How to Understand LMMSE Transceiver. Design for MIMO Systems From Quadratic Matrix Programming

Transcription

1 How to Understand LMMSE Transceiver 1 Design for MIMO Systems From Quadratic Matrix Programming arxiv: v4 [cs.it] 2 Mar 2013 Chengwen Xing, Shuo Li, Zesong Fei, and Jingming Kuang Abstract In this paper, a unified inear minimum mean-square-error (LMMSE) transceiver design framework is investigated, which is suitabe for a wide range of wireess systems. The unified design is based on an eegant and powerfu mathematica programming technoogy termed as quadratic matrix programming (QMP). Based on QMP it can be observed that for different wireess systems, there are certain common characteristics which can be expoited to design LMMSE transceivers e.g., the quadratic forms. It is aso discovered that evoving from a point-to-point MIMO system to various advanced wireess systems such as muti-ce coordinated systems, muti-user MIMO systems, MIMO cognitive radio systems, ampifyand-forward MIMO reaying systems and so on, the quadratic nature is aways kept and the LMMSE transceiver designs can aways be carried out via iterativey soving a number of QMP probems. A comprehensive framework on how to sove QMP probems is aso given. The work presented in this paper is ikey to be the first shot for the transceiver design for the future ever-changing wireess systems. Index Terms Quadratic matrix programming, transceiver designs, MSE, muti-ce, cooperative communications, cognitive radio. Chengwen Xing, Shuo Li, Zesong Fei and Jingming Kuang are with the Schoo of Information and Eectronics, Beijing Institute of Technoogy, Beijing, , China, Phone : (86) , (E-mai: chengwenxing@ieee.org, {sureee,feizesong, jmkuang}@bit.edu.cn). The materia in this paper was partiay presented at the Internationa Conference on Wireess Communications and Signa Processing (WCSP), Huangshan, China, Sep The corresponding author is Chengwen Xing.

2 2 I. INTRODUCTION In order to satisfy the ever-increasing wireess data rate requirements and to enabe high quaity and highy diversified wireess services, wireess research never stops to search new discoveries and deveopment in ideas, technoogies, systems and everything avaiabe. More and more avaiabe wireess resources are introduced into wireess systems. The scope of wireess designs has been extended to be muti-dimensiona such as tempora, frequency, spatia even coding. As a gift the muti-dimensiona wireess resources bring new chaenges into wireess system designs. To order to reaize the promised performance gains coming from these resources, some corresponding new technoogies need to be adopted, such as mutipe-carrier technoogy, mutipe-antenna technoogy and so on. Referring to the spatia resource, mutipe-input mutipe-output (MIMO) technoogy is a great success in both theoretica research and industria productions [1]. Aong with the evovement of wireess systems, MIMO becomes to be a fundamenta and important ingredient of compicated wireess systems e.g., cooperative communications, cognitive communications, physica ayer security communications, network coding based communications and so on. Athough MIMO technoogy has promised great potentias in diversity and mutipexing gains, a compicated transmit/receive beamforming or transceiver design is usuay needed [2]. Different from the simpe singe antenna case, for MIMO transmissions the resources shoud be carefuy aocated across spatia domain according to avaiabe channe state information (CSI) at transmitter or receiver or both [3]. For MIMO transceiver designs, there are various performance metrics such as capacity, bit error rate (BER), mean-square-error (MSE) and so on. Different performances represent the different preferences of the wireess designers. Meanwhie, because of a variety of wireess service requirements and wireess environments, different wireess systems have totay different network architectures and wireess interfaces. In the resuting transceiver designs, a of these facts are refected on the constraints and the variabes invoved in the considered optimization probems. In other words, for different wireess systems the transceiver design probems have different signa modes, different power constraints, different numbers of variabes and even different performance criteria. As a resut transceiver designs must be investigated case by case. From the theoretica research perspective, the theorists and researchers woud ike to find a unified design

3 3 which can revea some common nature of the transceiver designs. To the best our knowedge, up to date the transceiver designs have not been unified for both different performance metrics and different systems. Athough the transceiver designs with different performance metrics for different systems are totay different, the work on unifying transceiver designs never stops. In the existing work for unified inear MIMO transceiver designs, the widey used ogic is for a given wireess system inear transceiver designs with different performance metrics are unified into one kind of optimization probem [3], [4]. It is we-known that there are two guideines for inear MIMO transceiver designs, i.e., using majorization theory [3] and weighting operation [4]. For the majorization theory based guideine, the transceiver design ogic is to formuate different performance metrics as different functions of the diagona eements of the data detection MSE matrix at the destination. Then the objective functions are cassified into Schur-convex or Schurconcave functions. Reying on the fundamenta properties of Schur-convex/concave functions, the optima soutions can be derived. On the other hand, using weighting operations, the different performance metrics are optimized by soving a weighted MSE minimization probem with different weighting matrices. In this paper, in contrast to the existing work we give a unified transceiver design which aims at unifying the inear transceiver designs for different wireess systems with the same performance metric named as minimum mean-square-error (MMSE). It can be reveaed that for the beamforming designs in different wireess systems such as muti-ce coordinated beamforming design, muti-user MIMO beamforming design, cognitive MIMO beamforming design, ampifyand-forward MIMO reaying beamforming design and their corresponding robust transceiver designs with randomy distributed channe estimation errors and so on, the transceiver design probems can aways be soved by iterativey soving a series of matrix quadratic programming (QMP) probems that can be efficienty soved. It is true that our work focuses on iterative inear minimum mean-square-error (LMMSE) transceiver designs which may not be the optima strategy. This kind of transceiver design suffers from some we-know weaknesses coming from the MMSE objective or iterative design procedure itsef or both. We want to highight that iterative LMMSE designs sti have severa attractive properties to make them much powerfu in engineering appications, as they can be appied to a wide range of fieds. Furthermore, they can give a soution with satisfactory performance and they can aso act as a benchmark for other kinds of suboptima schemes.

