statement 
         [0030]    If we let γ i  be the target SINR for UE i, for i=1, 2, . . . , K and define SINR i   DL /γ i  as the SINR margin for UE i, the target SINRs for all UEs can be achieved if the worst-case SINR margin is greater than unity. 
         [0031]    Consider maximizing the worst-case SINR margin over all possible beamformers and transmit powers under per-transmitter power constraints (PTPC), i.e. find 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                     DL 
                   
                   = 
                   
                     
                       max 
                       
                         U 
                         , 
                         
                           p 
                           _ 
                         
                       
                     
                      
                     
                       
                         min 
                         
                           1 
                           ≤ 
                           i 
                           ≤ 
                           K 
                         
                       
                        
                       
                         
                           S 
                            
                           
                               
                           
                            
                           I 
                            
                           
                               
                           
                            
                           N 
                            
                           
                               
                           
                            
                           
                             
                               R 
                               i 
                               DL 
                             
                             ( 
                             
                               U 
                               , 
                               
                                 p 
                                 _ 
                               
                             
                             ) 
                           
                         
                         
                           γ 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     3 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       s 
                       . 
                       t 
                       . 
                       
                           
                       
                        
                       
                         p 
                         n 
                         AP 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             
                               S 
                               n 
                             
                           
                         
                          
                         
                           p 
                           i 
                         
                       
                       ≤ 
                       
                         P 
                         max 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∀ 
                         n 
                       
                       = 
                       1 
                     
                     , 
                     2 
                     , 
                     
                       … 
                        
                       
                           
                       
                        
                       N 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where C DL  is the maximum worst-case SINR margin, S n  is the set of UEs connected (e.g., wirelessly connected) to TP n, P n   AP  is the total transmit power for TP n and P max  is the maximum transmit power per TP. 
         [0032]    Downlink Power Allocation 
         [0033]    Given a fixed beamformer matrix 
         [0000]    
       
         
           
             
               
                 U 
                 ~ 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         u 
                         ~ 
                       
                       _ 
                     
                     1 
                   
                   , 
                   
                     
                       
                         u 
                         ~ 
                       
                       _ 
                     
                     2 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     
                       
                         u 
                         ~ 
                       
                       _ 
                     
                     K 
                   
                 
                 ] 
               
             
             , 
           
         
       
     
         [0000]    we find a downlink power allocation that maximizes the worst-case SINR margin. 
         [0034]    Let           be a transmit power vector with the minimum L1 norm (or sum power) ∥{tilde over (p)}∥ that maximizes the worst-case SINR margin while satisfying PTPC. Then it can be shown that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           C 
                           DL 
                         
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               I 
                                
                               
                                   
                               
                                
                               N 
                                
                               
                                   
                               
                                
                               
                                 
                                   R 
                                   i 
                                   DL 
                                 
                                 ( 
                                 
                                   
                                     U 
                                     ~ 
                                   
                                   , 
                                   
                                     
                                       p 
                                       _ 
                                     
                                     ~ 
                                   
                                 
                                 ) 
                               
                             
                             
                               γ 
                               i 
                             
                           
                            
                           
                               
                           
                            
                           for 
                            
                           
                               
                           
                            
                           i 
                         
                         = 
                         1 
                       
                     
                     , 
                     2 
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     K 
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       p 
                       ~ 
                     
                     n 
                     AP 
                   
                   = 
                   
                     
                       
                         P 
                         max 
                       
                        
                       
                           
                       
                        
                       forsome 
                        
                       
                           
                       
                        
                       n 
                     
                     = 
                     
                       n 
                       0 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    (i.e. the same SINR margin is achieved by all UEs and the margin is achieved by having at least one TP transmit at the maximum power). 
         [0035]    Equation Eq. (5) can be written in matrix notation as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         ~ 
                       
                       _ 
                     
                      
                     
                       
                         
                           C 
                           DL 
                         
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                       
                         - 
                         1 
                       
                     
                   
                   = 
                   
                     
                       
                         D 
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                        
                       
                         Ψ 
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                        
                       
                         
                           p 
                           ~ 
                         
                         _ 
                       
                     
                     + 
                     
                       
                         D 
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                        
                       
                         σ 
                         _ 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where Ψ(U) is a matrix with elements given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       { 
                       
                         Ψ 
                          
                         
                           ( 
                           U 
                           ) 
                         
