# Patent

## Title
Robust linear precoder designs for multi-cell downlink transmission

## Abstract

Methods and systems for optimizing the utilities of receiver devices in a wireless communication network are disclosed. Precoder design formulations that maximize a minimum worst-case rate or a worst-case sum rate are described for both full base station, cooperation and limited base station cooperation scenarios. In addition, optimal equalizers are also selected to optimize the worst-case sum rate.

## Background
RELATED APPLICATION INFORMATION
This application is a divisional of U.S. patent application Ser. No. 12/878,258, filed on Sep. 9, 2010, which in turn claims priority to U.S. Provisional Application No. 61/240,769, filed on Sep. 9, 2009, the content of which is incorporated herein by reference.


BACKGROUND
1. Technical Field
The present invention relates to optimizing the utility of devices in a wireless communication network and, more particularly, to the determination of precoders that optimize the network utility.
2. Description of the Related Art
The increasing demand for accommodating growing numbers of users within wireless networks has a tendency to limit the signal quality in such networks due to interference generated by serving a large number of users. A useful approach for mitigating the interference in downlink transmissions is to equip the base stations with multiple transmit antennas and employ transmit precoding. Such precoding exploits the spatial dimension to ensure that the signals intended for different users remain easily separable at their designated receivers. To enable precoded transmission, the base stations should acquire the knowledge of channel states or channel state information (CSI).
Many existing works have addressed the beamforming design problem, by assuming that each base station communicates with its respective terminals independently. In such a framework, inter-cell interference is simply regarded as additional background noise and the design of the beamforming vectors is performed on a per-cell basis only. Other works assume that both data and channel state information of all users could be shared in real-time, so that all base stations can act as a unique large array with distributed antenna elements. These works employ joint beamforming, scheduling and data encoding to simultaneously serve multiple co-channel users. Moreover existing works also assume that the channel state information is perfectly known to the sources.
SUMMARY
Exemplary embodiments of the present invention enable the robust optimization of user-utilities by designing precoders that account for inter-cell interference between users of different cells and intra-cell interference between users in a common cell. Furthermore, exemplary embodiments can also consider channel states and corresponding uncertainty regions of channels received by users that are intended for other users in the same cell or in other cells to ensure that the designed precoders provide optimal utilities.
One exemplary embodiment is directed to a method for optimizing the utility of receiver devices in a wireless communication network. In the method, information indicative of channel states corresponding to channels received by a set of receiver devices can be obtained. In addition, a precoding matrix can be determined by maximizing, for the set of receiver devices, a utility estimate corresponding to a minimum receiver rate within a set of rates corresponding to a bounded set of channel estimation errors determined by considering a set of channels including channels received by the set of receiver devices from base stations other than a base station servicing the set of receiver devices. Further, beamforming signals generated in accordance with the determined precoding matrix can be transmitted to the receiver devices.
An alternative exemplary embodiment is drawn towards a method for optimizing the utility of receiver devices in a wireless communication network. The method may include obtaining information indicative of channel states of channels received by a set of receiver devices. Additionally, a precoding matrix can be determined by maximizing, for the set of receiver devices, a utility estimate corresponding to a minimum receiver weighted-sum rate within a set of rates corresponding to a bounded set of channel estimation errors determined by considering a set of channels including channels received by the set of receiver devices from base stations other than a base station servicing the set of receiver devices. Thereafter, beamforming signals generated in accordance with the determined precoding matrix can be transmitted to the receiver devices.
Another exemplary embodiment is directed to a system for optimizing the utility of receiver devices, in a wireless communication network. The system may comprise a set of base stations that are configured to jointly apply a beamforming design to transmit beamforming signals to sets of receiver devices respectively served by the base stations. Here, a precoding matrix in the design for each base station can be determined by maximizing a utility estimate corresponding to a minimum receiver rate or a minimum receiver weighted-sum rate within a set of rates corresponding to a bounded set of channel estimation errors determined by considering a set of channels including channels received by the set of receiver devices from base stations other than a base station serving the set of receiver devices.
These and other features and advantages will become apparent from the following detailed description of illustrative embodiments thereof, which is to be read in connection with the accompanying drawings.



BRIEF DESCRIPTION OF DRAWINGS
The disclosure will provide details in the following description of preferred embodiments with reference to the following figures wherein:
FIG. 1 is a block diagram of an exemplary network communication system exhibiting inter-cell and intra-cell interference.
FIG. 2 is a block diagram of an exemplary network communication system exhibiting inter-cell and intra-cell interference, and illustrating uncertainty regions of received channels.
FIG. 3 is a block/flow diagram of an exemplary method for optimizing the utility of receiver devices in a wireless communication network by maximizing a worst-case minimum rate.
FIG. 4 is a block/flow diagram of an exemplary method for determining precoding matrices in accordance with a robust minimum rate optimization scheme in a full base station cooperation scenario.
FIG. 5 is a block/flow diagram of an exemplary method for successively determining precoding matrices in accordance with a robust minimum rate optimization scheme in a limited base station cooperation scenario.
FIG. 6 is a block/flow diagram of an exemplary method for greedily determining precoding matrices in accordance with a robust minimum rate optimization scheme in a limited base station cooperation scenario.
FIG. 7 is a block/flow diagram of an exemplary method for optimizing the utility of receiver devices in a wireless communication network by maximizing a worst-case weighted sum-rate.
FIG. 8 is a block/flow diagram of an exemplary method for determining precoding matrices in accordance with a robust weighted sum-rate optimization scheme.



DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
Exemplary embodiments of the present invention address the problem of designing beam forming vectors or beamformers for a multi-cell, multi-user wireless communication system. Coordinated information processing by the base stations of multi-cell wireless networks enhances the overall quality of communication in the network. Such coordinations for optimizing any desired network-wide quality of service (QoS) typically involve the acquisition and sharing of some channel state information (CSI) by and between the base stations. With perfect knowledge of channel states, the base stations can adjust their transmissions to achieve a network-wide QoS optimality. In practice, however, the CSI can be obtained only imperfectly. As a result, due to the uncertainties involved, the network is not guaranteed to benefit from a globally optimal QoS. Nevertheless, if the channel estimation perturbations are confined within bounded regions, the QoS measure will also lie within a bounded region. Therefore, by exploiting the notion of robustness in the worst-case sense, worst-case QoS guarantees for the network can be achieved and asserted.
Embodiments of the present invention described herein below can employ a model for noisy channel estimates that assumes that estimation noise terms lie within known hyper-spheres. Embodiments include linear transceivers that optimize a worst-case QoS measure in downlink transmissions. In particular, embodiments can maximize the worst-case weighted sum-rate of the network and the minimum worst-case rate of the network. Several centralized or fully cooperative and distributed (with limited cooperation) processes and systems, entailing different levels of complexity and information exchange among the base stations, described herein below can be employed to implement such transceiver designs.
Referring now in detail to the figures in which like numerals represent the same or similar elements and initially to FIG. 1, a portion of a wireless communication system 100 in which embodiments of the present invention can be implemented is illustrated. Each base-station (BS) or source 102, 104 servicing cells 106, 108, respectively, in the system 100 is modeled as having access to imperfect channel state information. All base stations can be equipped with multiple transmit antennas and each mobile user 110, 112 in the system can be equipped with a single receive antenna. Each base station in the system can employ linear beamforming to serve each one of its scheduled users or mobile devices on any resource slot. Here, each scheduled user is assumed to be served by only one base station. The association of each user to a particular base station (its serving base station) can be pre-determined. Further, each user can receive a useful (desired) signal 114 from its serving base station in addition to interference from its serving base station (116) as well as from adjacent base stations (118). The interference includes all signals transmitted in the same resource slot as the desired signal but which are intended for, the other users. It should be understood that a resource slot can be a time, frequency and/or code slot.
Each BS or source in the System 100 has access to channel state information that can be limited in that it is not global and imperfect in that it is subject to errors. In addition, some side information about the nature of errors can be available and limited information exchange (signaling) between base-stations is possible. According to aspects of the present principles, beam vectors can be designed for a cluster or set of coordinating base stations based on a given set of channel estimates, which ensure a robust performance. For example, in the case that the errors lie in a bounded region, the designed beam vectors can maximize a utility that is guaranteed for all possible errors, whereas in the case that the errors are realizations drawn from known distributions, the designed beam vectors can maximize a utility under probabilistic guarantees. As understood by those of skill in the art, a utility measure can be based on and directly correlated to a minimum receiver rate or a weighted sum of receiver rates.
FIG. 2 provides a more detailed system 200 in which embodiments can be implemented. As depicted in FIG. 2, all transmitters 202, 203 are equipped with multiple transmit antennas 212 and the users are each equipped with a single receive antenna 214. Every transmitter can be configured to employ linear beamforming to serve each one of its scheduled users. Each user, in turn, receives useful (desired) signals 206 from one transmitter and also receives interference 210 from its serving transmitter as well as adjacent transmitters. As noted above, the interference can include all signals transmitted in the same resource slot as the desired signal but which are intended for other users.
Each transmitter or source in system 200 employs an estimate of each one of its “outgoing channels,” which are the channels between a transmitter and all users. Each transmitter also employs an estimate of all the “incoming channels” seen by each one of the users served by it. In other words, the incoming channels are the channels seen by each served user from all transmitters. However, in practice, these estimates are imperfect. In particular, the true channel between any source and receiver is equal to the sum of the corresponding estimate and an error vector. According to exemplary aspects, the actual error vector is unknown to the source but the source is aware of the region, referred to as the uncertainty region, in which the error vector lies. In FIG. 2, the uncertainty regions are represented by ellipsoids 208. It should be noted that any reference to a “base station” herein can correspond to any one or more base stations in systems 100 and/or 200.
As indicated above, many existing works have addressed the beamforming design problem by assuming that each base station communicates with its respective terminals independently. Further, other works assume that both data and channel state information of all users could be shared in real-time, so that all base stations can act as a unique large array with distributed antenna elements and employ joint beamforming, scheduling and data encoding to simultaneously serve multiple co-channel users. In practice, however, only a much lower level of coordination among base stations is feasible and it is much more reasonable to assume that each user is served by only one base station. Moreover, as indicated above, existing works also assume that the channel state information is perfectly known to the sources, which unfortunately does not hold in practice.
Embodiments described herein apply to a more general model of multi-cell wireless networks, which hitherto has not been investigated for robust optimization, and treat the problem of joint robust transmission optimization for multiple or all cells. The significance of such multi-cell transmission optimization is that it incorporates the effects of inter-cell interferences, which are ignored when the cells optimize their transmissions independently. Furthermore, a practical constraint which forbids real-time data sharing among base stations can be incorporated so that each user can be served by only one base station.
Embodiments can design beam vectors for a cluster or set of coordinating transmitters (base stations) based on a given set of channel estimates and their uncertainty regions, which ensure a robust performance. In particular, the designed beam vectors can maximize a utility that is guaranteed for all possible errors within the respective uncertainty regions. For example, embodiments can design beam vectors by optimizing the worst-case weighted sum-rate or the minimum worst-case rate. For any given set of beams, user weights and channel estimates, the worst-case weighted sum-rate is the weighted sum-rate obtained when the errors assume their worst-possible values within the respective uncertainty regions. Thus, by maximizing the worst-case weighted sum-rate over the choice of beams, embodiments can obtain a robust choice of beam vectors that yields the best possible weighted sum-rate that is guaranteed for all possible errors within the respective uncertainty regions.
Transmission Model
A detailed transmission model is now provided to better illustrate aspects of the present principles. In the model, a multi-cell network includes M cells, where each cell has one base station that serves K users. The BSs are equipped with N transmit antennas and each user employs one receive antenna. Bm denotes the BS of the mth cell and Umk denotes the kth user in the mth cell for mε{1, . . . , M} and k ε{1, . . . , K}. Quasi-static flat-fading channels arc assumed and the downlink channel from Bn to Umk is denoted by hm,nk εC1×N.
Let xm=[xm1, . . . , xmK]T εCK×I denote the information stream of Bm, intended for serving its designated users via spatial multiplexing. It can be assumed that E[xmxmH]=I. Prior to transmission by Bm, the information stream xm is linearly processed by the precoding matrix Fm εCN×K. The kth column of Fm is denoted by wmk εCN×1, which is the beam carrying the information stream intended for user Umk. By defining fmk εC\{0} as the single-tap receiver equalizer deployed by Umk, the received post-equalization signal at Umk is given by




  
    
      
        
          
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where zmk: CN(0,1) accounts for the additive white complex Gaussian noise. It can be assumed that the users deploy single-user decoders for recovering their designated messages while suppressing the messages intended for other users as Gaussian interference. Therefore, the signal to interference-plus-noise ratio (SINR) of user Umk (with the optimal equalizer) is given by





  
    
      
        
          
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Also, M{tilde over (S)}Emk is defined as the mean square-error (MSE) of user Umk when it deploys the equalizer fmk and is given by





  
    
      
        
          
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MSEmk is further defined as the MSE corresponding to the minimum mean-square error (MMSE) equalizer which minimizes the MSE over all possible equalizers, i.e.,





  
    
      
        
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It can be assumed that user Umk knows its incoming channels, {hm,nk}n=1M, perfectly. In contrast, each BS is assumed to acquire, only noisy estimates of such channels corresponding to its designated receivers, i.e., Bm knows the channels {hm,nk}k,n imperfectly. {tilde over (h)}m,nk denotes the noisy estimate of the channel hm,nk available at Bm (and possibly other BSs via cooperation) and the channel estimation errors, which are unknown to the BSs, is defined as

Dm,nk=hm,nk−hm,nk−{tilde over (h)}m,nk, ∀m,nε{1, . . . , M}, and ∀kε{1, . . . , K}.  (5)

It can be assumed that such channel estimation errors are bounded and confined within an origin-centered hyper-spherical region of radius εm,nk, i.e., ∥Dm,nk∥2≦εm,nk. It is also noted that all the results derived in the sequel can be readily extended to the case where the uncertainty regions are bounded hyper-ellipsoids. In the sequel, for any matrix A, ∥A∥2 is used to denote matrix A's Frobenius norm.

