Cooperative Algorithms for MIMO Amplify-and-Forward Relay Networks
SSUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 1
Cooperative Algorithms for MIMOAmplify-and-Forward Relay Networks
Kien T. Truong,
Student Member, IEEE,
Philippe Sartori, andRobert W. Heath, Jr.*,
Fellow, IEEE
Abstract
Interference alignment is a signaling technique that provides high multiplexing gain in the interferencechannel. It can be extended to multi-hop interference channels, where relays aid transmission betweensources and destinations. In addition to coverage extension and capacity enhancement, relays increase themultiplexing gain in the interference channel. In this paper, three cooperative algorithms are proposedfor a multiple-antenna amplify-and-forward (AF) relay interference channel. The algorithms design thetransmitters and relays so that interference at the receivers can be aligned and canceled. The first algorithmminimizes the sum power of enhanced noise from the relays and interference at the receivers. The secondand third algorithms rely on a connection between mean square error and mutual information to solvethe end-to-end sum-rate maximization problem with either equality or inequality power constraints viamatrix-weighted sum mean square error minimization. Since we can find a globally optimal solutionin each iteration, the resulting iterative algorithms are convergent. Simulations show that the proposedalgorithms achieve higher end-to-end sum-rates and multiplexing gains that existing strategies for AFrelays, decode-and-forward relays, and direct transmission. The first algorithm outperforms the otheralgorithms at high signal-to-noise ratio (SNR) but performs worse than them at low SNR. Thanks topower control, the third algorithm outperforms the second algorithm at the cost of additional overhead.
Index Terms
Interference alignment, relay-aided interference alignment, two-hop interference channel, relay inter-ference channel, relay beamforming, joint source-relay design.
The authors are with the Department of Electrical and Computer Engineering, WNCG, 2501 Speedway Stop C0806, TheUniversity of Texas at Austin, Austin, TX, 78712-1687 (email: [email protected] and [email protected], phone: (512)686 8225, fax: (512) 471 6512). P. Sartori is with Huawei Technologies, Inc. (email: [email protected])This work was supported by a gift from Huawei Technologies, Inc. a r X i v : . [ c s . I T ] S e p UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 2
I. I
NTRODUCTION
Relay interference channels model networks where a stage of intermediate nodes, called relays, helpmultiple transmitters communicate with their receivers using shared radio resources [1]–[4]. Upcomingcellular standards are considering relay communication for coverage extension and capacity enhance-ment [5], [6]. Prior work, however, shows that single-antenna relays do not work well in the presenceof co-channel interference [7], [8]. In this paper, we consider multiple-antenna relay systems to takethe advantage of the interference management capability of multiple-input multiple-output (MIMO)communication. Many interference management strategies have been proposed for the MIMO single-hop interference channel [9]–[13]. Although these single-hop results can be applied separately for thetransmitter-relay hop and for the relay-receiver hop, even higher sum-rates can be achieved if the relaysare configured jointly [14], [15]. Obtaining the most from relay interference channels requires advancedinterference management strategies that jointly configure the transmitters, relays, and receivers.A general challenge to designing algorithms for the interference channel is that the sum capacity isunknown. The multiplexing gain of a network is a first-order approximation of its sum-capacity at highsignal-to-noise ratio (SNR) [16]. Interference alignment is a multiplexing gain maximizing signalingtechnique for the single-hop interference channel [13], achieving the maximum number of degrees offreedom. The idea is to arrange the transmitted signals such that interference is constrained within only aportion of the signal space observed by each receiver, leaving the remaining portion for interference-freedetection of the desired signal [9]. The maximum multiplexing gains achievable through interferencealignment, however, depend on the characteristics of the interference channels. For a symmetric MIMOinterference channel with constant channel coefficients, the maximum multiplexing gain is upper-boundedby the total number of antennas at a transmitter-receiver pair regardless the number of pairs [11], [12].Note that the bound is tight in certain cases and corresponds to the total available spatial dimensions ofa pair. Increasing the number of spatial dimensions in the network, using for example relays, is one wayto improve the maximum achievable multiplexing gain.Relays can be classified based on their signal processing operation, among which the most popular aredecode-and-forward (DF - the relays decode the received signals then re-encode before retransmitting)and amplify-and-forward (AF - the relays apply linear signal processing to the signal before forwarding).Without decoding the received signals, AF relays need no knowledge of the codebooks used by thetransmitters and likely have lower baseband complexity and fast signal processing. In addition, transparent
UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 3 to the modulation and coding of the signals, AF relays are more suitable for applications in heterogeneousnetworks comprising many nodes of different complexity or even standards [17]. In this paper, we focuson a half-duplex MIMO AF relay interference channel. Since half-duplex relays cannot transmit andreceive at the same time, they are more practical than full-duplex relays.Several interference management strategies designed specifically for the one-way AF relay interferencechannel have been proposed [15], [18]–[32]. Although relays cannot improve the multiplexing gains ofthe single-antenna fully-connected interference channel with time-varying or frequency-selective channelcoefficients [18], they are beneficial for reducing the number of independent channel extensions neededto align interference at the receivers [19]. Prior work often considers networks operating in specialcircumstances. It is assumed in [15], [20]–[24] that there are enough antennas at the relays to cancelall interference on the reception and then to nullify all interference on the retransmission, allowingmultiplexing gains to scale linearly with the number of users. Other prior work considers only smallnetworks with up to three pairs to derive some kind of closed-form strategies [25]–[27]. Prior workin [28] considers design problems with different objective functions including sum power minimizationand minimum SINR maximization. In addition, the algorithms in [28] are applicable only for single-antenna receivers. Prior work in [30]–[32] develops noncooperative resource allocation strategies for AFrelay networks while our work focuses on centralized algorithms for cooperative resource allocation. Inthis paper, we consider a general setting in the sense that we assume no special constraints on the numberof wireless nodes or the number of antennas at a node. The closest AF relay model to ours is consideredin [29], which is only for single-antenna transmitters and receivers. Further, they assume no crosslinksfrom relays to receivers, resulting in an oversimplified design problem.Sum-rate maximization problems are nonconvex and NP-hard, i.e., their global optima cannot be foundin a polynomial time. It is even challenging to find their good local optima corresponding to interferencealigned solutions using gradient-based algorithms from arbitrary initializations because those solutionshave very narrow regions of attraction [33]. Thus, we propose to formulate three new design problems thathave exactly the same constraints with sum-rate maximization problems but with better-behaved objectivefunctions to find high-quality solutions in terms of sum-rate maximization. Based on the observation thatinterference alignment solutions make total leakage power go to zero [19], [34], [35], we formulatea problem that aims at minimizing the sum power of interference and enhanced noise from the relays.Based on a relationship between achievable rates and mean squared error (MSE), we formulate two matrix-
UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 4 weighted sum-MSE minimization problems, either without power control or with power control. The keyis that they have the same stationary points as their corresponding end-to-end sum-rate maximizationproblems. Although the newly formulated optimization problems are still nonconvex and NP-hard, theymay be easier to solve. Next, we propose to adopt an alternating minimization approach [36], [37]to develop iterative algorithms for solving the newly formulated problems. In each iteration, all butone variable is fixed and we focus on designing the remaining variable by solving a single-variableoptimization problem obtained from the corresponding original multi-variable problems. Since we areable to find a global optimum for the single-variable optimization problem in each iteration, the proposedalgorithms are guaranteed to converge. Note that the power constraints at the relays depend on both thetransmit precoders at the transmitters and the processing matrices at the relays, adding more constraints tothe design problems. Thus, it is not straightforward to extend the methods used for the single-hop designproblems to solve the two-hop design problems. Our initial results in this paper were reported in [38].Compared with [38], this paper presents three different algorithms, has more discussion of convergenceand provides simulations that emphasize the achievable end-to-end sum-rates and multiplexing gains.We use Monte Carlo simulation to evaluate the average end-to-end sum-rates and multiplexing gainsachievable through the proposed algorithms. First, the numerical results confirm the convergence of theproposed algorithms as expected. Second, over the iterations of the total leakage minimization algorithm,the true interference dominates at the beginning but is canceled quickly; after that, the enhanced noisefrom the relays becomes dominant. This means that relay-aided interference alignment should take intoaccount the enhanced noise from the relays. Third, the total leakage minimization algorithm achieves lowerend-to-end sum-rates than the others at low-to-medium SNR values because it ignores the desired signalpower and noise power at the receivers. Nevertheless, the MSE-based algorithms result in unfairness,i.e., some users have much smaller rates than the others. Thus, the MSE-based algorithms achieve lowerend-to-end sum-rates and multiplexing gains than the total leakage minimization algorithm at high SNR.One reason for this is that the MSE-based algorithms may either turn off some data streams or nullifythe desired signals to some receivers. Fourth, for fixed numbers of antennas at the transmitters and at thereceivers, even with half-duplex loss, AF relays can provide larger end-to-end multiplexing gains than DFrelays or direct transmissions. Finally, the results show that AF relays provide larger average achievableend-to-end sum-rates than do DF relays. The proposed algorithms also provide higher achievable end-to-end sum-rates than the existing AF relaying strategies that do not align interference at the receivers.
