Patent Publication Number: US-8526992-B2

Title: Zero-forcing linear beamforming for coordinated cellular networks with distributed antennas

Description:
TECHNICAL FIELD 
     The concepts described herein may relate to methods and arrangements in a network. In particular, the concepts described herein may relate to methods and arrangements for distributed antennas (DA) in a coordinated cellular network. 
     BACKGROUND 
     In a coordinated cellular system, a DA arrangement may employ various schemes to control other-cell interference. In one approach, Costa-precoding (also known as dirty-paper coding (DPC)) followed by zero-forcing beamforming (ZFBF) may be employed to regulate downlink signals transmitted to users so as to minimize mutual interference among users of respective cell origins. In this scheme, a network produces the signals to be transmitted to different users in a pre-determined order (e.g., a signal for user  1  is produced first, a signal for user  2  is produced next, etc.). A significant constraint with a DPC-ZFBF scheme is that a signal transmitted to, for example, user (i), must not create interference at the antennas of all other users who preceded user (i) according to the pre-determined order. 
     A cellular environment may include, among other things, a network having (t) transmitters, and (m) mobile devices having (r) receiving antennas per mobile device. For example, the cellular environment may include users (i) (i=1, 2, . . . , m) corresponding to the mobile devices. Each mobile device may be in a different cell, and the transmitters can send independent messages to the mobile devices. One or more of the transmitters may not be co-located. 
     For purposes of discussion, assume there is an average total power constraint (P) at the transmitters. The Gaussian broadcast channel (GBC) may be an additive noise channel and each time sample can be represented by the following expression:
 
 y   i   =H   i   x+n   i  i=1, 2, . . . , m,  (1)
 
where (x) is a vector of size (t*1) that represents the total signal transmitted from all of the transmitters. Under a total average power constraint at the transmitters, it may be required that the E[x † x]&lt;P. Y i  is the output vector received by users (i). The output vector is a vector of size (r*1). H i  is a fixed matrix channel for users (i) whose size is (r*t). These channel matrices are fixed and known at the transmitters and the mobile devices. N i  is a Gaussian, circularly symmetric, complex-valued random noise vector with a zero mean and a covariance of σ 2   i I.
 
     For purposes of discussion, assume that the total number of transmit antennas equals the total number of receive antennas (i.e., t=r*m), and that (r) independent streams are transmitted to each mobile device. Thus, t=r*m independent streams may be transmitted in total. Additionally, let (x j ) denote the symbols of the j-th transmitted stream with power (q j ), and (Φ i )={j|x j  belongs to users (i)}. Associated with each transmitted stream (x j ) is a transmitted beamforming vector (V j ). Thus, the total transmitted signal may be represented by the following expression: 
     
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       i 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         x 
                         j 
                       
                       ⁢ 
                       
                         
                           V 
                           j 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Assume that (V j ) has unit norm. The signal transmitted for users (i) may be represented by the following expression: 
                       x   i     =       ∑     j   ∈     Φ   i         ⁢           ⁢       x   j     ⁢     V   j           ,           (   3   )               
and the covariance of the signal transmitted for users (i) may be represented by the following expression:
 
                       S   i     =       ∑     j   ∈     Φ   i         ⁢       V   i     ⁢     V   j   †     ⁢     q   j           ,           (   4   )               
and the covariance of the total transmitted signal x may be represented by the following expression:
 
     
       
         
           
             
               
                 
                   S 
                   = 
                   
                     
                       ∑ 
                       i 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         S 
                         i 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Beamforming Rates 
     A description of achievable rates with beamforming is provided. Under the assumptions that the symbols in different streams are independent and identically distributed (i.i.d.) Gaussian random variables that are chosen independently from each other, a maximum rate that can be delivered to users (i) may be calculated. For example, the received vector at users (i) can be represented by the following expression: 
     
       
         
           
             
               
                 
                   
                     y 
                     i 
                   
                   = 
                   
                     
                       
                         H 
                         i 
                       
                       ⁢ 
                       
                         x 
                         i 
                       
                     
                     + 
                     
                       
                         H 
                         i 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                         
                         ⁢ 
                         
                           x 
                           j 
                         
                       
                     
                     + 
                     
                       
                         n 
                         i 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The sum of the last two terms in expression (6) are colored Gaussian noise (e.g., the noise values are correlated in some fashion) for users (i) with a covariance R i =σ i   2 I+H i (Σ j≠1 S j )H i   † . Thus, the maximum data rate that can be delivered reliably to users (i) may be represented by the following expression: 
     
       
         
           
             
               
                 
                   
                     R 
                     i 
                     BF 
                   
                   = 
                   
                     log 
                     ⁢ 
                     
                       
                         
                           det 
                           ( 
                           
                             I 
                             + 
                             
                               
                                 
                                   H 
                                   i 
                                 
                                 ( 
                                 
                                   
                                     ∑ 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     S 
                                     j 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 H 
                                 i 
                                 † 
                               
                             
                           
                           ) 
                         
                         
                           det 
                           ⁡ 
                           
                             ( 
                             
                               I 
                               + 
                               
                                 
                                   
                                     H 
                                     i 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ∑ 
                                         
                                           j 
                                           = 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         S 
                                         j 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   H 
                                   i 
                                   † 
                                 
                               
                             