Problem Statement
Using the transmission model described above, a formal problem statement is now provided, which details the goal of optimizing network-wide performance measures through the design of precoding matrices {Fm} and/or receiver equalizers {fmk} achieved by various exemplary embodiments. Such optimization hinges on the accuracy of channel estimates available at the BSs. Due to the uncertainties about channels estimates, the notion of robust optimization in the worst-case sense is adopted. The solution of the worst-case robust optimization is feasible over the entire uncertainty region and provides the best guaranteed performance over all possible CSI errors.
Based on this notion of robustness, two rate optimization problems are considered. One pertains to maximizing the worst-case weighted sum-rate of the multi-cell network and the other one seeks to maximize the minimum worst-case rate in the network. Both optimizations are subject to individual power constraints for the BSs. Let Rmk denote the rate assigned to user Umk and the power budget for the BS Bm is denoted by Pm. Also, P is defined as the vector of power budgets, P=[P1, . . . , PM].
First the robust max-min rate problem, which aims to maximize the minimum worst-case rate of the network subject to the power budget P, is considered. Because the users can be assumed to deploy single-user decoders, Rmk can be defined in terms of the SINR of user Umk: Rmk=log(1+SINRmk). Therefore, the robust max-min rate problem can be posed as




  
    
      
        
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As the second problem, optimization of the worst-case weighted sum-rate of the network is considered. For a given set of positive weighting factors' {αmk}, where αmk is the weighting factor corresponding to the rate of user Umk, this problem is formalized as




  
    
      
        
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Solving these problems yields the design of the precoder matrices. It should be noted that for a given set of precoders {Fm} and channel realizations {hm,nk}, the MMSE equalization factors {fmk} can be obtained in closed form.
It should also be noted that the joint design of the optimal precoders may involve having each BS acquire global CSI, which necessitates full cooperation (full CSI exchange) among the BSs. Such centralized processes are employed in certain exemplary embodiments described herein below with the assumption that such full cooperation is feasible. In practice, however, full cooperation might not be implementable. In such cases, distributed processes, also described herein below, that entail limited cooperation among the BSs can be employed. Some of the exemplary embodiments that utilize distributed processes yield the same performance as their centralized counterparts.
Robust Max-Min Rate Optimization
Referring now to FIG. 3 with continuing reference to FIGS. 1 and 2, a method 300 for optimizing the utility of receiver devices in a wireless communication system is illustrated. In particular, the method 300 generally implements a robust max-min rate optimization and can be performed in a variety of ways, as discussed further herein below. Method 300 can be performed by a base station 102, 202 or a control center, which may itself be a designated base station 102, 202, if full cooperation between base stations is feasible. Alternatively, if only limited cooperation is feasible, then the method can be performed independently by each base station, as discussed further herein below.
The method 300 may begin at step 302 in which a base station or a control center may obtain a set of estimates of channel states corresponding to channels received by a set of receiver devices. These estimates can be received in the form of feedback from the receiver devices. The feedback from each receiver device can be limited to a few bits. Thus, each receiver device may have to quantize each one of the channel estimates available to it. Consequently, the base station or control center may have access to channel estimates that are corrupted by quantization errors. As indicated above, the channel state information may correspond to {hm,nk}n=1M, which includes the channel hm,mk received by the receiver from the base station Bm servicing the receiver in addition to the channels hm,nk (n≠m) received by the receiver from base stations Bn servicing cells other than cell m. In a fully cooperative scenario, each base station may communicate the respective channel state information for receivers or users in the corresponding cell to the other base stations in the network or to a control center. Alternatively, in a limited cooperation scenario, while each base station can obtain channel information {hm,nk}n=1M for receivers in its own cell, each base station can communicate prospective precoding matrices to other bases to optimize receiver utility in the network, as discussed in more detail herein below.
At step 304, a base station or control center can determine a precoding matrix for the set of receiver devices by maximizing a minimum worst case rate of the network. For example, as discussed above with regard to equation (6), the precoding matrices {Fm} for the base stations in the network can be determined in accordance with a power budget by solving the following:




  
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Thus, for base station m, the term





  
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is an example of a utility estimate corresponding to the minimum receiver rate





  
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within a set of rates corresponding to a bounded set of channel estimation errors {Dm,nk}. Furthermore, the estimate can be determined by considering a set of channels {hm,nk}n=1M including channels hm,nk (n≠m) received by the set of receiver devices from base stations Bn (n≠m) other than a base station Bm servicing the set of receiver devices. For example, as noted above and below with respect to various exemplary embodiments, {hm,nk}n=1M can be used to determine values of SINRmtk to solve equation (6).

It should be noted that, at step 304, in the full cooperation scenario, the control center can determine precoding matrices for each base station using channel state information for receivers in each cell serviced by the base stations and can assign the precoding matrices to the base stations to enable them to generate optimized, beam forming signals for transmission to the receiver. Alternatively, in the full cooperation scenario, each base station may independently determine their own precoding matrix using the same methods, where each base station can receive channel state information from each other base station, determine the precoding matrices for the entire network and apply the precoding matrix. Various exemplary embodiments that can be employed to implement step 304 are discussed further herein below.
At step 306, each base station can transmit beamforming signals generated in accordance with the determined precoding matrix to their own respective receiver devices in their respective cell.
Returning to step 304, one or more base stations can implement step 304 by solving equation (6) via power optimization or via MSE optimization. To better illustrate how embodiments can implement the power optimization approach, a solution to equation (6) is presented in which each base station is assumed to serve only one user in its respective cell.
Single-User Cells (K=1)
Under the assumption that each BS is serving one user, i.e., K=1, the downlink transmission model essentially becomes equivalent to a multi-user Gaussian interference channel with M transmitters and M respective receivers. For the ease or notation the superscript k is omitted in the subsequent analysis and discussions with regard to single-user cells. When K=1 the precoder of BS Bm consists of only one column vector which is referred to by wm. For the given channel estimates {{tilde over (h)}m,n}, sinrm is defined as




  
    sin
    ⁢
    
      
    
    ⁢
    
      r
      m
    
  
  ⁢
  
    =
    Δ
  
  ⁢
  
    
      min
      
        {
        
          D
          
            m
            ,
            n
          
        
        }
      
    
    ⁢
    
      SIN
      ⁢
      
        
      
      ⁢
      
        R
        m
      
    
  




as the worst-case (smallest) SINRm over the uncertainty regions.

By introducing a slack variable a>0, the epigraph form of the robust max-min rate optimization problem S(P) given in (6) is given by




  
    
      
        
          S
          ⁡
          
            (
            P
            )
          
        
        =
        
          {
          
            
              
                
                  max
                  
                    
                      {
                      
                        w
                        m
                      
                      }
                    
                    ,
                    a
                  
                
              
              
                a
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      m
                    
                  
                  ≥
                  a
                
              
              
                
                  
                    ∀
                    m
                  
                  ,
                
              
            
            
              
                
                  
                
              
              
                
                  
                    
                      
                      
                        w
                        m
                      
                      
                    
                    2
                    2
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  
                    m
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




The closed-form characterization of sinrm can be found by recalling (2) to obtain





  
    
      
        
          
            sin
            ⁢
            
              
            
            ⁢
            
              r
              m
            
          
          =
          
            
              
                min
                
                  {
                  
                    D
                    
                      m
                      ,
                      n
                    
                  
                  }
                
              
              ⁢
              
                
                  
                    
                    
                      
                        h
                        
                          m
                          ,
                          m
                        
                      
                      ⁢
                      
                        w
                        m
                      
                    
                    
                  
                  2
                
                
                  
                    
                      ∑
                      
                        n
                        ≠
                        m
                      
                    
                    ⁢
                    
                      
                        
                        
                          
                            h
                            
                              m
                              ,
                              n
                            
                          
                          ⁢
                          
                            w
                            n
                          
                        
                        
                      
                      2
                    
                  
                  +
                  1
                
              
            
            =
            
              
                
                  min
                  
                    D
                    
                      m
                      ,
                      m
                    
                  
                
                ⁢
                
                  
                    
                    
                      
                        h
                        
                          m
                          ,
                          m
                        
                      
                      ⁢
                      
                        w
                        m
                      
                    
                    
                  
                  2
                
              
              
                
                  
                    ∑
                    
                      n
                      ≠
                      m
                    
                  
                  ⁢
                  
                    
                      max
                      
                        D
                        
                          m
                          ,
                          n
                        
                      
                    
                    ⁢
                    
                      
                        
                        
                          
                            h
                            
                              m
                              ,
                              n
                            
                          
                          ⁢
                          
                            w
                            n
                          
                        
                        
                      
                      2
                    
                  
                
                +
                1
              
            
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  




where the second equality holds by noting that finding the worst-case SINRm can be decoupled into finding the worst-case (smallest) numerator term and the worst-case, (largest) denominator terms. In order to further simplify sinrm, the result of the following lemma can be used. The proof is omitted for brevity purposes.

Lemma 1 For any given h εC1×N, w εCN×1, ε εRt, and positive definite matrix Q; gmin and gmax defined as




  
    g
    min
  
  ⁢
  
    =
    Δ
  
  ⁢
  
    {
    
      
        
          
            
              
                min
                x
              
            
            
              
                
                  
                  
                    hw
                    +
                    xw
                  
                  
                
                2
              
            
          
          
            
              
                s
                .
                t
                .
              
            
            
              
                
                  
                    
                      xQx
                      H
                    
                  
                  ≤
                  ɛ
                
                ,
              
            
          
        
        ⁢
        
          

        
        ⁢
        and
        ⁢
        
          

        
        ⁢
        
          g
          max
        
      
      ⁢
      
        =
        Δ
      
      ⁢
      
        {
        
          
            
              
                max
                
                  x
                  .
                
              
            
            
              
                
                  
                  
                    hw
                    +
                    xw
                  
                  
                
                2
              
            
          
          
            
              
                s
                .
                t
                .
              
            
            
              
                
                  
                    
                      xQx
                      H
                    
                  
                  ≤
                  ɛ
                
                ,
              
            
          
        
      
    
  




are given by

gmin=|(|hw|−ε√{square root over (wHQ−1w)})+|2, and gmax=∥hw|+ε√{square root over (wHQ−1w)}|2,

where (x)+=max{0,x}, ∀xεR.

By recalling (9) and invoking the result of the Lemma 1 for the choice of Q=I, sinrm can be further simplified as




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            r
            m
          
        
        =
        
          
            
              
                
                
                  (
                  
                    
                      
                      
                        
                          
                            h
                            ~
                          
                          
                            m
                            ,
                            m
                          
                        
                        ⁢
                        
                          w
                          m
                        
                      
                      
                    
                    -
                    
                      
                        ɛ
                        
                          m
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          
                          
                            w
                            m
                          
                          
                        
                        2
                      
                    
                  
                  )
                
                
              
              2
            
            
              
                
                  ∑
                  
                    n
                    ≠
                    m
                  
                
                ⁢
                
                  
                    
                    
                      
                        
                        
                          
                            
                              h
                              ~
                            
                            
                              m
                              ,
                              n
                            
                          
                          ⁢
                          
                            w
                            n
                          
                        
                        
                      
                      -
                      
                        
                          ɛ
                          
                            m
                            ,
                            n
                          
                        
                        ⁢
                        
                          
                            
                            
                              w
                              n
                            
                            
                          
                          2
                        
                      
                    
                    
                  
                  2
                
              
              +
              1
            
          
          .
        
      
    
    
      
        (
        10
        )
      
    
  




The relevant scenarios are where ∀m,∥{tilde over (h)}m,m∥2>εm,m so that S(P)>0. Given the closed-form characterization of sinrm, solving S(P) can be facilitated by solving a power optimization problem defined as





  
    
      
        
          P
          ⁡
          
            (
            
              P
              ,
              a
            
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          {
          
            
              
                
                  min
                  
                    
                      {
                      
                        w
                        m
                      
                      }
                    
                    ,
                    b
                  
                
              
              
                b
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      m
                    
                  
                  ≥
                  a
                
              
              
                
                  
                    ∀
                    m
                  
                  ,
                
              
            
            
              
                
                  
                
              
              
                
                  
                    
                      
                        
                        
                          w
                          m
                        
                        
                      
                      2
                    
                    
                      
                        P
                        m
                      
                    
                  
                  ≤
                  b
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




The connection between S(P) and P(P, a) is established in the following useful theorem. The proof is simple and hence omitted for brevity.

Theorem 1 For any given power budget P, P(P,a) is strictly increasing and continuous in a at any strictly feasible a and is related to S(P) via

P(P,S(P))=1.

Strict monotonicity and continuity of P(P,a) in a at any strictly feasible a provides that there exists a unique a* satisfying P(P, a*)=1. Hence, taking into account Theorem 1 establishes that S(P) can be solved by finding a* that satisfies P(P, a*)=1. Due to monotonicity and continuity of P(P,a), finding a* can be implemented via a simple iterative bi-section search. Each iteration involves solving P(P,a) for a different value of a. P(P,a) can be cast as a convex problem with a computationally efficient solution.
Theorem 2 Problem P(P, a) can be posed as a semidefinite programming (SDP) problem.
The proof is omitted for brevity purposes.
To employ such a procedure for solving S(P), the BSs should be fully cooperative such that each BS can acquire estimates of all network-wide channel states.
Multi-User Cells (K>1)
Downlink transmissions serving more than one user in each cell (K>1) are now considered. The major difference between the analysis for multiuser cells and that of single-user cells arises from different characterizations of their corresponding worst-case SINR s. By defining sinrmk as the worst-case SINRmk in (2), sinrmk is obtained as




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            r
            m
            k
          
        
        =
        
          
            min
            
              {
              
                D
                
                  m
                  ,
                  n
                
                k
              
              }
            
          
          ⁢
          
            
              
                
                  
                  
                    
                      h
                      
                        m
                        ,
                        m
                      
                      k
                    
                    ⁢
                    
                      w
                      m
                      k
                    
                  
                  
                
                2
              
              
                
                  
                    ∑
                    
                      l
                      ≠
                      k
                    
                  
                  ⁢
                  
                    
                      
                      
                        
                          h
                          
                            m
                            ,
                            m
                          
                          k
                        
                        ⁢
                        
                          w
                          m
                          l
                        
                      
                      
                    
                    2
                  
                
                +
                
                  
                    ∑
                    
                      n
                      ≠
                      m
                    
                  
                  ⁢
                  
                    
                      ∑
                      l
                    
                    ⁢
                    
                      
                        
                        
                          
                            h
                            
                              m
                              ,
                              n
                            
                            k
                          
                          ⁢
                          
                            w
                            n
                            l
                          
                        
                        
                      
                      2
                    
                  
                
                +
                1
              
            
            .
          