4 4 Wireess systems change very fast e.g., from a point-to-point system to cognitive radio networks or cooperative networks. Athough we can describe some about what the future wireess systems is ike to be, unfortunatey we never know what they are exacty. However, the authors beieve that there is definitey something that wi not change. Quadratic forms which widey exist are most ikey to be kept in transceiver designs because most energy reated probems wi have quadratic forms. Inspired by this fact, the framework proposed in this paper may work as the first shot which we can do for the coming wireess systems. We aso want to highight that athough ony transceiver design is investigated in our work, there exist severa cosey reated research topics such as training design in channe estimation procedure [6] or signa reduction in sensor networks [7]. Taking signa reduction [7] as an exampe, it is exacty the forwarding matrix design for ampify-and-forward (AF) MIMO reaying systems [8]. In addition it is weknown that training design and transceiver design have the same nature. Then it is not surprising that the soution proposed in this paper can aso be appied to such kind of cosey reated topics. By the way, the main difference between this paper and its conference version [9] is that the detaied expanations, justifications and discussions are given at various points of the paper. In addition, the important numerica simuations are given in this journa version. This paper is organized as foows. In Section II, an interesting understanding of the transceiver designs from optimization theory is presented and it shows the evovement of transceiver designs is just the same as the procedure to make the optimization probems more compicated. In Section III, a concrete exampe of inear transceiver design is first given which shows the motivation for iterative agorithms. Meanwhie, the quadratic nature of LMMSE transceiver design is aso reveaed. How to expoit the quadratic nature is investigated in Section IV and the framework on QMP is discussed as we. In addition, the appications are specified. After that, an extension on robust designs is considered in Section V. The numerica resuts is finay presented in Section VI. Notations: The foowing notations are used throughout this paper. Bodface owercase etters denote vectors, whie bodface uppercase etters denote matrices. The notations Z T, Z and Z H denote the transpose, conjugate and conjugate transpose of the matrix Z, respectivey and Tr(Z) is the trace of the matrix Z. The symbo I M denotes an M M identity matrix, whie 0 M,N denotes an M N a zero matrix. The notation Z 1/2 is the Hermitian square root of the positive semi-definite matrix Z, such that Z 1/2 Z 1/2 = Z and Z 1/2 is aso a Hermitian matrix. The

5 5 symbo E denotes statistica expectation operation. The operation vec(z) stacks the coumns of the matrix Z into a singe vector. The symbo represents Kronecker product. II. MOTIVATIONS At the beginning, we woud ike to discuss why our attention is concentrated on inear minimum mean-square-error (LMMSE) transceiver designs in this paper. However for ceratin performance metrics such as bit error rate (BER) the performance of inear transceivers may be not as good as that of the noninear counterparts, inear transceivers are sti preferred by practica wireess systems due to their ow compexity. On the other hand, mean-square-error (MSE) is a widey used performance metric for estimation, detection and optimization agorithm designs in wireess systems. It shoud be pointed out that MSE acting as performance metric suffers from severa inherent drawbacks as it is not the utimate performance metric e.g., capacity and BER. Roughy speaking, MSE can be seen as an approximation of the utimate performance metrics, athough they have very cose reationships and particuary in some specia cases, they are even equivaent with each other. The tractabiity is the main advantage of MSE. For severa utimate performance metrics, their formuations may be too compicated to optimize. For engineers, the case where there is a soution is much better than that there is no soution. From the perspective of optimization theory, the LMMSE transceiver designs are in nature some specific optimization probems under different constraints. In genera, there are two kinds of variabes invoved in the optimization probems, i.e., precoder/beamforming matrices and equaizer matrices. The main difference between them is that the equaizers are usuay unconstrained. Whie for precoder/beamforming, the story is different as there are aways various kinds of constraints on the transmitters. The simpest MIMO communication system is the singe user point-to-point MIMO system with ony one power constraint. The signa mode is y = HFs+n where y is the received signa at the receiver and H is the channe matrix between the transmitter and receiver. The symbo F denotes the precoder matrix at the source, s is the transmitted signa and n is the additive noise at the receiver. The corresponding LMMSE transceiver design probem is formuated as min f(g,f) = E{ Gy s 2 } s.t. Tr(FF H ) P (1)

6 6 where G is the equaizer matrix at the receiver and P represents the maximum transmit power at the transmitter. In the foowing, we try to understand transceiver designs for various wireess systems evoving from the previous one for the point-to-point MIMO systems. There are ony two possibe directions to make the transceiver design probem (1) more compicated, i.e., enarging the set of variabes or enarging the sect of constraints. When there are more than one constraint, these constraints can be homogeneous or not (have the same physica meaning or not). As previousy discussed, the constraints are aways reated to transmitters. If the constraints are homogeneous, it means that there may exist many transmitters in the considered wireess system, such as muti-user MIMO (MU-MIMO) upink. In this case the constraints are described as the second order term of the matrices variabes is smaer than a threshod. Of course, the invoved constraints can be inhomogeneous. For exampe, in the cognitive radio, there are usuay two kinds of constraints. One is the power constraints and the other is interference constraints. In the atter one, the second order term of the matrices variabes is aso smaer than a threshod. Different from power constraints it describes that the caused interference in a certain direction must be ower than a threshod. In the foowing, we refer to the previous kind of constraints with second term smaer than a threshod as positive constraints. An interesting question is what about the constraint for which the quadratic term is arger than a threshod. It means in a certain direction the energy shoud be arger than a threshod. In a ong time, there is no such kind of wireess systems. Recenty, energy harvesting communications give a very important appication of this case [10]. In an energy harvesting communication, except a traditiona receiver, there aso exists an energy harvesting receiver which aims at harvesting the energy emitted by the transmitter to charge its own battery. As a resut, the transmitter shoud guarantee the energy harvested by the energy harvesting receiver is arger than a threshod. Simiar to the case of cognitive radio, in the foowing we refer to this cass of constraints with second term arger than a threshod as negative constraints. In concusion, different mathematica formuations of the constraints represent different communication system setups. In the foowing, we ist severa concrete and representative exampes to iustrate the reationships between various advanced wireess systems and the simpest pointto-point MIMO system. In particuar, we want to show how to change of the optimization probem (1) to become the corresponding optimization probem for an advanced wireess system. Case 1: When ony the number of unconstrained variabes increases, it corresponds to MU-