                             ) 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The beamforming rate region refers to all the m-tuples of rates {R i   BF } i=1   m  in expression (7) obtained by all possible combinations of beamforming weights V′ j s and powers q l s. The beamforming rate region is clearly a sub-set of the capacity region of the GBC. 
     Rates with Costa-Precoding and Beamforming 
     A description of achievable rates with Costa-precoding and beamforming is provided. As previously described, Costa-precoding entails an encoding of messages for different users at the transmitters according to a certain order. For purposes of discussion, assume that a message for a user  1  is encoded first, a message for a user  2  is encoded second, etc. Since a transmitter is used to form all of the transmitted messages, at the time the message for user (i) is to be formed (i.e., encoded), the messages from all users (j&lt;i) have been already formed. Thus, these messages are known by the transmitter prior to encoding the message for user (i). Further, the transmitter is assumed to know the channel to the i-th user. Accordingly, the interference that is seen by user (i) from the messages (or symbols) transmitted for users (j&lt;i) is known at the transmitter prior to encoding the message for user (i). Based on this known interference, Costa-precoding allows the transmitter to pre-code the symbols for user (i) to significantly reduce the impact of the interference from users (j&lt;i) at the mobile device of user (i). Additionally, Costa-precoding may not require any increase in the transmitted power. In one implementation. Costa-precoding may utilize, for example, low-density parity-check (LDPC) codes and superposition coding. 
     Consider a scalar point-to-point channel that is represented by the following expression:
 
 y=cx+s+z,   (8)
 
where (c) is a complex-valued channel coefficient that is known both to the transmitter and the mobile device: (s) and (z) are independent Gaussian noise with (s) known non-causally at the transmitter, but not known at the mobile device; and (z) is unknown to both the transmitter and the mobile device. For purposes of discussion, assume a total transmitted power constraint of E|x| 2 &lt;P 0 . Under this framework, the capacity of this channel is the same as the additive white Gaussian noise (AWGN) channel, expressed as y=cx+z. That is, possessing the transmitter side information of (s) at the transmitter may be equivalent to knowing (s) both at the transmitter and the mobile device. Thus, with Costa pre-coding, the mobile device can achieve the same rate as with the AWGN channel, and with the total transmit power remaining below (P).
 
     Returning to user (i), consider the received signal at the i-th user, represented by the following expression: 
     
       
         
           
             
               
                 
                   
                     y 
                     i 
                   
                   = 
                   
                     
                       
                         H 
                         i 
                       
                       ⁢ 
                       
                         x 
                         i 
                       
                     
                     + 
                     
                       
                         H 
                         i 
                       
                       ( 
                       
                         
                           ∑ 
                           
                             j 
                             &lt; 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           x 
                           j 
                         
                       
                       ) 
                     
                     + 
                     
                       
                         H 
                         i 
                       
                       ( 
                       
                         
                           ∑ 
                           
                             j 
                             &gt; 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           x 
                           j 
                         
                       
                       ) 
                     
                     + 
                     
                       
                         n 
                         i 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     At the transmitter, the messages are encoded for users (i) sequentially (e.g., in the order of (1, 2, . . . , m)). At the time of encoding the message for user (i), the second term in expression (9) is known to the transmitter since this term involves transmitted symbols from previously encoded users. However, for user (i), this second term can be considered interference that is known at the transmitter, but not at the mobile device. Hence, Costa pre-coding can be applied to transmissions from users (j&lt;i), and the effective channel seen by user (i) may be represented by the following expression 
     
       
         
           
             
               
                 
                   
                     y 
                     i 
                   
                   = 
                   
                     
                       
                         H 
                         i 
                       
                       ⁢ 
                       
                         x 
                         i 
                       
                     
                     + 
                     
                       
                         H 
                         i 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               j 
                               &gt; 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             x 
                             j 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         n 
                         i 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     If Costa pre-coding and beamforming at the transmitter is employed, the effective colored Gaussian noise seen by the i-th mobile device has a covariance σ i   2 I+H i (Σ j&gt;1 S j )H i   † , and the resulting maximum data rate that can be reliably transmitted to user (i) may be represented by the following expression: 
     
       
         
           
             
               
                 
                   
                     R 
                     i 
                     
                       BF 
                       - 
                       Costa 
                     
                   
                   = 
                   
                     log 
                     ⁢ 
                     
                       
                         
                           det 
                           ( 
                           
                             I 
                             + 
                             
                               
                                 
                                   H 
                                   i 
                                 
                                 ( 
                                 
                                   
                                     ∑ 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     S 
                                     j 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 H 
                                 i 
                                 † 
                               
                             
                           
                           ) 
                         
                         
                           det 
                           ⁡ 
                           
                             ( 
                             
                               I 
                               + 
                               
                                 
                                   
                                     H 
                                     i 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ∑ 
                                         
                                           j 
                                           &gt; 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         S 
                                         j 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   H 
                                   i 
                                   † 
                                 
                               
                             
                             ) 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The Costa-precoding/beamforming rate region is defined as all rate m-tuples {R i   BF-Costa } t=1   m  achieved by arbitrary choices of beamforming vectors (V j s), power allocation across streams (q j s), and precoding order. In this regard, the Costa-precoding/beamforming rate region may coincide with the capacity region of the GBC. In other words, Costa-precoding followed by linear beamforming can be used to achieve any point in the capacity region of the GBC subject to a total transmitted power constraint. 
     There is a difference between a Costa-precoding for a scalar-valued channel (e.g., as indicated in expression (8)), and a pre-coding for known vector interference in the vector channel (e.g., as indicated in expression (9)). However, using the fact that Minimum Mean Square Error (MMSE) receivers with successive cancellation are capacity-achieving for a Multiple-In Multiple-Out (MIMO) channel (with or without transmitter side channel state information), decomposition of the vector-valued MIMO channel in expression (9) may be achieved in several parallel Single-In Single-Out (SISO) channels (e.g., one SISO channel for each of the streams belonging to a given user). Since each of these SISO channels can be considered a scalar-valued channel, scalar Costa-precoding for known scalar interference can be applied to each of the SISO channels. 
     Zero-Forcing Beamforming Vectors for Costa-Precoding/Beamforming 
     In Costa-precoding (CP)-ZFBF, CP will be used to encode the symbols for user (i) such that user (i) will effectively see no interference from users (j&lt;i). The combined effect of using CP followed by ZFBF is that user (i) effectively sees no interference from signals transmitted to all other users. 
     For purposes of finding the optimal ZFBF vectors for CP-ZFBF, assume that the total transmitted power by all of the streams of user (i) is less than (P i ). In this regard, for user (i), finding (in) beamforming vectors such that: 
     Requirement (1)—The transmitted signal by user (i) deposits no interference at antennas of all users (j&lt;i); and 
     Requirement (2)—The data achieved by user (i) is maximized may be desirable. 
     Under one approach, a composite channel matrix may be defined, for example, the composite channel matrix may be represented by the following expression: 
     