        
      
    
    
      
        (
        12
        )
      
    
  




Unlike the single-user setup, when K>1, the uncertainty regions of the numerator and the summands of the denominator of sinrmk are not decoupled. Therefore, finding sinrmk cannot be decoupled into finding the worst-case numerator and the terms in the denominator independently. To the knowledge of the inventors, handling constraints on such worst-case SINR s even in single-cell downlink transmissions is not mathematically tractable and the robust design of linear precoders for these systems is carried out suboptimally. In the sequel suboptimal approaches are also proposed for solving the robust max-min rate optimization in multi-cell networks.

Two suboptimal approaches can be employed for solving S(P). In the first approach, a lower bound on the worst-case SINR is found and, in the formulation of S(P), each worst-case SINR is replaced with its corresponding lower bound. Similar to the single-user cells setup discussed above, this approximate, problem can be solved efficiently through solving a counterpart power optimization problem. In the second approach, the robust max-min rate optimization problem can be converted into a robust min-max MSE optimization problem and an upper bound on the maximum worst-case MSE, which in turn provides a lower bound on the minimum worst-case rate, can be found. Each approach is described herein below.
Solving Via Power Optimization
Returning to method 300, exemplary embodiments can determine the precoding matrices at step 304 by maximizing, at step 310, a slack variable corresponding to a lower bound of an estimate corresponding to the minimum worst-case receiver rate under a power constraint. For example, the worst-case value of SINRmk, which is denoted by sinrmk, is not mathematically tractable. Consequently, to solve (6), lower bounds on the worst-case SINR s are found as follows. sinrmk,1 is defined as




  
    
      
        sin
        _
      
      ⁢
      
        r
        m
        k
      
    
    ⁢
    
      =
      Δ
    
    ⁢
    
      
        
          min
          
            D
            
              m
              ,
              m
            
            k
          
        
        ⁢
        
          
            
            
              
                h
                
                  m
                  ,
                  m
                
                k
              
              ⁢
              
                w
                m
                k
              
            
            
          
          2
        
      
      
        
          
            max
            
              D
              
                m
                ,
                m
              
              k
            
          
          ⁢
          
            
              ∑
              
                l
                ≠
                k
              
            
            ⁢
            
              
                
                
                  
                    h
                    
                      m
                      ,
                      m
                    
                    k
                  
                  ⁢
                  
                    w
                    m
                    l
                  
                
                
              
              2
            
          
        
        +
        
          
            ∑
            
              n
              ≠
              m
            
          
          ⁢
          
            
              max
              
                D
                
                  m
                  ,
                  n
                
                k
              
            
            ⁢
            
              
                ∑
                l
              
              ⁢
              
                
                  
                  
                    
                      h
                      
                        m
                        ,
                        n
                      
                      k
                    
                    ⁢
                    
                      w
                      n
                      l
                    
                  
                  
                
                2
              
            
          
        
        +
        1
      
    
  
  ,




where, clearly, sinrmk≦sinrmk. By applying Lemma 1,





  
    
      
        
          
            
              sin
              _
            
            ⁢
            
              r
              m
              k
            
          
          =
          
            
              
                
                
                  
                    (
                    
                      
                        
                        
                          
                            
                              h
                              ~
                            
                            
                              m
                              ,
                              m
                            
                            k
                          
                          ⁢
                          
                            w
                            m
                            k
                          
                        
                        
                      
                      ⁢
                      
                        –ɛ
                        
                          m
                          ,
                          m
                        
                        k
                      
                      ⁢
                      
                        
                          
                          
                            w
                            m
                            k
                          
                          
                        
                        2
                      
                    
                    )
                  
                  +
                
                
              
              2
            
            
              
                
                  
                    
                      
                        max
                        
                          D
                          
                            m
                            ,
                            m
                          
                          k
                        
                      
                      ⁢
                      
                        
                          h
                          
                            m
                            ,
                            m
                          
                          k
                        
                        ⁢
                        
                          
                            
                              Y
                              
                                m
                                ,
                                k
                              
                            
                            ⁡
                            
                              (
                              
                                Y
                                
                                  m
                                  ,
                                  k
                                
                              
                              )
                            
                          
                          H
                        
                        ⁢
                        
                          
                            (
                            
                              h
                              
                                m
                                ,
                                m
                              
                              k
                            
                            )
                          
                          H
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        ∑
                        
                          n
                          ≠
                          m
                        
                      
                      ⁢
                      
                        
                          max
                          
                            D
                            
                              m
                              ,
                              n
                            
                            k
                          
                        
                        ⁢
                        
                          
                            h
                            
                              m
                              ,
                              n
                            
                            k
                          
                          ⁢
                          
                            
                              
                                F
                                n
                              
                              ⁡
                              
                                (
                                
                                  F
                                  n
                                
                                )
                              
                            
                            H
                          
                          ⁢
                          
                            
                              (
                              
                                h
                                
                                  m
                                  ,
                                  n
                                
                                k
                              
                              )
                            
                            H
                          
                        
                      
                    
                    +
                    1
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        13
        )
      
    
  




where Ym,k is defined as





  
    Y
    
      m
      ,
      k
    
  
  ⁢
  
    =
    Δ
  
  ⁢
  
    
      [
      
        
          
            w
            m
            1
          
          ⁢
          
            
          
          ⁢
          ...
        
        ⁢
        
          
        
        ,
        
          w
          m
          
            k
            -
            1
          
        
        ,
        
          w
          m
          
            k
            +
            1
          
        
        ,
        ...
        ⁢
        
          
        
        ,
        
          w
          m
          K
        
      
      ]
    
    .
  




By introducing the slack variable a and invoking the lower bounds on the SINR s given in (13), a lower bound on the robust max-min rate is obtained as follows:




  
    
      
        
          
            S
            1
          
          ⁡
          
            (
            P
            )
          
        
        ≡
        
          {
          
            
              
                
                  max
                  
                    
                      {
                      
                        F
                        m
                      
                      }
                    
                    ,
                    a
                  
                
              
              
                a
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    
                      
                        sin
                        ⁢
                        r
                      
                      _
                    
                    m
                    k
                  
                  ≥
                  a
                
              
              
                
                  
                    ∀
                    m
                  
                  ,
                  k
                  ,
                
              
            
            
              
                
                  
                
              
              
                
                  
                    
                      
                      
                        F
                        m
                      
                      
                    
                    2
                    2
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




To maximize the slack variable in step 304, the base station can, at step 312, iteratively solve a power optimization problem to obtain a value of the slack variable that solves the slack variable maximization problem by determining whether slack variable values result in a pre-determined solution of the power optimization problem that is correlated to the solution, to the slack variable maximization problem. For example, similar to the single-user scenario discussed above, solving S1(P) can be carried out by alternatively solving a power optimization problem in conjunction with a linear bi-section search. The power optimization of interest with per BS power constraints is given by




  
    
      
        
          
            P
            1
          
          ⁡
          
            (
            
              P
              ,
              a
            
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          {
          
            
              
                
                  max
                  
                    
                      {
                      
                        F
                        m
                      
                      }
                    
                    ,
                    b
                  
                
              
              
                b
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    
                      
                        sin
                        ⁢
                        r
                      
                      _
                    
                    m
                    k
                  
                  ≥
                  a
                
              
              
                
                  
                    ∀
                    m
                  
                  ,
                  k
                  ,
                
              
            
            
              
                
                  
                
              
              
                
                  
                    
                      
                        
                        
                          F
                          m
                        
                        
                      
                      2
                    
                    
                      
                        P
                        m
                      
                    
                  
                  ≤
                  b
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




The result in Theorem 1 can be extended for a multiuser cell setup (K>1) in order to establish the connection S1(P) and P1(P,a). The proof is simple and is omitted for brevity.

Theorem 3 For any given power budget P, P1(P,a) is strictly increasing and continuous in a at any strictly feasible a and is related to S1(P) via

P1(P,S1(P))=1.  (16)

Algorithm 1 of Table 1 below solves S1(P) by solving P1(P,a) combined with a bi-section line search. The optimality of Algorithm 1 and its convergence follows from the monotonicity and continuity of P1(P,a) at any feasible a. Similar to Theorem 2, it can be shown that P1(P,a) has a computationally efficient solution.
Theorem 4 Problem P1(P,a) can be posed as an SDP problem.
The proof of Theorem 4 is also omitted for brevity purposes.
It should be noted that when K=1, S1(P) is identical to S(P) and an optimal solution to the latter problem is obtained using Algorithm 1.







TABLE 1





Algorithm 1 - Robust Max-Min SINR Optimization


via Power Optimization (K ≧ 1)
















 1:
Input P and {{tilde over (h)}m,nk,εm,nk}


 2:
Initialize amin = 0 and amax = minm,k{Pm|(∥{tilde over (h)}m,mk∥2 − εm,mk)+|2}


 3:
 a0 ← amin


 4:
 Repeat


 5:
 Solve P1(P,a0) and obtain {Fm}


 6:
if P1(P,a0) ≦ 1


 7:
amin ← a0


 8:
Else


 9:
 amax ← a0


10:
end if


11:
a0 ← (amin + amax) / 2


12:
until amax − amin ≦ δ


13:
Output S1(P) = a0 and {Fm}









Referring now to FIG. 4, with continuing reference to FIG. 3 and Table 1, a detailed method 400 for maximizing the slack variable by solving a power optimization problem in step 312 according to an exemplary embodiment is illustrated. It should be noted that Algorithm 1 of Table 1 can be implemented in the method 400.
Method 400 can begin at step 402 in which the power constraint P, the channel estimates {{tilde over (h)}m,nk} available at the base stations in the network and their corresponding uncertainty region radii {εmk} can be input or received by a base station. Here, the power constraint P can be selected in accordance with design choice and base station capabilities while the channel estimations {tilde over (h)}m,nk, for example, can be obtained as quantized feedback from receiver devices as discussed above with respect to step 302 and can be communicated between the base stations.
At step 404, the base station can initialize amin and amax: amin=0 and




  
    a
    max
  
  =
  
    
      min
      
        m
        ,
        k
      
    
    ⁢
    
      
        {
        
          
            P
            m
          
          ⁢
          
            
              
              
                
                  (
                  
                    
                      
                        
                        
                          
                            h
                            ~
                          
                          
                            m
                            ,
                            m
                          
                          k
                        
                        
                      
                      2
                    
                    -
                    
                      ɛ
                      
                        m
                        ,
                        m
                      
                      k
                    
                  
                  )
                
                +
              
              
            
            2
          
        
        }
      
      .
    
  




At step 406, the base station can set a0 to amin.
At step 408, the base station can solve P1(P,a0) using (15) with a0 to obtain a prospective precoding matrix {Fm}.
At step 410, the base station can determine whether the value a0 of the slack variable a results in a pre-determined solution of the power optimization problem that is correlated to the solution to the slack variable maximization problem. For example, as noted above P1(P,a) is related to S1(P) via P1(P,S1(P))=1, where the pre-determined solution here is 1. Thus, the base station may determine whether P1(P,a0)≦1. If P1(P,a0)≦1, then the method may proceed to step 414 in which the base station can set amin to a0. Otherwise, the method may proceed to step 412, in which the base station can set amax to a0.
Thereafter, the method may proceed to step 416, in which the base station can set a0=(amin+amax)/2. Next, the method may proceed to step 418 in which the base station may determine whether amax−amin≦δ for some δ, which is selected based on design choice. If amax−amin≦δ, then the method may end at step 420, in which the base station may obtain the solution to the power optimization problem as S1(P)=a0 and {Fm}. Otherwise, the method may proceed to step 408 and repeat.
Solving Via MSE Optimization
As an alternative to solving via power,optimization, the precoding matrices may be determined using MSE optimization. For example, in method 300 of FIG. 3, exemplary embodiments can determine the precoding matrices at step 304 by minimizing, at step 320, a maximum mean square error within a set of mean square errors corresponding to the bounded set of channel estimation errors. Further, the minimization of the MSE can be implemented by minimizing a slack variable correlated to an upper bound of the mean square error. For example, the robust max-min rate optimization problem can be transformed into a robust min-max MSE optimization problem by using the fact that




  
    
      M
      ⁢
      
        
      
      ⁢
      S
      ⁢
      
        
      
      ⁢
      
        E
        m
        k
      
    
    =
    
      1
      
        1
        +
        
          
            SIN
            ⁢
            R
          
          m
          k
        
      
    
  
  ,




where, as indicated above, MSEmk is the MSE of user Umk when it deploys the MMSE equalizer. Consequently, the worst-case MSE corresponding to the worst-case SINR is given as





  
    
      
        
          
            max
            
              (
              
                D
                
                  m
                  ,
                  n
                
                k
              
              )
            
          
          ⁢
          
            M
            ⁢
            
              
            
            ⁢
            S
            ⁢
            
              
            
            ⁢
            
              E
              m
              k
            
          
        
        =
        
          
            1
            
              1
              +
              
                
                  min
                  
                    {
                    
                      D
                      
                        m
                        ,
                        n
                      
                      k
                    
                    }
                  
                
                ⁢
                
                  
                    SIN
                    ⁢
                    R
                  
                  m
                  k
                
              
            
          
          =
          
            
              1
              
                1
                +
                
                  
                    sin
                    ⁢
                    r
                  
                  m
                  k
                
              
            
            .
          
