Complete Power Reallocation for MU-MIMO under Per-Antenna Power Constraint
Sucheol Kim, Hyeongtaek Lee, Hwanjin Kim, Yongyun Choi, Junil Choi
aa r X i v : . [ c s . I T ] F e b Complete Power Reallocation for MU-MIMO underPer-Antenna Power Constraint
Sucheol Kim, Hyeongtaek Lee, Hwanjin Kim, Yongyun Choi, and Junil Choi
Abstract —This paper proposes a beamforming method undera per-antenna power constraint (PAPC). Although many beam-former designs with the PAPC need to solve complex optimizationproblems, the proposed complete power reallocation (CPR)method can generate beamformers with excellent performanceonly with linear operations. CPR is designed to have a simplestructure, making it highly flexible and practical. In this paper,three CPR variations considering algorithm convergence speed,sum-rate maximization, and robustness to channel uncertaintyare developed. Simulation results verify that CPR and its varia-tions satisfy their design criteria, and, hence, CPR can be readilyutilized for various purposes.
Index Terms —Per-antenna power constraint, multi-usermultiple-input multiple-output, computational complexity, robustbeamforming.
I. I
NTRODUCTION
Use of multiple antennas is a popular technique for wirelesscommunication systems [1]–[3]. It allows multi-user (MU)diversity gain to increase the sum-rate in multi-user multiple-input multiple-output (MU-MIMO) systems [4]–[6]. Dirtypaper coding (DPC) is a well-known transmit technique thatachieves the capacity of MU-MIMO downlink channel [7], [8].DPC, however, requires very high computational complexity toimplement in practice, and many linear precoders were devel-oped to lower the computational complexity by feasible levelwith suboptimal performance instead [9]–[12]. As an attractivelinear precoder, a zero-forcing (ZF) beamformer provides finebalance between the complexity and performance [9], [10].The performance and simplicity of ZF beamformer areguaranteed under the total transmit power constraint but notunder the per-antenna power constraint (PAPC). Includingthe beamformers in [9] and [10], many beamformers weredesigned under the total transmit power constraint [12], [13].In practice, however, power amplifiers, which have limit ontheir maximum power, are connected to each antenna [11],[14]. Therefore, the PAPC must be taken into account forpractical beamformer designs.It is not a simple problem to design beamformers withthe PAPC though. Even the ZF beamformer design becomesnon-trivial under the PAPC [11]. It is possible to simplydownscale the ordinary ZF beamformer to satisfy the PAPC;however, this simple approach usually does not exploit fullavailable power at each amplifier, resulting in significant per-formance degradation [15]. In [16], a beamforming algorithm
S. Kim, H. Lee, H. Kim, and J. Choi are with the School of ElectricalEngineering, Korea Advanced Institute of Science and Technology (e-mail: { loehcumik,htlee8459,jin0903,junil } @kaist.ac.kr).Y. Choi is with the Network Business, Samsung Electronics Co., LTD (e-mail: [email protected]). was proposed to maximize the total transmit power under thePAPC, but no practical communication metrics, e.g., sum-rate,minimum data rate, or beamforming gain, were consideredfor the beamformer design. By considering a relaxed PAPC, aprecoder in [17] is designed to maximize a weighted sum-rate.In many cases, beamformers were designed by solving one ormore optimization problems including the PAPC as a con-straint. In [18], semidefinite programming (SDP) was adoptedto maximize the sum-rate subject to the PAPC and a zerointerference constraint. The relaxation of rank-one constraintin SDP, however, requires to solve an additional optimizationproblem, making the design highly complex. The duality ofprimal- and dual-optimization problems was applied in [14]and [19] to maximize the data rate or to minimize the transmitpower under the PAPC. In [20], a normalized beamformer andits power distribution were updated iteratively where one isfixed during the update of another. Most of the beamformerdesigns with the PAPC, however, require non-linear operationsor solving complicated optimization problems without closedform solutions. The PAPC, which is a must in practice, revivesthe computational complexity problem that was once relaxedfor linear suboptimal beamformers.Another concern for beamformer designs is whether abase station (BS) has accurate downlink channel information.It would be reasonable that the BS has accurate channelinformation when the user equipments (UEs) are static, makingthe wireless channels vary slowly [21], [22]. This is even morefeasible for time division duplexing (TDD) systems exploitinguplink/downlink channel reciprocity [23]–[25]. We, however,can not expect all the UEs are always static, and uncertaintyinheres in the channel information of moving