7 7 MIMO downink transceiver designs [14], [15]. In the foowing, f( ) represents the sum MSE function whose specific formuation is determined by the corresponding system mode. The inear transceiver design for MU-MIMO downink is given as foows min f([g 1,,G K ],F) s.t. Tr(FF H ) P (2) where F is the beamforming matrix at the base station, and G k is the equaizer matrix at the k th mobie user. Additionay, P denotes the maximum transmit power at the base station. Case 2: When both the numbers of constrained variabes and their corresponding constraints increase and the constraints are independent with each other, the resut corresponds to MU- MIMO upink transceiver designs [5]. In this case, the optimization probem is formuated as min f(g,[f 1,,F K ]) s.t. Tr(F k F H k ) P k (3) where F k denotes the precoding matrix at the k th mobie user and the equaizer at the base station is denoted as G. In addition, P k denotes the maximum transmit power at the k th mobie use. Case 3: When both the numbers of constrained variabes and unconstrained variabes increase and the constraints are independent as we, this case corresponds to muti-ce transceiver designs [26]. The beamforming design probem for muti-ce cooperation reads as min f([g 1,,G K ],[F 1,,F K ]) s.t. Tr(F k F H k ) P k, (4) where G k is the equaizer at the k th base station and F k is the precoder matrix at the k th mobie termina. Moreover, P k is the maximum transmit power at the k th mobie termina. Case 4: Ony increase the number of constraints and keep the set of variabes unchanged. If the constraints are positive constraint, this case corresponds to cognitive radio (CR) transceiver designs. For CR, the transceiver design probem is formuated as min s.t. f(g,f) Tr(FF H ) P Tr(H S FF H H H S ) γ. (5)

8 8 where H S is the channe matrix between the secondary user node and the primary user node and γ denotes the aowabe interference threshod. Case 5: In contrast to Case 4, when ony increasing negative constraints, it corresponds to energy harvesting oriented transceiver designs. The energy harvesting beamforming design is given as min s.t. f(g,f) Tr(FF H ) P Tr(H P FF H H H P) γ, (6) where H P denotes the channe between the source node and the energy harvesting node. The physica meaning of the second constraint is in the information transmission, the source node wants to charge the energy harvesting node as we. It shoud be pointed out that the main difference between Cases 4 and 5 is that the increased constraint is negative or positive. Case 6: When both the number of the constrained variabes and the number of corresponding number of constraints increase and meanwhie the constraints are couped, this case corresponds to the ampify-and-forward (AF) MIMO reaying transceiver designs [17]. The transceiver design for two-hop AF MIMO reaying systems can be formuated as min f(g,[f 1,F 2 ]) s.t. Tr(F 1 F H 1 ) P 1 Tr(F 2 (H 1 F 1 F H 1H H 1 +σn 2 1 I)F H 2) P 2 (7) where F 1 is the source precoder at the source node and F 2 is the forwarding matrix at the reay. Furthermore, H 1 is the channe matrix between the source node and the reay node. In addition P 1 and P 2 are the maximum transmit power at the source and reay, separatey. Notice that the matrix σn 2 1 I is the noise covariance matrix at the reay and H 1 F 1 F H 1 HH 1 +σ2 n 1 I is the received signa correation matrix at the reay. It is obvious that the two constraints are couped with each other. Inspired by the formuation, for a more genera muti-hop mode, the transceiver design

9 9 probem becomes min f(g,[f 1,,F K ]) s.t. Tr(F 1 F H 1) P 1 Tr(F k R k 1 F H k) P k 2 k K R k 1 = H k 1 F k 1 R k 2 F H k 1 HH k 1 +σ2 n k 1 I 2 k K R 0 = I (8) where F k is the forwarding matrix at the k th node, P k is the corresponding maximum transmit power and H k is the k th hop channe matrix. The matrix σn 2 k 1 I is the covariance matrix of the additive noise at the (k 1) th reay and R k 1 is the received signa correation matrix at the (k 1) th reay. From Case 1 to Case 6, it can be concuded that the evovement of wireess communication systems is exacty the evovement of optimization probem becoming compicated. Of course, the story can continue and we wi have Case 7, Case 8 and so on. For engineers, physica meaning is more important than mathematics itsef. However, as engineering probems must be perceptibe in mathematics here based on these exampes we can say that physica meanings cannot be independent of mathematics which can hep us to predict what the future communication systems woud ike to be. In the foowing, we wi show in detai that for the above optimization probems when iterative agorithms are used, the considered optimization probem admits quadratic nature. As a resut, the quadratic matrix programming technoogy can be used. III. QUADRATIC NATURE OF THE TRANSCEIVER DESIGNS In this section, the quadratic nature of the aforementioned optimization probems is investigated. It is totay redundant to discuss it case by case. For simpicity we take a representative exampe to iustrate that quadratic matrix programming (QMP) probems are of great importance in LMMSE transceiver designs. Note that this exampe has been discussed in detai in our previous work [16]. Here, it ony provides a proogue of our work in this paper. First, we want to highight that the agorithm discussed in the foowing is not imited to this exampe, which has a much wider appication range. We aim at providing a comprehensive framework on LMMSE