                       
                       } 
                     
                     
                       i 
                       , 
                       k 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 u 
                                 _ 
                               
                               k 
                               H 
                             
                              
                             
                               R 
                               i 
                             
                              
                             
                               
                                 u 
                                 _ 
                               
                               k 
                             
                           
                         
                         
                           
                             i 
                             ≠ 
                             k 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               i 
                               = 
                               k 
                             
                             , 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       D 
                        
                       
                         ( 
                         U 
                         ) 
                       
                     
                     = 
                     
                       Diag 
                        
                       
                         { 
                         
                           
                             
                               γ 
                               1 
                             
                             
                               
                                 
                                   u 
                                   _ 
                                 
                                 1 
                                 H 
                               
                                
                               
                                 R 
                                 1 
                               
                                
                               
                                 
                                   u 
                                   _ 
                                 
                                 1 
                               
                             
                           
                           , 
                           
                             
                               γ 
                               2 
                             
                             
                               
                                 
                                   u 
                                   _ 
                                 
                                 2 
                                 H 
                               
                                
                               
                                 R 
                                 2 
                               
                                
                               
                                 
                                   u 
                                   _ 
                                 
                                 2 
                               
                             
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             
                               γ 
                               K 
                             
                             
                               
                                 
                                   u 
                                   _ 
                                 
                                 K 
                                 H 
                               
                                
                               
                                 R 
                                 K 
                               
                                
                               
                                 
                                   u 
                                   _ 
                                 
                                 K 
                               
                             
                           
                         
                         } 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     σ 
                     _ 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             σ 
                             1 
                             2 
                           
                           , 
                           
                             σ 
                             2 
                             2 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             σ 
                             K 
                             2 
                           
                         
                         ] 
                       
                       T 
                     
                     . 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
         [0036]    Assuming that n 0  is known a priori, then using equation Eq. (6) in Eq. (7) we get 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       max 
                     
                      
                     
                       
                         
                           C 
                           DL 
                         
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                       
                         - 
                         1 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           1 
                           _ 
                         
                         
                           n 
                           o 
                         
                         T 
                       
                        
                       
                         D 
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                        
                       
                         Ψ 
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                        
                       
                         
                           p 
                           ~ 
                         
                         _ 
                       
                     
                     + 
                     
                       
                         
                           1 
                           _ 
                         
                         
                           n 
                           o 
                         
                         T 
                       
                        
                       
                         D 
                          
                         
                           ( 
                           
                             U 
                             ~ 
                           
                           ) 
                         
                       
                        
                       
                         σ 
                         _ 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where 1 n     0    is a K×1 vector with ones in positions iεS n     0    and zeros in all other positions. Equations Eq. (7) and Eq. (11) can be combined into the set of equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Φ 
                         ( 
                         
                           
                             U 
                             ~ 
                           
                           , 
                           
                             n 
                             0 
                           
                         
                         ) 
                       
                        
                       
                         
                           
                             p 
                             ~ 
                           
                           _ 
                         
                         ext 
                       
                     
                     = 
                     
                       
                         
                           
                             C 
                             DL 
                           
                            
                           
                             ( 
                             
                               U 
                               ~ 
                             
                             ) 
                           
                         
                         
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             p 
                             _ 
                           
                           ~ 
                         
                         ext 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     12 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     Φ 
                      
                     
                       ( 
                       
                         U 
                         , 
                         n 
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               D 
                                
                               
                                 ( 
                                 U 
                                 ) 
                               
                             
                              
                             
                               Ψ 
                                
                               
                                 ( 
                                 U 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               D 
                                
                               
                                 ( 
                                 U 
                                 ) 
                               
                             
                              
                             
                               σ 
                               _ 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 1 
                                 _ 
                               
                               n 
                               T 
                             
                              
                             
                               D 
                                
                               
                                 ( 
                                 U 
                                 ) 
                               
                             
                              
                             
                               
                                 Ψ 
                                  
                                 
                                   ( 
                                   U 
                                   ) 
                                 
                               
                               / 
                               
                                 P 
                                 max 
                               
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 _ 
                               
                               n 
                               T 
                             
                              
                             
                               D 
                                
                               
                                 ( 
                                 U 
                                 ) 
                               
                             
                              
                             
                               
                                 σ 
                                 _ 
                               
                               / 
                               
                                 P 
                                 max 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    and           is a (K+1)×1 vector, given by 
         [0000]    
       
         
           
             
               
                 
                   p 
                   _ 
                 
                 ~ 
               
               ext 
             
             = 
             
               
                 [ 
                 
                   
                     p 
                     ~ 
                   
                   1 
                 
                 ] 
               
               . 
             