        
      
    
    
      
        (
        17
        )
      
    
  




Here, the term





  
    max
    
      {
      
        D
        
          m
          ,
          n
        
        k
      
      }
    
  
  ⁢
  
    M
    ⁢
    
      
    
    ⁢
    S
    ⁢
    
      
    
    ⁢
    
      E
      m
      k
    
  




corresponds to the maximum mean square error within a set of mean square errors corresponding to a bounded set of channel estimation errors {Dm,nk}. In addition, by recalling the problem S(P) given in (6) and taking into account the representation of MSEmk given in (4) and the worst-case MSE given in (17), the robust max-min rate optimization problem can be solved by equivalently solving





  
    
      
        
          
            S
            2
          
          ⁡
          
            (
            P
            )
          
        
        =
        
          {
          
            
              
                
                  
                    min
                    
                      {
                      
                        F
                        m
                      
                      }
                    
                  
                  ⁢
                  
                    max
                    
                      k
                      ,
                      m
                    
                  
                
              
              
                
                  
                    max
                    
                      {
                      
                        D
                        
                          m
                          ,
                          n
                        
                        k
                      
                      }
                    
                  
                  ⁢
                  
                    
                      min
                      
                        f
                        m
                        k
                      
                    
                    ⁢
                    
                      M
                      ⁢
                      
                        
                      
                      ⁢
                      
                        S
                        ~
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        E
                        m
                        k
                      
                    
                  
                
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    
                      
                      
                        F
                        m
                      
                      
                    
                    2
                    2
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  
                    m
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




An upper bound on S2(P), which in turn results in a lower bound on S(P), can be found and employed to facilitate solving (18). By invoking the inequality




  
    
      
        
          
            max
            
              {
              
                D
                
                  m
                  ,
                  n
                
                k
              
              }
            
          
          ⁢
          
            
              min
              
                f
                m
                k
              
            
            ⁢
            
              M
              ⁢
              
                
              
              ⁢
              
                S
                ~
              
              ⁢
              
                
              
              ⁢
              
                E
                m
                k
              
            
          
        
        ≤
        
          
            min
            
              f
              m
              k
            
          
          ⁢
          
            
              max
              
                {
                
                  D
                  
                    m
                    ,
                    n
                  
                  k
                
                }
              
            
            ⁢
            
              M
              ⁢
              
                
              
              ⁢
              
                S
                ~
              
              ⁢
              
                
              
              ⁢
              
                E
                m
                k
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




and introducing the slack variable aεR+, the following upper bound S2(P) on S2(P) can be obtained as follows:





  
    
      
        
          
            
              S
              _
            
            2
          
          ⁡
          
            (
            P
            )
          
        
        ≡
        
          {
          
            
              
                
                  min
                  
                    
                      {
                      
                        
                          F
                          m
                        
                        ,
                        
                          f
                          m
                          k
                        
                      
                      }
                    
                    ,
                    a
                  
                
              
              
                a
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    
                      max
                      
                        {
                        
                          D
                          
                            m
                            ,
                            n
                          
                          k
                        
                        }
                      
                    
                    ⁢
                    
                      M
                      ⁢
                      
                        
                      
                      ⁢
                      
                        S
                        ~
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        E
                        m
                        k
                      
                    
                  
                  ≤
                  
                    a
                    2
                  
                
              
              
                
                  
                    ∀
                    k
                  
                  ,
                  m
                  ,
                
              
            
            
              
                
                  
                
              
              
                
                  
                    
                      
                      
                        F
                        m
                      
                      
                    
                    2
                    2
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




It should be noted that the base station can implement the MSE minimization at step 320 by formulating and solving, at step 322, a generalized eigenvalue problem (GEVP). It is shown herein below that the problem in (20) is equivalent to a generalized eigenvalue problem (GEVP) which can be solved efficiently.
Theorem 5 The problem S2(P) can be optimized efficiently as a GEVP.
Theorem 5 can be proven as follows. By recalling (3) and further defining the slack variables {bm,nk}, the constraints (M{tilde over (S)}Emk≦a2) can be equivalently presented as follows.




   
  
    {
    
      
        
          
            
              
                
                  
                    ∑
                    n
                  
                  ⁢
                  
                    
                      (
                      
                        b
                        
                          m
                          ,
                          n
                        
                        k
                      
                      )
                    
                    2
                  
                
                +
                1
              
            
            ≤
            
              
                f
                m
                k
              
              ⁢
              a
            
          
        
        
          
            
              ∀
              m
            
            ,
            k
          
        
        
          
            
          
        
      
      
        
          
            
              
                
                
                  
                    
                      h
                      
                        m
                        ,
                        m
                      
                      k
                    
                    ⁢
                    
                      F
                      m
                    
                  
                  -
                  
                    
                      f
                      m
                      k
                    
                    ⁢
                    
                      e
                      k
                    
                  
                
                
              
              2
            
            ≤
            
              b
              
                m
                ,
                m
              
              k
            
          
        
        
          
            
              ∀
              m
            
            ,
            k
            ,
          
        
        
          
            
              ∀
              
                
                  
                    
                    
                      D
                      
                        m
                        ,
                        m
                      
                      k
                    
                    
                  
                  2
                
                ≤
                
                  ɛ
                  
                    m
                    ,
                    m
                  
                  k
                
              
            
            ,
          
        
      
      
        
          
            
              
                
                
                  
                    h
                    
                      m
                      ,
                      n
                    
                    k
                  
                  ⁢
                  
                    F
                    n
                  
                
                
              
              2
            
            ≤
            
              b
              
                m
                ,
                n
              
              k
            
          
        
        
          
            
              ∀
              k
            
            ,
            
              m
              ≠
              n
            
            ,
          
        
        
          
            
              ∀
              
                
                  
                    
                    
                      D
                      
                        m
                        ,
                        n
                      
                      k
                    
                    
                  
                  2
                
                ≤
                
                  ɛ
                  
                    m
                    ,
                    n
                  
                  k
                
              
            
            ,
          
        
      
    
  




where ek denotes a length K unit vector having a one in its kth position and zeros elsewhere. Without loss of generality, we have assumed fmkεR+ as multiplying the vectors wmk with any unit-magnitude complex scalar will not change the objective or the constraints of the problem S2(P). Next, by applying the Schur Complement lemma the constraints ∥hm,mkFm−fmkek∥2≦bm,mk for all ∥Dm,mk∥2≦εm,mk, can be equivalently stated as





  
    
      [
      
        
          
            
              b
              
                m
                ,
                m
              
              k
            
          
          
            
              
                
                  (
                  
                    
                      
                        h
                        ~
                      
                      
                        m
                        ,
                        m
                      
                      k
                    
                    +
                    
                      D
                      
                        m
                        ,
                        m
                      
                      k
                    
                  
                  )
                
                ⁢
                
                  F
                  m
                
              
              -
              
                
                  f
                  m
                  k
                
                ⁢
                
                  e
                  k
                
              
            
          
        
        
          
            
              
                
                  
                    (
                    
                      F
                      m
                    
                    )
                  
                  H
                
                ⁢
                
                  
                    (
                    
                      
                        
                          h
                          ~
                        
                        
                          m
                          ,
                          m
                        
                        
                          k
                          ,
                          k
                        
                      
                      +
                      
                        D
                        
                          m
                          ,
                          m
                        
                        k
                      
                    
                    )
                  
                  H
                
              
              -
              
                
                  f
                  m
                  k
                
                ⁢
                
                  e
                  k
                  H
                
              
            
          
          
            
              
                b
                
                  m
                  ,
                  m
                
                k
              
              ⁢
              I
            
          
        
      
      ]
    
    ±
    0
  
  ,
  
    

  
  ⁢
  
    ∀
    
      
        
          
          
            D
            
              m
              ,
              m
            
            k
          
          
        
        2
      
      ≤
      
        ɛ
        
          m
          ,
          m
        
        k
      
    
  
  ,




which are equivalently given by




  
    
      T
      m
      k
    
    ⁢
    
      =
      Δ
    
    ⁢
    
      
        [
        
          
            
              
                
                  b
                  
                    m
                    ,
                    m
                  
                  k
                
                -
                
                  λ
                  
                    m
                    ,
                    m
                  
                  k
                
              
            
            
              
                
                  
                    
                      h
                      ~
                    
                    
                      m
                      ,
                      m
                    
                    k
                  
                  ⁢
                  
                    F
                    m
                  
                
                -
                
                  
                    f
                    m
                    k
                  
                  ⁢
                  
                    e
                    k
                  
                
              
            
            
              0
            
          
          
            
              
                
                  
                    
                      (
                      
                        F
                        m
                      
                      )
                    
                    H
                  
                  ⁢
                  
                    
                      (
                      
                        
                          h
                          ~
                        
                        
                          m
                          ,
                          m
                        
                        k
                      
                      )
                    
                    H
                  
                
                -
                
                  
                    f
                    m
                    k
                  
                  ⁢
                  
                    e
                    k
                    H
                  
                
              
            
            
              
                
                  b
                  
                    m
                    ,
                    m
                  
                  k
                
                ⁢
                I
              
            
            
              
                -
                
                  
                    
                      ɛ
                      
                        m
                        ,
                        m
                      
                      k
                    
                    ⁡
                    
                      (
                      
                        F
                        m
                      
                      )
                    
                  
                  H
                
              
            
          
          
            
              0
            
            
              
                
                  -
                  
                    ɛ
                    
                      m
                      ,
                      m
                    
                    k
                  
                
                ⁢
                
                  F
                  m
                
              
            
            
              
                
                  λ
                  
                    m
                    ,
                    m
                  
                  k
                
                ⁢
                I
              
            
          
        
        ]
      
      ±
      0
    
  




  
    
      ∀
      m
    
    ,
    
      k
      .
    
  



Similarly it can be shown that the constraints ∥hm,nkFn∥2≦bm,nk holding for all ∥Dm,nk∥2≦εm,nk are equivalently given by




  
    
      U
      
        m
        ,
        n
      
      k
    
    ⁢
    
      =
      Δ
    
    ⁢
    
      
        [
        
          
            
              
                
                  b
                  
                    m
                    ,
                    n
                  
                  k
                
                -
                
                  λ
                  
                    m
                    ,
                    n
                  
                  k
                
              
            
            
              
                
                  
                    h
                    ~
                  
                  
                    m
                    ,
                    n
                  
                  k
                
                ⁢
                
                  F
                  n
                
              
            
            
              0
            
          
          
            
              
                
                  
                    (
                    
                      F
                      n
                    
                    )
                  
                  H
                
                ⁢
                
                  
                    (
                    
                      
                        h
                        ~
                      
                      
                        m
                        ,
                        n
                      
                      k
                    
                    )
                  
                  H
                
              
            
            
              
                
                  b
                  
                    m
                    ,
                    n
                  
                  k
                
                ⁢
                I
              
            
            
              
                -
                
                  
                    
                      ɛ
                      
                        m
                        ,
                        n
                      
                      k
                    
                    ⁡
                    
                      (
                      
                        F
                        n
                      
                      )
                    
                  
                  H
                
              
            
          
          
            
              0
            
            
              
                
                  -
                  
                    ɛ
                    
                      m
                      ,
                      n
                    
                    k
                  
                
                ⁢
                
                  F
                  n
                
              
            
            
              
                
                  λ
                  
                    m
                    ,
                    n
                  
                  k
                
                ⁢
                I
              
            
          
        
        ]
      
      ±
      0
    
  




  
    
      ∀
      
        m
        ≠
        n
      
    
    ,
    
      k
      .
    
  



Finally, it is noted that the constraint √{square root over (Σn(bm,nk)2+1)}≦fmka is equivalent to Vmk+fmkaI±0, ∀m, k, where





  
    
      V
      m
      k
    
    ⁢
    
      =
      Δ
    
    ⁢
    
      [
      
        
          
            0
          
          
            
              b
              m
              k
            
          
          
            1
          
        
        
          
            
              
                (
                
                  b
                  m
                  k
                
                )
              
              H
            
          
          
            0
          
          
            0
          
        
        
          
            1
          
          
            0
          
          
            0
          
        
      
      ]
    
  
  ,
  
    ∀
    m
  
  ,
  
    k
    .
  




Consequently, the problem S2(P) is equivalent to





   
  
    {
    
      
        
          
            min
            
              
                {
                
                  
                    F
                    m
                  
                  ,
                  
                    f
                    m
                    k
                  
                
                }
              
              ,
              b
              ,
              λ
              ,
              a
            
          
        
        
          a
        
        
          
            
          
        
      
      
        
          
            s
            .
            t
            .
          
        
        
          
            
              V
              m
              k
            
            +
            
              
                
                  f
                  m
                  k
                
                ⁢
                aI
              
              ±
              0
            
          
        
        
          
            
              ∀
              m
            
            ,
            k
          
        
      
      
        
          
            
          
        
        
          
            
              
                T
                m
                k
              
              ±
              0
            
            ,
          
        
        
          
            
              ∀
              k
            
            ,
            m
            ,
          
        
      
      
        
          
            
          
        
        
          
            
              
                U
                
                  m
                  ,
                  n
                
                k
              
              ±
              0
            
            ,
          
        
        
          
            
              ∀
              
                m
                ≠
                n
              
            
            ,
            k
          
        
      
      
        
          
            
          
        
        
          
            
              
                
                
                  F
                  m
                
                
              
              2
              2
            
            ≤
            
              P
              m
            
          
        
        
          
            
              ∀
              m
            
            ,
          
        
      
    
  




which is a standard form of GEVP.

Limited Cooperation
In certain embodiments, distributed methods can be employed for networks that do not support full CSI exchange between the BSs. One such distributed method is denoted as Algorithm 2 and is provided herein below in Table 2 Algorithm 2 can be utilized in a network in which limited information exchange between the BSs is employed. Here, each BS can design its precoders independently of others. The cost incurred for enabling such distributed processing is the degraded performance compared with the centralized processes discussed herein above.