       
         
           
             
               
                 
                   
                     H 
                     _ 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               H 
                               1 
                             
                           
                         
                         
                           
                             
                               H 
                               2 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               H 
                               m 
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Next, let  H =GQ by employing a QR decomposition of  H . In one implementation, a Gram-Schmidt orthogonalization to the rows of  H  may be employed, where Q is a unitary matrix and G is a lower triangular matrix. The beamforming vector for the j-th stream may then be chosen as the j-th column of the unitary matrix Q † . Due to the Gram-Schmidt process, the i-th column of matrix Q †  is orthogonal to the first (i−1) rows of H (i.e. V; will be orthogonal to all the antennas of the users coming before the user associated with the i-th stream). 
     The beamforming vectors obtained under this approach may result in no interference deposited by user (i) on all the users (j&lt;i). That is, the beamforming weights, under this approach, may satisfy requirement (1), as described above. However, the beamforming vectors do not satisfy requirement (2). 
     That is, the beamforming vectors for the first user can be chosen freely. Thus, the rate delivered to the first user is maximized by choosing these beamforming vectors to be the singular vectors of matrix H 1  having (r) largest singular values. However, the beamforming vectors obtained through the QR decomposition of  H  are completely different from the optimal beamforming vectors for user  1 . 
     Under another approach, a matrix Q whose rows are the channels to the users (j&lt;i) may be represented by the following expression: 
     
       
         
           
             
               
                 
                   
                     Q 
                     i 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               H 
                               1 
                             
                           
                         
                         
                           