10 10 transceiver design. Our discussions are not imited to any specific communication system. We try to revea the nature of LMMSE transceiver designs and answer the questions why QMP shoud aways be chosen and how to sove the transceiver design optimization probems using QMP. A. An exampe: The considered exampe is a mixture of Case 3 and Case 6. Here, a dua-hop AF reaying network is investigated. As shown in Fig. 1, there are mutipe source nodes, reay nodes and destination nodes. Furthermore, different sources can have different numbers of transmit antennas and data streams to transmit. It is denoted that the number of transmit antennas of the i th source is N S,i. It is aso assumed that for each source node there may be more than one corresponding destination node. There are aso mutipe reay nodes in the network, and the j th reay has M R,j receive antennas and N R,j transmit antennas. At the first hop, the source nodes transmit data to the reay nodes. The received signa x j at the j th reay node is x j = H sr,ij k (P iks ik )+ i [H sr,j k (P ks k )] +n 1,j. (9) where s ik is the data vector transmitted by the i th source node to the k th destination with the covariance matrix R sik = E{s ik s H ik }. When the ith source node does not want to transmit signa to the k th destination, s ik is a a-zero vector. At the source, before transmission the signa is mutipied a precoder P ik under a transmit power constraint k Tr(P ikr sik P H ik ) P s,i, where P s,i is the maximum transmit power at the i th source node. The matrix H sr,ij is the MIMO channe matrix between the i th source node and the j th reay node. Symbo n 1,j is the additive Gaussian noise with the covariance matrix R n1,j. At the j th reay node, the received signa x j is mutipied by a precoder matrix F j, under a power constraint Tr(F j R xj F H j ) P r,j where R xj = E{x j x H j } and P r,j is the maximum transmit power. Then the resuting signa is transmitted to the destination. The received signa at the k th

11 11 destination y k can be written as y k = j (H rd,jkf j x j )+n 2,i = [H rd,jkf j j (H sr,jp k s k )] + [H rd,jif j j (H sr,j (P ms m ))] m k + j (H rd,jkf j n 1,j )+n 2,k. (10) where H rd,jk is the MIMO channe matrix between the j th reay and the k th destination, and n 2,k is the additive Gaussian noise vector at the k th destination with covariance matrix R n2,k. The optimization probem of inear minimum mean-square-error (LMMSE) transceiver design can be formuated as [16] min k MSE k = E{ G k y k [s T 1k,,sT N sk ]T 2 } s.t. Tr(F j R xj F H j ) P r,j j E r k Tr(P ikr sik P H ik ) P s,i i E s (11) where [s T 1k,,sT N sk ]T is the desired signa to be recovered at the k th destination. Additionay E r and E s denote the set of reay nodes and the set of source nodes, respectivey. The optimization probem (11) is a very genera probem which incudes the foowing scenarios as its specia cases. Muti-user MIMO upink transceiver design [17]: Mutipe muti-antenna mobie users communicate with a muti-antenna base station. Muti-user MIMO downink transceiver design [17]: A muti-antenna base station communicates with mutipe muti-antenna mobie users. Muti-ce coordinated beamforming design: Mutipe muti-antenna base stations communicate cooperativey with mutipe muti-antenna mobie users. Two-way AF MIMO reaying LMMSE transceiver design [18]: Two-way AF MIMO reaying can be taken as a soft combination of upink and downink beamforming designs. Athough, the optimization probem (11) ony considers one-way reaying systems. The extension from one-way to two-way is straightforward when an iterative optimization framework is used.

12 12 B. Iterative Agorithms As in the optimization (11) there are too many variabes to be optimized and meanwhie the nonconvex nature of the optimization probem (11) makes it very compicated, generay it is difficut to find the cosed-form gobay optima soutions. In order to design the transceivers, severa suboptima soutions are usuay proposed. Iterative agorithm is one of the most widey used and important suboptima soutions. In an iterative agorithm, the variabes are optimized sequentiay. It can be interpreted that iterative agorithms use iterative procedure to soften the hardness of the origina optimization probems as in the iterative procedure the couping reationships among the invoved variabes can be removed first. We admit that iterative agorithms suffer from some we-known weaknesses. First, the fina soution is greaty affected by the initia vaue seection. Second, the convergence of an iterative agorithm must be guaranteed. If not, the agorithm may be meaningess. Third, in genera even with proved convergence there is no guarantee that the fina soution is gobay optima. However, iterative agorithms sti have two important characteristics making them preferabe. First, it can be appied to a much wide area of transceiver designs ranging from a point-topoint system to a distributed network. Second, it can act as a performance benchmark for other suboptima soutions. Actuay iterative agorithms are widey adopted in transceiver designs or beamforming designs for MIMO systems no matter you ove it or hate it [13]. When iterative agorithms are adopted to sove the optimization probem (11), in each iteration one variabe is optimized and the others are fixed, and then the probem admits quadratic nature. C. Quadratic nature of the LMMSE transceiver designs Data detection MSE is an integration over the signas and noises. From its name, it is obvious that MSE is a certain quadratic formuation with respect to each invoved variabe. Moreover, in this paper, we concentrate our attention to the case where the variabes are matrices, as in MIMO systems the variabes to be optimized are usuay compex matrices. Inspired by these facts, to characterize a quadratic function with a compex matrix variabe a kind of functions termed as quadratic matrix (QM) functions X is first defined as f (X) = Tr(D X H A X)+2R{Tr(B H X)}+c (12)