           
         
       
     
         [0037]    From Eq. 12, it can be observed that the SINR margin C DL (Ũ) is a reciprocal eigen-value of the non-negative matrix Φ(Ũ, n 0 ) with           as the corresponding eigen-vector such that 
         [0000]    
       
         
           
             
               
                 { 
                 
                   
                     
                       p 
                       ~ 
                     
                     _ 
                   
                   ext 
                 
                 } 
               
               
                 K 
                 + 
                 1 
               
             
             = 
             1. 
           
         
       
     
         [0000]    It can be shown that the maximum eigen-value and its associated eigen-vector are strictly positive and correspond to the inverse SINR margin and the optimum power allocation, respectively. Note that the SINR targets are achievable only if the maximum eigen-value is less than or equal to 1. 
         [0038]    TP with maximum transmit power 
         [0039]    In the derivation given above, it was assumed that the TP (n 0 ) transmitting at the maximum allowed power (P max ) is known. In practice n 0  can be found as follows. Solve the power allocation problem in (12) for {circumflex over (n)} 0 ε{1, 2, . . . N} where {circumflex over (n)} 0  is a hypothetical (i.e., assumed) value of n 0 , until a value is found for which the power allocation satisfies PTPC Eq. (4). It can be shown that PTPC can be satisfied for only one value of {circumflex over (n)} 0 , unless two or more values result in the same power allocation vector. 
         [0040]    Beamformers 
         [0041]    In the above derivation, it was assumed that the beamformers (Û) are known a priori. The beamformers can be found by considering a virtual dual uplink multiple access channel (MAC) which is the dual of the downlink channel given by ( 1 ). The system model (received signal) for the dual uplink MAC for UE i is given by 
         [0000]        y   i   UL   =H   i   H   x   i   UL   +w   UL   Eq. (14) 
         [0000]    where W UL  is a zero-mean additive white Gaussian noise (AWGN) process with identity covariance matrix and x i   UL  is the signal transmitted from UE i, given by 
         [0000]        x   i   UL =(√{square root over ( q   i /σ i   2 )})[ S   i,1   UL   ,S   i,2   UL   , . . . , S   i,M     1     UL ] T   Eq. (15) 
         [0000]    where q i  is the transmit power of UE i. Assuming that the linear filter u i  is used at the TPs to detect the symbols transmitted by UE i, the SINR for UE i for the dual uplink MAC is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       S 
                        
                       
                           
                       
                        
                       I 
                        
                       
                           
                       
                        
                       N 
                        
                       
                           
                       
                        
                       
                         
                           R 
                           i 
                           UL 
                         
                         ( 
                         
                           U 
                           , 
                           
                             q 
                             _ 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           q 
                           i 
                         
                          
                         
                           
                             u 
                             _ 
                           
                           i 
                           H 
                         
                          
                         
                           R 
                           i 
                           ′ 
                         
                          
                         
                           
                             u 
                             _ 
                           
                           i 
                         
                       
                       
                         
                           
                             ∑ 
                             
                               k 
                               ≠ 
                               i 
                             
                           
                            
                           
                             
                               q 
                               k 
                             
                              
                             
                               
                                 u 
                                 _ 
                               
                               i 
                               H 
                             
                              
                             
                               R 
                               k 
                               ′ 
                             
                              
                             
                               
                                 u 
                                 _ 
                               
                               i 
                             
                           
                         
                         + 
                         1 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     16 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     R 
                     i 
                     ′ 
                   
                   = 
                   
                     
                       R 
                       i 
                     
                     / 
                     
                       
                         σ 
                         i 
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     17 
                     ) 
                   
                 
               
             
           
         
       
     
         [0042]    It can be shown that the downlink and the virtual dual uplink MAC have the same SINR achievable regions. Thus, given downlink SINR targets y i , i=1, 2, . . . , K and downlink SINR margin C DL (Ũ) achieved by means of downlink power allocation           and beamforming vectors Ũ, there exists a power allocation           for the dual uplink MAC with sum power 
         [0000]    
       