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The organization of the remainder of this paper is as follows. Section II describes the system model.Section III formulates the end-to-end sum-rate maximization problems and presents our proposed ap-proach. Section IV develops three cooperative algorithms that aim at finding high-quality solutions of thesum-rate maximization problems. Section V evaluates numerically the proposed algorithms. Section VIconcludes this paper and suggests future research.Notation: We use normal letters (e.g., a ) for scalars, lowercase and uppercase boldface letters (e.g., h and H ) for column vectors and matrices. I N and 0¯ N are the identity matrix and all-zero matrices of size N × N . ν n min ( A ) gives the eigenvectors corresponding to the n smallest eigenvalues of A . For a matrix A , A T is the transpose matrix, (cid:107) A (cid:107) F the Frobenious norm, A ∗ the conjugate transpose, and tr( A ) the trace. vec( A ) denotes the vec operator to transform A into a while vec − ( a ) denotes the inverseoperator. ⊗ is the Kronecker product. E [ · ] is the statistical expectation operator. () ( n ) denotes iterationindex. () T is used for transmitters’ parameters, () R for receivers’, and () X for relays’.II. S YSTEM M ODEL
Consider a relay interference channel where M half-duplex AF relays aid the one-way communicationbetween K pairs of transmitters and receivers, as illustrated in Fig. 1. Each transmitter has data for onlyone receiver and each receiver is served by only one transmitter. Each pair is assigned a unique index k ∈ K (cid:44) { , · · · , K } . Transmitter k has N T ,k antennas while receiver k has N R ,k antennas for k ∈ K .Similarly, each relay is assigned a unique index m ∈ M (cid:44) { , · · · , M } . Relay m has N X ,m antennasfor m ∈ M . The half-duplex relays cannot transmit and receive at the same time, thus the transmissionprocedure consists of two stages. In the first stage, the transmitters send data to the relays. In the secondstage, the relays apply linear processing to the received signals and forward to the receivers. We assumethe direct channels between the transmitters and the receivers are ignored by the second-stage receivers. TX RX Relay 1TX TX K Relay 2Relay M RX RX K G G G K,1 G K,M H H H M,1 H M,K F K F W K W U U M U W Fig. 1. A relay interference channel where M half-duplex AF relays aid the communication of K transmitter-receiver pairs. UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 6
We denote H m,k ∈ C N X ,m × N T ,k as the matrix channel from transmitter k to relay m and G k,m ∈ C N R ,k × N X ,m as the matrix channel from relay m to receiver k for k ∈ K and m ∈ M . We assume thatperfect and instantaneous knowledge of H m,k and G k,m for k ∈ K and m ∈ M is available at a centralprocessing unit. Although this is a strict requirement, our results are still valuable since they show thesubstantial gains that can be achieved through coordination. Our results can be used as a benchmark forfuture work that makes more practical CSI assumptions.Let s k ∈ C d k × be the transmit symbol vector at transmitter k , where d k ≤ min { N T ,k , N R ,k } is thenumber of data streams from transmitter k to receiver k for k ∈ K . The transmit symbols are independentidentically distributed (i.i.d.) such that E ( s k s ∗ k ) = I d k . Transmitter k uses a linear transmit precoder F k ∈ C N T ,k × d k to map s k to its transmit antennas. Let p maxT ,k be the maximum transmit power. The actualtransmit power at transmitter k is p T ,k = tr( F ∗ k F k ) . Let n X ,m be spatially white, additive Gaussian noiseat relay m with covariance E ( n X ,m n ∗ X ,m ) = σ ,m I N X ,m for m ∈ M . With perfect synchronization, relay m observes the following signal y X ,m = K (cid:88) k =1 H m,k F k (cid:124) (cid:123)(cid:122) (cid:125) H m,k s k + n X ,m . (1)Let U m ∈ C N X ,m × N X ,m be the processing matrix at relay m . The transmit signal at relay m is given by x X ,m = U m y X ,m = K (cid:88) k =1 U m H m,k s k + U m n X ,m . (2)Relay m actually uses the following transmit power p X ,m = K (cid:88) k =1 tr( U m H m,k H ∗ m,k U ∗ m ) + σ ,m tr( U m U ∗ m ) . (3)There are two possible types of power constraints at the relays: i) a set of individual power constraintsat the relays and ii) a sum power constraint at all the relays. Individual relay power constraints are oftenconsidered in the cellular system literature [39], [40]. While a sum power constraint is often consideredin the ad hoc network literature to extend the lifetime of battery-powered relays [41], [42]. Let p maxX ,m bethe maximum transmit power at relay m and p maxX be the maximum sum transmit power at all the relays.When power control is considered, the individual relay power constraints are p X ,m ≤ p maxX ,m , ∀ m ∈ M ;whereas the sum relay power constraint is (cid:80) Mm =1 p X ,m ≤ p maxX . Without power control, the inequalitiesin the power constraint expressions are replaced by equalities. The following sections focus on thesum power constraint at the relays. Section IV-D discusses the applicability of individual relay powerconstraints while Section V simulates the impact of individual relay power constraints on achievable UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 7 end-to-end sum-rates.Let n R ,k be spatially white, additive Gaussian noise at receiver k with covariance E ( n R ,k n ∗ R ,k ) = σ ,k I N R ,k . We denote G k,m = G k,m U m . Receiver k observes the following signal y k = M (cid:88) m =1 G k,m x X ,m + n R ,k (4) = K (cid:88) q =1 M (cid:88) m =1 G k,m H m,q (cid:124) (cid:123)(cid:122) (cid:125) T k,q s q + M (cid:88) m =1 G k,m n X ,m + n R ,k , (5)where T k,q is the effective end-to-end channel from transmitter q to receiver k for k, q ∈ K . Applyinga linear receive filter W k ∈ C N R ,k × d k to y k , receiver k obtains ¯ y k = W ∗ k T k,k s k (cid:124) (cid:123)(cid:122) (cid:125) desired signal + K (cid:88) q =1 q (cid:54) = k W ∗ k T k,q s q (cid:124) (cid:123)(cid:122) (cid:125) interference + M (cid:88) m =1 W ∗ k G k,m n X ,m (cid:124) (cid:123)(cid:122) (cid:125) enhanced noise from relays + W ∗ k n R ,k (cid:124) (cid:123)(cid:122) (cid:125) local noise . (6)The pre-processing interference-plus-noise covariance matrix at receiver k is R k = K (cid:88) q =1 q (cid:54) = k T k,q T ∗ k,q + M (cid:88) m =1 σ ,m G k,m G ∗ k,m + σ ,k I d k . (7)For notational convenience, we denote { F } (cid:44) { F k } Kk =1 , { U } (cid:44) { U m } Mm =1 and { W } (cid:44) { W k } Kk =1 .We also denote U m,k (cid:44) U m H m,k and W k,m (cid:44) W ∗ k G k,m for k ∈ K and m ∈ M . Table I summarizesthe notation of equivalent channel gain matrices used in the paper for k, q ∈ K and m ∈ M . TABLE IN
OTATION OF EQUIVALENT CHANNEL MATRICES USED IN THE PAPER FOR k, q ∈ K
AND m ∈ M . Equivalent channel matrix Definition H m,k H m,k F k G k,m G k,m U m U m,k U m H m,k W k,m W ∗ k G k,m T k,q (cid:80) Mm =1 G k,m H m,q = (cid:80) Mm =1 G k,m U m H mq F q III. P
ROBLEM F ORMULATION AND P ROPOSED A PPROACH
We formulate the end-to-end sum-rate maximization problem in Section III-A and propose an approachto solving it in Section III-B.