TABLE 2





Algorithm 2-Distributed Robust Max-Min SINR Optimization
















 1:
for m = 1, . . . , M do


 2:
 Input Pm


 


 3:
 



  
    
      B
      m
    
    ⁢
    
      
    
    ⁢
    initializes
    ⁢
    
      
    
    ⁢
    
      F
      m
    
  
  =
  
    
      
        
          P
          m
        
        K
      
      ⁡
      
        [
        
          
            
              
                
                  
                    
                      (
                      h
                    
                    ~
                  
                  
                    m
                    ,
                    m
                  
                  1
                
                )
              
              H
            
            
              
                
                
                  
                    h
                    ~
                  
                  
                    m
                    ,
                    m
                  
                  1
                
                
              
              2
            
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            
              
                (
                
                  
                    h
                    ~
                  
                  
                    m
                    ,
                    m
                  
                  K
                
                )
              
              H
            
            
              
                
                
                  
                    h
                    ~
                  
                  
                    m
                    ,
                    m
                  
                  K
                
                
              
              2
            
          
        
        ]
      
    
    ⁢
    
      
    
    ⁢
    and
  






 



broadcasts Wm = FmFmH


 4:
end for


 5:
 Using Fm, {Wn}n≠m, each Bm computes sinrml, ∀ l


 6:
Repeat


 7:
 for m = 1, . . . , M do


 8:
 Bm solves S1,m(P) and obtains Fm*;


 9:
 Bm broadcasts Wm* = Fm* (Fm*)H


10:
 Each Bn, n ≠ m, computes sinrnl*, ∀ l based on Wm*, Fn 



 and {Wj}j≠m,n


11:
  if minl{ sinrnl*} < minl{ sinrnl} then Bn sends an error 



  message to Bm


12:
  if Bm receives no error message then it sets Fm ← Fm* 



  and broadcasts an update message


13:
  Upon receiving the update message each Bn, n ≠ m, 



  sets Wm ← Wm* and updates sinrnl, ∀ l


14:
 end for


15:
until no further precoder update is possible


16:
Output {Fm}









An underlying notion of the distributed algorithm in Table 2 is to successively update the precoder of one BS at-a-time while keeping the rest unchanged. More specifically, at the mth iteration, all precoders {Fn}m≠n are fixed and only BS Bm updates its precoder by maximizing the worst-case smallest rate of the mth cell. By recalling (14), the optimization problem solved by Bm is given by




  
    
      
        
          
            S
            
              1
              ,
              m
            
          
          ⁡
          
            (
            P
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          {
          
            
              
                
                  max
                  
                    
                      F
                      m
                    
                    ,
                    a
                  
                
              
              
                a
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    
                      sin
                      _
                    
                    ⁢
                    
                      r
                      m
                      k
                    
                  
                  ≥
                  a
                
              
              
                
                  ∀
                  k
                
              
            
            
              
                
                  
                
              
              
                
                  
                    
                      
                      
                        F
                        m
                      
                      
                    
                    2
                    2
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  
                
              
            
            
              
                
                  
                
              
              
                
                  
                    F
                    n
                  
                  ⁢
                  
                    
                  
                  ⁢
                  are
                  ⁢
                  
                    
                  
                  ⁢
                  fixed
                
              
              
                
                  
                    for
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                  ≠
                  m
                
              
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




Similar to the approach discussed above with regard to solving via power optimization, it can be readily verified that S1,m(P) can be solved through the following power optimization problem,





  
    
      
        
          
            P
            
              1
              ,
              m
            
          
          ⁡
          
            (
            
              P
              ,
              a
            
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          {
          
            
              
                
                  min
                  
                    
                      F
                      m
                    
                    ,
                    b
                  
                
              
              
                b
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    
                      sin
                      _
                    
                    ⁢
                    
                      r
                      m
                      k
                    
                  
                  ≥
                  a
                
              
              
                
                  ∀
                  k
                
              
            
            
              
                
                  
                
              
              
                
                  
                    
                      
                        
                        
                          F
                          m
                        
                        
                      
                      2
                    
                    
                      
                        P
                        m
                      
                    
                  
                  ≤
                  b
                
              
              
                
                  
                    ∀
                    m
                  
                  ,
                
              
            
            
              
                
                  
                
              
              
                
                  
                    F
                    n
                  
                  ⁢
                  
                    
                  
                  ⁢
                  are
                  ⁢
                  
                    
                  
                  ⁢
                  fixed
                
              
              
                
                  
                    for
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                  ≠
                  
                    m
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        22
        )
      
    
  




which is connected to the original problem S1,m(P) as follows.

Corollary 1 For any given power budget P, P1,m(P,a) is strictly increasing and continuous in a at any strictly feasible a and is related to S1,m(P) via

P1,m(P,S1,m(P))=1 for m=1, . . . , M.  (23)

To compute sinrmk in (13), it is clear that Bm needs to know Wn=FnFnH, n≠m. Using {Wn}, Bm can solve S1,m(P) optimally and obtains Fm* through solving P1,m(P,a) in conjunction with a linear bi-section search, as shown above in line 8 of Algorithm 2. It should be noted that solving S1,m(P) optimizes the minimum worst-case rate locally in the mth cell and does not necessarily lead to a boost in the network utility function. As a result, in an exemplary embodiment, Bm is permitted to update its precoder to Fm* only if such update results in a network-wide improvement, as shown in lines 9-13 of Algorithm 2.
The successive updates of the precoders can continue until no precoder can be further updated unilaterally. The convergence to such point is guaranteed by noting that Algorithm imposes the constraint that B can update its precoder only if it results in network-wide improvement.
Referring now to FIG. 5 with continuing reference to FIG. 3, a base station may determine a precodino matrix at step 304 by implementing a method 500 for successively determining matrices in a network in which base stations have limited cooperation. Algorithm 2 described above can be employed in the method 500, which in turn can be performed at each base station. The method 500 may begin at step 502 in which the base station can perform initialization and can broadcast an indication of the preliminary precoding matrix. For example, as indicated above in Table 2, the base station Bm may employ a power constraint Pm as an input. Further, the base station Bm can initialize its precoder matrix




  
    
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          2
        
      
    
    ]
  




and can broadcast Wm=FmFmH. The base station can obtain the estimated channels as discussed above with respect to step 302 to compute the preliminary precoder matrix Fm.

At step 504, the base station can fix the precoding matrices for the other base stations in the network and can compute a rate indication for users in its cell. For example, each other base station can perform the initialization and can broadcast indications of precoders for its estimated channels as discussed above with regard to step 502. Here, the base station can receive and compile values of Wn broadcast by each other base station Bn. In addition, the base station may compute a rate indication for users in its cell. For example, as provided above in Algorithm 2, using Fm, {Wn}n≠m, the base station Bm can compute sinrml, ∀l using equation 13.
At step 506, the base station can maximize a slack variable corresponding to a lower bound of the minimum worst case rate under a power constraint to determine the precoding matrix. For example, as indicated in Algorithm 2, the base station may solve Sl,m(P) in (21) to obtain Fm*, where the slack variable a is a lower bound on sinrmk and where ∥Fm∥22≦Pm. Similar to the process discussed above with regard to Algorithm 1, the base station may iteratively solve a power optimization problem to obtain a value of the slack variable that solves the slack variable maximization problem by determining whether slack variable values result in a pre-determined solution of the power optimization problem that is correlated to the solution to the slack variable maximization problem. For example, the base station may solve (21) using the power optimization problem (22) and the predetermined solution (23).
At step 508, the base station can transmit the determined precoding matrix to the other base stations in the network. For example, the base station Bm can broadcast Wm*=Fm*(Fm*)H, as indicated above in Algorithm 2.
At step 510, the base station can determine whether the precoding matrix Fm* results in a lower minimum receiver rate at one or more of the other base stations. For example, as indicated in Algorithm 2, each other base station Bn can compute sinrnl*, ∀l using (13) based on Wm*, Fn and {Wj}j≠m,n, where Fn is determined previously by each base station Bn by performing the initialization as mentioned above with regard to step 504 and {Wj}j≠m,n is obtained as a result of broadcasts performed by other base stations as mentioned above with regard to step 504. Moreover, as provided in Algorithm 2, each base station can determine whether minl{ sinrnl*<minl{ sinrnl}, where sinrnl is computed in the same manner discussed above with regard to the determination of sinrml in step 504, and, if so, can send an error message to base station Bm. In turn, if the base station Bm does not receive an error message, then the base station can presume that its precoder matrix Fm* does not, result in a lower minimum receiver rate at another base station. If the base station Bm does receive an error message, then the base station can set presume that its precoder matrix Fm* does result in a lower minimum receiver rate at another base station.
If the base station determines that its precoding matrix does not result in a lower minimum receiver rate at one or more base stations, then the method can proceed to step 512 in which the base station can apply the determined precoding matrix to the beamforming signals. For example, the base station Bm can apply Fm* to the beam forming signals. Thereafter, the method may repeat for another time frame, using Fm* and any Fn* selected by other base stations performing the method 500 in lieu of the corresponding preliminary precoders determined in accordance with step 502. It should also be noted that at step 512, the base station Bm can broadcast an update message indicating that Fm* has been applied by base station Bm. In turn, the other base stations Bn can receive the update message and can set Wm to Wm*.
If the base station determines that its precoding matrix results in a lower minimum receiver rate at one or more base stations, then the method can proceed to step 514 in which the base station can apply a current precoding matrix to the beamforming signals. For example, the current precoding matrix may be the precoding matrix Fm that can be determined at step 502. Alternatively, the current precoding matrix may be Fm* that was selected in a previous iteration of method 500. Thereafter, the method may repeat for another time frame, using the current precoding matrix and any Fn* or Wn* selected by other base stations performing the method 500 in lieu of the corresponding preliminary precoders previously determined in accordance with, for example, step 502.
It should be noted that in accordance with exemplary aspects, a variation of Algorithm 2 can be employed. In this variation, instead of fixing {1, . . . , M} as the order of processing, where the order in which the BSs attempt to update their precoders is fixed, as done in Algorithm 2 described above, a greedy approach can be applied. In particular, at each iteration of the process, each BS can compute its precoder, assuming precoders of other BSs to be fixed. Then, in a bidding phase, each BS can broadcast its choice and only the choice which maximizes the network minimum worst-case rate is accepted by all BSs.
Referring now to FIG. 6 with continuing reference to FIGS. 3 and 5, a base station may determine a precoding matrix at step 304 by implementing a method 600 for greedily determining matrices in a network in which base stations have limited cooperation. Steps 602-608 are identical to corresponding, respective steps 502-508 of method 500 except that the method steps 602-608 are performed simultaneously by each base station in the network (or a subset of base stations in the network).
At step 610, each base station Bn can determine whether its determined precoding matrix, such as Fn*, maximizes a lowest minimum receiver rate among the base stations. For example, each base station Bm may independently determine sinrnl*, ∀l, n for each precoding matrix Fn* received from each base station Bn and for its corresponding precoding matrix Fn=m*. Thereafter, using the values of sinrnl*, ∀l, n, each base station can determine which precoding matrix Fn* maximizes the network minimum worst-case rate and can apply that precoding matrix in lieu of the corresponding precoding matrix Fn to update its corresponding sinrml.
If the base station Bm determines that its determined precoding matrix maximizes a lowest minimum receiver rate among the base stations, then the method can proceed to step 612 in which the base station can apply the determined precoding matrix to the beamforming signals. For example, the base station Bm can apply Fm* to the beam forming signals. Thereafter, the method may repeat for another time frame, using Fm* and any Fn* selected by other base stations performing the method 600 in lieu of the corresponding preliminary precoders determined in accordance with step 602.
If the base station determines that its precoding matrix does not maximize a lowest minimum receiver rate among the base stations, then the method can proceed to step 614 in which the base station can apply a current precoding matrix to the beamforming signals. For example, the current precoding matrix may be the precoding matrix Fm that can be determined at step 602. Alternatively, the current precoding matrix may be Fm* that was selected in a previous iteration of method 600. Thereafter, the method may repeat for another time frame, using the current precoding matrix and any Fn* selected by other base stations performing the method 600 in lieu of the corresponding preliminary precoders previously determined in accordance with, for example, step 602.
According to other exemplary aspects, another distributed algorithm that can optimally solve the optimization problem in (14) can be employed by base stations. Here, the distributed process may solve the optimization problem in (14) by introducing more auxiliary variables and using dual decomposition. This process involves a higher level of inter-BS signaling than the previous distributed procedures discussed above.
First, (14) can be rewritten as




  
    
      
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                                (
                                
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                          ⁢
                          
                            
                              (
                              
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                          (
                          
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                        2
                      
                    
                  
                  
                    
                      
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                        m
                        ≠
                        n
                      
                      ,
                    
                  
                
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




To solve (24), a bi-section search over a can be employed in which for any fixed a, the following problem can be solved





  
    
      
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                      k
                      ,
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      
                        
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                            m
                          
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                        2
                        2
                      
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                        m
                      
                    
                  
                  
                    
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                      m
                    
                  
                
                
                  
                    
                      
                        
                          max
                          
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                              m
                              ,
                              n
                            
                            k
                          
                        
                        ⁢
                        
                          
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                              m
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                              n
                            
                            k
                          
                          ⁢
                          
                            
                              
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                                n
                              
                              ⁡
                              
                                (
                                
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                                  n
                                
                                )
                              
                            
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                          ⁢
                          
                            
                              (
                              
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                                  m
                                  ,
                                  n
                                
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                              )
                            
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                      ≤
                      
                        
                          (
                          
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                        2
                      
                    
                  
                  
                    
                      
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                          n
                        
                      
                      ,
                      
                        k
                        .
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




Thus, returning to FIG. 3, precoding matrices may be determined by maximizing a slack variable at step 310, in which the slack variable is given by in a (24). In turn, step 310 can be implemented by iteratively minimizing, at step 350, a sum of norms of precoding matrices using fixed values of the slack variable in accordance with, for example, (25). In addition, step 350 can be performed by a base station using a dual composition approach.
For example, for each BS m variables βm,nk,m, βn,mj,m which denote its copies of βm,nk, βn,mj, respectively, can be defined. Also, let b(m) be the vector formed by collecting all such variables. Then, (25) can be written as




  
    
      
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                        ≥
                        a
                      
                    
                    
                      
                        
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                        ,
                        k
                        ,
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      
                        
                          max
                          
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                              m
                              ,
                              n
                            
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                        ⁢
                        
                          
                            h
                            
                              n
                              ,
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                            j
                          
                          ⁢
                          
                            
                              
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                                m
                              
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                                (
                                
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                                  m
                                
                                )
                              
                            
                            H
                          
                          ⁢
                          
                            
                              (
                              
                                h
                                
                                  n
                                  ,
                                  m
                                
                                j
                              
                              )
                            
                            H
                          
                        
                      
                      ≤
                      
                        
                          (
                          
                            β
                            
                              n
                              ,
                              m
                            
                            
                              j
                              ,
                              m
                            
                          
                          )
                        
                        2
                      
                    
                  
                  
                    
                      
                        ∀
                        j
                      
                      ,
                      
                        n
                        ≠
                        m
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      
                        
                          
                          
                            F
                            m
                          
                          
                        
                        2
                        2
                      
                      ≤
                      
                        P
                        m
                      
                    
                  
                  
                    
                      ∀
                      m
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      
                        β
                        
                          m
                          ,
                          n
                        
                        
                          k
                          ,
                          m
                        
                      
                      =
                      
                        β
                        
                          m
                          ,
                          n
                        
                        
                          k
                          ,
                          n
                        
                      
                    
                  
                  
                    
                      
                        ∀
                        k
                      
                      ,
                      
                        m
                        ≠
                        
                          n
                          .
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




Similar to the proof of Theorem 4, it can be shown that (26) is equivalent to a (convex) SDP and thus strong duality holds for (26) provided Slater's condition is also satisfied. Thus, dual variables {λm,nk} can be defined and I can denote the vector formed by collecting all such variables. The following partial Lagrangian is considered





  
    
      
        
          L
          ⁡
          
            (
            
              
                {
                
                  
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                    m
                  
                  ,
                  
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                      (
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                }
              
              ⁢
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                    m
                    ,
                    n
                  
                  k
                
                ⁡
                
                  (
                  
                    
                      β
                      
                        m
                        ,
                        n
                      
                      
                        k
                        ,
                        m
                      
                    
                    -
                    
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                        m
                        ,
                        n
                      
                      
                        k
                        ,
                        n
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




and the dual function





  
    
      
        
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          ⁡
          
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                              (
                              m
                              )
                            