UEs. Therefore,beamformers need to be robust to the channel uncertainty. Thebeamformer in [26] was designed for each UE to minimize thetotal transmit power ensuring a quality of service (QoS) withthe SDP relaxation under an interference constraint. In [27],the signal-to-leakage-plus-noise ratio (SLNR) was maximizedwith a probabilistic constraint on the leakage power. In [28],the transmit power was minimized under QoS constraintsassuming the channel uncertainty as a bounded error. Recently,in [29], a deep learning technique was used to generate robustbeamformers directly from inaccurate channel information.These beamformers do not satisfy the PAPC, making them im-practical though. Both the PAPC and robustness to the channeluncertainty were taken into account for the beamformer designin [30]; however, it requires to solve multiple complicatedoptimization problems.In general, linear beamformers lose their benefit of lowcomplexity when the PAPC and the channel uncertainty are taken into account. In this paper, we propose a novel lowcomplexity beamformer design, dubbed as complete powerreallocation (CPR), which satisfies the PAPC. Proposed CPRiteratively updates a beamformer to fully exploit the transmitpower in each antenna while the update can be conductedby linear operations with low complexity. CPR also requiresonly a finite number of iterations regardless of its designparameters, whereas most of iterative beamformer designalgorithms may suffer from a convergence issue depending ontheir design parameters [20], [26], [30]. With its simple struc-ture, CPR easily can have different variations to aim certainpurposes. As concrete examples, we explain three variationsof CPR, each for the convergence speed of algorithm, sum-rate maximization, and robustness to the channel uncertaintyof moving UEs.The rest of paper is organized as follows. In Section II,system and channel models are described. In Section III, thedetails of proposed CPR and its variations are explained.The numerical results to evaluate the proposed techniques areshown in Section IV, and the conclusion follows in Section V. Notations : A matrix and a vector are written in a bold facecapital letter and a bold face small letter. For a matrix or avector, its transpose, Hermitian, and element-wise conjugateare represented as ( · ) T , ( · ) H , and ( · ) ∗ . A † means the pseudo-inverse of matrix A . The a -th column and b -th row of matrix A are denoted as ( A ) (: ,a ) and ( A ) ( b, :) . The b -th component ofvector a is remarked as ( a ) ( b ) . I a is the a × a identity matrix,and a represents the a × all zero vector. The concatenationof matrices is expressed as [ A , B ] where A and B have thesame number of rows. ⊙ represents the Hadamard product.A diagonal matrix, of which the diagonal elements are theelements of a , is denoted as diag( a ) . Real { a } and Imag { a } are the real and imaginary part of complex number a , and thephase of a is represented as ∠ a . The uniform distribution inthe range [ a, b ] is written as U[ a, b ] . The complex multivariatenormal distribution with mean vector µ and covariance matrix Σ is represented as CN ( µ , Σ ) . For a set I , the function idx I ( i ) , i ∈ I gives the index of i in the set I , and itscardinality is written as C ( I ) . A subspace that is spanned bythe columns of matrix A is denoted as S ( A ) . The function proj( a , S ( A )) projects a vector a on the subspace S ( A ) .II. S YSTEM AND CHANNEL MODELS
We consider an MU-MIMO system with a BS equipped with M antennas and K UEs each with single antenna. Assumingthe block-fading model, the received signal at the k -th UEduring the i -th fading block is y k,i = h H k,i f k,i s k,i + K X ℓ =1 ℓ = k h H k,i f ℓ,i s ℓ,i + n k,i , (1)where h k,i ∈ C M × is the channel vector between the BS andthe k -th UE, f k,i ∈ C M × is the beamforming vector of BS tosupport the k -th UE, s k,i is the i -th transmit symbol for the The transmit symbol s k,i is able to vary during a fading block in practice.To make notation simple, we neglect this fact since the transmit symbol isirrelevant to linear precoder designs. k -th UE satisfying E [ | s k,i | ] = 1 , and n k,i ∼ CN (0 , σ ) isthe noise of which variance is the same for all time instances.For the beamforming vectors f k,i , we consider practicalpower constraints as K X k =1 k f k,i k ≤ P tot , (2) max m ∈{ , ··· ,M } K X k =1 (cid:12)(cid:12) ( f k,i ) ( m ) (cid:12)(cid:12) ≤ P ant , (3)where (2) is the total power constraint with the maximumtotal transmit power P tot , and (3) is the per-antenna powerconstraint (PAPC) with the maximum antenna transmit power P ant . For each transmission, the two power constraints needto be satisfied simultaneously. In this paper, we consider themaximum antenna transmit power as P ant = P tot /M , andthis let the PAPC in (3) be a sufficient condition for the totaltransmit power constraint in (2).The channel vector of each UE is modeled as [31], [32] h k, = R k g k, , (4) h k,i = η k h k,i − + q (1 − η k ) R k g k,i , i ≥ , (5)where R k = E h h k,i h H k,i i is the spatial correlation matrix, g k,i ∼ CN ( M , β k I M ) is the innovation process of the chan-nel vector of the k -th UE at the i -th fading block, β k modelsthe large-scale fading effect, and η k is the temporal correlationcoefficient. At the i -th fading-block, the overall channel matrixbecomes H i = [ h ,i , · · · , h K,i ] by concatenating the channelvectors of all UEs. For the long-term second order statistics R k and η k , we adopt the exponential correlation model andthe Jakes’ model, respectively [33]. The exponential spatialcorrelation matrix is given by R k = r k · · · r ( M − k r ∗ k · · · r ( M − k ... ... . . . ... ( r ∗ k ) ( M − ( r ∗ k ) ( M − · · · , (6)where r k ∈ C satisfies | r k | < and ≤ ∠ r k < π . TheJakes’ model for the temporal correlation is given as η k = J (2 πf D,k t ) , (7)where J ( · ) is the -th order Bessel function, f D,k is theDoppler frequency, and t is the channel instantiation interval.The Doppler frequency of a static UE is zero, and the channelvector is invariant over i with η k = 1 . For a moving UE,the temporal correlation becomes less than one, and thecorresponding channel vector varies over time.III. B EAMFORMER DESIGNS
The motivation and concept of the proposed CPR methodare introduced in Section III-A. Then, the details of CPR andits variations are explained from Section III-B to Section III-Eaiming for different purposes or circumstances. To explainCPR clearly, we assume all UEs are static from Section III-Ato Section III-D, and the block-fading index i of channels and beamformers is omitted for legibility. In Section III-E, weconsider several moving UEs and restore the time index i todistinguish the outdated and the current channels. A. Motivation and concept of CPR
For a single UE multiple-input single-output (MISO) sys-tem, a beamformer that maximizes the beamforming gain isthe matched beamformer. Under the power constraints in (2)and (3), the matched beamformer can be designed as f sMB = p P ant h (cid:12)(cid:12) ( h ) ( m max ) (cid:12)(cid:12) , (8) m max = argmax m ∈{ , ··· ,M } (cid:12)(cid:12) ( h ) ( m ) (cid:12)(cid:12) , (9)where h ∈ C M × is the MISO channel. When only the totaltransmit power constraint is considered, the ordinary matchedbeamformer f MB = √ P tot h k h k maximizes the beamforminggain | h H f MB | . However, the beamformer in (8), which isadditionally constrained by the PAPC, is a scaled-down versionof the ordinary matched beamformer, and the transmit powerof the beamformer f sMB is k f sMB k = P ant k h k (cid:12)(cid:12) ( h ) ( m max ) (cid:12)(cid:12) ≤ k f MB k = P tot , (10)where the equality scarcely holds when all the elements of f MB have the same magnitude. The decrease of transmit powerof beamformer f sMB depends on the distribution of magni-tude of elements | ( f MB ) ( m ) | . With widespread of magnitude | ( f MB ) ( m ) | , some antennas may need to suppress their transmitpower significantly, resulting in a low beamforming gain.The scaled-down beamformer f sMB usually has far lesstransmit power than the total transmit power, i.e., k f sMB k
We now consider the MU-MIMO system in which all UEsare static. To support multiple UEs, it is possible to adoptthe same approach in Section III-A to the beamformer foreach UE. The power constraints in (2) and (3), however,jointly affect beamformers for all UEs, and the power increaseof beamformer of a specific UE may reduce the availablepower of beamformers for other UEs. In addition, the matchedbeamformer discussed in Section III-A may result in largerinterference for other UEs with increased transmit power. We,hence, need an effective beamformer to handle the inter-userinterference and to distribute the transmit power over UEswhile satisfying the power constraints in (2) and (3).Proposed CPR develops a beamformer by combining mul-tiple effective beamformers that can manage interference,e.g., ZF beamformer, SLNR beamformer [12], or regularizedzero-forcing (RZF) beamformer [34]. In this paper, the ZFbeamformer is adopted as a concrete example to explain CPR.The outline of ZF-based CPR is shown in Algorithm 1. Forthe given initial beamformer F and algorithm parameter p ,CPR iteratively adds up extra beamformers. At each iteration,the extra beamformer c W n is obtained for the antenna subset I ( p ) n , of which elements have power less than pP ant . Afterthe dimension of the extra beamformer c W n is properly set to form W n , it is added to the previous beamformer F n − afteradjusting the power distribution with the coefficient matrix A n = diag( a n ) . In this subsection, A n is designed forthe equal power distribution over UEs while another powerdistribution strategy for unequal power distribution will bediscussed in Section III-D. For the equal power distribution,we set a n as ( a n ) ( k ) = α (EP ,n ) (cid:13)(cid:13) ( W n ) (: ,k ) (cid:13)(cid:13) , k ∈ { , · · · , K } , (15)where α (EP ,n ) is the coefficient to satisfy the PAPC.Under the PAPC, α (EP ,n ) is designed to maximize the sum-rate of the updated beamformer asargmax ˆ α n ∈ C K X k =1 log (cid:18) σ (cid:12)(cid:12)(cid:12) h H k ( F n − + W n A n ) (: ,k ) (cid:12)(cid:12)(cid:12) (cid:19) subject to K X k =1 (cid:13)(cid:13) ( F n − + W n A n ) (: ,k ) (cid:13)(cid:13) ≤ P tot , max m ∈{ , ··· ,M } K X k =1 (cid:12)(cid:12) ( F n − + W n A n ) ( m,k ) (cid:12)(cid:12) ≤ P ant , ∠ (cid:0) h H k ( F n − ) (: ,k ) (cid:1) = ∠ (cid:0) h H k ( W n A n ) (: ,k ) (cid:1) ,k ∈ { , · · · , K } , A n = diag( a n ) , ( a n ) ( k ) = ˆ α n (cid:13)(cid:13) ( W n ) (: ,k ) (cid:13)(cid:13) , k ∈ { , · · · , K } , (16)where the third constraint aligns the product of the channelvector with the previous beamformer h H k ( F n − ) (: ,k ) and thatwith the n -th extra beamformer h H k ( W n A n ) (: ,k ) . The twobeamformers ( F n − ) (: ,k ) and ( W n ) (: ,k ) provide positive realvalues when they are multiplied with channel vector h k , andthis restricts the coefficient ˆ α n in (16) also to be a real value.Then, the data rate maximization of the k -th UE becomes thesame as the maximization of the coefficient ˆ α n as argmax ˆ α n ∈ R log (cid:18) σ (cid:12)(cid:12)(cid:12) h H k ( F n − + W n A n ) (: ,k ) (cid:12)(cid:12)(cid:12) (cid:19) = argmax ˆ α n ∈ R (cid:12)(cid:12)(cid:12) h H k ( F n − ) (: ,k ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) h H k ( W n A n ) (: ,k ) (cid:12)(cid:12)(cid:12) + 2 · Real n h H k ( F n − ) (: ,k ) (cid:16) h H k ( W n A n ) (: ,k ) (cid:17) ∗ o ( a ) = argmax ˆ α n ∈ R ˆ α n k ( W n ) (: ,k ) k (cid:12)(cid:12)(cid:12) h H k ( W n ) (: ,k ) (cid:12)(cid:12)(cid:12) + 2 ˆ α n k ( W n ) (: ,k ) k h H k ( F n − ) (: ,k ) (cid:16) h H k ( W n ) (: ,k ) (cid:17) ∗ | {z } positive real number = argmax ˆ α n ∈ R ˆ α n , (17)where ( a ) is by the alignment constraint in (16). Since theobjective function in (17) is independent of the UE index k ,we can design the single variable ˆ α n to maximize the datarate of each UE, which directly maximizes the sum-rate.By replacing the objective function in (16) with the mag-nitude of coefficient | ˆ α n | , we can obtain the optimal solution Algorithm 1
CPR algorithm
Initialize Set initial beamformer: F = [ f , · · · , f K ] = [ M , · · · , M ] Set parameter < p ≤ and stopping criteria Beamformer update For ≤ n ≤ M − K + 1 Find antenna set: I ( p ) n = ( m : K X k =1 (cid:12)(cid:12) ( f k ) ( m ) (cid:12)(cid:12) < pP ant ) If C ( I ( p ) n ) < M : break End if Calculate an extra beamformer for the antenna set: c W n = (cid:16) b H H n (cid:17) † ∈ C C ( I ( p ) n ) × K b H n = ( H ) ( I ( p ) n , :) Adjust beamformer dimension: ( W n ) ( m, :) = ( ( c W n ) (idx I ( p ) n ( m ) , :) , m ∈ I ( p ) n T K , m / ∈ I ( p ) n Set coefficient matrix: A n = diag( a n ) Combine beamformers: F n = F n − + W n A n If one of stopping criteria is satisfied: F CPR = F n break End if
End for in a closed form. To find the solution α (EP ,n ) , we first findthe possible values ˆ α n,m that satisfy the PAPC of each m -thantenna with equality K X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( F n − ) ( m,k ) + ˆ α n,m (cid:13)(cid:13) ( W n ) (: ,k ) (cid:13)(cid:13) ( W n ) ( m,k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = P ant , (18)which can be reformulated as a second order complex equa-tion as (19) on the top of the next page, where w n = (cid:2)(cid:13)(cid:13) ( W n ) (: , k − , · · · , (cid:13)(cid:13) ( W n ) (: ,K ) k − (cid:3) . The root of the sec-ond order equation is in (20), and we set θ m to align the twobeamformers F n − and W n as in (21). Then, the solutionof the problem (16), which needs to satisfy the PAPC for allantennas, is obtained as α (EP ,n ) = argmin ˆ α n ∈{ ˆ α n, , ··· , ˆ α n,m } | ˆ α n | . (22) | ˆ α n,m | (cid:13)(cid:13) ( W n ) ( m, :) ⊙ w n (cid:13)(cid:13) + 2 · Real n ˆ α n,m ( F n − ) ∗ ( m, :) (cid:0) ( W n ) ( m, :) ⊙ w n (cid:1) T o + (cid:13)(cid:13) ( F n − ) ( m, :) (cid:13)(cid:13) − P ant = 0 (19) ˆ α n,m = 1 (cid:13)(cid:13)(cid:13) ( W n ) ( m, :) ⊙ w n (cid:13)(cid:13)(cid:13) − ( F n − ) ( m, :) (cid:16) ( W n ) ( m, :) ⊙ w n (cid:17) H + e jθ m s(cid:12)(cid:12)(cid:12)(cid:12) ( F n − ) ( m, :) (cid:16) ( W n ) ( m, :) ⊙ w n (cid:17) H (cid:12)(cid:12)(cid:12)(cid:12) − (cid:13)(cid:13)(cid:13) ( W n ) ( m, :) ⊙ w n (cid:13)(cid:13)(cid:13) (cid:18)(cid:13)(cid:13)(cid:13) ( F n − ) ( m, :) (cid:13)(cid:13)(cid:13) − P ant (cid:19)! (20) θ m = sin − Imag (cid:26) ( F n − ) ( m, :) (cid:16) ( W n ) ( m, :) ⊙ w n (cid:17) H (cid:27)s(cid:12)(cid:12)(cid:12)(cid:12) ( F n − ) ( m, :) (cid:16) ( W n ) ( m, :) ⊙ w n (cid:17) H (cid:12)(cid:12)(cid:12)(cid:12) − (cid:13)(cid:13)(cid:13) ( W n ) ( m, :) ⊙ w n (cid:13)(cid:13)(cid:13) (cid:18)(cid:13)(cid:13)(cid:13) ( F n − ) ( m, :) (cid:13)(cid:13)(cid:13) − P ant (cid:19) (21)With the maximum antenna power P ant = P tot /M , the up-dated beamformer satisfies the total transmit power constraintwhen it satisfies the PAPC as K X k =1 (cid:13)(cid:13) ( F n − + W n A n ) (: ,k ) (cid:13)(cid:13) = M X m =1 K X k =1 (cid:12)(cid:12) ( F n − + W n A n ) ( m,k ) (cid:12)(cid:12) ≤ M X m =1 P ant = P tot . (23)The design of coefficient matrix A n in (16) makes at leastone antenna in the antenna set I ( p ) n to use the maximumantenna power P ant at each algorithm iteration. As the pro-posed algorithm iterates, then, the number of antennas thatuse the maximum antenna power increases. Since Line 5in Algorithm 1 requires the cardinality of antenna set tobe no smaller than the number of UEs C ( I ( p ) n ) ≥ K , themaximum algorithm iteration is restricted by M − K + 1 .Under the maximum algorithm iteration, the parameter p inAlgorithm 1 affects the algorithm convergence speed and thetransmit power of final beamformer F CPR . A small value of p would increase the magnitude of ( a n ) ( k ) , speeding up theconvergence of CPR algorithm. As a simple example, Fig. 1represents the per-antenna transmit power of a beamformer F = [ f , f ] with M = 4 BS antennas and K = 2 UEs. Thesecond antenna is using the maximum antenna power P ant , andif we take a small p = p and F n − = F , the antenna set foriteration in Line 4 of Algorithm 1 becomes I ( p ) n = { , } . Thecorresponding extra beamformer after dimension adjustment is W n = ( c W n ) (1 , :) T K ( c W n ) (2 , :) T K , (24)where c W n ∈ C C ( I ( p n ) × K is the extra beamformerbefore the adjustment. The expected magnitude of ( a n ) ( k ) is approximately proportional to the gap Fig. 1: Transmit power per-antenna with M = 4 , K = 2 . P ant − max m ∈I ( p n { ( | ( f ) ( m ) | + | ( f ) ( m ) | } , whichrepresents the amount of available power not in use. On thecontrary, if we take relatively large p = p and F n − = F ,the antenna set becomes I ( p ) n = { , , } , and the magnitudeof ( a n ) ( k ) is approximately proportional to a smaller gap P ant − max m ∈I ( p n { ( | ( f ) ( m ) | + | ( f ) ( m ) | } than the case of p = p . With small p = p , large magnitude of coefficients | ( a n ) ( k ) | raises the transmit power increment at each iterationand consequently accelerates the convergence speed ofCPR algorithm by rapidly increasing the transmit powerclose to P tot . With small value p = p , however, it isnot possible to exploit the potential power of 4-th antenna P ant − ( | ( f ) (4) | + | ( f ) (4) | ) . Hence, we can balance thetotal transmit power of the final beamformer F CPR and theconvergence speed of CPR algorithm by selecting a proper p .While the maximum iteration number of CPR is M − K + 1 as stated in Line 3 of Algorithm 1, we can setspecific stopping criteria considering the power constraints in(2) and (3). For example, Algorithm 1 can be stopped when99% of total transmit power is used or 90% of antennasare exploiting the full antenna power P ant . Depending onthe beamformers used in CPR, we may need to considerother stopping criteria. For CPR with the ZF beamformer, inAlgorithm 1, we included Line 5 to ensure the full columnrank of matrix b H n in Line 7, which can be satisfied when C ( I ( p ) n ) ≥ K as long as all UEs experience independentchannels. It is obvious this criterion can be met faster with small value p = p than with large value p = p .While CPR can accommodate any linear beamformers byadjusting Line 7, the use of ZF beamformer provides a benefitthat the final beamformer F CPR has a higher beamforminggain than the initial beamformer without additional interfer-ence. This is shown in the following lemma.
Lemma 1.
For an antenna set I ( p ) n , a beamformer F n − ∈ C M × K , and two positive integers M and K , if an extrabeamformer c W n ∈ C C ( I ( p ) n ) × K is a ZF beamformer ofchannel ( H ) ( I ( p ) n , :) , then the beamformer F n = F n − + W n A n , (25) gives a strictly increased beamforming gain without additionalinterference for all UEs where A n and W n are defined inAlgorithm 1.Proof: See Appendix ILemma 1 holds for any initial beamformers as long as c W n is set to be the ZF beamformer. This means that any well-designed beamformer can be improved further by Algorithm 1,unless the beamformer satisfies the stopping criteria of Algo-rithm 1. In Algorithm 1, the initial beamformer is set as a zeromatrix, and the following remark states that this initializationlet the final beamformer F CPR be a ZF beamformer.
Remark.
When the initial beamformer F is set to be azero matrix [ M , · · · , M ] , the updated beamformers F n inAlgorithm 1 are ZF beamformers with increasing beamforminggains as the algorithm iterates. Under the maximum iteration number M − K +1 , Lemma 1assures the convergence of Algorithm 1. C. Fast convergence CPR (FC-CPR)
In practice, it would be better to increase the transmit poweras much as possible at early iterations to speed up CPRprocess. In this subsection, we propose fast convergence CPR(FC-CPR), which is a variation of CPR to have less numberof CPR iterations. FC-CPR exploits CPR based on the factthat 1) CPR converges faster with small p in Algorithm 1and 2) the increase of transmit power in each CPR processis approximately proportional to the gap between P ant andthe maximum of current transmit powers allocated to theantennas to be updated as explained in the previous subsection.As in Algorithm 2, FC-CPR sets the initial parameter as asmall value p = p init and increases p by ∆ p whenever CPRconverges. A small p makes the size of the antenna set I ( p ) n inLine 4 of Algorithm 1 small and allocates much power onlyto the antennas in I ( p ) n , letting CPR converges faster. Withsmall p , however, the antennas not included in the set I ( p ) n may not fully exploit their available power. FC-CPR, therefore,increases p whenever CPR converges, and then CPR operateswith another small set of antennas. This let FC-CPR dealwith most of antennas exploiting their full power. FC-CPRalso allows to handle a small dimensional channel matrix ( H ) ( I ( p ) n , :) in Line 7 of Algorithm 1 making overall processless complex. FC-CPR stops when p ≥ p max with a predefined p max . Algorithm 2
FC-CPR algorithm
Initialize Set initial beamformer: F = [ f , · · · , f K ] = [ M , · · · , M ] Set parameters < p = p init < p max , < p max ≤ , and ∆ p > Beamformer update For ≤ n ′ ≤ M − K + 1 Run CPR with initial beamformer F and p If p < p max p = p + ∆ p F = F CPR Else F FCCPR = F CPR break End if End for
Although FC-CPR exploits CPR multiple times, the totaliteration number of FC-CPR is usually smaller than that ofCPR. Note that α (EP ,n ) in (15) guarantees that at least oneantenna in the set I ( p ) n is allocated with the maximum antennapower P ant per iteration, and the stopping criteria in Line 5of Algorithm 1 also holds for FC-CPR. Therefore, the totaliteration number of FC-CPR is always smaller than M − K +1 ,and the large amount of transmit power increment per iterationlet FC-CPR converge faster than CPR. This is numericallyshown in Section IV. D. Power distribution strategy of CPR
We have considered a n in (15) to achieve equal powerdistribution over UEs. It is possible to have non-uniform powerdistribution strategies by setting a n as ( a n ) ( k ) = α n ( b ) ( k ) (cid:13)(cid:13) ( W n ) (: ,k ) (cid:13)(cid:13) , k ∈ { , · · · , K } , (26)where b ∈ C K × determines the power distribution acrossUEs. Since b is fixed over all CPR iterations, a n is stillupdated by a single variable α n where α n can be obtainedby solving (16) after substituting ( a n ) ( k ) = ˆ α n k ( W n ) (: ,k ) k with ( a n ) ( k ) = ˆ α n ( b ) ( k ) k ( W n ) (: ,k ) k .We adopt the water-filling power distribution strategy, whichis known as an optimal solution to maximize the sum-rate,as a representative example. The conventional water-fillingproblem, however, does not consider the PAPC, and wepropose a strategy that mimics the water-filling solution. First,we set b WF as the water-filling solution of following problemmaximize ˆ b ∈ C K × K X k =1 log | (ˆ b ) ( k ) h H k p k | σ + (cid:12)(cid:12)(cid:12)P ℓ = k (ˆ b ) ( ℓ ) h H k p ℓ (cid:12)(cid:12)(cid:12) subject to p k = (cid:16)(cid:0) H H (cid:1) † (cid:17) (: ,k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ( H H ) † (cid:17) (: ,k ) (cid:13)(cid:13)(cid:13)(cid:13) , k ∈ { , · · · , K } , K X k =1 | (ˆ b ) ( k ) | ≤ P tot , (27)where the problem has a well-known closed form solution[35]. We let the optimal solution of (27) as b WF that is givenas ( b WF ) ( k ) = σ max ( , s λ − (cid:12)(cid:12) h H k p k (cid:12)(cid:12) ) , (28)where λ is determined by the total transmit power constraint k b WF k ≤ P tot . We then use b WF and a n in (26) forthe beamformer update. Similarly, other power distributionstrategies, e.g., maxmin power allocation to ensure QoS, canbe taken into account with proper setting of b . E. CPR for robust beamforming
Until now, we have considered the MU-MIMO systemwith static UEs to clearly explain CPR. In this subsection,we consider the original system with several moving UEsand recover the subscript i that represents the block-fadingindex to distinguish the outdated and current channels. Then,the channels of K s static UEs are fixed over i , and thechannels of K m moving UEs vary over i , which makes theBS need to design beamformers without the knowledge ofcurrent channel H i . The outdated channel information at theBS usually reduces the data rate of moving UEs by both theincrease of interference and the decrease of desired signalpower. For the robustness to the effect of channel uncertainty,we propose CPR with candidate channels (CPR-cc) as anothervariation of CPR. Without loss of generality, we set the first K m UEs are moving among K = K m + K s UEs.CPR-cc uses an extended channel H i − that is a concatena-tion of outdated channel H i − and candidate channel matrices C k,i − ∈ C M × N c , k ∈ { , · · · , K m } for moving UEs H i − = [ H i − , C ,i − , · · · , C K m ,i − ] , (29)where N c is the number of candidate channels for each movingUE , and the candidate channel matrix C k,i − has columnseach of which is a prediction of h k,i based on the previouschannels { h k,b } { b
In this section, we first analyze the CPR beamformer andits variations, and then compare some of CPR variations withother beamforming methods in [17], [18], and [30]. In [17],a beamformer is designed to maximize a specific weightedsum-rate with a relaxed PAPC P max = 2 P tot /M . For a faircomparison, the resulting beamformer in [17] is additionallyscaled down to satisfy the PAPC in (3). The beamformingmethods in [18] and [30] solve multiple optimization problemsto design beamformers. The beamforming method in [18]maximizes the sum-rate with zero interference constraints andthe PAPC, and the beamforming method in [30] maximizes thesum signal-to-interference-plus-noise ratio (SINR) with imper-fect channel information and the PAPC. A simply normalizedZF beamformer is also presented as a reference, which iscalculated as F nZF = [ f nZF , , · · · , f nZF ,K ] , (35) f nZF ,k = ν f ZF ,k k f ZF ,k k , k ∈ { , · · · , K } , (36) [ f ZF , , · · · , f ZF ,K ] = (cid:0) H H i − (cid:1) † , (37)where ν is the normalization factor to satisfy the PAPC. Weset ν to have the maximum magnitude as ν = min m ∈{ , ··· ,M } p P ant K X k =1 (cid:12)(cid:12)(cid:12)(cid:12) ( f ZF ,k ) ( m ) k f ZF ,k k (cid:12)(cid:12)(cid:12)(cid:12) ! − . (38)We consider M transmit antennas at the BS and K m movingUEs and K s static UEs. The channel instantiation intervalis set as t = 40 ms, and the velocity of moving UEs areassumed to be the same as v . The Doppler frequency thenbecomes f D,k = vf c /c with carrier frequency f c = 2 . GHzand the speed of light c = 3 · m/s. For the channelmodel, we set the large-scale fading term β k = z k /x k where x k ∼ U[1 , models the pathloss effect and z k models theshadowing effect following a log-normal distribution withstandard deviation dB [32]. Each ∠ r k for channel correlationmatrix is uniformly distributed in [0 , π ) with | r k | = 0 . as in[31]. The magnitude of transmit symbol is set to be | s k,i | = 1 ,and the BS signal-to-noise ratio (SNR) is P tot σ . FC-CPR isdesigned with p init = 0 . , ∆ p = 0 . , and p max = 0 . ,and CPR is designed with p = p init or p = p max where
10 20 30 40 50 6018192021222324
Fig. 2: Sum-rate over algorithm iteration with M = 64 , K m = 0 , K s = 8 , SNR = 0 dB.each value is optimized numerically. For CPR-cc, two outdatedchannels H i − and H i − are considered as N c = 2 candidatechannels of moving UEs, and the sum-rate is obtained as K X k =1 log | h H k,i f k | σ + (cid:12)(cid:12)(cid:12)P ℓ = k h H k,i f ℓ (cid:12)(cid:12)(cid:12) , (39)where the beamformers f k , k ∈ { , · · · , K } are designedbefore estimating the current channel H i . A. Algorithm convergence
CPR iteratively updates a beamformer, and p balancesthe maximum performance and the convergence speed. Toimprove both the convergence speed and the maximum per-formance, we proposed FC-CPR, and numerical results aredepicted in Fig. 2 to assess the sum-rate over algorithm itera-tions. Depending on the power distribution strategy, (EP) and(WF) are denoted to represent the equal power distribution andthe water-filling power distribution, respectively. Regardlessof distribution strategy, CPR with small p = p init quicklyconverges but provides a low sum-rate. On the contrary, CPRwith large p = p max slowly converges to a high sum-rate witharound 50 iterations. FC-CPR achieves both the high sum-rateand fast convergence by approaching the high sum-rate of CPRwith large p = p max near the 8-th iteration. By the virtue ofits simple structure, CPR can be operated with large p = p max under a moderate complexity. Hence, in the next subsection,only the results of CPR with large p = p max are depicted forthe readability of figures.Another factor that differs the maximum sum-rate is thepower distribution strategy. The water-filling distribution givesa higher sum-rate than equal power distribution, and it isnatural since the water-filling is designed to raise the sum-rate. When the minimum data rate has a high priority, however,the equal power distribution would be better than the water-filling distribution. The two distribution strategies are shownas examples, and a proper strategy can be set depending on aspecific purpose. -10 -5 0 5 10 15 200510152025 (a) K m = 0 , K s = 4 -10 -5 0 5 10 15 200510152025303540 (b) K m = 0 , K s = 8 Fig. 3: Sum-rate over SNR with M = 16 . B. Data rate comparison
We consider two scenarios for data rate comparison, onewith all static UEs and the other with a moving UE. The firstscenario is considered in Fig. 3 without the results of CPR-ccand the beamformer in [30], which are designed to compensatethe channel uncertainty. The sum-rate of CPR (WF) and thebeamformer in [18] is similarly high and shows a remarkablegain compared to the normalized ZF beamformer. This clearlyshows that CPR (WF) can achieve the same data rate with [18],which is based on complicated optimization problems, withsignificantly less complexity. The sum-rate of CPR (EP) arebetween the simply normalized ZF beamformer and the abovebeamformers. The beamformer in [17] is designed to maximizea specific weighted sum-rate, which weights the data rate ofeach UE with the inverse of squared norm of channel vector,and the sum-rate with equal weights for UEs are lower thansimply normalized ZF beamformer due to the relaxed PAPC.The simulation results of the second scenario are in Fig. 4with the velocity of moving UE km/h. The beamformersin Fig. 3 still have the same trend in Fig. 4 while the sum-rate of beamformer in [30] is close to that of CPR (WF). -10 -5 0 5 10 15 2005101520 (a) M = 16 -10 -5 0 5 10 15 200510152025 (b) M = 32 Fig. 4: Data rate over SNR with K m = 1 , K s = 3 , v =3 km / h . -10 -5 0 5 10 15 200.20.40.60.811.21.41.6 Fig. 5: Data rate of moving UE over SNR with M = 16 , K m = 1 , K s = 3 , v = 3 km / h .The sum-rate of CPR-cc (WF) is little lower than that ofCPR (WF) since CPR-cc is to improve the performance of Fig. 6: Data rate of moving UE over velocity with M = 16 , K m = 1 , K s = 3 , SNR = 10 dB.moving UE, not the sum-rate. The simple use of outdatedchannel degraded the sum-rate of CPR-cc (WF), and the useof better channel prediction method, which is out of scopeof this paper, would be able to raise the sum-rate of CPR-cc (WF) as well. The robustness is assessed by the data rateof moving UE in Fig. 5. The beamformer in [30], which is arobust version of the beamformer in [18], provides higher datarate than the beamformer in [18]. As SNR increases, however,the data rate of the beamformer in [30] falls below that ofCPR (WF), i.e., even CPR (WF) is more robust to the channeluncertainty than the beamformer in [30]. Although the sum-rate of CPR-cc (WF) is little lower than that of CPR (WF),CPR-cc (WF) gives the highest data rate for the moving UE,providing robustness to the channel uncertainty.In Fig. 6, the data rate of moving UE is assessed accordingto the velocity v . Over the velocity, the data rate of movingUE decreases, while CPR-cc (WF) still gives the highest datarate, and CPR (WF) outperforms the beamformer in [30] forall range of velocity of interest.V. C ONCLUSION
We proposed a beamformer design to effectively exploittransmit power under the PAPC. CPR develops a beamformerby iteratively adding up extra beamformers. All the processesare conducted by linear operations without solving intri-cate optimization problems, which ensures low computationalcomplexity. In addition, the simple structure of CPR makesit highly flexible and allows CPR to be easily adapted tovarious scenarios. As examples, we designed several variationsof CPR in the perspective of convergence speed, sum-ratemaximization, and robustness for the channel uncertainty. Thesimulation results verified that CPR and its variations satisfytheir design purposes.A
PPENDIX
I: P
ROOF OF L EMMA k -th UE is (cid:12)(cid:12) h H k ( F n ) (: ,k ) (cid:12)(cid:12) = (cid:12)(cid:12) h H k ( F n − + W n A n ) (: ,k ) (cid:12)(cid:12) = (cid:12)(cid:12) h H k ( F n − ) (: ,k ) (cid:12)(cid:12) + 2 · Real n(cid:0) h H k ( F n − ) (: ,k ) (cid:1) ∗ ( a ) ( k ) h H k ( W n ) (: ,k ) o + (cid:12)(cid:12) ( a ) ( k ) h H k ( W n ) (: ,k ) (cid:12)(cid:12) a ) > (cid:12)(cid:12) h H k ( F n − ) (: ,k ) (cid:12)(cid:12) , (40)where ( a ) is by the coefficients ( a n ) ( k ) that are designedin (15) to align the products of channel and two beam-formers ∠ (cid:0) h H k ( F n − ) (: ,k ) (cid:1) = ∠ (cid:0) ( a n ) ( k ) h H k ( W n ) (: ,k ) (cid:1) . Withthe alignment, Real n(cid:0) h H k ( F n − ) (: ,k ) (cid:1) ∗ ( a ) ( k ) h H k ( W n ) (: ,k ) o equals to (cid:0) h H k ( F n − ) (: ,k ) (cid:1) ∗ ( a ) ( k ) h H k ( W n ) (: ,k ) ≥ . In ad-dition, c W n is the ZF beamformer and provides a positivebeamforming gain.Next, the interference of the ℓ -th beamformer ( F n ) (: ,ℓ ) tothe k -th UE channel h k is h H k ( F n ) (: ,ℓ ) = h H k ( F n − ) (: ,ℓ ) | {z } existing interference + ( a ) ( ℓ ) h H k ( W n ) (: ,ℓ ) | {z } additional interference , (41)where k ∈ { , · · · , K } , ℓ ∈ { , · · · , K } , and k = ℓ . Theadditional interference disappears as ( a ) ( ℓ ) h H k ( W n ) (: ,ℓ ) = ( a ) ( ℓ ) M X m =1 ( h k ) ∗ ( m ) ( W n ) ( m,ℓ ) = ( a ) ( ℓ ) X m ∈I ( p ) n ( h k ) ∗ ( m ) ( W n ) ( m,ℓ ) + ( a ) ( ℓ ) X m/ ∈I ( p ) n ( h k ) ∗ ( m ) ( W n ) ( m,ℓ ) = ( a ) ( ℓ ) ( h k ) H( I ( p ) n ) ( c W n ) (: ,ℓ )( a ) = 0 , (42)where ( a ) is by the fact that the beamformer c W n is the ZFbeamformers of ( H ) ( I ( p ) n , :) .R EFERENCES[1] D. Gesbert, M. Shafi, Da-shan Shiu, P. J. Smith, and A. Naguib, “Fromtheory to practice: an overview of MIMO space-time coded wirelesssystems,”
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