         
           
             
                
               
                 
                   q 
                   ~ 
                 
                 _ 
               
                
             
             = 
             
                
               
                 
                   p 
                   ~ 
                 
                 _ 
               
                
             
           
         
       
     
         [0000]    such that 
         [0000]      SINR i   UL ( Ũ )=γ i   C   DL ( Ũ ), for i=1,2, . . . K  Eq. (18) 
         [0000]    (i.e., the same SINRs are achieved in the dual uplink MAC). Substituting the above equations in (16) and simplifying, the uplink power allocation           can be obtained as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       q 
                       _ 
                     
                     ~ 
                   
                   = 
                   
                     
                       Diag 
                        
                       
                         ( 
                         
                           σ 
                           _ 
                         
                         ) 
                       
                     
                      
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   C 
                                   DL 
                                 
                                  
                                 
                                   ( 
                                   
                                     U 
                                     ~ 
                                   
                                   ) 
                                 
                               
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 D 
                                  
                                 
                                   ( 
                                   
                                     U 
                                     ~ 
                                   
                                   ) 
                                 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                           - 
                           
                             
                               Ψ 
                               T 
                             
                              
                             
                               ( 
                               
                                 U 
                                 ~ 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       
                         1 
                         _ 
                       
                       . 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     19 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where 1 is a column vector of ones. 
         [0043]    The SINR targets are achieved in both links by the same beamforming vectors/receive filters. Thus, the solution of the downlink beamforming problem can be obtained by solving the uplink filtering problem. 
         [0044]    For a given uplink power allocation          , the optimal unit-norm receive filters maximizing (16) are given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           u 
                           _ 
                         
                         ~ 
                       
                       i 
                     
                     = 
                     
                       
                         
                           e 
                           max 
                         
                         ( 
                         
                           
                             R 
                             i 
                             ′ 
                           
                           , 
                           
                             
                               Q 
                               i 
                             
                              
                             
                               ( 
                               
                                 
                                   q 
                                   _ 
                                 
                                 ~ 
                               
                               ) 
                             
                           
                         
                       
                       
                          
                         
                           
                             e 
                             max 
                           
                           ( 
                           
                             
                               R 
                               i 
                               ′ 
                             
                             , 
                             
                               
                                 Q 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   
                                     q 
                                     _ 
                                   
                                   ~ 
                                 
                                 ) 
                               
                             
                           
                            
                         
                       
                     
                   
                   , 
                   
                     1 
                     ≤ 
                     i 
                     ≤ 
                     K 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     20 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where e max (A,B) is the generalized eigen-vector of the matrices A and B corresponding to the largest eigen-value, and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Q 
                       i 
                     
                      
                     
                       ( 
                       
                         
                           q 
                           _ 
                         
                         ~ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           
                             k 
                             = 
                             1 
                           
                           , 
                           
                             k 
                             ≠ 
                             i 
                           
                         
                         K 
                       
                        
                       
                         
                           
                             q 
                             ~ 
                           
                           k 
                         
                          
                         
                           R 
                           k 
                           ′ 
                         
                       
                     
                     + 
                     
                       I 
                       . 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     21 
                     ) 
                   
                 
               
             
           
         
       
     
         [0045]    Referring now to  FIG. 2 ,  FIG. 2  is a flow chart illustrating a process  200 , according to some embodiments, for determining a beam forming vector for use in transmitting data from a TP  104  (e.g., TP  104 ( a )) to a UE  102  (e.g., UE  102 ( a ) connected to the TP  104 . A controller  106  (see  FIG. 1 ), which is in communication with TPs  104 , may perform process  200 . 
         [0046]    Process  200  begins in step  202 , where controller  106  receives channel information from the TP  104 . For example, in step  202  controller  106  may receive from the TP  104  the following information: (1) signal information pertaining to a transmit correlation matrix associated with the UE  102 , (2) noise information, and (3) interference information pertaining to a transmit correlation matrix associated with at least one other UE using the TP  104  (e.g., UE  102 ( b )). The signal information includes a transmit correlation matrix or information from which controller  106  can derive the transmit correlation matrix. 
         [0047]    In step  204 , controller  106  initializes one or more variables. For example, in step  204  controller  106  may initialize a set of variables representing a set of uplink (UL) transmit (Tx) powers, a variable representing a worst-case signal-to-interference-plus-noise ratio (SINR) margin, and a counter variable (j) (e.g., j may be set to a value of 1). 
         [0048]    In step  206 , controller  106  determines a set of beam forming vector using the signal information, noise information, interference information, and the set of UL Tx powers. For example, in step  206 , controller  106  determines the beam forming vector using equation 20. 
         [0049]    In step  208 , controller  106  determines a set of DL Tx powers and a worst-case SINR margin using the set of beam forming vectors determined in step  206 . 
         [0050]    In step  210 , controller determines the value of j and determines whether j is set to a value of 1. If it is, controller  106  proceeds to step  212 , otherwise controller proceeds to step  216 . 
         [0051]    In step  212 , controller  106  updates the variables representing the UL Tx powers using the maximum worst-case SINR margin determined in step  208 . For example, in step  206 , controller  106  updates the variables representing the UL Tx powers using equation 19. 
         [0052]    In step  214 , j is incremented. After step  214 , controller repeats steps  206 - 208 . That is, controller (a) determines a new set of beam forming vectors using, among other things, the variables representing the UL Tx powers, which were updated in step  212 , and (b) determines a new set of DL Tx powers and a new maximum worst-case SINR margin using the newly determined beam forming vectors. 
         [0053]    In step  216 , controller  106  determines whether the maximum worst-case SINR margin has converged. For example, if we let C DL (j) equal the most recently determined maximum worst-case SINR margin and we let C DL (j−1) equal the previous maximum worst-case SINR margin that was determined in step  208 , then, in step  216 , controller  106  determines whether the maximum worst-case SINR margin has converged by dividing C DL (j) by C DL (j−1) and determining whether the result is less than or equal to a predetermined threshold. The predetermined threshold may be set equal to 1+ε. If in step  216  controller  106  determines that the maximum worst-case SINR margin has not converged, then process  200  proceeds back to step  212 , otherwise it proceeds to step  218 . 
         [0054]    In step  218 , the newly a determined DL Tx power and beam forming vector for UE  102 ( a ) are used to transmit data to UE  102 ( a ). 
         [0055]    Referring now to  FIG. 3 ,  FIG. 3  is a flow chart illustrating an iterative process  300 , according to some embodiments, for performing step  208 . Process  300  may begin in step  302 , where controller  106  initializes a variable n 0 . For example, in step  302  n 0  may be set to a value of 1. In step  304 , controller  106  uses n 0  and the most recent set of beam forming vectors determined as a result of performing step  206  to determine a set of downlink transmit powers. For example, in step  304 , controller  106  determines the set of downlink transmit powers using equation 12, n 0  and the set of beam forming vectors. 
         [0056]    In step  306 , controller  106  uses the determined set of downlink transmit powers to determine whether a per-transmitter power constraint (PTPC) is satisfied. For example, in step  306 , controller  106  determines whether equation (4) is true. If the PTPC is satisfied, process  300  ends, otherwise process  300  proceeds to step  308 . In step  308 , n 0  is incremented (i.e., n 0 =n 0 +1). After step  308 , process  30 C returns to step  304 . 
         [0057]    Referring now to  FIG. 4 ,  FIG. 4  is a functional block diagram of controller  106  according to some embodiments of the invention. As shown, controller  106  may comprise a data processing system  402  (e.g., one or more microprocessors), a data storage system  405  (e.g., one or more non-volatile storage devices) and computer software  408  stored on the storage system  406 . Configuration parameters  410  may also be stored in storage system  406 . Controller  106  also includes transmit/receive (Tx/Rx) circuitry  404  for transmitting data to and receiving data from TPs  104  and transmit/receive (Tx/Rx) circuitry  405  for transmitting data to and receiving data from, for example, network  110 . Software  408  is configured such that when processor  402  executes software  408 , controller  106  performs steps described above (e.g., steps described above with reference to the flow charts). For example, software  408  may include: (1) computer instructions for. 
         [0058]    While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example only, and not limitation. Thus, the breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments. 
         [0059]    Additionally, while the processes described above and illustrated in the drawings are shown as a sequence of steps, this was done solely for the sake of illustration. Accordingly, it is contemplated that some steps may be added, some steps may be omitted, the order of the steps may be re-arranged, and some steps may be performed in parallel.