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A. End-to-end Sum-Rate Maximization
For tractable analysis, we assume Gaussian signaling is used. For a given { F } and { U } , the achievablerate for the k -th transmitter-receiver pair is maximized by using the linear MMSE receive filter [43] W MMSE k = ( T k,k T ∗ k,k + R k ) − T k,k , (8)where R k is given in (7). Thus, we only need to focus on the design of { F } and { U } . Note that thecorresponding maximum achievable rate is given by [43] R k (cid:0) { F } , { U } (cid:1) = log det (cid:0) I d k + T ∗ k,k R − k T k,k (cid:1) . (9)The sum of the end-to-end achievable rates is defined as R sum (cid:0) { F } , { U } (cid:1) = − K (cid:88) k =1 log det (cid:16) E MMSE k ( { F } , { U } ) (cid:17) . (10)The end-to-end sum-rate maximization problem without power control is formulated as follows ( OP - noPC) : min { F } , { U } − R sum (cid:0) { F } , { U } (cid:1) s.t. p T ,k = p maxT ,k , k = 1 , · · · , K, (11) M (cid:88) m =1 p X ,m = p maxX . (12) Remark 1:
The counterpart problem with power control can be obtained by replacing the equalities inthe constraints by the inequalities. We denote it as ( OP - PC) . Power control may improve the end-to-endsum-rates at the expense of additional overhead because the central unit needs to inform the transmittersabout both the norm and the shape of designed transmit precoders.
Remark 2:
The following ( F ,k , W ,k , U ,m ) for k ∈ K and m ∈ M satisfies the constraints of both( OP - PC ) and ( OP - noPC) F ,k = (cid:115) p maxT ,k d k I N T ,k × d k , k ∈ K , (13) W ,k = (cid:114) d k I N R ,k × d k , k ∈ K , (14) U ,m = (cid:112) αp maxX I N X ,m × N X ,m , m ∈ M , (15)where α = (cid:16) (cid:80) Kk =1 p maxT ,k d k (cid:80) Mm =1 tr( I d k × N T ,k H ∗ m,k H m,k I N T ,k × d k ) + (cid:80) Mm =1 N X ,m σ ,m (cid:17) − . Remark 3: ( OP - PC ) and ( OP - noPC) are nonconvex and NP-hard. Moreover, even the smallestconfiguration of the MIMO AF relay interference channel with K = M = 2 and N T = N X = N R = 2 requires the determination of twelve complex variables for the transmit precoders and relay processing UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 9 matrices, which makes even a brute force approach challenging.
B. Proposed Approach
Instead of directly solving for the globally optimal solutions of ( OP - PC ) and ( OP - noPC) , we aimat finding their high-quality solutions with reasonable computational complexity. To do this, in SectionIII-B1 and Section III-B2, we formulate two classes of new optimizations problems that have exactly thesame constraints as ( OP - PC ) or ( OP - noPC) but with different objective functions.
1) Total Leakage Minimization:
This section presents an approach for interference alignment in theAF relay interference channel, which is inspired by those for the single-hop interference channel in [19],[34], [35]. The underlying observation for this approach is that when interference alignment is feasible,the sum power of the interference at all the receivers, also known as the leakage, is zero. From (6),there are three groups of unwanted signals at each receiver: i) interference, ii) enhanced noise from therelays, and iii) local noise. We denote I ( { F } , { U } , { W } ) as the total leakage power of the AF relayinterference channel. By evaluating the expectation and exploiting the independence of transmit signals s k for k ∈ K and using the equality (cid:107) A (cid:107) F = tr( AA ∗ ) , we obtain I ( { F } , { U } , { W } ) = K (cid:88) k =1 K (cid:88) q =1 q (cid:54) = k tr( W ∗ k T k,q T ∗ k,q W k ) . (16)In our opinion, the high SNR regime of the relay interference channel corresponds to high transmitpower at both the transmitters and the relays. As a result, in addition to eliminating completely interfer-ence, we also need to eliminate the enhanced relay noise; otherwise, the enhanced relay noise power scaleswith the desired signal power, preventing the system from achieving high multiplexing gain. We denote N ( { U } , { W } ) as the sum power of enhanced noise from the relays. By evaluating the expectations andexploiting the independence of the noise vectors at the relays, we obtain N ( { U } , { W } ) = K (cid:88) k =1 M (cid:88) m =1 σ ,m tr( W ∗ k G k,m G ∗ k,m W k ) . (17)Note that scaling down transmit power at either the transmitters or relays decreases the total leakagepower at the receivers. For example, if ( { (1 /a ) F } , { U } , { W } ) is used instead of ( { F } , { U } , { W } ) where { (1 /a ) F } = { (1 /a ) F , · · · , (1 /a ) F K } and a > , then both the actual transmit power at thetransmitters and the total leakage power decrease a > times. Thus, equality power constraints atthe transmitters and relays are required to obtain a meaningful design problem. This means that powercontrol should not be considered in the context of total leakage power minimization. In other words, UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 10 we do not use the total leakage minimization approach to find solutions to ( OP - PC) . Also to obtain ameaningful design problem, we add the orthonormal constraints on W k as W ∗ k W k = I d k for k ∈ K .Without such constraints, we can always use zero matrices as the receive filters to get zero total leakagepower. Consequently, to find high-quality solutions of ( OP - noPC) , we propose to solve the followingproblem ( T L ) : min { F } , { U } , { W } I ( { F } , { U } , { W } ) + N ( { U } , { W } ) s.t. p T ,k = p maxT ,k , k ∈ K , (18) M (cid:88) m =1 p X ,m = p maxX , (19) W ∗ k W k = I d k , k ∈ K . (20)Note that ( T L ) is nonconvex and in general is NP-hard. Also, ( T L ) does not take into account thedesired signal power and local noise at the receivers. Remark 4:
The total leakage minimization problem formulated in [29] for an AF relay network is asimplified version of (
T L ). It is assumed in [29] that the transmitters and receivers are equipped witha single antenna. Each pair is aided by a dedicated multiple-antenna AF relay. The formulation in [29]does not consider power constraints at the relays. In addition, it is assumed that there are no cross-linksfor the transmissions from relays to receivers, i.e. G k,q =
0¯ for all k, q ∈ K and k (cid:54) = q . As a result, forfixed { F } and { W } , the algorithm in [29] can determine each U m separately.
2) Sum Mean Squared Error Minimization:
The section presents another approach that is based on arelationship between the achievable rates and MSE values at the receivers with Gaussian signaling [44].This is inspired by prior work on the MIMO broadcast channel [45], MIMO interference channel [33],MIMO interference broadcast channel [46], and two-way relay channel [47], [48]. Let E k ( { E } , { U } , W k ) be the MSE matrix at receiver k . After some manipulation, we obtain E k ( { E } , { U } , W k ) = W ∗ k ( T k,k T ∗ k,k + R k ) W k − W ∗ k T k,k − T ∗ k,k W k + I d k . (21)Note that the MSE at receiver k , defined as M SE k = tr( E k ( { E } , { U } , W k )) , is minimized by thelinear MMSE receive filter W MMSE k . Moreover, it is well-established that [33], [44]–[49] R k (cid:0) { F } , { U } (cid:1) = − log det( E k ( { F } , { U } , W MMSE k )) . (22)We introduce auxiliary weight matrix variables { V } (cid:44) ( V , · · · , V K ) that are square ( V k ∈ C d k × d k )and positive semidefinite for k ∈ K . The weight matrices V k are just auxiliary variables for the UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 11 optimization technique and have no actual physical meaning. Define the matrix-weighted sum of MSEvalues as follows
W M SE sum ( { F } , { U } , { W } , { V } ) = K (cid:88) k =1 (cid:18) tr (cid:16) V k E k (cid:0) { F } , { U } , { W } (cid:1)(cid:17) − log det (cid:0) V k (cid:1)(cid:19) . (23)Then, we formulate the following weighted sum-MSE minimization problem ( WMSE - noPC) : min { F } , { U } , { W } , { V } W M SE sum (cid:0) { F } , { U } , { W } , { V } (cid:1) s.t. p T ,k = p maxT ,k , k ∈ K (24) M (cid:88) m =1 p X ,m = p maxX . (25)Similarly, we formulate ( WMSE - PC) by replacing the equalities in ( WMSE - noPC) . Using the samesteps in [46], [49], we can show that ( WMSE - noPC) and ( OP - noPC) have exactly the same stationarypoints if we use the linear MMSE receivers and choose the following matrix weights V opt k (cid:0) { F } , { U } , W MMSE k (cid:1) = E − k (cid:0) { F } , { U } , W MMSE k ) , (26) = I d k + T ∗ k,k R − k T k,k . (27)This observation is also true for ( WMSE - PC) and ( OP - PC) . Thus, instead of directly solving ( OP - noPC) (or ( OP - PC) ), we can focus on finding high-quality solutions to its corresponding weighted sum-MSEminimization problem, which has a better-behaved objective function.
Remark 5:
The matrix-weighted sum-MSE value
W M SE sum ( { F } , { U } , { W } , { V } ) is convex withrespect to F k for k ∈ K if we always choose V k according to (26). Indeed, we can check that M SE k isconvex with respect to F q for all k, q ∈ K . By construction, V opt k (cid:0) { F } , { U } , W MMSE k (cid:1) is a Hermitianand positive semidefinite matrix for k ∈ K . Then, by definition W M SE sum ( { F } , { U } , { W } , { V } ) isalso convex with respect to F k for k ∈ K . IV. A LGORITHMS
The problems ( T L ) , ( WMSE - noPC) , and ( WMSE - PC) formulated in Section III-B are nonconvexand in general are NP-hard. In this section, rather than attempting solving for their globally optimalsolutions, we adopt an alternating minimization approach [36] to develop iterative algorithms for findingtheir high-quality solutions. In each iteration, we alternatively fix all but one variable and determinethe remaining variable by solving a single-variable optimization problem. The optimization problemin each iteration is always feasible since it has the outcome of the previous iteration as a feasible
UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 12 point. After initialization, the algorithms are repeated until a convergent point is reached. Section IV-Apresents the algorithm for solving (
T L ), which is denoted as Algorithm 1. Two algorithms for solving ( WMSE - noPC) and ( WMSE - PC) are presented in Section IV-B and Section IV-C. They are denotedas Algorithm 2 and Algorithm 3, respectively.
A. Algorithm for Total Leakage Minimization ( T L ) There are three classes of design subproblems in Algorithm 1: i) receiver filter design, ii) relayprocessing matrix design, and iii) transmit precoder design.
1) Receive Filter Design for (
T L ): We can rewrite the cost function as (cid:80) Kk =1 tr( W ∗ k Z k W k ) , where Z k = (cid:80) Kq =1 q (cid:54) = k T k,q T ∗ k,q + (cid:80) Mm =1 σ ,m G k,m G ∗ k,m . Since W k for k ∈ K are decoupled in the cost function,they can be determined separately and in parallel by solving ( T L - W k ) : W k = arg min X ∈ C N R ,k × dk : X ∗ X = I dk tr( X ∗ Z k X ) . It follows from [50] that a global optimum of ( T L - W k ) is W k = ν d k min ( Z k ) .
2) Relay Processing Matrix Design for ( T L ) : We focus on determining U m for some m ∈ M bysolving the following single-variable optimization problem ( T L - U m )min X ∈ C N X ,m × N X ,m K (cid:88) k =1 K (cid:88) q =1 (cid:54) = k tr (cid:16) X H m,q H ∗ m,q X ∗ W ∗ k,m W k,m (cid:17) + σ ,m K (cid:88) k =1 tr( X ∗ W ∗ k,m W k,m X )+ K (cid:88) k =1 K (cid:88) q =1 q (cid:54) = k M (cid:88) n =1 n (cid:54) = m tr (cid:18) X H m,q H ∗ n,q U ∗ n W ∗ k,n W k,m (cid:19) + K (cid:88) k =1 K (cid:88) q =1 q (cid:54) = k M (cid:88) n =1 n (cid:54) = m tr (cid:18) W ∗ k,m W k,n U n H n,q H ∗ m,q X ∗ (cid:19) s.t. tr (cid:18) X (cid:16) K (cid:88) k =1 H m,k H ∗ m,k + σ ,m I N X ,m (cid:17) X ∗ (cid:19) = η U ,m , (28)where η U ,m = p maxX − (cid:80) Mn =1 n (cid:54) = m (cid:80) Kk =1 tr (cid:0) U n H n,k H ∗ n,k U ∗ n (cid:1) − (cid:80) Mn =1 n (cid:54) = m σ ,n tr( U n U ∗ n ) . Because of thespecial form of the first term in the cost function of ( T L - U m ) , it is not straightforward to use themethods for the single-hop interference channel like those in [35] to solve ( T L - U m ) .We propose to transform ( T L - U m ) into a more readily solvable form by introducing a new variable UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 13 u m = vec( U m ) ∈ C N ,m × . We define the following matrices that are independent of u m A ,m = K (cid:88) k =1 (cid:18) K (cid:88) q =1 q (cid:54) = k H m,q H ∗ m,q + σ ,m I N X ,m (cid:19) T ⊗ (cid:0) W ∗ k,m W k,m (cid:1) , (29) a ,m = vec (cid:18) K (cid:88) k =1 K (cid:88) q =1 q (cid:54) = k M (cid:88) n =1 n (cid:54) = m W ∗ k,m W k,n U n H n,q H ∗ m,q (cid:19) , (30) A ,m = (cid:18) K (cid:88) k =1 H m,k H ∗ m,k + σ ,m I N X ,m (cid:19) T ⊗ I N X ,m . (31)Note that with probability one, A ,m is Hermitian and positive definite while A ,m is Hermitian andpositive semidefinite. Then, we use the following equalities, tr( ABA ∗ C ) = (vec( A )) ∗ ( B T ⊗ C ) vec( A ) , tr( A ∗ BA ) = tr( AIA ∗ B ) = (vec( A )) ∗ ( I ⊗ B ) vec( A ) and tr( AB ∗ ) = (vec( B )) ∗ vec( A ) [51], totransform both the cost function and the constraint of ( T L - U m ) into quadratic expressions of u m . Thequadratically constrained quadratic program (QCQP) for designing u m is ( T L - u m ) : min x ∈ C N ,m × x ∗ A ,m x + a ∗ ,m x + x ∗ a ,m s.t. x ∗ A ,m x = η U ,m . (32)This is a QCQP with a single equality quadratic constraint. It is nonconvex as well.In solving ( T L - u m ) , we introduce a new variable Y = u m (cid:16) u ∗ m (cid:17) = u m u ∗ m u m u ∗ m ∈ C ( N ,m +1) × ( N ,m +1) . It follows that Y is a rank-one Hermitian positive semidefinite matrix with bottomright entry equal to 1. We can rewrite ( T L - u m ) equivalently as ( T L - u m u ∗ m ) : min Y ∈ C ( N ,m +1) × ( N ,m +1) tr A ,m a ,m a ∗ ,m Y s.t. tr A ,m N ,m × × N ,m Y = η U ,m + 1 , (33) tr N ,m × N ,m N ,m × × N ,m Y = 1 , (34) Y (cid:23) , rank( Y ) = 1 . (35)While the cost function and all other constraints are convex, the rank constraint is nonconvex. This rankconstraint is actually the main difficulty in solving ( T L - u m u ∗ m ) . Dropping this rank constraint, however,we obtain a relaxed version of ( T L - u m u ∗ m ) , which is a convex optimization problem and also known UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 14 as a semidefinite relaxation (SDR) of ( T L - u m u ∗ m ) . Note that a complex-valued separable homogeneousQCQP with n constraints is guaranteed to have a global optimum with rank r ≤ √ n [52]. Therefore,having n = 1 constraints, ( T L - u m u ∗ m ) is guaranteed to have a rank-one global optimum. The SDR of ( T L - u m u ∗ m ) can be solved, to any arbitrary accuracy, in a numerically reliable and efficient manner byreadily available software packages, e.g., the convex optimization toolbox CVX [53]. It is not guaranteed,however, that solving the SDR by the available software packages provides a desired rank-one globaloptimum of the SDR. Fortunately, we can construct a rank-one global optimum of the SDR from theresulting general-rank global optimum using the rank-reduction procedure in [52], which is an extension ofthe purification technique in [54]. The key idea in each step of the procedure is to modify the eigenvaluesof the general-rank global optima to remove the largest eigenvalue. Each step of the procedure gives usanother global optima with the same eigenvectors but with one fewer nonzero eigenvalues. We notice thatthe last entry of the column vector obtained by the decomposition of the rank-one global optimum [55]may be a complex number with modulus of 1. By multiplying the resulting column vector with theconjugate of its last entry, we obtain a desired column vector in the form of ( u m T , which correspondsto another rank-one global optimum of ( T L - u m u ∗ m ) . We then use the vec − operator to get a globallyoptimal solution U m of ( T L - U m ) from the resulting u m .