                          
                        
                        }
                      
                    
                    ⁢
                    
                      
                        ∑
                        m
                      
                      ⁢
                      
                        
                          
                          
                            F
                            m
                          
                          
                        
                        2
                        2
                      
                    
                  
                  +
                  
                    
                      ∑
                      
                        m
                        ≠
                        n
                      
                    
                    ⁢
                    
                      
                        ∑
                        k
                      
                      ⁢
                      
                        
                          λ
                          
                            m
                            ,
                            n
                          
                          k
                        
                        ⁡
                        
                          (
                          
                            
                              β
                              
                                m
                                ,
                                n
                              
                              
                                k
                                ,
                                m
                              
                            
                            -
                            
                              β
                              
                                m
                                ,
                                n
                              
                              
                                k
                                ,
                                n
                              
                            
                          
                          )
                        
                      
                    
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                  
                    
                  
                  ⁢
                  
                    
                      
                        
                          
                            
                              
                                
                                
                                  
                                    (
                                    
                                      
                                        
                                        
                                          
                                            
                                              h
                                              ~
                                            
                                            
                                              m
                                              ,
                                              m
                                            
                                            k
                                          
                                          ⁢
                                          
                                            w
                                            m
                                            k
                                          
                                        
                                        
                                      
                                      -
                                      
                                        
                                          ɛ
                                          
                                            m
                                            ,
                                            m
                                          
                                          k
                                        
                                        ⁢
                                        
                                          
                                            
                                            
                                              w
                                              m
                                              k
                                            
                                            
                                          
                                          2
                                        
                                      
                                    
                                    )
                                  
                                  +
                                
                                
                              
                              2
                            
                            
                              
                                
                                  
                                    
                                      
                                        max
                                        
                                          D
                                          
                                            m
                                            ,
                                            m
                                          
                                          k
                                        
                                      
                                      ⁢
                                      
                                        
                                          h
                                          
                                            m
                                            ,
                                            m
                                          
                                          k
                                        
                                        ⁢
                                        
                                          
                                            
                                              Y
                                              
                                                m
                                                ,
                                                k
                                              
                                            
                                            ⁡
                                            
                                              (
                                              
                                                Y
                                                
                                                  m
                                                  ,
                                                  k
                                                
                                              
                                              )
                                            
                                          
                                          H
                                        
                                        ⁢
                                        
                                          
                                            (
                                            
                                              h
                                              
                                                m
                                                ,
                                                m
                                              
                                              k
                                            
                                            )
                                          
                                          H
                                        
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    
                                      
                                        ∑
                                        
                                          n
                                          ≠
                                          m
                                        
                                      
                                      ⁢
                                      
                                        
                                          (
                                          
                                            β
                                            
                                              m
                                              ,
                                              n
                                            
                                            
                                              k
                                              ,
                                              m
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                    +
                                    1
                                  
                                
                              
                            
                          
                          ≥
                          a
                        
                      
                      
                        
                          
                            ∀
                            k
                          
                          ,
                          m
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          
                            max
                            
                              D
                              
                                n
                                ,
                                m
                              
                              k
                            
                          
                          ⁢
                          
                            
                              h
                              
                                n
                                ,
                                m
                              
                              j
                            
                            ⁢
                            
                              
                                
                                  F
                                  m
                                
                                ⁡
                                
                                  (
                                  
                                    F
                                    m
                                  
                                  )
                                
                              
                              H
                            
                            ⁢
                            
                              
                                (
                                
                                  h
                                  
                                    n
                                    ,
                                    m
                                  
                                  j
                                
                                )
                              
                              H
                            
                          
                        
                        ≤
                        
                          
                            (
                            
                              β
                              
                                n
                                ,
                                m
                              
                              
                                j
                                ,
                                m
                              
                            
                            )
                          
                          2
                        
                      
                    
                    
                      
                        
                          ∀
                          j
                        
                        ,
                        
                          n
                          ≠
                          m
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          
                            
                            
                              F
                              m
                            
                            
                          
                          2
                          2
                        
                        ≤
                        
                          P
                          m
                        
                      
                    
                    
                      
                        ∀
                        m
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




The dual problem splits into M smaller problems of the form





  
    
      
        {
        
          
            
              
                
                  
                    min
                    
                      {
                      
                        
                          F
                          m
                        
                        ,
                        
                          b
                          
                            (
                            m
                            )
                          
                        
                      
                      }
                    
                  
                  ⁢
                  
                    
                      
                      
                        F
                        m
                      
                      
                    
                    2
                    2
                  
                
                +
                
                  
                    ∑
                    
                      n
                      :
                      
                        n
                        ≠
                        m
                      
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          ∑
                          k
                        
                        ⁢
                        
                          
                            λ
                            
                              m
                              ,
                              n
                            
                            k
                          
                          ⁢
                          
                            β
                            
                              m
                              ,
                              n
                            
                            
                              k
                              ,
                              m
                            
                          
                        
                      
                      -
                      
                        
                          ∑
                          j
                        
                        ⁢
                        
                          
                            λ
                            
                              n
                              ,
                              m
                            
                            j
                          
                          ⁢
                          
                            β
                            
                              n
                              ,
                              m
                            
                            
                              j
                              ,
                              m
                            
                          
                        
                      
                    
                    )
                  
                
              
            
          
          
            
              
                s
                .
                t
                .
                
                  
                
                ⁢
                
                  
                    
                      
                        
                          
                            
                              
                              
                                
                                  (
                                  
                                    
                                      
                                      
                                        
                                          
                                            h
                                            ~
                                          
                                          
                                            m
                                            ,
                                            m
                                          
                                          k
                                        
                                        ⁢
                                        
                                          w
                                          m
                                          k
                                        
                                      
                                      
                                    
                                    -
                                    
                                      
                                        ɛ
                                        
                                          m
                                          ,
                                          m
                                        
                                        k
                                      
                                      ⁢
                                      
                                        
                                          
                                          
                                            w
                                            m
                                            k
                                          
                                          
                                        
                                        2
                                      
                                    
                                  
                                  )
                                
                                +
                              
                              
                            
                            2
                          
                          
                            
                              
                                max
                                
                                  D
                                  
                                    m
                                    ,
                                    m
                                  
                                  ′
                                
                              
                              ⁢
                              
                                
                                  h
                                  
                                    m
                                    ,
                                    m
                                  
                                  k
                                
                                ⁢
                                
                                  
                                    
                                      Y
                                      
                                        m
                                        ,
                                        k
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        Y
                                        
                                          m
                                          ,
                                          k
                                        
                                      
                                      )
                                    
                                  
                                  H
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      h
                                      
                                        m
                                        ,
                                        m
                                      
                                      k
                                    
                                    )
                                  
                                  H
                                
                              
                            
                            +
                            
                              
                                ∑
                                
                                  n
                                  ≠
                                  m
                                
                              
                              ⁢
                              
                                
                                  (
                                  
                                    β
                                    
                                      m
                                      ,
                                      n
                                    
                                    
                                      k
                                      ,
                                      m
                                    
                                  
                                  )
                                
                                2
                              
                            
                            +
                            1
                          
                        
                        ≥
                        a
                      
                    
                    
                      
                        ∀
                        k
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      
                        
                          max
                          
                            D
                            
                              n
                              ,
                              m
                            
                            j
                          
                        
                        ⁢
                        
                          
                            h
                            
                              n
                              ,
                              m
                            
                            j
                          
                          ⁢
                          
                            
                              
                                F
                                m
                              
                              ⁡
                              
                                (
                                
                                  F
                                  m
                                
                                )
                              
                            
                            H
                          
                          ⁢
                          
                            
                              (
                              
                                h
                                
                                  n
                                  ,
                                  m
                                
                                j
                              
                              )
                            
                            H
                          
                        
                      
                      ≤
                      
                        
                          (
                          
                            β
                            
                              n
                              ,
                              m
                            
                            
                              j
                              ,
                              m
                            
                          
                          )
                        
                        2
                      
                    
                  
                  
                    
                      
                        ∀
                        j
                      
                      ,
                      
                        n
                        :
                        
                          n
                          ≠
                          m
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                    
                      F
                      m
                    
                    
                  
                  2
                  2
                
                ≤
                
                  
                    P
                    m
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




Using the arguments provided in the preceding sections, each of the smaller problems in (29) can be shown to be equivalent to an SDP. Invoking the strong duality, the primal optimal solution can be recovered by solving the dual problem maxl{g(I)}. The latter problem can be also solved in a distributed manner via the sub-gradient method. In particular, suppose {{circumflex over (b)}(m)} are the optimized variables obtained upon solving the decoupled optimization problems in (29). Then the dual variables can be updated using a sub-gradient as λm,nk→λm,nk+μ({circumflex over (β)}m,nk,m−{circumflex over (β)}m,nk,n), ∀k,m≠n, where μ is a positive step size parameter. Updating λm,nk involves exchanging {circumflex over (β)}m,nk,m, {circumflex over (β)}m,n between BSs m,n, respectively. Finally, it should be noted that a speed-up can be obtained at the cost of some sub-optimality by forcing equality after a few steps of the sub-gradient method. In particular, both βm,nk,m, βm,nk,n can be set to be equal to




  
    
      
        β
        ^
      
      
        m
        ,
        n
      
      
        k
        ,
        m
      
    
    +
    
      
        β
        ^
      
      
        m
        ,
        n
      
      
        k
        ,
        n
      
    
  
  2




for all k,m≠n and then the M decoupled problems in (26) can be concurrently optimized over {Fm}. The current choice of a can be declared feasible, if and only if all of the problems are feasible.

Robust Weighted Sum-Rate Optimization
Referring now to FIG. 7 with continuing reference to FIGS. 1-3, an exemplary method 700 for optimizing the utility of receiver devices in a wireless communication system is illustrated. Flere, the method 700 generally implements a robust weighted sum-rate optimization and can be performed in a variety of ways, as discussed further herein below. Similar to method 300, method 700 can be performed by a base station 102, 202 or a control center if full cooperation between base stations is feasible. Alternatively, if only limited cooperation is feasible, then the method can be performed independently by each base station, as discussed further herein below.
The method 700 may begin at step 702 in which a base station or a control center may obtain a set of estimates of channel states corresponding to channels received by a set of receiver devices. These estimates can be received in the form of feedback from the receiver devices. The feedback from each receiver device can be limited to a few bits. Thus, each receiver device may have to quantize each one of the channel estimates available to it. Consequently, the base station or control center has access to channel estimates that can be corrupted by quantization errors. As indicated above, the channel state information may correspond to {hm,nk}n=1M, which includes the channel hm,mk received by the receiver from the base station Bm servicing the receiver in addition to the channels hm,nk (n≠m) received by the receiver from base stations Bn servicing cells other than cell m. In a fully cooperative scenario, each base station may communicate the respective channel state information for receivers or users in the corresponding cell to the other base stations in the network or to a control center. Alternatively, in a limited cooperation scenario, while each base station can estimate channel information {hm,nk}n=1M for receivers in its own cell, each base station can communicate prospective precoding matrices and equalizer matrices to other bases to optimize receiver utility in the network, as discussed in more detail herein below.
At step 704, a base station or control center can determine a precoding matrix for the set of receiver devices by maximizing a worst-case weighted sum rate of the network. For example, as discussed above with regard to equation (7), the precoding matrices {Fm} for the base stations in the network can be determined in accordance with a power budget by solving the following:




  
    R
    ⁡
    
      (
      P
      )
    
  
  ⁢
  
    =
    Δ
  
  ⁢
  
    {
    
      
        
          
            max
            
              {
              
                F
                m
              
              }
            
          
        
        
          
            
              min
              
                {
                
                  D
                  
                    m
                    ,
                    n
                  
                  k
                
                }
              
            
            ⁢
            
              
                ∑
                
                  m
                  =
                  1
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  K
                
                ⁢
                
                  
                    α
                    m
                    k
                  
                  ⁢
                  
                    R
                    m
                    k
                  
                
              
            
          
        
        
          
            
          
        
      
      
        
          
            s
            .
            t
            .
          
        
        
          
            
              
                
                
                  F
                  m
                
                
              
              2
              2
            
            ≤
            
              P
              m
            
          
        
        
          
            ∀
            
              m
              .
            
          
        
      
    
  




Here the term




  
    min
    
      {
      
        D
        
          m
          ,
          n
        
        k
      
      }
    
  
  ⁢
  
    
      ∑
      
        m
        =
        1
      
      M
    
    ⁢
    
      
        ∑
        
          k
          =
          1
        
        K
      
      ⁢
      
        
          α
          m
          k
        
        ⁢
        
          R
          m
          k
        
      
    
  




is an example of the minimum receiver weighted sum-rate which is within a set of rates corresponding to a bounded set of channel estimation errors {Dm,nk,}. The weights αmk can be employed to implement a user-priority scheme, where the rates Rmk of users Umk with a higher priority are given higher corresponding weights αmk. Furthermore, the estimate can be determined by considering a set of channels {hm,nk}n=1M including channels hm,nk (n≠m) received by the set of receiver devices from base stations Bn (n≠m) other than a base station Bm servicing the set of receiver devices. For example, as noted below with respect to various exemplary embodiments, {hm,nk}n=1M can be used to determine values of M{tilde over (S)}Emk to solve equation (7).

At step 706, each base station can transmit beamforming signals generated in accordance with the determined precoding matrix to their own respective receiver devices in their respective cell.
It should be noted that, at step 704, in the full cooperation scenario, a control center can determine precoding matrices for each base station using channel state information for receivers in each cell serviced by the base stations and can assign the precoding matrices to the base stations to enable them to generate optimized, beamforming signals for transmission to the receiver. Alternatively, in the full cooperation scenario, each base station may independently determine their own precoding matrix using the same methods, where each base station can receive channel state information from other base stations, determine the precoding matrices for the entire network and apply a corresponding precoding matrix for its own beamforming signals. In addition, one or more base stations can implement step 704 by solving equation (7) via MSE optimization. Exemplary embodiments that solve (7) using MSE optimization in a full cooperation scenario and a limited cooperation scenario are described herein below.
Full Cooperation
By recalling the definitions in (5) and taking into account that the uncertainty regions corresponding to SINRmk and SINRnl for m≠n or l≠k are disjoint, finding the worst-case SINR for each user can be carried out independently of the rest. Hence, by recalling that the worst-case SINR of user Umk is denoted by sinrmk, the problem R(P) in (7) is given by




  
    
      
        
          R
          ⁡
          
            (
            P
            )
          
        
        =
        
          {
          
            
              
                
                  max
                  
                    {
                    
                      F
                      m
                    
                    }
                  
                
              
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    M
                  
                  ⁢
                  
                    
                      ∑
                      
                        k
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      
                        α
                        m
                        k
                      
                      ⁢
                      
                        log
                        ⁡
                        
                          (
                          
                            1
                            +
                            
                              
                                sin
                                ⁢
                                r
                              
                              m
                              k
                            
                          
                          )
                        
                      
                    
                  
                
              
              
                
                  
                
              
            
            
              
                
                  s
                  .
                  t
                  .
                