13 13 where A = A H C n n, B C n r, c R, D = D H C r r. In addition, R{ } denotes the rea part. It can be seen that a QM function consists of three terms which are second-order term, first-order term and zero-order term. If the foowing conditions are satisfied, not matter what the system is, the MSE with inear transceivers is a QM function with respect to each variabe, separatey. (1). The considered system is a inear system. Linearity is defined based on the foowing two properties: (a.1) The received signa at the destination is a inear function of the transmit signa when a design variabes are fixed. (a.2) The received signa at the destination is a inear function with respect to each variabe when the signa and the other design variabes are a fixed. (2). The desired signas are independent of the noises. It means that when the transmit signa vector is denoted by s and the equivaent noise vector is v, the foowing equaity must hod E{sv H } = 0. (13) Moreover, the constraints in the transceiver designs for wireess systems are usuay QM functions as we. This is because the invoved constraints are usuay reated with energy, which definitey have quadratic terms e.g., transmit power, interference to primary users, and so on. Therefore, it is of great importance to investigate the optimization probems consisting of QM functions in both objective function and constraint functions. This kind of optimization probems is named as quadratic matrix programming (QMP) probems. It can be observed that in each iteration, the optimization probem (11) becomes a QMP probem. Athough in [19], a definition of quadratic matrix programming is given, in this paper we first revise the definition given in [19] in order to accommodate more cases e.g., the probem (11). As a resut, our definition is more genera and has a wider range of appications. A standard QMP probem is defined as Type 1 QMP: min X Tr(D 0 X H A 0 X)+2R{Tr(B H 0X)}+c 0 s.t. Tr(D i X H A i X)+2R{Tr(B H i X)}+c i 0,i I Tr(D j X H A j X)+2R{Tr(B H j X)}+c j = 0,j E X C n r (14)

14 14 where A = A H C n n, B C n r, c R, D = D H C r r, {0} I E. These assumptions are essentia to guarantee that the objective function and constraint functions are rea-vaued functions, as it is meaningess to minimize a compex-vaued function. The main difference between our definition and that given in [19] is that in [19] D = I whie in our definition they can be arbitrary Hermitian matrices. In the foowing section, the important characteristics of QMP probems wi be discussed, based on which a comprehensive framework on how to sove it is aso given. In the seque the Type 1 QMP probems are abbreviated to be T-1-QMP probems. IV. FUNDAMENTALS OF QMP In this section, the fundamenta properties of QMP are investigated. It is obvious that quadratic matrix programming (QMP) is a specia case of quadraticay constrained quadratic programming (QCQP) which is a very famous and widey used [23]. Obviousy the QMP probems have much better properties (e.g., Kronecker structure) than traditiona QCQP probems, which can be further expoited to sove the considered optimization probems more efficienty. This is exacty the motivation of the research on QMP [19], [20]. We embark on our investigation from the T-1-QMP probems in (14), which are the most genera probems. Genera QMP: Based on the properties of Kronecker product and the foowing definitions Ω DT A vec(b ), {0} I E (15) vec H (B ) c the optimization probem (14) is equivaent to min Tr(Ω 0 Z) s.t. Tr(Ω i Z) 0, Tr(Ω j Z) = 0 Z = [vec T (X) 1] T [vec H (X) 1]. (16) If the constraintrank(z) = 1 is reaxed (it is a we-known semi-definite reaxation (SDR) [20]), we have the foowing semi-definite programming (SDP) probem [22], which can be efficienty

15 15 soved by interior point poynomia agorithms where Z is a Hermitian matrix. min Z Tr(Ω 0 Z) s.t. Tr(Ω i Z) 0, Tr(Ω j Z) = 0 [Z] NNs+1,NN s+1 = 1, Z 0, (17) Appications: Generay speaking, for iterative LMMSE transceiver designs for the previousy considered systems in each iteration the variabes can aways be soved using the soution for the genera QMP probem, e.g, muti-ce transceiver designs, CR transceiver designs, energy harvesting transceiver designs, AF MIMO reaying transceiver designs and so on. Convex QMP: When A and D are both positive semi-definite matrices and the invoved constraints are ony inequaity constraints, the QMP probem (14) is convex [21]. Convexity may be the most favorabe property for an optimization probem and convex optimization probems can usuay be efficienty soved. In the seque, it is reveaed that for convex QMP probems, it does not need the previous SDR to compute the optima soutions. In the foowing, two approaches to soving convex QMP probems are proposed. SDP Based Agorithm: Using the properties of Kronecker product Tr(AB) = vec H (A H )vec(b), the QM function can be reformuated as Tr(D H 2 X H A XD 1 2 )+2R{Tr(B H X)}+c = Tr(D H 2 X H A H 2 A 1 2 XD 1 2 )+2R{Tr(B H X)}+c = vec H (X)(D 2 A H 2 )(D T 2 A 1 2 )vec(x) +2R{vec H (B )vec(x)}+c 0, (18) based on which and together with Schur compement emma, the optimization probem (14) can