3) Transmit Precoder Design for (
T L ): We now focus on designing F k for some k ∈ K by solvingthe following single-variable optimization problem ( T L - F k ) : min X ∈ C N T ,k × dk tr (cid:32) X ∗ (cid:18) K (cid:88) q =1 q (cid:54) = k M (cid:88) m =1 M (cid:88) n =1 U ∗ m,k W ∗ q,m W q,n U n,k (cid:19) X (cid:33) s.t. tr( X ∗ X ) = p T ,k (36) tr (cid:32) X ∗ (cid:16) M (cid:88) m =1 U ∗ m,k U m,k (cid:17) X (cid:33) = η F ,k , (37)where η F ,k = p maxX − (cid:80) Kq =1 q (cid:54) = k (cid:80) Mm =1 tr (cid:16) F ∗ q U ∗ m,q U m,q F q (cid:17) − (cid:80) Mm =1 σ ,m tr (cid:16) U m U ∗ m (cid:17) . Note that ( T L - F k ) is non-convex and in general is NP-hard. Since ( T L - F k ) has two equality constraints, the use of theLagrange multiplier method requires a more complicated 2-D search.Similar to Section IV-A2, we propose a method for transforming ( T L - F k ) into an equivalent optimiza-tion problem and for solving for its global optimum. We start by defining a new variable f k = vec( F k ) ∈ UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 15 C N T ,k d k × . We also define the following matrices which are independent of f k B ,k = I d k ⊗ (cid:18) K (cid:88) q =1 q (cid:54) = k M (cid:88) m =1 M (cid:88) n =1 U ∗ m,k W ∗ q,m W q,n U n,k (cid:19) , (38) B ,k = I d k ⊗ (cid:18) M (cid:88) m =1 U ∗ m,k U m,k (cid:19) . (39)Both B ,k and B ,k are Hermitian positive definite matrices. Using tr( A ∗ BA ) = (vec( A )) ∗ ( I ⊗ B ) vec( A ) [51], we transform ( T L - F k ) into the following single-variable optimization problem ( T L - f k ) : min x ∈ C N T ,kdk × x ∗ B ,k x s.t. x ∗ x = p maxT ,k , (40) x ∗ B ,k x = η F ,k . (41)Note that ( T L - f k ) is a complex-valued homogeneous QCQP with two equality quadratic constraints.Nevertheless, ( T L - f k ) is still nonconvex and NP-hard [52], [56].In solving ( T L - f k ) , we introduce a new variable Y = xx ∗ . Note that Y = xx ∗ requires that Y be arank-one Hermitian positive semidefinite matrix. In addition, since a ∗ Ba = tr( Baa ∗ ) for any matrix B and any vector a [57], we obtain an equivalent optimization problem of ( T L - f k ) as follows ( T L - f k f ∗ k ) : min Y ∈ C N T ,kdk × N T ,kdk tr( B ,k Y ) s.t. tr( Y ) = p maxT ,k , (42) tr( B ,k Y ) = η F ,k , (43) Y (cid:23) , rank( Y ) = 1 . (44)Similar to solving ( T L - u m u ∗ m ) , we adopt the SDP method for solving ( T L - f k f ∗ k ) . Since ( T L - f k ) is acomplex-valued separable homogeneous QCQP with n = 2 constraints, it is guaranteed that ( T L - f k f ∗ k ) has a global optimum of rank r = 1 ≤ √ n . We can use readily available software packages, e.g.,the convex optimization toolbox CVX [53], to solve for a general rank global optimum of the SDRof ( T L - f k f ∗ k ) . Next, we can always construct a rank-one global optimum of the SDR from any of itsgeneral-rank global optimum, e.g., by using the rank reduction procedure in [52]. The decomposition ofthe rank-one global optimum [55] gives us the desired f k . Finally, we use the vec − operator to get aglobally optimal solution F k of ( T L - F k ) from the resulting f k . UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 16
B. Algorithm for Sum MSE Minimization without Power Control ( WMSE - noPC) The design subproblems in the iterations of Algorithm 2 belong to one of the following four categories.
1) Matrix Weight Design for ( WMSE - noPC) : Since the matrix weights V opt k for k ∈ K areindependent of each other, they can be updated in parallel based on (26).
2) Receive Filter Design for ( WMSE - noPC) : Recall that this approach requires the receivers usethe linear MMSE receive filters W MMSE k given in (8). The receive filters can be updated in parallel.
3) Relay Processing Matrix Design for ( WMSE - noPC) : We focus on the design of U m for some m ∈ M . By substituting (7) and (21) into (23) and removing the terms independent of U m , we obtain theobjective function of the design problem for U m . Let ( WMSE - noPC - U m ) denote the design problemof U m . After some manipulation and using the fact that V k is Hermitian, we obtain the formulation of ( WMSE - noPC - U m ) as min X ∈ C N x ,m × N X ,m K (cid:88) k =1 K (cid:88) q =1 tr (cid:16) X H m,q H ∗ m,q X ∗ W ∗ k,m V k W k,m (cid:17) + σ ,m K (cid:88) k =1 tr( X ∗ W ∗ k,m V k W k,m X ) − K (cid:88) k =1 tr( H m,k V ∗ k W k,m X ) + K (cid:88) k =1 K (cid:88) q =1 M (cid:88) n =1 n (cid:54) = m tr( H m,q H ∗ n,q U ∗ n W ∗ k,n V k W k,m X ) − K (cid:88) k =1 tr( X ∗ W ∗ k,m V k H ∗ m,k ) + K (cid:88) k =1 K (cid:88) q =1 M (cid:88) n =1 n (cid:54) = m tr( X ∗ W ∗ k,m V ∗ k W k,n U n H n,q H ∗ m,q ) s.t. tr (cid:18) X ∗ (cid:16) K (cid:88) k =1 H m,k H ∗ m,k + σ ,m I N X ,m (cid:17) X (cid:19) = η U ,m . (45)Note that ( WMSE - noPC - U m ) differs from ( T L - U m ) mainly due to the appearance of V k in thecost function. We introduce a new variable u m = vec( U m ) and define the following matrices C ,m = K (cid:88) k =1 (cid:18) K (cid:88) q =1 H m,q H ∗ m,q + σ ,m I N X ,m (cid:19) T ⊗ (cid:16) W ∗ k,m V k W k,m (cid:17) , (46) c ,m = vec (cid:18) − K (cid:88) k =1 W ∗ k,m V k H ∗ m,k + M (cid:88) n =1 n (cid:54) = m K (cid:88) k =1 K (cid:88) q =1 W ∗ k,m V k W k,n U n H n,q H ∗ m,q (cid:19) . (47)Using the same manipulation as in Section IV-A2 and denoting C ,m = A ,m , we obtain the followingequivalent optimization problem ( WMSE - noPC - u m ) : min x ∈ C N ,m × x ∗ C ,m x + c ∗ ,m x + x ∗ c ,m s.t. x ∗ C ,m x = η U ,m . (48)Note that ( WMSE - noPC - u m ) has exactly the same form as ( T L - u m ) , thus we can apply the same UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 17 method used for solving ( T L - u m ) to find a globally optimal solution u m of ( WMSE - noPC - u m ) . Wethen use the vec − operator to get a globally optimal solution U m of ( WMSE - noPC - U m ) from theresulting u m .
4) Transmit Precoder Design for ( WMSE - noPC) : We define the following matrices for the designof F k for some k ∈ K D ,k = K (cid:88) q =1 M (cid:88) m =1 M (cid:88) n =1 U ∗ m,k W ∗ q,m V q W q,n U n,k , (49) D ,k = M (cid:88) m =1 U ∗ m,k W ∗ k,m V ∗ k , (50) D ,k = M (cid:88) m =1 U ∗ m,k U m,k . (51)After some manipulation, we obtain the following single-variable optimization problem ( WMSE - noPC - F k ) : min X ∈ C N T ,k × dk tr( X ∗ D ,k X ) − tr( D ∗ ,k X ) − tr( D ,k X ∗ ) s.t. tr( X ∗ X ) = p maxT , (52) tr( X ∗ D ,k X ) = η T ,k . (53)Recall that we define f k = vec( F k ) . We introduce a new variable Y = f k (cid:16) f ∗ k (cid:17) = f k f ∗ k f k f ∗ k .It follows that Y is a rank-one Hermitian positive semidefinite matrix with the bottom right entry equalto 1. Then, we transform ( WMSE - noPC - F k ) equivalently into the following problem ( WMSE - noPC - f k f ∗ k ) : min Y ∈ C ( N T ,kdk +1) × ( N T ,kdk +1) tr I d k ⊗ D ,k − vec( D ,k ) − (vec( D ,k )) ∗ Y s.t. tr( Y ) = p maxT + 1 , (54) tr I d k ⊗ D ,k N T ,k d k × × N T ,k d k Y = η T ,k + 1 , (55) tr N T ,k d k × N T ,k d k N T ,k d k × × N T ,k d k Y = 1 , (56) Y (cid:23) , rank( Y ) = 1 . (57)Note that ( WMSE - noPC - f k f ∗ k ) has the same form as ( T L - u m u ∗ m ) but with one more constraint. Since ( WMSE - noPC - f k f ∗ k ) has n = 3 constraints (excluding the rank-one constraint), its SDR obtained UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 18 by relaxing the rank-one constraint is exact [52]. Thus, we can use the same steps as those in solv-ing ( T L - u m u ∗ m ) to find the desired column vector f k corresponding to a rank-one global optimumof ( WMSE - noPC - f k f ∗ k ) . We then use the vec − operator to get a globally optimal solution F k of ( WMSE - noPC - F k ) from the resulting f k . C. Algorithm for Sum MSE Minimization with Power Control ( WMSE - PC)
In this section, we discuss briefly how to solve ( WMSE - PC) . Using the same steps as in SectionIV-B, we can develop Algorithm 3 for finding high-quality solutions of ( WMSE - PC) . The details ofAlgorithm 3 are provided in [38], thus here we only compare and contrast the steps of Algorithm 3 andthose of Algorithm 2. First, the matrix weight and receive filter designs for ( WMSE - PC) are exactlythe same as those for ( WMSE - noPC) . Second, the relay processing matrix design for ( WMSE - PC) can be solved by the Lagrangian multiplier method with the only difference is that the multiplier mustbe nonnegative. Finally, the optimization problem for the transmit precoder design for ( WMSE - PC) isobtained by replacing the equality constraints in ( WMSE -noPC- F k ) by the corresponding inequalityconstraints. Fortunately, the resulting optimization problem is convex with respect to F k . In particular,it follows from Remark 5 that the objective function of the problem ( WMSE -PC- F k ) is convex withrespect to F k . In addition, since D ,k is a Hermitian and positive semidefinite matrix, then we can easilycheck that the constraints of the resulting problem are also convex with respect to F k . Thus, any availablesoftware package for convex optimization could be used to solve for its unique global optimum F k . D. Discussion
In this section, we discuss the proposed algorithms in the following aspects: i) the convergence, ii) thequality of the solution, and iii) the assumption on power constraints at the relays.In terms of convergence, we are able to find a global optimum of the single-variables minimizationproblem in each iteration. Thus, the cost function of the original multi-variable optimization problemis non increasing after each iteration [36], [37]. This guarantees that all the proposed algorithms areconvergent. Note that the authors of [49] adopt the alternating minimization to develop an iterativealgorithm for solving a weighted sum-MSE minimization problem for the single-hop MIMO interferencebroadcast channel. That optimization problem has a differentiable objective function and a set of separableconstraints in the main variables. Using the results from the general optimization [58], the authors of [49]
UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 19 are able to claim that their proposed alternating minimization algorithm converges to a stationary pointof the corresponding weighted sum-MSE minimization problem, which is also a stationary point of theassociated sum-rate maximization problem Nevertheless, the optimization problems in the paper, ( T L ) , ( WMSE - noPC) , and ( WMSE - PC) , have non-separable constraints due to the impact of transmitterprecoders on the transmit power constraints at the relays. Thus, we are currently unable to make anystrong claim about whether or not our proposed iterative algorithms always converge to a stationary pointof the corresponding optimization problems.All the proposed algorithms are not guaranteed to reach a global optimum of the corresponding multi-variable optimization problem. The quality of the resulting solution depends on the initialization. Oneway to improve the performance of the proposed algorithms is to use multiple initializations, selectingthe one with the best performance at the expenses of running time.The proposed algorithms in the current form are applicable only under the assumption of sum-powerconstraint at the relays. If individual power constraints at the relays are considered, we must formulatethe corresponding optimization problems. For example, as in Section III, to find high-quality solutionsof the sum-rate maximization problem with individual power inequality constraints at the relays, we canformulate the following weighted sum-MSE minimization problem ( WMSE - PC -ind ) : min { F } , { U } , { W } , { V } W M SE sum (cid:0) { F } , { U } , { W } , { V } (cid:1) s.t. p T ,k ≤ p maxT ,k , k = 1 , · · · , K, (58) p X ,m ≤ p maxX ,m , m = 1 , · · · , M. (59)In some cases, we can use the same steps as in the previous sections to develop new alternativeminimization based algorithms for solve the counterpart problems with individual power constraintsat the relays, like ( WMSE - PC -ind ) . Note that the main difference between the sum power constraintcase and the individual power constraint case is the extra constraints in the transmit precoder design.Specifically, with per-relay power constraints, the number of quadratic constraints of the resulting QCQPfor the transmit precoder design is ( M + 1) instead of 2 as with the sum-power constraint. The extraconstraints make it impossible to use the same SDP method to a globally optimal solution of the transmitprecoder design problems for individual relay power constraints when there are more than two relays,i.e., M ≥ . Developing new methods to solve the counterpart problems to ( T L ) or ( WMSE -noPC ) when M ≥ is left for future work. When power control is considered, however, we can still use the UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 20 same steps as in the previous sections to solve for high-quality solutions to ( WMSE - PC -ind ) . Indeed,with power control, the single-variable optimization problems for designing the relay processing matricesand transmit precoders in solving ( WMSE - PC -ind ) are convex [38]. Thus, we are always able to findtheir global optimum. V. S IMULATIONS
This section presents Monte Carlo simulation results to investigate the average end-to-end sum-rateperformance and to gain insights into the achieved multiplexing gains of the proposed algorithms. Weconsider only symmetric systems, which are denoted as ( N R × N T , d ) K + N M X , where N R ,k = N R , N T ,k = N T , d k = d and N X ,m = N X for k ∈ K , m ∈ M . The power values are normalized such that σ R ,k = σ X ,m = 1 , p maxT ,k = P , p maxX ,m = P , and p maxX = M P for k ∈ K , m ∈ M . The channel realizationsare flat in time and frequency. The channel coefficients are generated as i.i.d. zero-mean unit-variancecomplex Gaussian random variables. No path loss is assumed in the simulations, thus the average powerof all cross-links on the same hop is the same. The plots are produced by averaging over 1000 randomchannel realizations. For each channel realization, the initial transceivers are chosen randomly subject tothe power constraints at the transmitters and relays. The same initializations are used where applicable.Each iteration updates either one transmitter or one relay and then all the receive filters. The same orderof relays or transmitters selected for updating is used where applicable, for example, in the comparisonof the proposed algorithms. We use the CVX toolbox [53] to solve convex problems.For comparison, we consider the dedicated DF relay interference channel where one DF relay isdedicated to aid one and only one transmitter-receiver pair, i.e., K = M . Using equal time-sharing, theend-to-end achievable rate of a pair is defined as half of the minimum between the achievable rate fromthe transmitter to the associate DF relay and that from the relay to the receiver. We are interested onlyin the performance of DF relays when spatial interference alignment strategies, like those in [34], [49],are applied on two hops. Although other interference alignment techniques, like asymmetric complexsignaling [59], may improve the performance of DF relays, their impacts on DF relays are left for futurework. Based on [12], we derive an upper-bound on the achievable end-to-end multiplexing gain of thededicated DF relay interference channel ( N R × N T , d ) K + N K X as . ∗ min (cid:110)(cid:106) K ( N X + N T ) K +1 (cid:107) , (cid:106) K ( N R + N X ) K +1 (cid:107)(cid:111) .Individual power constraints at the relays are considered in the comparison of AF relays and DF relays.We also assume the transmit power at a transmitter or a relay in DF relay systems is equal to P . UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 21
1) Convergence:
Fig. 2 illustrates the convergence behavior of the proposed algorithms. Fig. 2(a)provides the analysis of the sum power of post-processed leakage signals of Algorithm 1 for a randomchannel realization of the (4 × , + 4 system. We observe that the sum power of leakage signalsdecreases monotonically over iterations. Interestingly, the interference and the enhanced relay noisechange their roles during the process of Algorithm 1. The interference is dominant at the beginning,however, it can be aligned and then cancelled quickly in a few iterations. After this point, the enhancedrelay noise becomes dominant - its sum power is thousands times larger than the interference sum power.Unfortunately, given that many spatial dimensions have been devoted to deal with interference, it becomeschallenging for Algorithm 1 to align and cancel the enhanced relay noise power. Intuitively, the enhancedrelay noise can be thought of as a source of single-hop interference from “virtual uncoordinated relays”that impacts directly the receivers. Thus, we need to take into account both the interference and enhancedrelay noise in the design of interference alignment strategies for the AF relay interference channel. Fig.2(b) provides the values of W M SE sum achieved by Algorithm 2 and by Algorithm 3 over iterations for achannel realization of the (2 × , + 2 system. We observe that W M SE sum values for both algorithmsare non increasing over iterations. Although the convergence speeds of the proposed algorithms are quitefast for these configurations, they might be slow for networks with large values of K or d . −4 −3 −2 −1 Iteration index P o w e r v a l ue s i n li nea r sc a l e Sum power of leakage (intf. & relay noise)Sum power of interferenceSum power of enhanced relay noise (a) Total leakage power at the receivers over iterations ofAlgorithm 1 for a channel realization of (4 × , + 4 . M a t r i x − w e i gh t ed s u m − M SE v a l ue s Algorithm 3Algorithm 2 (b) Matrix-weighted sum-MSE values over iterations ofAlgorithm 2 and Algorithm 3 for a channel realization of (2 × , + 2 .Fig. 2. Convergence behavior of the proposed algorithms.
2) Comparison of the Proposed Algorithms:
In this experiment, we simulate the average achievableend-to-end sum-rates for the (2 × , +2 system under the sum power constraint at the relays as shown UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 22 A v e r age a c h i e v ab l e end − t o − end s u m − r a t e s [ bp s / H z ] Algorithm 3Algorithm 2Algorithm 1
Fig. 3. Comparison of the average achievable end-to-end sum-rates of the proposed algorithms for the (2 × , + 2 system. in Fig. 3. We consider a sum relay power constraint. Thanks to power control, Algorithm 3 outperformsAlgorithm 2 in this experiment. Both Algorithm 2 and Algorithm 3 outperform Algorithm 1 at low-to-medium SNR values because Algorithm 1 does not take into account the desired signal and noise at thereceivers while the other do. Interestingly, at high SNR values, Algorithm 1 outperforms both Algorithm2 and Algorithm 3. Especially, Algorithm 1 can achieve a higher multiplexing gain than do the other.Zooming in on per-user achievable end-to-end rates, we find that for Algorithm 2 and Algorithm 3, someusers have much smaller rates than do the others; they even turn off some data streams. This unfairnesslimits the maximum end-to-end multiplexing gains achievable by the two algorithms. Thus, Algorithm 1is more suitable than the others for investigating the maximum achievable end-to-end multiplexing gainsof MIMO AF relay networks.
3) Sum Power Constraints vs. Individual Power Constraints at Relays:
In the previous experiments,we consider sum-power constraints at the relays. In this experiment, we consider the impacts of individualpower constraints. Note that any feasible point satisfies the individual power constraints at the relays alsosatisfies the corresponding sum-power constraint. Based on the discussion in Section IV-D, we focuson the case where power control is considered as it allows for the use of any number of relays. Fig. 4shows the achievable end-to-end sum-rates as functions of the transmit power at a base station or a relayfor both types of power constraints at the relays for the following three systems: i) (4 × , + 4 , ii) (2 × , + 2 , and iii) (1 × , + 2 . We observe that Algorithm 3 for the sum-power constraintcase slightly outperforms its counterpart algorithm for the individual power constraint case in terms of UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 23 maximizing average achievable end-to-end sum-rates. This gain is due to having more freedom in powerallocation in the sum-power constraint case as relays may transmit at a higher value than the maximumtransmit power at a relay in the individual power constraint case. This means that extra constraintsadded by the individual power constraints at the relays have little impact on the end-to-end sum-rateperformance of the proposed algorithms. In the following experiments, we use only the counterpartversion of Algorithm 3 that is designed for individual relay power constraints, which we refer to as‘modified Algorithm 3’. A v e r age a c h i e v ab l e end − t o − end s u m − r a t e s [ bp s / H z ] Individual power constraints at relaysSum power constraint at relays (4 ×
4, 2) + 4 (2 ×
2, 1) + 2 (1 ×
1, 1) + 2 Fig. 4. Achievable end-to-end sum-rates with both types of power constraints at the relays for the (1 × , +2 , (2 × , +2 ,and (4 × , + 4 systems.
4) Comparison with Existing Strategies:
In these experiments, we simulate several existing transceiverdesign strategies for the relay interference channel. For fair comparison, in this experiment and theremaining experiments, we consider the individual relay power constraints. Specifically, we simulate twostrategies for the AF relay case. One is the AF TDMA distributed beamforming (BF), where all therelays help only one transmitter-receiver pair at a time (which is an extension of the design in [42] formultiple-antenna receivers). Another is the dedicated relay BF where each AF relay is devoted to aidingone and only one transmitter-receiver pair. This means that interference is ignored and we apply thejoint source-relay design in [60], [61] independently for the two-hop channels from the transmitters totheir associated receivers. We also three strategies for the DF relay case that correspond to independentapplications of single-hop strategies on two hops. The single-hop strategies include the following: i)selfish (SF) beamforming (i.e., each transmitter aims at maximizing the achievable rate to its associatedreceiver), ii) interference alignment strategy based on total leakage (TL) minimization [34], and iii) the
UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 24 iteratively weighted MSE sum-rate (SR) maximization strategy [49]. T [dB] A c h i e v ab l e end − t o − end s u m − r a t e s [ bp s / H z ] AF, modified Algorithm3AF, TDMA distributed BFAF, dedicated relay BFDF, SR & SRDF, TL & TLDF, BF & BF
Fig. 5. Achievable end-to-end sum-rates for the (2 × , + 2 system. Fig. 5 shows the results for the (2 × , +2 system. Recall that Algorithm 3 outperforms all the otherin all regions. It achieves an end-to-end multiplexing gain of 2 (which is equal to half of the total numberof data streams). Note that we do not claim that this is the maximum degrees of freedom of this system.More complicated designs, for example those that can take advantage of symbol extensions [26], mayachieve higher end-to-end multiplexing gains. Unaware of interference, the dedicated relay strategies forboth AF relay and DF relay cases achieve zero multiplexing gains. While the multiplexing gain achievedby the DF TL & TL strategy is zero, that by the DF SR & SR strategy is nonzero. The reason isthat interference alignment is not feasible for the configuration on the two hops, interference cannotbe completely eliminated using the TL algorithm. Although the SR algorithm is able to turn off somedata streams, one data stream on each hop in this case, to make interference alignment feasible. Note,however, that it may turn off data streams of different pairs on two hops. Thus, on average the DF SR & SR strategy achieves an end-to-end multiplexing gain less than 1.5 (half of the number of remainingdata streams when interference alignment is feasible). Finally, thanks to orthogonalization transmission,the AF TDMA distributed BF can achieve an end-to-end multiplexing gain of 0.5.Another experiment focuses on comparing Algorithm 3 with the the minimum-SINR maximizationalgorithm in [28]. Note that the algorithm in [28] is applicable only for single-antenna receivers. Fig.6 shows the end-to-end achievable sum-rates of Algorithm 3 and the minimum-SINR maximizationalgorithm for two configurations (1 × , + 2 and (1 × , + 2 . We observe that the end-to-end UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 25 −25 −20 −15 −10 −5 0 5 10 15012345678 Transmit power at a base station or a relay [dB] A v e r age a c h i e v ab l e end − t o − end s u m − r a t e s [ bp s / H z ] (1 × + 2 (1 × + 2 modifiedAlgorithm 3Min-SINR max [12] Fig. 6. Comparison of achievable end-to-end sum-rates of Algorithm 3 (shown by the solid lines) and the minimum-SINRmaximization algorithm in [28] (shown by the dashed lines) for two configurations (1 × , + 2 and (1 × , + 2 . sum-rate performance of the minimum-SINR maximization algorithm increases at low transmit power (i.e.,in the noise-limited regime) and saturates at high transmit power (i.e., the interference-limited regime).Thus, the algorithm achieves a end-to-end multiplexing gain of zero. This is reasonable since it is notdesigned specifically for interference management. Thanks to its capability of interference management,Algorithm 3 still achieves non-zero end-to-end multiplexing gains and provides large end-to-end sum-rategains over the minimum-SINR maximization algorithm in the interference-limited regime. Our algorithm,however, performs worse than the the minimum-SINR maximization algorithm in the noise-limited regimewhere interference becomes a negligible issue.
5) Maximum Achievable Multiplexing Gains:
We fix N R = N T = 2 and d = 1 . Fig. 7 shows theachievable end-to-end multiplexing gains achieved by using Algorithm 1 as a function of K for N X = 3 and N X = 5 for AF relays, DF relays, and direct transmission. We notice that with these values of N X and when K is small, due to the half-duplex loss, both the AF relay and DF relay cases achievelower multiplexing gains than the direct transmission. While the DF relay case cannot outperform thedirect transmission, the AF relay case can achieve higher multiplexing gains when there are more than 6users. Thus, we can claim that AF relays help increase the achievable end-to-end multiplexing gains ofinterference channels. In addition, we observe that there exist upper-bounds on the achievable end-to-endmultiplexing gains for all the simulated cases - AF relays, DF relays, and direct transmission. Theoreticalinvestigation of the upper-bounds is left for future work. UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 26 A c h i e v ab l e end - t o - end m u l t i p l e x i ng ga i n AF relays, N X = 5DF relays, N X = 5AF relays, N X = 3DF relays, N X = 3Direct transmission Fig. 7. Achievable end-to-end multiplexing gains as functions of K for the (2 × , K + N K X systems.
10 20 30 4071013161921 Transmit power at a base station or a relay [dB] A v e r age a c h i e v ab l e end − t o − end s u m − r a t e s [ bp s / H z ] Increasing N N = 1 N = 2 N = 5N = 10N = 20
Fig. 8. Achievable end-to-end sum-rates of the opportunistic approach for the (2 × , +2 system with N = { , , , , } .
6) Opportunistic Approach:
The end-to-end sum-rate performance of the stationary points found byAlgorithm 2 and Algorithm 3 depend significantly on the initializations. The opportunistic approachproposes to use multiple initializations and then chooses the one with the highest end-to-end sum-rates.Let N denote the number of random initializations. Fig. 8 shows the average end-to-end sum-rates forseveral values of N achieved by Algorithm 3 in the (2 × , + 2 system. For this setting, at a transmitpower of 30dB, the gain provided by the opportunistic approach over the non-opportunistic approach is6.4 % for N = 2 , 13.2 % for N = 5 , 16.9 % for N = 10 , and 20.6 % for N = 20 . Note that the higher thevalue of N , the larger the average achievable end-to-end sum-rates. Also, the additional gains obtainedby using an extra random initialization decreases in N . Nevertheless, the benefits of this opportunisticapproach come at the expense of longer running time. UBMITTED TO IEEE TRANSACTIONS ON SIGNAL PROCESSING 27
VI. C
ONCLUSIONS AND F UTURE W ORK
We developed three cooperative algorithms for joint designs of the transmitters, relays, and receiversof the MIMO AF relay interference channel. Algorithm 1 aims at minimizing the sum power of theinterference signals and the enhanced noise from the relays. Based on a relationship between MSE andmutual information, Algorithm 2 (Algorithm 3) is able to find a stationary point of the end-to-end sum-ratemaximization problems with equality (inequality) power constraints. Simulations show that thanks to theconsideration of the desired signal power and the noise power at the receivers, Algorithm 2 and Algorithm3 outperform Algorithm 1 at low-to-medium SNR. Nevertheless, they perform worse than Algorithm 1at high SNR due to unfairness in rate allocation among users. The multiplexing gains achievable by theproposed algorithms provide lower bounds on the total number of degrees of freedom in MIMO AF relaynetworks, which remains unknown. Also, the use of AF relays results in higher end-to-end multiplexinggains than both the use of DF relays and the direct transmission.A major limitation of our algorithms is that global CSI is required to implement them in their presentform. Naturally this is challenging to achieve in a distributed system. We believe the results are stillvaluable, however, because they provide a benchmark for developing algorithms that relax the global CSIassumptions. Future work should focus on developing cooperative algorithms that require less overhead,have faster convergence speed, allow for lower implementation complexity, and account for channelestimation error R
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