                             
                               H 
                               2 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               H 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     In view of the above, the beamforming vectors for user (i) belong to the null space of matrix Q i . Thus, the beamforming vectors for user (i) can be set as the projection of the rows of H i  on the null space of Q i  that are closest to the rows of H i . That is, the beamforming vectors for user (i) may be chosen as those vectors in the null space of matrix Q i  that are closest to the rows of H i . 
     The beamforming vectors obtained under this approach may result in no interference deposited by user (i) on all the users (j&lt;i). That is, the resulting beamforming vectors may satisfy requirement 1. However, these beamforming vectors do not satisfy requirement 2. 
     As a result of the foregoing, neither of the approaches for choosing the beamforming vectors for CP-ZFBW maximize the data rate that will be delivered to the users. Accordingly, an approach for maximizing the data rate delivered to user (i), while, at the same time, making sure that user (i) does not create interference to users (j&lt;i), would be beneficial. 
     SUMMARY 
     It is an object to obviate at least some of the above disadvantages and to improve the operation of a network. 
     According to one aspect, in a distributed antenna system ( 100 ) that includes a plurality of transmitters ( 115 ) and a controller ( 110 ), a method, performed by the controller ( 110 ), may be characterized by performing dirty-paper coding on downlink transmissions to users based on an order of the users, calculating beamforming vectors to provide that each of the downlink transmissions associated with each of the users does not interfere with all other users, and maximizing, based on the calculated beamforming vectors, a data rate subject to a power constraint of the distributed antenna system ( 100 ). 
     According to yet another aspect, a controller ( 110 ) for coordinating data transmissions by antennas ( 310 ) of a distributed antenna system ( 100 ), the controller ( 110 ) may be characterized by a memory ( 220 ) to store instructions, and a processor ( 200 ) to execute the instructions. The processor ( 200 ) may execute instructions to select beamforming vectors associated with the distributed antenna system ( 100 ) for transmission of data to users, where the beamforming vectors do not cause data transmissions to interfere with each other, and to maximize, based on the selection of beamforming vectors, a data rate. 
     According to yet another aspect, a computer-readable medium may contain instructions executable by a controller ( 110 ) associated with base stations ( 115 ) of a distributed antenna system ( 100 ), the computer-readable medium may be characterized by one or more instructions for calculating beamforming vectors related to downlink transmissions to users located in a plurality of cells based on an order of the users, and one or more instructions for maximizing a data rate subject to a transmit power constraint of the distributed antenna system ( 100 ). 
     As a result of the foregoing, maximum data rates may be delivered to users in a coordinated network with a distributed antenna system ( 100 ). Additionally, maximum data rates may be delivered while maintaining various power constraints. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1 . is a diagram of an exemplary environment that may be associated with concepts described herein; 
         FIG. 2  is a diagram illustrating exemplary components that may correspond to the controller depicted in  FIG. 1 ; 
         FIG. 3  is a diagram illustrating exemplary components that may correspond to the transmitters depicted in  FIG. 1 ; 
         FIG. 4  is a diagram illustrating exemplary components that may correspond to the mobile devices depicted in  FIG. 1 ; and 
         FIGS. 5 and 6  are flow diagrams related to processes associated with the concepts described herein. 
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description refers to the accompanying drawings. The same reference numbers in different drawings may identify the same or similar elements. Also, the following description does not limit the invention. The term “component,” as used herein, is intended to be broadly interpreted to include software, hardware, or a combination of hardware and software. 
       FIG. 1  illustrates an exemplary environment  100 . In one implementation, environment  100  may correspond to a wireless environment. While not illustrated, environment  100  may include other systems and/or networks. 
     As illustrated in  FIG. 1 , environment  100  may include, among other things, a network  105  including a controller  110  and transmitters  115 - 1  to  115 -T (collectively referred to as transmitters  115 , and in some instances, individually as transmitter  115 ), and mobile devices  120 - 1  to  120 -M (collectively referred to as mobile devices  120 , and in some instances, individually as mobile device  120 ) including receiving antennas  125 - 1  to  125 -X (collectively referred to as receiving antennas  125 , and in some instances, individually as receive antenna  125 ). 
     Network  105  may include, for example, a wireless network. In one implementation, network  105  may include a cellular network that includes a distributed antenna system (DAS). 
     Controller  110  may include one or more devices that coordinate the operation of the DAS. For example, controller  110  may coordinate the transmission of data via transmitters  115 . Controller  110  may store downlink channel information for every transmit antenna to every mobile device  120  in the DAS. 
     Transmitters  115  may include devices for communicating with mobile devices  120 . In one implementation, each transmitter  115  may correspond to a base station. Transmitters  115  may connect to controller  110  via, for example, a wired connection. Collectively, transmitters  115  form the DAS. Transmitters  115  may not be co-located. 
     Mobile devices  120  may include mobile terminals by which users may receive data from transmitters  115  in environment  100 . Mobile devices  120  may include, for example, a mobile phone, a personal digital assistant (PDA), a mobile computer, a laptop, and/or another type of handset or communication device. In other instances, mobile devices  120  may include a vehicle-mounted terminal. Each mobile device  120  may include one or more receiving antennas  125  to establish and maintain a radio link with one or more transmitters  115 . For example, each mobile device  120  may include (r) receiving antennas  125 . Each mobile device  120  may be in a different cell of environment  100 . 
     Although  FIG. 1  illustrates an exemplary environment  100 , in other implementations, fewer, additional, or different devices may be employed. Additionally, or alternatively, one or more devices of environment  100  may perform one or more functions described as being performed by one or more other devices of environment  100 . 
       FIG. 2  is a diagram illustrating exemplary components controller  110 . For example, controller  110  may include a processing system  200  having a beamforming vector calculator  205  and a power distribution calculator  210 , a communication interface  215 , and a memory  220 . Controller  110  may include additional and/or different components than the components illustrated in  FIG. 2 . 
     Processing system  200  may control the operation of controller  110 . For example, processing system  200  may include a general-purpose processor, a microprocessor, a data processor, a co-processor, a network processor, an application specific integrated circuit (ASIC), a controller, a programmable logic device, a chipset, a field programmable gate array (FPGA), or any other component or group of components that may interpret and execute instructions. 
     Beamforming vector calculator  205  may calculate beamforming vectors based on, for example, the exemplary expressions described below. Power distribution calculator  210  may calculate distribution of power among transmitted streams based on, for example, the exemplary expressions described below. 
     In one implementation, beamforming vector calculator  205  and/or power distribution calculator  210  may be implemented as software. In another implementation, beamforming vector calculator  205  and/or power distribution calculator  210  may be implemented as hardware. In still other implementations, beamforming vector calculator  205  and power distribution calculator  210  may be implemented as a combination of hardware and software. 