16 16 be reformuated as min s.t. t I (D T 2 0 A )vec(x) 0 ((D T 2 0 A )vec(x)) H 2R(vec H (B 0 )vec(x))+t I (D T 2 i A 1 2 i )vec(x) 0. (19) ((D T 2 i A 1 2 i )vec(x)) H 2R(vec H (B i )vec(x)) c i Notice that in our work, the variabes are compex matrices. For some optimization tooboxes, maybe ony rea variabes are permitted. In that case, ony a minor transformation is needed, which is where ṽ is defined as I N v v H a 0 I 2N ṽ ṽ T a 0 (20) ṽ = [Rea(v) T Imag(v) T ] T. (21) Furthermore, if A i and D i are both positive definite matrices (stronger than positive semidefinite matrices), the optimization probem can be further transformed into a more efficienty sovabe convex optimization probem e.g., second order conic programming (SOCP) probems. SOCP Based Agorithm: Notice that when A i and D i are both positive definite, the QM functions in both the objective function and constraints can be reformuated as Tr(D H/2 X H A XD 1/2 )+2R{Tr(B H X)}+c [ ] = A 1 2 XD 1 2 +A 1 2 B D c i F Tr(A 1 B D 1 B H ) (22) where F denotes Frobenious norm. Therefore, the optimization probem (14) can be reformuated as a standard SOCP probem which reads as min P k,t s.t. t [ [ A XD A B 0 D A 1 2 i XD 1 2 i +A 1 2 i B i D 1 2 i ] F t ] F Tr(A 1 i B i D 1 i B H i ) c i. (23)

17 17 Appications: Convex-QMP is suitabe for muti-ce transceiver designs and AF MIMO reaying transceiver designs. Based on the previous discussions it can be concuded that the better structure the stronger soution. In the remaining part of this section, we wi take a further step to concentrate our attention to the QMP probems which have the foowing specia structure Type 2 QMP: min X Tr(X H A 0 X)+2R{Tr(B H 0 X)}+c 0 s.t. Tr(X H A i X)+2R{Tr(B H i X)}+c i 0,i I Tr(X H A j X)+2R{Tr(B H j X)}+c j = 0,j E X C n r. (24) The Type 2 QMP probems are aso usuay encountered in the LMMSE transceiver designs for wireess communications [27]. It is worth investigating its properties detaiedy. For the notationa simpicity, in the foowing the T-2-QMP probems are referred to as the Type 2 QMP probems. A. Properties of T-2-QMP 1) T-2-QMP without Constraints: At the first gance, we discuss the case without constraint which reads as min X Tr(X H A 0 X)+2R{Tr(B H 0X)}+c 0 (25) where A 0 > 0. This case corresponds to inear minimum mean square error (LMMSE) equaizer design, which is aso named as LMMSE estimator design. In this case, as previousy discussed, the optimization probem is convex and the optima soution is exacty the soution making the differentiation of the objective equation equa 0 i.e., A 0 X = B 0. Specificay, the optima soution has the foowing cosed-form soution X opt = A 1 0 B 0. (26) This soution is a very strong soution, which is aso the optima soution of weighted MSE minimization probem independent of weighting matrices.

18 18 Weighted MSE is a direct generaization of sum MSE. Considering weighted MSE minimization, the optimization probem becomes to be min X Tr(W w X H A 0 X)+2R{Tr(W H wb H 0X)}+c 0 (27) where W w 0 is the weighting matrix. Foowing the same ogic as previousy discussed, the optima soutions must satisfy A 0 XW w = B 0 W w. (28) Actuay, this condition is a sufficient condition for the optima soution as the optimization probem is convex. Because W w can be i-rank, the optima soution is not unique. Notice that the foowing soution satisfying the previous condition (28) X opt = A 1 0 B 0. (29) This concusion is important as it shows that X opt is a dominating estimator. It is why even for capacity achieving transceiver designs, LMMSE equaizer is optima. Concusion 1: Without constraints, the optima soution X opt of the T-2-QMP probems has a cosed form. Notice that X H opt is just the Wiener fiter. It is we-known for a inear system with Gaussian noise, LMMSE equaizer is exacty the optima equaizer in the sense of both inear equaizers and noninear equaizers [25]. To the best of our knowedge, this soution can be appied to a inear equaizer designs in wireess systems. 2) T-2-QMP with One Constraint: After discussing the case without constraints, we take a step further to focus on the case where there is ony one constraint for the considered QMP probem. This case corresponds to the scenario when there is ony one transmit power constraint. Here we focus on the foowing T-2-QMP probem min X Tr(X H A 0 X)+2R{Tr(B H 0X)}+c 0 s.t. Tr(X H A 1 X) P, (30) where A > 0. For the probem we considered, the feasibe set is not empty. In this scenario, soving the matrix variabe can be reduced to sove an unknown scaar variabe. The computationa dimensionaity and compexity are both significanty reduced. In this foowing, we wi discuss this in detai.

19 19 For constrained optimization probems, if certain reguarity conditions are satisfied, Karush- Kuhn-Tucker (KKT) are the necessary conditions for the optima soutions and then KKT conditions can provide very important information to hep us find the optima soutions. When there is one constraint, inear independence of constraint quaification (LICQ) can be easiy proved, which is a famous reguarity condition [11], [12]. In this case, the condition for LICQ to hod is that the optima soution X is not a zero matrix. In practica wireess systems, this is aways true as when transmitter matrix is a zero, there is no information to be transmitted and of course it is not the optima soution. Therefore, KKT conditions are the necessary conditions for the optima soutions [11], [12]. The corresponding Lagrange function of the optimization probem (30) is expressed as L(X) = Tr(X H A 0 X)+2R{Tr(B H 0X)}+c 0 +µ(tr(x H A 1 X) P), (31) where µ 0 is the Lagrange mutipier. Based on (31), the KKT conditions of the optimization probem (30) can be directy derived to be [21] (A 0 +µa 1 )X = B 0 (32) µ(tr(x H A 1 X) P) = 0 (33) Tr(X H A 1 X) P (34) µ 0. (35) In this case with a singe constraint, the optima soution has the foowing semi-cosed-form soution X = (A 0 +µa 1 ) 1 B 0 (36) in which the ony unknown variabe is a scaar Lagrange mutipier. Substituting (36) into the constraint of (30), we have Tr(X H A 1 X) =Tr(B H 0(A 0 +µa 1 ) 1 A 1 (A 0 +µa 1 ) 1 B 0 ) =Tr(B H 0 A (A A 0 A µi) 2 A B 0 ) g(µ). (37)