              
              
                
                  
                    
                      
                      
                        F
                        m
                      
                      
                    
                    2
                    2
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




The problem R (P) as posed above, is not a convex problem. Optimal precoder design based on maximizing the weighted sum-rate even when the BSs have perfect CSI is intractable. To the best of the knowledge of the inventors, even in this case only techniques yielding locally optimal solutions can be used. According to exemplary aspects of the present invention, a suboptimal solution can be found by obtaining a conservative approximation of the problem R (P). This approximation provides a lower bound on R (P).

To start the set of functions {Smk(u): R→R is defined as




  
    
      
        S
        m
        k
      
      ⁡
      
        (
        u
        )
      
    
    ⁢
    
      =
      Δ
    
    ⁢
    
      
        
          α
          m
          k
        
        ⁢
        u
      
      -
      
        
          
            α
            m
            k
          
          
            1
            +
            
              
                sin
                ⁢
                r
              
              m
              k
            
          
        
        ⁢
        
          exp
          ⁡
          
            (
            
              u
              -
              1
            
            )
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      for
      ⁢
      
        
      
      ⁢
      m
    
    =
    1
  
  ,
  ...
  ⁢
  
    
  
  ,
  M
  ,
  
    k
    =
    1
  
  ,
  ...
  ⁢
  
    
  
  ,
  K
  ,
  
    
      and
      ⁢
      
        
      
      ⁢
      u
    
    ∈
    
      R
      .
    
  




It can be readily verified that





  
    
      
        
          
            
              max
              
                u
                ∈
                R
              
            
            ⁢
            
              
                S
                m
                k
              
              ⁡
              
                (
                u
                )
              
            
          
          =
          
            
              α
              m
              k
            
            ⁢
            
              log
              ⁡
              
                (
                
                  1
                  +
                  
                    
                      sin
                      ⁢
                      r
                    
                    m
                    k
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        and
        ⁢
        
          

        
        ⁢
        
          
            u
            *
          
          =
          
            
              arg
              ⁢
              
                
                  max
                  
                    u
                    ∈
                    R
                  
                
                ⁢
                
                  
                    S
                    m
                    k
                  
                  ⁡
                  
                    (
                    u
                    )
                  
                
              
            
            =
            
              
                log
                ⁡
                
                  (
                  
                    1
                    +
                    
                      
                        sin
                        ⁢
                        r
                      
                      m
                      k
                    
                  
                  )
                
              
              +
              1
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            ∀
            k
          
          ,
          
            m
            .
          
        
      
    
    
      
        (
        31
        )
      
    
  




Therefore, by incorporating the slack variables u=[umk] and substituting the objective function of R (P) with its equivalent term





  
    
      ∑
      
        in
        =
        1
      
      M
    
    ⁢
    
      
        ∑
        
          k
          =
          1
        
        K
      
      ⁢
      
        
          max
          
            u
            m
            k
          
        
        ⁢
        
          
            S
            m
            k
          
          ⁡
          
            (
            
              u
              m
              k
            
            )
          
        
      
    
  
  ,




the problem R (P) is equivalently given by





  
    
      
        
          R
          ⁡
          
            (
            P
            )
          
        
        =
        
          {
          
            
              
                
                  max
                  
                    
                      {
                      
                        F
                        m
                      
                      }
                    
                    ,
                    u
                  
                
              
              
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      M
                    
                    ⁢
                    
                      
                        ∑
                        
                          k
                          =
                          1
                        
                        K
                      
                      ⁢
                      
                        
                          α
                          m
                          k
                        
                        ⁢
                        
                          u
                          m
                          k
                        
                      
                    
                  
                  -
                  
                    
                      
                        α
                        m
                        k
                      
                      
                        1
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          
                            r
                            m
                            k
                          
                        
                      
                    
                    ⁢
                    
                      exp
                      ⁡
                      
                        (
                        
                          
                            u
                            m
                            k
                          
                          -
                          1
                        
                        )
                      
                    
                  
                
              
            
            
              
                
                  
                    s
                    .
                    t
                    .
                    
                      
                    
                    ⁢
                    
                      
                        
                        
                          F
                          m
                        
                        
                      
                      2
                      2
                    
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




For any fixed u the intermediate problem {tilde over (R)}(P,u) is defined, where {tilde over (R)}(P,u) yields the optimal precoders {Fm} corresponding to the given u and power budget P. Since for a given u the term ΣmΣk αmkumk becomes a constant, the following is obtained




  
    
      
        
          
            R
            ~
          
          ⁡
          
            (
            
              P
              ,
              u
            
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          {
          
            
              
                
                  min
                  
                    {
                    
                      F
                      m
                    
                    }
                  
                
              
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    M
                  
                  ⁢
                  
                    
                      ∑
                      
                        k
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      
                        
                          α
                          m
                          k
                        
                        
                          1
                          +
                          
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            
                              r
                              m
                              k
                            
                          
                        
                      
                      ⁢
                      
                        exp
                        ⁡
                        
                          (
                          
                            
                              u
                              m
                              k
                            
                            -
                            1
                          
                          )
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    s
                    .
                    t
                    .
                    
                      
                    
                    ⁢
                    
                      
                        
                        
                          F
                          m
                        
                        
                      
                      2
                      2
                    
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




The problem {tilde over (R)}(P,u) can now be transformed into a weighted sum of the worst-case MSE s as follows,





  
    
      
        
          
            R
            ~
          
          ⁡
          
            (
            
              P
              ,
              u
            
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          {
          
            
              
                
                  min
                  
                    {
                    
                      F
                      m
                    
                    }
                  
                
              
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    M
                  
                  ⁢
                  
                    
                      ∑
                      
                        k
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      
                        α
                        m
                        k
                      
                      ⁢
                      
                        exp
                        ⁡
                        
                          (
                          
                            
                              u
                              m
                              k
                            
                            -
                            1
                          
                          )
                        
                      
                      ⁢
                      
                        
                          max
                          
                            {
                            
                              D
                              
                                m
                                ,
                                n
                              
                              k
                            
                            }
                          
                        
                        ⁢
                        
                          
                            min
                            
                              f
                              m
                              k
                            
                          
                          ⁢
                          
                            M
                            ⁢
                            
                              S
                              ~
                            
                            ⁢
                            
                              E
                              m
                              k
                            
                          
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    s
                    .
                    t
                    .
                    
                      
                    
                    ⁢
                    
                      
                        
                        
                          F
                          m
                        
                        
                      
                      2
                      2
                    
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




Next, for any given u an upper bound on {tilde over (R)}(P,u) can be found, where, by recalling (32) and (34), the upper bound on {tilde over (R)}(P,u) is a lower bound on R(P). By invoking the inequality in (19) and defining fm=[fmk]k, an upper bound on {tilde over (R)}(P,u) can be found as follows,




  
    
      
        
          
            R
            _
          
          ⁡
          
            (
            
              P
              ,
              u
            
            )
          
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          {
          
            
              
                
                  min
                  
                    {
                    
                      
                        F
                        m
                      
                      ,
                      
                        f
                        m
                      
                    
                    }
                  
                
              
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    M
                  
                  ⁢
                  
                    
                      ∑
                      
                        k
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      
                        α
                        m
                        k
                      
                      ⁢
                      
                        exp
                        ⁡
                        
                          (
                          
                            
                              u
                              m
                              k
                            
                            -
                            1
                          
                          )
                        
                      
                      ⁢
                      
                        
                          max
                          
                            {
                            
                              D
                              
                                m
                                ,
                                n
                              
                              k
                            
                            }
                          
                        
                        ⁢
                        
                          M
                          ⁢
                          
                            
                          
                          ⁢
                          
                            S
                            ~
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            E
                            m
                            k
                          
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    s
                    .
                    t
                    .
                    
                      
                    
                    ⁢
                    
                      
                        
                        
                          F
                          m
                        
                        
                      
                      2
                      2
                    
                  
                  ≤
                  
                    P
                    m
                  
                
              
              
                
                  ∀
                  m
                
              
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




{tilde over (R)}(P,u) can itself be sub-optimally solved by using the alternating optimization (AO) principle and optimizing {fm} and {Fm} in an alternating manner. By deploying AO {Fm} can be optimized while keeping {fm} fixed and vice versa. Since the objective is bounded and it decreases monotonically at each iteration, the AO procedure is guaranteed to converge. In the following theorem it is shown that solving R(P,u) at each step of the AO procedure is a convex problem with a computationally efficient solution.

Theorem 6 For arbitrarily fixed {Fm}, the problem R(P,u) can be optimized over {fm} efficiently as an SDP. Similarly, for arbitrarily fixed {fm}, the problem R(P,u) can be optimized over {Fm} efficiently as another SDP.
The proof of Theorem 6 is as follows. It is first shown that for any given and fixed {fmk}, the problem R(P,u) is equivalent to an SDP. qmk is defined as qmk=αmkexp(umk−1)/|fmk|2. Accordingly,




  
    
      ∑
      
        m
        =
        1
      
      M
    
    ⁢
    
      
        ∑
        
          k
          =
          1
        
        K
      
      ⁢
      
        
          α
          m
          k
        
        ⁢
        
          exp
          ⁡
          
            (
            
              
                u
                m
                k
              
              -
              1
            
            )
          
        
        ⁢
        M
        ⁢
        
          
        
        ⁢
        
          S
          ~
        
        ⁢
        
          
        
        ⁢
        
          E
          m
          k
        
      
    
  
  =
  
    
      ∑
      
        m
        =
        1
      
      M
    
    ⁢
    
      
        ∑
        
          k
          =
          1
        
        K
      
      ⁢
      
        
          
            (
            
              
                
                  q
                  m
                  k
                
                ⁢
                
                  
                    
                    
                      
                        
                          h
                          
                            m
                            ,
                            m
                          
                          k
                        
                        ⁢
                        
                          w
                          m
                          k
                        
                      
                      -
                      
                        f
                        m
                        k
                      
                    
                    
                  
                  2
                
              
              +
              
                
                  ∑
                  
                    l
                    ≠
                    k
                  
                
                ⁢
                
                  
                    q
                    m
                    l
                  
                  ⁢
                  
                    
                      
                      
                        
                          h
                          
                            m
                            ,
                            m
                          
                          l
                        
                        ⁢
                        
                          w
                          m
                          k
                        
                      
                      
                    
                    2
                  
                
              
              +
              
                
                  ∑
                  
                    n
                    ≠
                    m
                  
                
                ⁢
                
                  
                    ∑
                    l
                  
                  ⁢
                  
                    
                      q
                      n
                      l
                    
                    ⁢
                    
                      
                        
                        
                          
                            h
                            
                              n
                              ,
                              m
                            
                            l
                          
                          ⁢
                          
                            w
                            m
                            k
                          
                        
                        
                      
                      2
                    
                  
                
              
            
            )
          
          
            }
            
              ≡
              
                g
                ⁡
                
                  (
                  
                    w
                    m
                    k
                  
                  )
                
              
            
          
        
        .
      
    
  




Clearly, the optimization of R(P,u) now decouples into M optimization problems of the form





  
    
      
        {
        
          
            
              
                min
                
                  
                    F
                    m
                  
                  ,
                  
                    b
                    m
                  
                
              
            
            
              
                b
                m
              
            
          
          
            
              
                s
                .
                t
                .
              
            
            
              
                
                  
                    max
                    
                      {
                      
                        D
                        
                          n
                          ,
                          m
                        
                        ′
                      
                      }
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        k
                        =
                        1
                      
                      K
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        
                          w
                          m
                          k
                        
                        )
                      
                    
                  
                
                ≤
                
                  b
                  m
                
              
            
          
          
            
              
                
                  
                    
                    
                      F
                      m
                    
                    
                  
                  2
                  2
                
                ≤
                
                  P
                  m
                
              
            
            
              
                ∀
                
                  m
                  .
                
              
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




It can be verified that the constraints can be equivalently expressed as finitely many Linear Matrix Inequalities (LMIs) so that the optimization problem is equivalent to an SDP. Next, suppose {Fm} are arbitrarily fixed. Then R(P,u) reduces to





  
    min
    
      {
      
        f
        m
        1
      
      }
    
  
  ⁢
  
    
      ∑
      
        m
        =
        1
      
      M
    
    ⁢
    
      
        ∑
        
          k
          =
          1
        
        K
      
      ⁢
      
        
          α
          m
          k
        
        ⁢
        
          exp
          ⁡
          
            (
            
              
                u
                m
                k
              
              -
              1
            
            )
          
        
        ⁢
        
          
            max
            
              {
              
                D
                
                  m
                  ,
                  n
                
                k
              
              }
            
          
          ⁢
          
            M
            ⁢
            
              
            
            ⁢
            
              S
              ~
            
            ⁢
            
              
            
            ⁢
            
              E
              m
              k
            
          
        
      
    
  




The above optimization problem decouples into KM smaller problems of the form





  
    
      
        
          min
          
            {
            
              f
              m
              k
            
            }
          
        
        ⁢
        
          
            max
            
              {
              
                D
                
                  m
                  ,
                  n
                
                1
              
              }
            
          
          ⁢
          
            M
            ⁢
            
              
            
            ⁢
            
              S
              ~
            
            ⁢
            
              
            
            ⁢
            
              E
              m
              k
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




Substituting gmk=1/fmk in (37), the optimization can instead be performed over gmk and the latter optimization problem can be readily shown to be equivalent to an SDP.








TABLE 3





Algorithm 3 - Robust Weighted Sum-rate Optimization


















1:
Input P and {{tilde over (h)}m,nk, εm,nk}



2:
Initialize Fm, fm for all m and u



3:
Repeat



4:
 Solve R(P, u) by optimizing over {Fm} and {fmk} in an 




 alternating manner;



5:
 Update umk ← 1 − log(max{Dm,nk}M{tilde over (S)}Emk), ∀k, m



6:
 until convergence



7:
Output {Fm}










Algorithm 3 provided in Table 3 above summarizes steps that can be employed for sub-optimally solving R (P). Algorithm 3 is constructed based on the connection between the objective functions of R (P) and R(P,u). At each iteration of Algorithm 3 for a fixed u, R(P,u) is solved by using the AO principle as discussed above and a new set of precoders and equalizers is obtained. The minimum rate achieved by using this set of precoders and equalizers provides a lower bound on R (P). This set of precoders and equalizers is also deployed for computing the worst-case MSE s and updating u as umk=1−log(max{Dm,nk} M{tilde over (S)}Emk), ∀k, m. It should be noted that the worst-case MSE s can be computed using the techniques described above with respect to the proof of Theorem 5.
Since R (P) is bounded from above, so is any lower bound on it. Therefore, the utility function of Algorithm 3 is bounded and increases monotonically in each iteration. Thus, convergence of Algorithm 3 is guaranteed.
With reference now to FIG. 8 with continuing reference to FIG. 7, an exemplary method 800 for determining precoding matrices and equalizers that optimize the utility of receivers in a wireless communication network in accordance with an exemplary embodiment is illustrated. Algorithm 3 can be employed in the method 800 to determine the precoders and the equalizers. Further, the method 800 can be performed by one or more base stations, including a control center, and can be performed to implement step 704 of the method 700. Alternatively, each base station may implement a distributed variation of the method 800, as discussed further herein below.
Method 800 can begin at step 802, in which a base station may obtain powers P, channel estimates {tilde over (h)}m,nk, uncertainty regions εm,nk and user-weights αmk. The values of these parameters can be communicated between the base stations.
At step 804, the base station can initialize the slack variables u, the choice of precoding matrices Fm and the equalizers fm for all m base stations using the variables obtained at step 802.
At step 806, the base station may obtain the worst-case weighted MSE formulation. For example, the base station may obtain the formulation (35).
At step 808, the base station can iteratively solve the formulation over a set of precoding matrices and a set of equalizers by alternately optimizing the formulation for the set of precoding matrices with a fixed equalizer matrix and optimizing the formulation for the set of equalizer matrices with a fixed precoding matrix until convergence of the formulation to obtain a new set of optimal precoding matrices and a new set optimal equalizer matrices for the base stations. For example, holding u fixed, the base station may solve or optimize R(P,u) over the set of precoding matrices {Fm} and the equalizers {fm} in an alternating manner by solving corresponding SDPs until the formulation converges. The new precoding matrices {Fm} and the equalizers {fm} can be obtained to update the slack variables and to determine whether the formulation has converged to the maximum MSE.
At step 810, the base station may update the slack variables with the precoding matrices and new equalizers. For example, the base station may update the slack variables u by setting the precoders and equalizers to the new precoding matrices and new equalizers and holding them fixed. The slack variable may be updated by using the closed from expression, where umk can be set to 1−log(max{Dm,nk} M{tilde over (S)}Emk) for all m, k.
At step 812, the base station may compute the worst-case weighted sum rate with the new sets of precoding matrixes and equalizer matrices. For example, as indicated above, the procedure set forth in the proof of Theorem 5 can be employed to find the worst-case weighted sum rate.
At step 814, the base station can determine whether the worst case rated sum-rate has converged. If the worst-case rated sum-rate has converged, then the method may proceed to step 806 and repeat. Otherwise the method may proceed to step 816, in which the base station can output the choice of precoding matrices {Fm} and equalizer matrices {fm} that lead to the convergence of the worst-case rated sum rate.
In accordance with an exemplary embodiment, a control center can distribute the precoding matrices {Fm} and equalizers {fm} to the corresponding base stations, each of which may apply its respective precoding matrix Fm and equalizer fm to generate beamforming signals for transmission to its set of receiver devices. For example, each base station may apply its precoding matrix Fm and equalizer fm to generate beamfortning signals for transmission in step 706 of method 700.
Distributed Implementation
An advantage of the AO based approach employed to sub-optimally solve R (P) in the previous section is that it is amenable to a distributed implementation. In particular, it should be noted that for fixed u, {fm}, the optimization over {Fm} decouples into M smaller problems (36) that can be solved concurrently by the M BSs. Similarly, for fixed u, {Fm}, the optimization over {fm} decouples into KM smaller problems (37) that can be solved concurrently. Finally, for a given {Fm,fm} the elements of u can also be updated concurrently. Consequently, Algorithm 3 and method 800 can indeed be implemented in a distributed fashion with appropriate information exchange among the BSs.
For example, similar to methods 500 and 600 of FIGS. 5 and 6, respectively, each base station m may successively determine its corresponding precoding matrix and equalizers using Algorithm 3 or method 800 by employing a fixed set of channel estimations and a fixed set of equalizers for base stations n≠m. For example, each base station m can receive channels {hn,ml} and/or precoding matrices {Fn≠m} and {fnl} from the other base stations in step 802 and may optimize its precoding matrices Fm and equalizers fm in an alternating manner using (36) and (37) respectively by holding fm and Fm fixed respectively, and holding u fixed as discussed above with regard to step 808. Thereafter, as discussed above with respect to step 810, the base station m can update u using the closed form expression. Subsequently, the base station m can iteratively determine the precoding matrices Fm and equalizers fm using alternating optimization and can update the slack variable u until the worst-cast weighted sum rate has converged, as discussed above with regard to steps 812 and 814 of method 800.
In addition, after obtaining the optimized precoding matrices Fm and equalizers fm, similar to method 500 of FIG. 5, the base station m can determine whether the optimized precoding matrices Fm and equalizers fm result in a higher MSE at any of the other base stations by broadcasting of the precoding matrices Fm and equalizers fm and by employing error messages. In addition, the base station can apply the optimized precoding matrices Fm and equalizers fm in step 706 to transmit beamforming signals to its set of receivers if the optimized precoding matrices and equalizers do not result in a higher MSE at any of the other base stations.
In an alternative implementation, after obtaining the optimized precoding matrices Fm and equalizers fm, similar to method 600 of FIG. 6, the base stations may update the precoders and matrices in a greedy manner using a bidding procedure. For example, similar to step 608, each base station may transmit its corresponding optimized precoding matrices and equalizers to each other base station and the base stations may determine that only the precoding matrix Fm and equalizers fm resulting in a lowest network-wide MSE will be applied. Thus, the corresponding base station m can apply the precoding matrix Fm and equalizers fm to transmit beamforming signals to its receivers while the remaining base stations n≠m can update its values of the precoders and equalizers employed at base station m. The process may thereafter be repeated for a following time frame.
SLIR Optimization
Another adequate albeit sub-optimal approach for selecting the beamforming vectors considers the worst-case signal-to-leakage-interference-plus-noise ratio (SLINR) can be employed. The SLINR metric has been shown to be an effective metric over networks with perfect CSI. In particular, the worst-case SLINR corresponding to user k in cell M is given by,




  
    
      
        
          slinr
          m
          k
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            min
            
              {
              
                D
                
                  m
                  ,
                  n
                
                k
              
              }
            
          
          ⁢
          
            
              
                
                
                  
                    h
                    
                      m
                      ,
                      m
                    
                    k
                  
                  ⁢
                  
                    w
                    m
                    k
                  
                
                
              
              2
            
            
              
                
                  ∑
                  
                    j
                    ≠
                    k
                  
                
                ⁢
                
                  
                    
                    
                      
                        h
                        
                          m
                          ,
                          m
                        
                        j
                      
                      ⁢
                      
                        w
                        m
                        k
                      
                    
                    
                  
                  2
                
              
              +
              
                
                  ∑
                  
                    n
                    ≠
                    m
                  
                
                ⁢
                
                  
                    ∑
                    l
                  
                  ⁢
                  
                    
                      
                      
                        
                          h
                          
                            n
                            ,
                            m
                          
                          l
                        
                        ⁢
                        
                          w
                          m
                          k
                        
                      
                      
                    
                    2
                  
                
              
              +
              1
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




The Uncertainty regions in the numerator and denominator arc decoupled so that





  
    
      
        
          slinr
          m
          k
        
        =
        
          
            
              min
              
                {
                
                  D
                  
                    m
                    ,
                    m
                  
                  k
                
                }
              
            
            ⁢
            
              
                
                
                  
                    h
                    
                      m
                      ,
                      m
                    
                    k
                  
                  ⁢
                  
                    w
                    m
                    k
                  
                
                
              
              2
            
          
          
            
              
                ∑
                
                  j
                  ≠
                  k
                
              
              ⁢
              
                
                  max
                  
                    {
                    
                      D
                      
                        m
                        ,
                        m
                      
                      f
                    
                    }
                  
                
                ⁢
                
                  
                    
                    
                      
                        h
                        
                          m
                          ,
                          m
                        
                        j
                      
                      ⁢
                      
                        w
                        m
                        k
                      
                    
                    
                  
                  2
                
              
            
            +
            
              
                ∑
                
                  n
                  ≠
                  m
                
              
              ⁢
              
                
                  ∑
                  l
                
                ⁢
                
                  
                    max
                    
                      {
                      
                        D
                        
                          n
                          ,
                          m
                        
                        l
                      
                      }
                    
                  
                  ⁢
                  
                    
                      
                      
                        
                          h
                          
                            n
                            ,
                            m
                          
                          l
                        
                        ⁢
                        
                          w
                          m
                          k
                        
                      
                      
                    
                    2
                  
                
              
            
            +
            1
          
        
      
    
    
      
        (
        39
        )
      
    
  




If a per-user power profile {Pmk} has been given, then the beamforming vectors can be independently designed by solving





  
    
      
        {
        
          
            
              
                
                  max
                  
                    w
                    m
                    k
                  
                
                ⁢
                
                  
                
                ⁢
                
                  slinr
                  m
                  k
                
              
            
          
          
            
              
                
                  
                    s
                    .
                    t
                    .
                    
                      
                    
                    ⁢
                    
                      
                        
                        
                          w
                          m
                          k
                        
                        
                      
                      2
                      2
                    
                  
                  ≤
                  
                    
                      P
                      m
                      k
                    
                    ⁢
                    
                      ∀
                      k
                    
                  
                
                ,
                
                  m
                  .
                
              
            
          
        
      
    
    
      
        (
        40
        )
      
    
  




The maximization problem in (40) can be exactly solved by alternatively solving a power optimization problem in conjunction with a linear bi-section search as described in (14) and (15).
Exemplary embodiments of the present invention discussed herein above provide improved precoder designs that optimize user-rates by accurately accounting for inter-cell interference and intra-cell interference. By considering channel states and corresponding uncertainty regions of interference channels received by users, the precoders of exemplary embodiments can guarantee minimum utility levels for all possible errors in the uncertainty regions.
It should be understood that embodiments described herein may be entirely hardware, entirely software or including both hardware and software elements. In a preferred embodiment, the present invention is implemented in hardware and software, which includes but is not limited to firmware resident software, microcode, etc.
Embodiments may include a computer program product accessible from a computer-usable or computer-readable medium providing program code for use by or in connection with a computer or any instruction execution system. A computer-usable or computer readable medium may include any apparatus that stores, communicates, propagates, or transports the program for use by or in connection with the instruction execution system, apparatus, or device. The medium can be magnetic, optical, electronic, electromagnetic, infrared, or semiconductor system (or apparatus or device) or a propagation medium. The medium may include a computer-readable storage medium such as a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disk and an optical disk, etc.
A data processing system suitable for storing and/or executing program code may include at least one processor coupled directly or indirectly to memory elements through a system bus. The memory elements can include local memory employed during actual execution of the program code, bulk storage, and cache memories which provide temporary storage of at least some program code to reduce the number of tunes code is retrieved from bulk storage during execution. Input/output or I/O devices (including but not limited to keyboards, displays, pointing devices, etc.) may be coupled to the system either directly or through intervening I/O controllers.
Network adapters may also be coupled to the system to enable the data processing system to become coupled to other data processing systems or remote printers or storage devices through intervening private or public networks. Modems, cable modem and Ethernet cards are just a few of the currently available types of network adapters.
Having described preferred embodiments of a system and method (which are intended to be illustrative and not limiting), it is noted that modifications and variations can be made by persons skilled in the art in light of the above teachings. It is therefore to be understood that changes may be made in the particular embodiments disclosed which are within the scope of the invention as outlined by the appended claims. Having thus described aspects of the invention, with the details and particularity required by the patent laws, what is claimed and desired protected by Letters Patent is set forth in the appended claims.

## Claims

1. A wireless communications system, comprising:
a first set of one or more mobile stations;
a first base station that serves the first set of mobile stations; and
a second base station coordinated with the first base station,
wherein each of the first set of mobile stations obtains first channel state information corresponding to the first base station and second channel state information corresponding to the second base station, and
wherein each of the first set of mobile stations transmits to the first base station a first indication of the first channel state information and a second indication of the second channel state information.

2. The wireless communications system of claim 1,
wherein the first base station decides a precoder according to the first and second indications received from the first set of mobile stations.

3. The wireless communications system of claim 1, further comprising:
a second set of one or more mobile stations,
wherein each of the second set of mobile stations obtains third channel state information corresponding to the first base station and fourth channel state information corresponding to the second base station, and
wherein each of the second set of mobile stations transmits to the first base station a third indication of the third channel state information and a fourth indication of the fourth channel state information.

4. The wireless communications system of claim 3,
wherein the first base station decides a precoder according to the first and second indications received from the first set of mobile stations and the third and fourth indications received from the second set of mobile stations.

5. The wireless communications system of claim 1,
wherein at least one of the first channel state information and the second channel state information comprises a channel estimate.

6. The wireless communications system of claim 1,
wherein at least one of the first channel state information and the second channel state information is quantized at each of the first set of mobile stations.

7. The wireless communications system of claim 3,
wherein at least one of the third channel state information and the fourth channel state information comprises a channel estimate.

8. The wireless communications system of claim 3,
wherein at least one of the third channel state information and the fourth channel state information is quantized at each of the second set of mobile stations.

9. A method implemented in a mobile station used in a wireless communications system including a first base station, and a second base station coordinated with the first base station, the method comprising:
obtaining first channel state information corresponding to the first base station serving the mobile station and second channel state information corresponding to the second base station; and
transmitting to the first base station a first indication of the first channel state information and a second indication of the second channel state information,
wherein the first base station serves a first set of one or more mobile stations including the mobile station.

10. The method of claim 9,
wherein the first base station decides a precoder according to the first and second indications received from the mobile station.

11. The method of claim 9,
wherein at least one of the first channel state information and the second channel state information comprises a channel estimate.

12. The method of claim 9, further comprising:
quantizing at least one of the first channel state information and the second channel state information.