     Communication interface  215  may include any transceiver-like mechanism that enables controller  110  to communicate with other devices and/or systems. For example, communication interface  215  may include a radio interface, an optical interface, an Ethernet interface, a coaxial interface, or some other type of interface for wired or wireless communication. In other words, communication interface  215  may allow for wired and/or wireless communication. Communication interface  215  may contain a group of communication interfaces to handle multiple traffic flows. 
     Memory  220  may include any type of unit that stores data and instructions related to the operation and use of controller  110 . For example, memory  220  may include a storing unit, such as a random access memory (RAM), a dynamic random access memory (DRAM), a static random access memory (SRAM), a synchronous dynamic random access memory (SDRAM), a ferroelectric random access memory (FRAM), a read only memory (ROM), a programmable read only memory (PROM), an erasable programmable read only memory (EPROM), an electrically erasable programmable read only memory (EEPROM), and/or a flash memory. 
     Although  FIG. 2  illustrates exemplary components of controller  110 , in other implementations, one or more components of controller  110  may be a component of a device other than controller  110 . Additionally, or alternatively, the functionality associated with beamforming vector calculator  205  and power distribution calculator  210 , as to be described more fully below, may be employed in a distributed fashion between or among more than one device of environment  100 . 
       FIG. 3  is a diagram illustrating exemplary components of transmitter  115 - 1 . Transmitters  115 - 2  through  115 -T may be similarly configured. As illustrated, transmitter  115 - 1  may include a processing system  300 , transceivers  305 , antennas  310 , and a communication interface  315 . Transmitter  115 - 1  may include additional and/or different components than the components illustrated in  FIG. 3 . 
     Processing system  300  may control the operation of transmitter  115 - 1 . Processing system  300  may also process information received via transceivers  305  and communication interface  315 . Processing system  300  may include, for example, a general-purpose processor, a microprocessor, a data processor, a co-processor, a network processor, an application specific integrated circuit (ASIC), a controller, a programmable logic device, a chipset, a field programmable gate array (FPGA), or any other component or group of components that may interpret and execute instructions. 
     Transceivers  305  may be associated with antennas  310  and include transceiver circuitry for transmitting and/or receiving symbol sequences in a network, such as network  105 , via antennas  310 . Antennas  310  may include one or more directional and/or omni-directional antennas. 
     Communication interface  315  may include any transceiver-like mechanism that enables transmitter  115 - 1  to communicate with other devices and/or systems. For example, communication interface  315  may include a radio interface, an optical interface, an Ethernet interface, a coaxial interface, or some other type of interface for wired or wireless communication. In other words, communication interface  315  may allow for wired and/or wireless communication. Communication interface  315  may contain a group of communication interfaces to handle multiple traffic flows. 
     Although  FIG. 3  illustrates exemplary components of transmitter  115 - 1 , in other implementations, one or more components of transmitter  115 - 1  may be a component of a device other than transmitter  115 - 1 . 
       FIG. 4  is a diagram illustrating exemplary components of mobile device  120 - 1 . Mobile devices  120 - 2  through  120 -M may be similarly configured. As illustrated, mobile device  120 - 1  may include an antenna assembly  400 , a communication interface  405 , a processing system  410 , a memory  415 , and a user interface  420 . Mobile device  120 - 1  may include additional and/or different components than the components illustrated in  FIG. 4 . 
     Antenna assembly  400  may include one or more antennas to transmit and receive wireless signals over the air. Communication interface  405  may include, for example, a transmitter that may convert baseband signals from processing system  410  to radio frequency (RF) signals and/or a receiver that may convert RF signals to baseband signals. 
     Processing system  410  may control the operation of mobile device  120 - 1 . For example, processing system  410  may include a general-purpose processor, a microprocessor, a data processor, a co-processor, a network processor, an application specific integrated circuit (ASIC), a controller, a programmable logic device, a chipset, a field programmable gate array (FPGA), or any other component or group of components that may interpret and execute instructions. 
     Memory  415  may include any type of unit that stores data and instructions related to the operation and use of mobile device  120 - 1 . For example, memory  415  may include a storing unit, such as a random access memory (RAM), a dynamic random access memory (DRAM), a static random access memory (SRAM), a synchronous dynamic random access memory (SDRAM), a ferroelectric random access memory (FRAM), a read only memory (ROM), a programmable read only memory (PROM), an erasable programmable read only memory (EPROM), an electrically erasable programmable read only memory (EEPROM), and/or a flash memory. 
     User interface  420  may include mechanisms for inputting information to mobile device  120 - 1  and/or for outputting information from mobile device  120 - 1 . Examples of input and output mechanisms may include a speaker, a microphone, control buttons, a keypad, a display and/or a vibrator to cause mobile device  120 - 1  to vibrate. 
     Although  FIG. 4  illustrates exemplary components of mobile device  120 - 1 , in other implementations, one or more components of mobile device  120 - 1  may be a component of a device other than mobile device  120 - 1 . 
     A description of processes that may be employed for determining the signals that may be transmitted to mobile devices jointly will be described. Based on the processes described below, data rates obtained by the mobile devices may be maximized under two constraints on the transmitted power. The first constraint may limit the total transmitted power from all of the transmitting antennas of the DAS. The second constraint may limit the power transmitted from each individual antenna of the DAS. 
       FIG. 5  is a diagram illustrating an exemplary process  500  that may be employed when calculating downlink transmissions. In one implementation, beamforming vector calculator  205  and/or power distribution calculator  210  of controller  110  may perform one or more of the operations of process  500 . In other implementations, process  500  may be performed by another device or group of devices including or excluding controller  110 . For purposes of discussion, users (i) (i=1, 2, . . . , m) may operate in mobile devices  120  having (r) receiving antennas  125  per mobile device  120 . Transmitters  115  may include (t) transmitters and all of transmitters  115  may not be co-located. 
     Process  500  may begin with performing dirty-paper coding based on an order of users (block  505 ). In one implementation, DPC may be employed to pre-code symbols and/or messages associated with users (i). In one implementation, the pre-coding of the symbols and/or messages associated with users (i) may be performed based on an order of users (i). In this instance, for example, DPC may be employed to pre-code symbols for user (i) so to eliminate the impact of interference from users (j&lt;i) at mobile device  120  of user (i). 
     Beamforming vectors may be calculated for each user (block  510 ). For example, beamforming vectors may be calculated based on the following expressions provided below. 
     For purposes of discussion, the symbols transmitted on the (r) streams of user (i) may be expressed as (x i,1 , x i,2 , . . . x i,r ), and the power of the symbol x i,j  may be expressed as (q i,j ). Additionally, the beamforming vector used to transmit symbol x i,j  may be expressed as V i,j . 
     In the CP-ZFBF framework, the first user is free to use any beamforming vectors. Given the channel of the first user in matrix H 1 , the choice for the beamforming vectors of user  1  may be the singular vectors of matrix H 1  that form the columns of the unitary matrix W 1 , which may be represented by the following expression:
 