20 20 It has been proved that g(µ) is a decreasing function with respect to µ [26], and the vaue of µ satisfying the KKT conditions can be found by using a simpe one-dimensiona search such as bisection search. Based on this concusion and the KKT conditions given previousy, the vaue of µ can be computed to be 0 if g(0) P µ =. (38) Sove g(µ) = P Otherwise It can be seen that the soution satisfying KKT conditions is unique. As the KKT conditions are the necessary conditions for the optima soutions. As a resut, the unique soution satisfying the KKT conditions is exacty the optima soution. This is of great importance. In this case, the unknown variabe is simpified from a matrix to a scaar. In other words, the number of variabes is significanty reduced and the corresponding computationa compexity wi be significanty reduced. Concusion 2: With ony one constraint, the T-2-QMP probem has a semi-cosed-form soution with an unknown scaar variabe. This soution is appicabe to downink MU-MIMO beamforming design at the base station and ampifying matrix design for the dua-hop AF MIMO reaying transceiver designs (incuding both one-way and two-way). Remark: We cannot argue that KKT conditions are necessary conditions for the optima soutions without any prior conditions. In Boyd s cassica textbook [21], it never states that KKT conditions are necessary optimaity conditions for any optimization probems. There are severa cases in which KKT conditions are not necessary optimaity conditions [11]. 3) T-2-QMP with more than one constraint: For T-2-QMP probems with more than one constraint, soving the optimization probems must aso rey on interior point agorithms. As a T- 2-QMP probem has much better structures comparing to a genera QMP probem discussed in the previous section, it exhibits more stronger convexity property which can be expoited to sove the optimization probem. As discussed in [20], the origina optimization probem is first transformed into its homogenized probem which can be efficienty soved. First, the homogenized QM function of the QM function defined previousy is denoted by f H i f H (Y;Z) =Tr(Y H A Y)+2R{Tr(Z H B H Y)} + c r Tr(ZH Z). (39)

21 21 Then introducing the foowing operators, M (f ) = A B H B c r I r the homogenized optimization probem of (24) is formuated as min Tr(M(f 0 )[Y;Z][Y;Z] H ) s.t. Tr(M(f i )[Y;Z][Y;Z] H ) α i,i I Tr(M(f j )[Y;Z][Y;Z] H ) = α j,j E (40) Z H Z = I r Y C n r. (41) Notice that the optima soution of (24)X opt equasx opt = Y opt Z H opt. DefiningU [Y;Z][Y;Z]H, after reaxing the rank constraint on U, we have the foowing optimization probem min U Tr(M(f 0 )U) s.t. Tr(M(f i )U) α i,i I Tr(M(f j )U) = α j,j E [U] n+1:n+r,n+1:n+r = I r U 0. (42) To recover X from U, an agorithm based on rank reduction has been discussed in detai in [19]. When the number of the constraints are ess than 2r, this reaxation is tight [20]. Comparing (42) with (19), it can be observed that the SDP probem for T-2-QMP probems has a much ower dimension than that for T-1-QMP. It is because the T-2-QMP probems have a better structure to be expoited. In other words, it can be concuded that T-2-QMP probems have much stronger convexity than T-1-QMP probems. Appications: The soution of the T-2-QMP probem can be appied to AF MIMO reaying transceiver design at the source node with cognitive radio interference constraints. B. Discussions As mentioned at the beginning of this section, QMP is a specia case of quadraticay constrained quadratic programming (QCQP) discussed in [23], it is important to compare the QMPbased agorithms with the QCQP-based agorithms given in [23]. Due to the fact that QMP

22 22 is a specia case of QCQP, QMP probems have better structures and enjoy better properties. For exampe, for the genera T-2-QMP probems, they have stronger duaity in semidefinite reaxation than the QCQP probems discussed in [23]. Particuary, for the case when there is ony one constraint, using the QMP-based agorithm, the optima soution can be computed by using a bisection search instead of soving a SDP probem. On the other hand, soving T-2- QMP probems, QMP-based agorithms have a much smaer dimension QCQP-based agorithms. Based on the compexity anaysis in [22], if the matrix variabe X is an M M matrix, for T-2- QMP probems using QMP-based agorithm in (42) the compexity is O(M 3.5 n(1/ǫ)) where ǫ is the precision. Whie using the QCQP-based agorithm in [23] the compexity is O(M 7 n(1/ǫ)). It can be seen that the QMP-based agorithms have a great advantage in terms of computationa compexity. In addition, it is aso very interesting to compare the QMP-based agorithms with the brute force iterative agorithms in which matrix variabes are just taken as muti-dimensiona vector variabes and then brute force agorithms such as neura network agorithms are used to compute them. The main advantage of the QMP-based agorithms is that for QMP-based agorithms some nature of the optimization probems is reveaed and this is the reason why in certain cases even with a constraint, the soution has a semi-cosed-form soution. For the genera cases, the QMPbased agorithms can expoit the probem structure to improve the precision of the fina soution and acceerate the convergence speed of the agorithm. V. ROBUST TRANSCEIVER DESIGNS BASED ON QMP From the practica viewpoint, due to the imited ength of training sequences and time varying nature of wireess channes, channe estimation errors are aways inevitabe. Channe errors wi significanty decreases system performance. It is we-estabished that robust transceiver designs or beamforming designs can mitigate this negative effects [26], [27]. A question naturay arises that whether the previousy discussed QMP-based agorithms can be appied to so-caed robust transceiver designs. This is exacty the focus of this section. When channe errors are considered, the channe state information can be written as [27] H = H + H (43)