H 1 =U 1 Λ 1 W 1   † ,  (14)
 
where U 1 , Λ 1 , and W 1  are the Singular Value Decomposition (SVD) of the matrix H 1 . Accordingly, only the diagonal entries of matrix Λ 1  are non-zero, and the non-zero entries of matrix Λ 1  may be expressed as, for example, (λ 1,1 , λ 1, 2 , . . . λ 1,r ). In this case, user  1  then effectively sees (in) parallel AWGN channels, as represented by the following expression:
 
 y   1,t =λ 1,l   x   1,l   +z   1,l  l=1 . . . r,  (15)
 
since the transmissions to all other users may be formed to deposit no interference at the antenna of user  1 . The effective noises z 1,l ′s may be i.i.d Gaussian with a variance of σ 1   2 . The beamforming weights for user  1  may be the first (r) columns of W 1 . Hence, the transmitted signal for user  1  may be represented by the following expression:
 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       1 
                     
                     = 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         r 
                       
                       ⁢ 
                       
                         
                           V 
                           
                             1 
                             , 
                             l 
                           
                         
                         ⁢ 
                         
                           x 
                           
                             1 
                             , 
                             l 
                           
                         
                       
                     
                   
                   , 
                   and 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     
                       1 
                       , 
                       l 
                     
                   
                   = 
                   
                     
                       
                         W 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           : 
                           
                             , 
                             l 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Next, the optimal beamforming weights for user (i) may be calculated. Recall, that the transmitted signal for user (i) may be formed as a summation, which may be represented by the following expression: 
     
       
         
           
             
               
                 
                   
                     x 
                     i 
                   
                   = 
                   
                     
                       ∑ 
                       
                         l 
                         = 
                         1 
                       
                       r 
                     
                     ⁢ 
                     
                       
                         V 
                         
                           i 
                           , 
                           l 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           
                             i 
                             , 
                             l 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     The beamforming vectors for user (i) may be chosen such that the transmission of (x t ) does not deposit any interference on receiving antennas  125  of mobile devices (j&lt;i)  120 . That is, (x t ) may satisfy the constraint, which may be represented by the following expression:
 
H j x t =0 j=1, 2, . . . , i−1.  (19)
 
     Based on expression (19), (x) may be in the null space of the matrix Q i . That is, the beamforming vectors for user (i) may belong to the null space of matrix Q i . In one implementation, the Gram-Schmidt procedure may be used to determine an orthonormal basis for the null space of matrix Q. Further, this basis may include at most N i =t−(i−1)*r vectors. For purposes of discussion, a matrix B i  may be formed, whose columns correspond to these basis vectors. For example, matrix B i  may have a size of (t*N i ). Accordingly, every vector satisfying expression (19) may be expressed as a linear combination of the columns of matrix B i . In such an instance, (x i ) may be represented by the following expression:
 
x i =B i   x   t ,  (20)
 
where (  x   t ) is any vector of size (N i ·1). Additionally, note that ∥x t ∥=∥  x   i ∥ since B t  may include orthonormal columns.
 
     Restricting the symbols for the i-th user to the form given in expression (20), and assuming that CP for users (j&lt;i) has been applied, the effective channel seen by the i-th user may be represented by the following expression:
 
 y   t   =H   t   B   i     x     i   +z   i ,  (21)
 
since the transmissions to all mobile devices (j&gt;i)  120  may be formed to deposit no interference on receiving antenna  125  of user (i).
 
     Subject to a power constraint on the vector signal transmitted for user (i) (e.g., subject to the constraint E∥x i ∥ 2 &lt;P i ), a determination may be made to find a set of beamforming vectors for user (i), (V i,1 , . . . , V i,r ), that maximize the data rate delivered to user (i), and the resulting transmitted signal (x i ) of expression (3) satisfies expression (19). 
     Since ∥x t ∥=∥  x   t ∥, and since (x i ) satisfying expression (19) may be of the form B t   x   t  for some  x   i , the above problem is equivalent to finding  x   i  in expression (21) that maximizes the mutual information between y i  and  x   i . To solve this equivalent problem. H i B t  may be represented in a SVD forming according to the following expression:
 
H i B t =U l Λ l W i   † ,  (22)
 
where the non-zero diagonal elements of Λ may be denoted by (λ t,1 , λ t,2 , . . . , λ t,r ). In this regard, an optimal choice for  x   t  may be represented by the following expression:
 
                     =       ∑     l   =   1     r     ⁢         W   i     ⁡     (     :     ,   l       )       ⁢     x     i   ,   l             ,           (   23   )               
where the corresponding optimal x i  may be represented by the following expressions:
 
     
       
         
           
             
               
                 
                   
                     x 
                     i 
                   
                   = 
                   
                     
                       B 
                       i 
                     
                     ⁢ 
                   
                 
               
               
                 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     24 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     = 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         r 
                       
                       ⁢ 
                       
                         
                           B 
                           i 
                         
                         ⁢ 
                         
                           
                             W 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             
                               : 
                               
                                 , 
                                 l 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           x 
                           
                             i 
                             , 
                             l 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     From expression (25) the optimal beamforming vectors for the i-th user may be represented by the following expression:
 
 V   t,l   =B   t   W   i (:,l) l=1, 2, . . . r.  (26)
 
     Based on the beamforming vectors indicated in expression (26), the effective channel seen by the i-th user decomposes into m parallel AWGN channels that may be represented by the following expression:
 
 y   i,l =λ i,l   2   x   i,l   +z   i,l  l=1, 2, . . . , r,  (27)
 
where z i,j  are i.i.d. Gaussian noise with a variance of σ i   2 .
 
     Recall that the x i,t ′s may be independent Gaussian symbols transmitted for user (i), and the variance of x i,l  is q i,l . Thus, the total power transmitted from the DAS in terms of q i,l s may be represented by the following expressions: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                           ⁢ 
                           
                             
                                
                               x 
                                
                             
                             2 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           E 
                           ⁢ 
                           
                             { 
                             
                               
                                  
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     m 
                                   
                                   ⁢ 
                                   
                                     x 
                                     i 
                                   
                                 
                                  
                               
                               2 
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             m 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 1 
                               
                               r 
                             
                             ⁢ 
                             
                               
                                 q 
                                 
                                   i 
                                   , 
                                   l 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         ( 
                         28 
                         ) 
                       
                     
                   
                   
                     
                       
                         ( 
                         29 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     Returning to  FIG. 5 , power may be distributed among user streams based on a total transmit power (block  515 ). For example, the distribution of power may be calculated based on the expressions provided below. 
     Since the optimal beamforming vectors for all users have been determined, the distribution of total transmit power, P, among the various streams may be determined. That is, a criteria may be specified for choosing q i,l ′s subject to certain constraints (e.g. subject to total transmitted power from the DAS being less than P). 
     Given a set of (g i,l : i=1, . . . , m; l=1, . . . , r), the resulting data rate achieved by the i-th user may be represented by the following expression: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       i 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             1 
                           
                           r 
                         
                         ⁢ 
                         
                           
                             log 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       λ 
                                       
                                         i 
                                         , 
                                         l 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       q 
                                       
                                         i 
                                         , 
                                         l 
                                       
                                     
                                   
                                   
                                     σ 
                                     i 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           i 
                         
                       
                       = 
                       1 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     m 
                     . 
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     The throughput of the network may be maximized (block  520 ). For example, one performance criteria may be to maximize the overall system throughput (or sum rate), which may be represented by the following expressions: 
     
       
         
           
             
               
                 
                   
                     
                       maximize 
                     
                     
                       
                         
                           
                             ∑ 
                             
                               = 
                               1 
                             
                             m 
                           
                           ⁢ 
                           
                             R 
                           
                         
                         = 
                         
                           ⁢ 
                           
                             log 
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   
                                     λ 
                                     
                                       i 
                                       , 
                                       l 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     q 
                                     
                                       i 
                                       , 
                                       l 
                                     
                                   
                                 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         subject 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         to 
                         ⁢ 
                         
                           : 
                         
                       
                     
                     
                       
                         
                           
                             ∑ 
                             
                               i 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             q 
                             
                               i 
                               , 
                               l 
                             
                           
                         
                         &lt; 
                         
                           P 
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     This sum rate maximization may be considered as a water-filling problem, and the solution may be found based on the following expressions: 
     
       
         
           
             
               
                 
                   
                     q 
                     
                       i 
                       , 
                       l 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         1 
                         λ 
                       
                       - 
                       
                         
                           σ 
                           i 
                           2 
                         
                         
                           λ 
                           
                             i 
                             , 
                             l 
                           
                           2 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           1 
                           λ 
                         
                         - 
                         
                           
                             σ 
                             i 
                             2 
                           
                           
                             λ 
                             
                               i 
                               , 
                               l 
                             
                             2 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     P 
                     . 
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     The minimum data rate for each user may be maximized (block  525 ). For example, as an alternative criteria, the minimum rate obtained by any of the users (or maximize the minimum common rate) may be maximized, which may represented by the following expressions: 
     
       
         
           
             
               
                 
                   max 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   min 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     R 
                     i 
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                     subject 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     to 
                     ⁢ 
                     
                       : 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           , 
                           l 
                         
                       
                       ⁢ 
                       
                         q 
                         
                           i 
                           , 
                           l 
                         
                       
                     
                   
                   &lt; 
                   
                     P 
                     . 
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     The optimization problem in expression (27) may be alternatively expressed as: 
     
       
         
           
             
               
                 max 
               
               
                 a 
               
               
                 
                     
                 
               
               
                 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     37 
                     ) 
                   
                 
               
             
             
               
                 
                   subject 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   to 
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   
                     a 
                     - 
                     
                       R 
                       i 
                     
                   
                   &lt; 
                   0 
                 
               
               
                 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   m 
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                     
                 
               
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         , 
                         l 
                       
                       
                           
                       
                     
                     ⁢ 
                     
                       q 
                       
                         i 
                         , 
                         l 
                       
                     
                   
                   &lt; 
                   
                     P 
                     . 
                   
                 
               
               
                 
                     
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     Since R t  is a convex function of q i,l ′s, the optimization problem in expression (39) is a proper convex optimization problem with a unique global optimum. Further, a convex optimization method may be used to solve this problem. These methods may include, for example, the primal-dual interior-point method, the barrier method, gradient-descent methods, and/or the Newtwon-type methods for solving Via conditions. Given that the transmissions to different users do not interfere with each other, it may be concluded that maximization of the minimum rate may lead to all users obtaining equivalent rates. 
     Although  FIG. 5  illustrates an exemplary process  500 , in other implementations, fewer, different, or additional operations may be performed. 
       FIG. 6  is a diagram illustrating an exemplary process  600  that may be employed when calculating downlink transmissions. In one implementation, beamforming vector calculator  205  and power distribution calculator  210  of controller  110  may perform one or more of the operations of process  600 . In other implementations, process  600  may be performed by another device or group of devices including or excluding controller  110 . 
     Process  600  may begin by performing dirty-paper coding based on an order of users (block  605 ). In one implementation, DPC may be employed to pre-code symbols and/or messages associated with users (i). In one implementation, pre-coding of the symbols and/or messages associated with users (i) may be performed based on an order of users (i). In this instance, for example. DPC may be employed to pre-code symbols for user (i) so to eliminate the impact of interference from users (j&lt;i) at mobile device  120  of user (i). 
     Beamforming vectors may be calculated for each user (block  610 ). For example, beamforming vectors may be calculated based on the expressions previously described in connection to block  510  of  FIG. 5 . Accordingly, a description of the operations associated with block  610  has been omitted. 
     The power may be distributed among user streams based on a per-antenna transmit power (block  615 ). For example, the distribution of power among user streams based on a per-antenna transmit power may be calculated based on the following expressions provided below. 
     Assume that the total transmitted power from the k-th transmitted antenna (e.g., associated at a transmitter  115 ) must be less than P k . In such an instance, the total power transmitted from the k-th antenna may be expressed as a linear function of q t,l ′s. That is, the per-antenna constraint associated with transmit antenna k may be represented by the following expression: 
                         ∑     i   =   1     m     ⁢       ∑     l   =   1     r     ⁢                V     i   ,   l       ⁡     (   k   )            2     ⁢     q     i   ,   l             &lt;     P   k       ,           (   40   )               
where V i,l (k) denotes the k-th element of the vector V i,l . Additionally, all of the coefficients multiplied with q i,l ′s in expression (40) are known, scalar-valued constraints since the beamforming vectors for all of the users have already been determined.
 
     The minimum data rate for each user may be maximized (block  620 ). For example, the minimum data rate for each user may be calculated based on the following expressions provided below. 
     Maximizing a minimum data rate based on a per-antenna power constraint may be represented by the following expressions: 
     
       
         
           
             
               
                 max 
               
               
                 a 
               
               
                 
                     
                 
               
               
                 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     41 
                     ) 
                   
                 
               
             
             
               
                 
                   subject 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   to 
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   
                     a 
                     - 
                     
                       R 
                       i 
                     
                   
                   &lt; 
                   0 
                 
               
               
                 
                   
                     i 
                     = 
                     1 
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   m 
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
             
               
                 
                     
                 
               
               
                 
                   
                     
                       
                         ∑ 
                         m 
                       
                       
                         i 
                         = 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         r 
                       
                       ⁢ 
                       
                         
                           
                              
                             
                               
                                 V 
                                 
                                   i 
                                   , 
                                   j 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           q 
                           
                             i 
                             , 
                             l 
                           
                         
                       
                     
                   
                   &lt; 
                   
                     P 
                     k 
                   
                 
               
               
                 
                   
                     k 
                     = 
                     1 
                   
                   , 
                   
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       l 
                       ⁢ 
                       
                           
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     Based on expression (43), the maximization of the minimum rate subject to a per-antenna power constraint may be considered a convex optimization problem that may be solved employing a convex optimization method. These methods may include, for example, the primal-dual interior-point method, the barrier method, gradient-descent methods, and/or the Newtwon-type methods for solving KKT conditions. 
     The throughput of the network may be maximized (block  625 ). In one embodiment, the overall system throughput (or the sum rate), subject to a per-antenna power constraint, may be represented by the following expressions: 
     
       
         
           
             
               
                 
                   
                     maximize 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         m 
                       
                       ⁢ 
                       
                         R 
                         i 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       log 
                       ⁡ 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               
                                 λ 
                                 
                                   i 
                                   , 
                                   l 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 q 
                                 
                                   i 
                                   , 
                                   l 
                                 
                               
                             
                             
                               σ 
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
             
               
                 
                   
                     subject 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     to 
                     ⁢ 
                     
                       : 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         m 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             1 
                           
                           r 
                         
                         ⁢ 
                         
                           
                             
                                
                               
                                 
                                   V 
                                   
                                     i 
                                     , 
                                     l 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                                
                             
                             2 
                           
                           ⁢ 
                           
                             q 
                             
                               i 
                               , 
                               l 
                             
                           
                         
                       
                     
                   
                   &lt; 
                   
                     P 
                     k 
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     Based on expressions (44) and (45), the maximization of the overall system throughput subject to a per-antenna constraint may be obtained based on a convex optimization algorithm since both the optimization criteria and the constraints are convex functions of q i,l ′s. The convex optimization may be solved employing various methods, such as, the primal-dual interior point method, the barrier method, gradient-decent methods, or the Newton-type methods for solving the KKT conditions. 
     Although,  FIG. 6  illustrates an exemplary process  600 , in other implementations, fewer, different, or additional operations may be performed. 
     Conclusion 
     The foregoing description of implementations provides illustration, but is not intended to be exhaustive or to limit the implementations to the precise form disclosed. Modifications and variations are possible in light of the above teachings or may be acquired from practice of the teachings. 
     In addition, while series of blocks have been described with regard to processes illustrated in  FIG. 5  and  FIG. 6 , the order of the blocks may be modified in other implementations. Further, non-dependent blocks may be performed in parallel. Further, one or more blocks may be omitted. 
     It will be apparent that aspects described herein may be implemented in many different forms of software, firmware, and hardware in the implementations illustrated in the figures. The actual software code or specialized control hardware used to implement aspects does not limit the invention. Thus, the operation and behavior of the aspects were described without reference to the specific software code—it being understood that software and control hardware can be designed to implement the aspects based on the description herein. 
     Even though particular combinations of features are recited in the claims and/or disclosed in the specification, these combinations are not intended to limit the invention. In fact, many of these features may be combined in ways not specifically recited in the claims and/or disclosed in the specification. 
     It should be emphasized that the term “comprises” or “comprising” when used in the specification is taken to specify the presence of stated features, integers, steps, or components but does not preclude the presence or addition of one or more other features, integers, steps, components, or groups thereof. 
     No element, act, or instruction used in the present application should be construed as critical or essential to the implementations described herein unless explicitly described as such. Also, as used herein, the article “a” and “an” are intended to include one or more items. Where only one item is intended, the term “one” or similar language is used. Further, the phrase “based on” is intended to mean “based, at least in part, on” unless explicitly stated otherwise. As used herein, the term “and/or” includes any and all combinations of one or more of the associated list items.