23 23 where H is the estimatedh and H is the corresponding channe estimation error, respectivey. The kronecker correation mode is widey used for channe estimation errors [26], [27] H = Σ 1 2 H W, Ψ 1 2. (44) where Σ and Ψ are the row and coumn correation matrices, respectivey. The inner matrix H W, is a random matrix with i.i.d Gaussian random eements with zero mean and unit variance. Take the simpest point-to-point MIMO system as exampe to iustrate the impact of random matrix integrations. For the point-to-point MIMO system, the data MSE at the destination equas to [3] E{Tr(GHFF H H H G H ) 2R{Tr(GHF)}+σn 2 Tr(GGH )} (45) where the expectation operation at the outside is due to channe estimation errors. This equation is a QM function with respect to F or G. As a QM function consists of zero order term, first order term and second order term of the variabes, in the foowing the matrix integrations over them are discussed separatey. Zero-term is a constant and it is obvious that its integration with respect to any variabe is itsef. Notice that the channe estimation errors are independent of the signa and the noise and their means are a zero. Based on these facts we directy have the foowing resut for the first order term E{H X} = H X. (46) The integration over the second order term is a itte bit compicated. In order to make it cear, a preiminary resut on compex matrix integration is given first. Compex matrix integration: For two M N random compex matrices Q and W, if they satisfy E{vec(Q)vec H (W)} = A B, (47) the foowing equaity hods Σ = E{QRW H } = BTr(RA T ) (48) Proof: See Appendix A.

24 24 Based on the Kronecker product mode (44) and the preiminary resut we have the foowing equation E{H XX H H H } = H XX H HH +Tr(XX H Ψ )Σ. (49) It is obvious that the expectation of a second-order term is aso a second-order term. The main difference compared to the perfect case is that there is a residua part Tr(XX H Ψ )Σ caused by channe error. Based on the resuts on the expectation on the zero term, first order term and the second order term we have the foowing concusion. Concusion 3: For LMMSE transceiver designs, expectations of channe estimation errors keep the quadratic nature of the origina QMP probems. Then it is not surprising that QMP technoogy can aso be used in robust transceiver designs. Remark: In the reference [24], ony the matrix operations for rea matrix variates are presented. Stricty speaking, it is not rigorous to directy use the resuts in that book [24] or simpy repace the symbo T by the symbo H in the invoved matrix operations. Here for competeness we give a detaied proof about compex matrix integrations to make sure our resuts are rigorous. VI. NUMERICAL RESULTS In this simuation part, in order to assess the effectiveness of the proposed soution, two different exampes are shown. In the first exampe, there are two pairs of source and destination. Moreover, there are two reays faciitating the communications between the sources and their corresponding destinations. The direct inks between the sources and destinations are negected due to deep fading. In Exampe 1, the source nodes ony transmit signas and the destination nodes ony receive signas. In the second exampe, there are two sources to exchange information assisted by two reays. In order to improve the spectra efficiency, the famous physica ayer network coding strategy named two-way reaying is adopted. Specificay, in the first time sot, two source terminas send their information to the reays and then the reays broadcast the fitered received signas to the two terminas. After that each termina removes its own transmitted signa in the first time sot first and then recovers its desired signa. In both the two exampes, a nodes are equipped with mutipe antennas. At each source node, two independent data streams, each with independent quadrature phase-shift keying (QPSK) symbos, are transmitted. Each point in the foowing figures is an average over 500

25 25 independent channe reaizations. Furthermore, the famous Matab toobox CVX [28] is used in this paper to sove the standard convex optimization probems. Exampe 1: In Exampe 1 for simpicity a nodes are equipped with N t antennas. In the first hop, the noise covariance matrices at the two reays are defined as R n1,1 and R n1,2, respectivey. Without oss of generaity, it is assumed that R n1,1 = R n1,2 = σn 2 1 I Nt. Simiary, in the second hop, the noise covariance matrices at different destination are defined as R n2,1 = R n2,2 = σn 2 2 I Nt. The signa-to-noise ratios (SNRs) for the source-reay inks are defined to be E sr,k = P s,k /N t σn 2 1, and are fixed to be E sr,k = 20dB. The SNR for each reay-destination ink is defined as E rd,k = P r,k /N t σn 2 2. For iterative agorithms, there is a we-known criterion for the initia points seection. It states that the initia vaue shoud be cose to the optima soution. However, this criterion seems meaningess as the optima soution is usuay unknown. Fig. 2 shows the tota data detection MSEs of the proposed agorithm with different initia precoder matrices at the source and reay when N t = 4. In our simuation settings, three kinds of initia vaues are seected to make a comparison, i.e., fu rank identity matrix with the power constraints satisfied, fu rank identity matrix without the constraints satisfied, diagona matrices with rank of 3 and satisfying the power constraints. It can be observed that the initia vaues being fu rank are much better than that with ower rank. The reason is fu rank initia vaues can provide a arger avaiabe set for the foowing optima vaue search than the ower rank initia vaues. Furthermore, for the fu rank initia vaues, the one satisfying constraints is better than that without satisfying constraints. As for most of practica transceiver designs, the optima soutions aways occur on the boundary of the constraints. As a resut the initia vaues satisfying constraints seem to be much coser than those without satisfying constraints and then they have better performance. Fig. 3 shows the performance advantage of the proposed agorithm over the simpest uniform power aocation scheme in terms of averaged MSE in the two different cases N t = 2 and N t = 4. In uniform power aocation agorithm, the precoder matrices at the sources and reay are proportiona to the identity matrices which are scaed by factors to make the equaities in the power constraints. In concusion we can say that the proposed iterative agorithm can act as a better benchmark agorithm compared with the naive uniform power aocation scheme. Exampe 2: