Design of Cooperative MIMO Wireless Sensor Networks with Partial Channel State Information

Date of publication xxxx yy, 2020, date of current version June 8, 2020.

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Design of cooperative MIMO wireless sensor networks with partial channel state information

DONATELLA DARSENA1, (Senior Member, IEEE), GIACINTO GELLI2, (Senior Member, IEEE) AND FRANCESCO VERDE2, (Senior Member, IEEE) 1Department of Engineering, Parthenope University, Naples I-80143, Italy (e-mail: [email protected]) 2Department of Electrical Engineering and Information Technology, University Federico II, Naples I-80125, Italy [e-mail: (gelli, f.verde)@unina.it] Corresponding author: Francesco Verde (e-mail: [email protected]).

ABSTRACT Wireless sensor networks (WSNs) play a key role in automation and consumer electronics applications. This paper deals with joint design of the source precoder, relaying matrices, and destination equalizer in a multiple-relay amplify-and-forward (AF) cooperative multiple-input multiple-output (MIMO) WSN, when partial channel-state information (CSI) is available in the network. In particular, the considered approach assumes knowledge of instantaneous CSI of the first-hop channels and statistical CSI of the second-hop channels. In such a scenario, compared to the case when instantaneous CSI of both the first- and second-hop channels is exploited, existing network designs exhibit a significant performance degradation. Relying on a relaxed minimum-mean-square-error (MMSE) criterion, we show that strategies based on potential activation of all antennas belonging to all relays lead to mathematically intractable optimization problems. Therefore, we develop a new joint relay-and-antenna selection procedure, which determines the best subset of the available antennas possibly belonging to different relays. Monte Carlo simulations show that, compared to conventional relay selection strategies, the proposed design offers a significant performance gain, outperforming also other recently proposed relay/antenna selection schemes.

INDEX TERMS Amplify-and-forward relays, multiple-input multiple-output (MIMO), partial channel state information, wireless sensor networks.

1 I. INTRODUCTION AND SYSTEM MODEL destination, with the assistance of NC half-duplex relays. We assume that there is no direct link between the source and the destination, due to high path loss values or obstructions, and arXiv:2003.01611v2 [eess.SP] 10 Jun 2020 ITH the advent of massive Internet of Things and we denote with NS, NR, and ND, respectively, the numbers of W massive machine-type communications, especially in antennas at the source, relays, and destination. The received the domain of 5G consumer electronics, there is a need to fur- signal at the destination can be expressed as ther enhance physical-layer performance of wireless sensor r = C b + v (1) networks (WSNs). In this respect, amplify-and-forward (AF) relaying is an effective way to improve transmission reliabil- ity over fading channels, by taking advantage of the broadcast 1 The fields of complex and numbers are denoted with C and R, respectively; matrices [vectors] are denoted with upper [lower] case boldface nature of wireless communications [1], [2], especially when letters (e.g., A or a); the field of m × n complex [real] matrices is denoted m×n m×n m m m× m× the network nodes are equipped with multiple-input multiple- as C [R ], with C [R ] used as a shorthand for C 1 [R 1]; output (MIMO) transceivers [3]–[5]. the superscripts ∗, T, H, −1, and † denote the conjugate, the transpose, the conjugate transpose, the inverse, and the Moore-Penrose generalized inverse of a matrix, respectively; {A}i j indicates the (i + 1, j + 1)th element m×n m of A ∈ C , with i ∈ {0,1,...,m − 1} and j ∈ {0,1,...,n − 1}; 0m ∈ R , m×n m×m O ∈ R , and Im ∈ R denote the null vector, the null matrix, and the We consider a one-way cooperative MIMO WSN aimed n×n identity matrix, respectively; tr(A) denotes the trace of A ∈ C ; rank(A) is N m×n at transmitting a symbol block b ∈ C B from a source to a the rank of A ∈ C ; finally, the operator E[·] denotes ensemble averaging.

VOLUME xx, 2020 1 Darsena et al.: Design of cooperative MIMO wireless sensor networks with partial channel state information

N ×N where C , GGFFFHHHFF 0 ∈ C D B is the dual-hop channel the source-to-relay channels, such a method requires that the matrix and v , GGFF w + n is the equivalent noise vector at selected relay sends instantaneous CSI of the corresponding the destination. The composite matrices source-to-relay channel to the destination for optimal decod- ing. Moreover, the optimization problem in [29] does not H [H T,H T,...,H T ]T ∈ (NCNR)×NS (2) , 1 2 NC C admit a closed-form solution and is solved by using a line ND×(NCNR) G , [G1,G2,...,GNc ] ∈ C (3) search algorithm. It has been shown in [30] that P-CSI relay selection ap- collect the first- (backward) and second-hop (forward) proaches for MIMO nodes, based only on the instantaneous MIMO channel coefficients of all the relays, respectively, knowledge of H, do not fully exploit the diversity arising whereas the diagonal blocks F ∈ NR×NR of i C from the presence of multiple relays. Besides instantaneous knowledge of H, statistical CSI of the second-hop matrix G F , diag(F 1,F 2,...,F NC ) (4) is used in [14], [31] to perform relay/antenna selection for NS×NB denote the relaying matrices, and F 0 ∈ C represents a MIMO AF cooperative network. However, the solutions NCNR the source precoding matrix. Finally, w ∈ C and n ∈ developed in [14], [31] still exhibit a significant performance ND C gather the noise samples at all the relays and at the degradation compared to designs based on F-CSI. destination, respectively. The vector r is subject to linear In this paper, we present new optimization methods for equalization at the destination through the equalizing matrix multiple-relay cooperative MIMO WSNs with P-CSI, i.e., NB×ND ˆ D ∈ C , hence yielding an estimate b , DDrr of the knowledge of the instantaneous value of H and the statistical source block b, whose entries are then subject to minimum- properties of G. Our design does not rely on F-CSI as in distance (in the Euclidean sense) detection. Increase in spec- [7]–[13], [15]–[17], and needs the same amount of P-CSI tral efficiency can be obtained by considering two-way exploited in [24]–[28], [31]. In this scenario, we consider relaying [6], which is based on establishing bidirectional a relaxed joint minimum-mean-square-error (MMSE) opti- connections between two or more terminals using one or mization of the source precoder F 0, the AF relaying matrices several half-duplex relays. in F, and the destination equalizer D, with a power constraint To achieve the expected gains, channel state information at the source [32] and a sum-power constraint at the relays (CSI) is required at the network nodes, i.e, source, AF relays, [10]. Specifically, capitalizing on our preliminary results and destination. Full CSI (F-CSI) is invoked in many papers [14], the novel contributions can be summarized as follows: dealing with optimization of one-way (see, e.g., [7]–[15]) and 1) We prove that the MMSE-based design attempting to two-way (see, e.g., [16], [17]) cooperative MIMO networks. activate all possible antennas of all relays leads to a Specifically, with reference to the system model (1), F-CSI is mathematically intractable optimization problem. tantamount to requiring: (i) instantaneous knowledge of the 2) We provide the proofs of the results reported in [14], by first-hop channel matrix H; (ii) instantaneous knowledge of enlightening that single relay selection [14], [24]–[28] the second-hop channel matrix G; (iii) instantaneous knowl- is suboptimal in the considered P-CSI scenario. edge of the dual-hop channel matrix C. While the dual-hop 3) We develop a new joint antenna-and-relay selection channel matrix C can be directly estimated at the destination algorithm, which is shown to significantly outperform by training, separate acquisition of the first- and second- the relay/antenna selection approaches [14], [26], [31] hop matrices H and G is more complicated to achieve, both in terms of average symbol error probability (ASEP). in terms of communication resources and signal overhead, The paper is organized as follows. Section II introduces the especially in multiple-relay WSNs. Moreover, since channel basic assumptions and discusses their practical implications. estimation errors occur in practical situations, robust opti- The proposed designs are developed in Section III. Sec- mization designs are needed [18], [19], which further com- tion IV reports simulation results in terms of ASEP, whereas plicate system deployment. In resource-constrained WSNs, Section V draws some conclusions. the use of partial CSI (P-CSI) can extend network lifetime and reduce the complexity burden. II. BASIC ASSUMPTIONS AND PRELIMINARIES Relay selection is a common strategy to reduce signal- The symbol block b in (1) is modeled as a circularly sym- H ing overhead and system design complexity in single-input metric complex random vector, with E[bbbb ] = INB . The single-output (SISO) cooperative WSNs [20]–[23]. Design entries of H and G are assumed to be unit-variance circularly of SISO relay selection procedures providing diversity gains symmetric complex Gaussian (CSCG) random variables. The – even when F-CSI is not available – has been addressed in noise vectors w and n are modeled as mutually independent [24]–[28]. Such methods rely on P-CSI, since selection of CSCG random vectors, statistically independent of (b,H,G), H H the best relay is based only on instantaneous knowledge of with E[wwww ] = INCNR and E[nnnn ] = IND , respectively. the source-to-relay channels. However, the diversity order of Hereinafter, we assume that C in (1) and the following the methods developed in these papers does not scale in the conditional covariance matrix of v, given G, number of relays N . For SISO nodes, a P-CSI relay selection C K [vvvvH |G] = GGFFFFF HGH + I (5) scheme has been proposed in [29], yielding full diversity vvvv , E ND order NC. However, besides the instantaneous knowledge of have been previously acquired at the destination during a

2 VOLUME xx, 2020 training session. Under such assumptions, it is known (see, Closed-form evaluation of the cost function in (9) is cumber- 2 e.g., [32]) that, for fixed matrices F 0 and F, the matrix some; however, under a1) and a2), it can be observed that D minimizing the trace of the conditional mean square er- h − i ˆ ˆ H I F HHHF HGHG F H F 1 ror (MSE) matrix E(F 0,F,D) , E[(b − b)(b − b) |H,G], tr INB + F 0 H F G GGFFHHFF0 H G given and , is the Wiener filter h H H H H −1i < tr F 0 H F G GGFFFHHHFF 0 (10) H H −1 Dmmse = C (CCCC + K vvvv) . (6) where the difference between the left- and right-hand Optimization of F 0 and F is carried out under the as- sides tends to zero as the minimum eigenvalue of sumption that only P-CSI is available at the source and H H H H F 0 H F G GGFFFHHHFF 0 is significantly larger than one. This the relays. Specifically, the source and the relays perfectly happens in the high signal-to-noise ratio (SNR) region, H i know the first-hop channel matrix , but the th relay has i.e., when PS and PD are sufficiently large. Relying on only knowledge of the second-order statistics (SOS) of its (10), we pursue a further relaxation of (8) by replacing n h −1i o own second-hop channel matrix Gi. These assumptions are tr I + F HH HF HGHGGFFFHHHFF H in (9) with EG NB 0 0 justified since, in some systems, the relays may be able to n h − i o F HHHF HGHG F H F 1 H exchange information among themselves before transmission its upper bound EG tr F 0 H F G GGFFHHFF0 H , [33]. In this case, knowledge of H at the relays is realistic which can be evaluated in closed-form as stated by the [7]–[14], [20]–[22], [24], [25], [27]–[30]. Moreover, since following Lemma. the SOS of Gi vary much more slowly than the instantaneous values of Gi, the feedback overhead from the destination to Lemma 1: Let us assume that: a3) ND > NB. Then, under the relays is significantly reduced, compared to [29]. a1), a2), and a3), it results that −1 n h H H H H −1i o tr(R ) III. THE PROPOSED P-CSI-BASED DESIGN EG tr F 0 H F G GGFFFHHHFF 0 H = ND − NB To obtain F 0 and F, we minimize the statistical average (with (11) respect to G) of the trace of the following matrix: H H H NB×NB where R , F 0 H F FFHHHFF 0 ∈ C . H −1 −1 E(F 0,F) , E(F 0,F,Dmmse) = (INB +C K vvvv C) (7) Proof: See Appendix A. under suitable power constraints. To this aim, we assume that: a1) F 0 is full-column rank, i.e., rank(F 0) = NB ≤ NS; At this point, evaluation of the expectation in the second a2) GGFFFHH is full-column rank, i.e., rank(GGFFFHH) = NS ≤ ND. constraint of (9) is in order. In this respect, one has It is noteworthy that assumption a2) necessarily requires tr(F HGHGGFF)|H = tr GHGFFFF H = trF HF that the matrices FFHH and H are full-column rank, i.e., EG EG (12) rank(FFHH) = rank(H) = N ≤ N N . Such assumptions en- S C R where we have also used the cyclic property [34] of the trace sure that C is full-column rank as well. Specifically, we operator. Therefore, under a1), a2), and a3), the optimization consider the following optimization problem: problem (9) can be simplified as follows n h H −1 −1i o min EG tr INB +C K vvvv C H subject to (s.to) h H H H −1i F 0,F min tr F 0 H F FFHHHFF 0 F F H H H F 0,F tr(F F ) ≤ P and G tr(GGFFFKKzzzz F G )|H ≤ P 0 0 S EG D s.to tr(F F H) ≤ and trF H F ≤ . (13) (8) 0 0 PS PD

H H H At this point, a comment regarding the constraints in (8) where K zz [zzzz |H] = HHFF F H + I is the condi- zzzz , E 0 0 NCNR and (13) is in order. The constraint tr(F F H) ≤ P in (8) tional (given H) covariance matrix of the vector z ∈ NCNR 0 0 S C and (13) limits the average transmitted power of the source collecting the signals received by all the relays, with P > 0 S and it is standard in the design of linear MIMO transceivers and P > 0 denoting the power threshold at the source and D [32]. Regarding the second constraint in (8), we observe that, at the destination, respectively. The constraint on the received H H given H and G, P(H,G) tr(GGFFFKKzzzz F G ) represents the power at the destination automatically limits the power ex- , average received power at the destination. It is noteworthy penditure at the relays. Since problem (8) is nonconvex, we that P(H,G) is typically limited in those scenarios where a consider its relaxed version: target performance has to be achieved and per-node fairness n h −1i o H H H H H is not of concern [3], [7]. The constraint tr F F ≤ PD in min EG tr INB + F 0 H F G GGFFFHHHFF 0 H F 0,F (13), which has been obtained by averaging a relaxed version H s.to tr(F 0 F 0 ) ≤ PS and of P(H,G) with respect to the probability distribution of G, H H ﬁxes a limit on the total average power transmitted by the EG tr(GGFFFFF G )|H ≤ PD (9) where we have used the expression of C and the 2 H −1 −1 H −1 The proof follows easily from the facts [34] that the trace of A is equal to inequalities tr[(INB + C K vvvv C) ] ≥ tr[(INB + C C) ] the sum of its eigenvalues and, if λ is an eigenvalue of a nonsingular matrix H H H H −1 −1 and tr(GGFFFKKzzzz F G ) ≤ tr(GGFFFFF G )tr(K zzzz) [34], [35]. A, then λ is an eigenvalue of A .

VOLUME xx, 2020 3 Darsena et al.: Design of cooperative MIMO wireless sensor networks with partial channel state information

3 T relays, so-called sum-power constraint [10]. where we have defined ω , [ω(1),ω(2),...,ω(NB)] ∈ N N To solve (13), we use the following Lemma. R B , δ i , [δi(1),δi(2),...,δi(NS)] ∈ R S , for i ∈ n×n {1,2,...,N }, Lemma 2: For a positive definite matrix A ∈ C , the C following inequality holds: NB NC 1 m f0 ω,{δ i}i=1 , 1 ∑ NC A−1 `=1 2 tr(A ) ≥ ∑ (14) ω(`)λ (∆N + `) δi(∆N + `) {A}`` h ∑ `=1 i=1 where {A} is the `th diagonal entry of A and the inequality (18) `` N N −N ≥ is achieved if A is diagonal. with ∆ , S B 0. All the inequality constraints in (17) are linear. However, it is shown in Appendix B that, when Proof: See [38, p. 65]. NC > 1, the cost function (18) is the sum of NB functions n+1 As a consequence of Lemma 2, the minimum value of the that are neither strictly convex nor strictly concave on R+ . H cost function in (13) is achieved if F 0 AAFF 0 is diagonal, with Hence, trying to solve (17) with the available optimization H H N ×N A , H F FFHH ∈ C S S . In what follows, we consider three tools leads to poor performance in multiple-relay WSNs. different approaches to achieve the desired diagonalization of H F 0 AAFF 0: the first one is based on the SVD of the composite B. DESIGN BASED ON THE SVD OF THE INDIVIDUAL matrix H = [H T,H T,...,H T ]T and it results in a (possible) 1 2 NC FIRST-HOP CHANNEL MATRICES selection of all the relays; the second one relies on the SVDs A simple design can be developed by setting F = O , H H H i NR×NR of the individual matrices H 1,H 2,...,H NC , thus leading to ? a single-relay selection; the last one exploits the SVDs of for each i ∈ {1,2,...,NC}−i . Basically, such a choice leads to a single-relay selection scheme [14], which imposes that row-based partitions of H and it can be interpreted as a joint ? antenna-and-relay selection scheme. only one relay (i.e., that for i = i ) is recruited to transmit and all the remaining ones keep silent in the second hop.

A. DESIGN BASED ON THE SVD OF THE COMPOSITE Herein, we assume that H i is full-column rank, i.e., FIRST-HOP CHANNEL MATRIX rank(H i) = NS ≤ NR, for each i ∈ {1,2,...,NC}. Let One can attempt to recruit all the relays in the second hop U O T V H H U h,i [ONS×(NR−NS),Λh,i] V h,i (19) of the cooperative scheme by diagonalizing F 0 AAFF 0 through H U O T V H H be the SVD of H , where the SVD H = U h [ONS×(NCNR−NS),Λh] V h of H, where i (N N )×(N N ) N ×N the matrices U h ∈ C C R C R and V h ∈ C S S are Λh,i , diag[λh,i(1),λh,i(2),...,λh,i(NS)] (20) unitary, and Λh , diag[λh(1),λh(2),...,λh(NS)] gathers the corresponding nonzero singular values arranged in increasing contains the singular values of H i, arranged in increasing N ×N order. By substituting the SVD of H in A, it follows by direct order, and the unitary matrices U h,i ∈ C R R and V h,i ∈ H NS×NS inspection that F 0 AAFF 0 is diagonal if (see, e.g., [39]) C collect the corresponding left and right singular H H 1/2 vectors, respectively. In this case, one has A = H i? F i? F i? H i? F 0 = V h,right Ω (15) and, by substituting the SVD of H i? in this matrix equation, 1/2 † F H A F F i = Qi ∆i U h,right,i (16) one has that the diagonalization of F 0 AAFF0 is ensured by N ×N 1/2 where V h,right ∈ C S B contains the NB rightmost columns F 0 = V h,i?,right Ω (21) from V , the matrices Ω diag[ω(1),ω(2),...,ω(N )] 1/2 H h , B F i? = Qi? ∆ U h,i?,right (22) and ∆i , diag[δi(1),δi(2),...,δi(NS)] are determined in a ? NR×NS ? NS×NB second step, for i ∈ {1,2,...,NC}, the arbitrary matrix where U h,i ,right ∈ C and V h,i ,right ∈ C contain NR×NS H the N and N rightmost columns from U ? and V ? , Qi ∈ obeys Qi Qi = IN , provided that NS ≤ NR, S B h,i h,i C S N ×N T T T T (NCNR)×NS Q ? R S U h,right [U ,U ,...,U ] ∈ respectively, Qi ∈ C is an arbitrary matrix obeying , h,right,1 h,right,2 h,right,NC C H Q ? Q ? I contains the NS rightmost columns from U h, with the matrix Qi Qi = INS , Ω has been deﬁned in Subsection III-A, and NR×NS ∆ [ ( ), ( ),..., (N )] U h,right,i ∈ C being full-column rank. ∆ , diag δ 1 δ 2 δ S . To fully specify the solution Using (15) and (16), problem (13) ends up to of (13) in the case of single-relay selection, optimization of Ω, ∆, and i? is accomplished in two steps. NB ? NC First, for a given i ∈ {1,2,...,NC}, by substituting (21) min f0 ω,{δ i}i=1 s.to ∑ ω(`) ≤ PS, ω(`) > 0, NC and (22) in (13), one obtains the scalar optimization problem ω,{δ i}i=1 `=1 with linear inequality constraints: NC NS H −1 and δi(`) U U , ,i ≤ PD, δi(`) > 0 ∑ ∑ h,right,i h right `` NB i=1 `=1 min f (i?,ω,δ ) s.to ω(`) ≤ , ω(`) > 0, (17) 1 ∑ PS ω,δ `=1 N 3Design with per-relay power constraints can be solved by properly S reformulating the problem into an equivalent optimization with a sum-power and ∑ δ(`) ≤ PD, δ(`) > 0 (23) constraint [36], [37]. `=1

4 VOLUME xx, 2020 (q ) (q ) with S 1 6= S 2 for q1 6= q2. Additionally, we use the notation (q) (q) NS Ni ∈ {0,1,...,NB} to indicate the number of pairs of S ? 1 (q) f1 (i ,ω,δ ) , ∑ 2 (24) having the same second entry: in other words, N represents λ ? (`)ω(`)δ(`) i `=NS−NB+1 h,i the number of antennas activated on the ith relay according where ω has been previously defined in Subsection III-A and NC (q) ω to the qth selection. It results that ∑i=1 Ni = NB. δ [δ(1),δ(2),...,δ(N )]T ∈ NS . Since f (i?,ω,δ ) is a , S R 1 The selected antennas generate a first-hop channel matrix convex function (see Appendix B), the optimization problem ? ? (q) (q) (q) (23) is convex and, thus, its solution ω (i ) and δ (i ) can (q) T T T T NB×NB opt opt H , [(H 1 ) ,(H 2 ) ,...,(H N ) ] ∈ C (27) be found by using efficient numerical techniques [40]. For C instance, if one resorts to interior point methods, convergence and a relaying matrix arbitrarily close to the optimal solution is achieved in a (q) (q) (q) (q) NB×NB F , diag(F ,F ,...,F N ) ∈ C (28) number of iterations that is proportional to the logarithm of 1 2 C (q) (q) (q) the problem dimension [41], with a complexity per iteration (q) N ×NB (q) N ×N with H i ∈ C i and F i ∈ C i i . By convention, dictated by the cost M of computing a Newton direction [42]. (q) (q) (q) ? if Ni = 0, then H i and F i are empty matrices. Second, the optimal value iopt of i is obtained as With reference to the qth selection, we formulate a new ? ? ? iopt arg min f1 i ,ω opt(i ),δ opt(i ) (25) , ? optimization problem, for q ∈ {1,2,...,Q − 2}, which is i ∈{1,2,...,NC} formally obtained from (13) by replacing H and F with which allows one to single out the best relay among the H (q) and F (q), respectively, whose cost function achieves its NC available ones. The solution of (25) can be obtained by minimum value if F H A(q) F is diagonal (see Lemma 2), ? 0 0 solving (23) for each i ∈ {1,2,...,NC}, with an overall (q) (q) H (q) H (q) (q) with A , (H ) (F ) F H . For i ∈ {1,2,...,NC}, complexity O [NC M log(NB + NS)]. (q) (q) (q) (q) H let H i = U h,i [O (q) (q) ,Λh,i ](V h,i ) be the SVD of In the SISO configuration, i.e., when NB = NS = NR = Ni ×(NB−Ni ) H(q) ND = 1, and√ when there is no precoding at the source, i.e., the (nonempty) matrix H i , which is assumed to be full- (q) (q) (q) (q) (q) F 0 ≡ f0 = PS, one gets F i? = diag(0,...,0, fi? ,0,...,0) row rank, i.e., rank(H ) = N , where U ∈ Ni ×Ni 2 i i h,i C and (25) boils down to iopt argmax ? {|hi| }, with (q) , i ∈{1,2,...,NC} and V ∈ NB×NB are unitary, and the diagonal matrix h denoting the channel coefficients between the source and h,i C i (q) (q) (q) (q) (q) the ith relay. According to [43], such a scheme has a full Λh,i , diag[λh,i (1),λh,i (2),...,λh,i (Ni )] collects the cor- diversity order equal to NC. However, as we will see in responding nonzero singular values arranged in increasing H (q) Section IV, such a design suffers from a diversity loss in a order. In this case, the diagonalization of F 0 A F 0 can be MIMO setting, i.e., when NS,NR,ND > 1. obtained by resorting to the following structures (q) H −1 1/2 F 0 = [(V ) ] Ω (29) C. DESIGN BASED ON THE SVD OF ROW-BASED h,right (q) 1/2 (q) H PARTITIONS OF THE COMPOSITE FIRST-HOP CHANNEL F i = Qi ∆i (U h,i ) (30) MATRIX where In the considered cooperative MIMO WSN, there are NC (q) (q) (q) (q) N ×N relays equipped with NR antennas, which amounts to a total V [V ,V ,...,V ] ∈ B B (31) h,right , h,right,1 h,right,2 h,right,NC C number of NC NR distributed antennas. Here, we propose to (q) (q) (q) 4 NB×Ni choose the best NB = NS antennas out of the NC NR ones. with V h,right,i ∈ C gathering the Ni rightmost Such antennas can either be physically located on a single (q) (q) (q) columns from V , Q ∈ Ni ×Ni is an arbitrary unitary relay, or be spatially distributed over different relays, thus h,i i C matrix, Ω and ∆ have been defined in Subsection III-A. accomplishing a joint antenna-and-relay selection scheme. i To optimize Ω, ∆ , and q, we resort to a two-step procedure Let S , {(n,i),∀n ∈ {1,2,...,NR},∀i ∈ {1,2,...,NC}} i collect all the NC NR antenna elements in the network, with as in the previous subsection. By substituting (29)–(30) in (q) (q) the generic (ordered) pair (n,i) uniquely identifying the nth (13) (with H and F in lieu of H and F, respectively), antenna located on the ith relay. The number of distinct for a given q ∈ {1,2,...,Q − 2}, one gets the convex opti- subsets of S that have exactly NB elements is given by the mization problem (see Appendix B) with linear inequality binomial coefficient Q NCNR.5 By excluding the trivial constraints: , NB choice /0and the degenerate case (discussed in Subsec- S NC min f2 q,ω,{δ i}i=1 s.to tion III-A) , we denote with NC ω,{δ i}i=1 (q) n (q) (q) (q) (q) (q) (q) o S (n ,i ),(n ,i ),...,(n ,i ) (26) NB −1 , 1 1 2 2 NB NB h (q) H (q) i ω(`) (V ) V ≤ PS, ω(`) > 0, ∑ h,right h,right `` the selected subset of S , with q ∈ {1,2,...,Q − 2}, obeying `=1 NC NS 4 Our design can be simply extended to the case NS ≥ NB. and ∑ ∑ δi(`) ≤ PD, δi(`) > 0 (32) 5The empty set /0and the set S are considered as subsets of S as well. i=1 `=1

VOLUME xx, 2020 5 Darsena et al.: Design of cooperative MIMO wireless sensor networks with partial channel state information

100 100 [26] (P-CSI) [26] (P-CSI) [31] [31] [26] (F-CSI) 1-R Selection -1 1-R Selection 10 [26] (F-CSI) [15] (F-CSI) [15] F-CSI 10-1

10-2

10-2 10-3 ASEP ASEP 10-4 10-3

10-5

10-4 10-6

10-5 10-7 4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20 SNR [dB] SNR [dB]

Figure 1: ASEP versus SNR (Example 1: N = 1 and NC = 2). Figure 2: ASEP versus SNR (Example 1: N = 1 and NC = 3). with link quality of both the first- and second-hop transmissions. Besides the single-relay selection method described in Sub- NC NB 1 f q,ω,{δ }NC (33) section III-B, referred to as “1-R Selection", and the joint 2 i i=1 , ∑ ∑ h i2 i=1 `=1 (q) antenna-and-relay selection scheme developed in Subsec- λh,i (`) ωi(`)δi(`) tion III-C, referred to as “JAR Selection", we also report the i−1 (q) performance of [26, CSI Assumptions I and II] in the case where ωi(`) , ω ∑m=1 Nm + ` , whereas ω and δ i have been defined in Subsection III-A. Similarly to problem (23), of single-antenna nodes (i.e., N = 1) and that of [31] for NC both single- and multiple-antennas nodes (i.e., N ∈ {2,3}). the solution ω opt(q) and {δ i,opt(q)}i=1 of (32) can be found by using, e.g., interior point methods [40]. Finally, the best As a reference lower bound, we additionally include in all the plots the ASEP curves of the F-CSI design proposed in value qopt of q is found by solving [15], whose design relies on the additional knowledge of NC G qopt , arg min f2 q,ω opt(q),{δ i,opt(q)}i=1 (34) the ith second-hop channel matrix Gi at the ith relay, for q∈{1,2,...,Q−2} i ∈ {1,2,...,NC}. This F-CSI method exhibits a theoretical which determines the best NB-dimensional subset of the diversity order equal to NC NR − NB + 1 [15]. 3 available NC NR antennas. The solution of (34) can be ob- The ASEP has been evaluated by carrying out 10 indepen- tained by solving (32) for each q ∈ {1,2,...,Q − 2}, with dent Monte Carlo trials, with each run using independent sets an overall complexity O [(Q − 2)M log(NB + NB NC)], which of channel realizations and noise, and an independent record is larger than that required to select the best relay (see of 106 source symbols. Subsection III-B), especially for large number of relays. When NB = NS = NR = 1, the optimization problems A. EXAMPLE 1: SINGLE-ANTENNA NODES (23)-(25) and (32)-(34) yield the same solution and, thus, We report in Figs. 1 and 2 the ASEP performance of the con- the design (32)-(34) exhibits full diversity order NC, too. sidered schemes as a function of the SNR, for single-antenna However, we will show in the next section that, when nodes (i.e., N = 1) and two different values of the number of N ,N ,N > 1 (MIMO WSN), the proposed joint antenna- B S R relays NC ∈ {2,3}. We would like to remember that, in the and-relay selection scheme ensures a significant performance case of N = 1, the two approaches “1-R Selection" and “JAR improvement with respect to single-relay selection, in terms Selection" are equivalent and, thus, only the performance of of both diversity order and coding gain. the “1-R Selection" method are reported. Results clearly show that no diversity is achieved by [26] IV. NUMERICAL RESULTS (CSI Assumption II corresponding to P-CSI) and [31], irre- In this section, to assess the performance of the considered P- spective of the number of relays. On the other hand, the “1-R CSI designs, we present the results of Monte Carlo computer Selection" scheme exhibits the same diversity order of the F- simulations, aimed at evaluating the ASEP of the correspond- CSI methods proposed in [26] (CSI Assumption I) and [15], ing cooperative systems, transmitting quadrature phase-shift- which linearly increases with NC. This fact allows the “1-R keying (QPSK) symbols. We set N , NB = NS = NR = ND Selection" design to significantly outperform both [26] (P- in all the forthcoming examples, with N ∈ {1,2,3}. We also CSI) and [31], which rely on the same amount of CSI. Re- assume that PS = PfD = Ptot. Consequently, the SNR is markably, the “1-R Selection" scheme performs comparably defined as SNR , Ptot, which measures the per-antenna to [26] (F-CSI) in the case of NC = 2 relays. Compared to

6 VOLUME xx, 2020 100 100 [31] [31] 1-R Selection 1-R Selection JAR Selection JAR Selection -1 -1 10 [15] (F-CSI) 10 [15] (F-CSI)

10-2 10-2

10-3 10-3 ASEP ASEP 10-4 10-4

10-5 10-5

10-6 10-6

10-7 10-7 4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20 SNR [dB] SNR [dB]

Figure 3: ASEP versus SNR (Example 2: N = 2 and NC = 2). Figure 5: ASEP versus SNR (Example 2: N = 3 and NC = 2).

100 100 [31] [31] 1-R Selection 1-R Selection JAR Selection -1 JAR Selection 10 [15] (F-CSI) -1 10 [15] (F-CSI)

10-2 10-2

10-3 10-3

ASEP -4 10 ASEP 10-4

10-5 10-5

10-6 10-6

-7 10 -7 4 6 8 10 12 14 16 18 20 10 4 6 8 10 12 14 16 18 20 SNR [dB] SNR [dB] Figure 4: ASEP versus SNR (Example 2: N = 2 and N = 3). C Figure 6: ASEP versus SNR (Example 2: N = 3 and NC = 3). single-relay selection, the performance improvement of the N , as in [15] which, however, requires F-CSI. F-CSI approaches – arising from the additional instantaneous C knowledge of the second-hop matrix G – becomes more and V. CONCLUSIONS more apparent when the number of relays NC increases. We studied the problem of designing multi-relay AF coop- B. EXAMPLE 2: MULTIPLE-ANTENNA NODES erative WSNs, based on the knowledge of the instantaneous Figs. 3, 4, 5, and 6 show the ASEP performance of the values of the first-hop MIMO channel matrix and statistical considered designs as a function of the SNR, for two different characterization of the second-hop one (partial CSI scenario). multi-antenna configurations N ∈ {2,3} and two different In this case, antenna/relay selection schemes arise necessarily to formulate mathematically tractable design problems, values of the number of relays NC ∈ {2,3}, respectively. It is apparent from these plots that, in a multi-antenna which can be solved by using standard convex optimization deployment, the “1-R Selection" approach and [31] perform tools. We have shown that, in a MIMO setting, the selection comparably, both exhibiting a diversity loss with respect to of the best relay is suboptimal and large performance im- the F-CSI design [15]. As claimed, especially in the high provements can be obtained by selecting the best antennas SNR regime, the proposed “JAR Selection" design signif- distributed over multiple relays. Numerical simulations have icantly outperforms both the “1-R Selection" scheme and shown that the proposed joint antenna-and-relay selection [31], under the same amount of P-CSI. Such a performance approach significantly outperforms existing schemes, which gap remarkably scales up as the number of antennas at the exploit the same amount of P-CSI. nodes increases from N = 2 to N = 3. Interestingly, the . diversity order of the “JAR Selection" scheme increases with

VOLUME xx, 2020 7 Darsena et al.: Design of cooperative MIMO wireless sensor networks with partial channel state information

APPENDIX A PROOF OF LEMMA 1 for all the points belonging to its domain. Preliminarily, we remember that C = GGFFFHHHFF 0 is full- Using standard calculus concepts, it can be verified that column rank if a1) and a2) hold. It can be shown (see, ∂ 2 2 e.g., [44]) that, conditioned on H, the kth diagonal entry 2 f = 3 (40) H −1 H −1 ∂x Ax (y1 + y2 + ··· + yn) {(C C) }kk of the matrix (C C) follows an inverse- ∂ 2 1 Gamma distribution, with shape parameter α , ND − NB + 1 −1 f = 2 2 (41) and scale parameter βk , 1/{R }kk, where R is defined in ∂x∂y j Ax (y1 + y2 + ··· + yn) the lemma statement. Thus, the probability density function ∂ 2 2 H −1 f = . of the random variable {(C C) } , given H, reads as 3 (42) kk ∂yi ∂y j Ax(y1 + y2 + ··· + yn) − 1 1 −α−1 xβ 2 n+1 pk(x) = x e k (35) We note that all the entries of f are nonnegative on R+ . ( ) α O Γ α βk In the particular case of n = 1, it is readily seen that the 2 2×2 where the gamma function Γ(α) = (α −1)! since ND −NB is determinant of O f ∈ R is given by a non-negative integer number [45]. Therefore, one has 3 det( 2 f ) = > 0 (43) O 2 4 4 NB A x y1 h H −1 i hn H −1o i EG tr C C H = EG C C H ∑ kk which shows that, when n = 1, f is a strictly convex function k=1 2 N on R+. Therefore, since the sum of convex functions is 1 B 1 Z +∞ − 1 −α xβk convex [40], the cost functions (23) or (32) are convex. = ∑ α lim x e dx . (36) Γ(α) k=1 βk δ→0 δ On the other hand, when n > 1, by resorting to the Lapla- After some calculations, eq. (36) can be rewritten as cian determinant expansion by minors, it results that n+1 N −1 2 j+1 2 h −1 i B β det( f ) = (−1) { f } M (44) CH C H = k ( − ,( )−1) O ∑ O 1 j 1 j EG tr ∑ lim γ α 1 δ βk j=1 k=1 Γ(α) δ→0 n×n 2 Γ(α − 1) NB 1 NB tr(R−1) where M1 j ∈ R is a so-called minor of O f , obtained = −1 = {R−1} = 2 ∑ βk ∑ kk by taking the determinant of O f with row 1 and column Γ(α) k=1 α − 1 k=1 ND − NB j crossed out. It can be verified that M1 j is zero, for each (37) 2 j ∈ {1,2,...,n + 1}. Thus, the determinant of O f is zero where we have exploited the definition of the incomplete at each point belonging to the domain of f if n > 1. This is R x s−1 −t 2 gamma function γ(s,x) , 0 t e dt [45] and its asymp- sufficient to infer that O f is neither positive nor negative totic property Γ(s) = limx→+∞ γ(s,x). definitive, which implies in its turn that, when n > 1, f is n+1 neither strictly convex nor strictly concave on R+ . APPENDIX B HESSIAN OF THE COST FUNCTION (15) Let us check convexity of a generic summand of the cost References function (18). To this end, it is sufficient to study the mul- [1] J.N. Laneman, D.N.C. Tse, and G.W. Wornell, “Cooperative diversity in tivariate function wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, pp. 3062–3080, Dec. 2004. 1 [2] R. Yuan, T. Zhang, J. Huang, and L. Sun, “Opportunistic cooperation f (x,y1,y2,...,yn) , (38) Ax(y1 + y2 + ··· + yn) and optimal power allocation for wireless sensor networks," IEEE Trans. Consumer Electron., vol. 56, pp. 1898–1904, Aug. 2010. with A > 0, x > 0, and yi > 0, for each i ∈ {1,2,...,n}. The [3] S. Cui, A.J. Goldsmith, and A. Bahai, “Energy-efficiency of MIMO and domain of f is therefore given by n+1, which is a convex cooperative MIMO techniques in sensor networks,” IEEE J. Select. Areas R+ Commun., vol. 22, pp. 1089–1098, Aug. 2004. set. 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VOLUME xx, 2020 9 Darsena et al.: Design of cooperative MIMO wireless sensor networks with partial channel state information

DONATELLA DARSENA (M’06-SM’16) received the Dr. Eng. degree summa cum laude in telecommunications engineering in 2001, and the Ph.D. degree in electronic and telecommunications engineering in 2005, both from the Univer- sity of Napoli Federico II, Italy. From 2001 to 2002, she worked as embedded system designer in the Telecommunications, Peripherals and Auto- motive Group, STMicroelectronics, Milano, Italy. Since 2005, she has been an Assistant Professor with the Department of Engineering, University of Napoli Parthenope, Italy. Her research interests are in the broad area of signal processing for communications, with current emphasis on multicarrier modulation systems, space-time techniques for cooperative and cognitive communications, green communications for IoT. Dr. Darsena has served as an Associate Editor for the IEEE COMMUNICATIONS LETTERS from December 2016 to July 2019. Since August 2019 she has been a Senior Area Editor for IEEE COMMUNICATIONS LETTERS and Associate Editor for IEEE ACCESS since October 2018.

GIACINTO GELLI (M’18-SM’20) was born in Napoli, Italy, on July 29, 1964. He received the Dr. Eng. degree summa cum laude in electronic engineering in 1990, and the Ph.D. degree in computer science and electronic engineering in 1994, both from the University of Napoli Federico II. From 1994 to 1998, he was an Assistant Pro- fessor with the Department of Information Engi- neering, Second University of Napoli. Since 1998 he has been with the Department of Electrical Engineering and Information Technology, University of Napoli Federico II, ﬁrst as an Associate Professor, and since November 2006 as a Full Professor of Telecommunications. He also held teaching positions at the University Parthenope of Napoli. His research interests are in the broad area of signal and array processing for communications, with current emphasis on multicarrier modulation systems and space-time techniques for cooperative and cognitive communications systems.

FRANCESCO VERDE (M’10-SM’14) was born in Santa Maria Capua Vetere, Italy, on June 12, 1974. He received the Dr. Eng. degree summa cum laude in electronic engineering from the Sec- ond University of Napoli, Italy, in 1998, and the Ph.D. degree in information engineering from the University of Napoli Federico II, in 2002. Since December 2002, he has been with the Univer- sity of Napoli Federico II. He ﬁrst served as an Assistant Professor of signal theory and mobile communications and, since December 2011, he has served as an Asso- ciate Professor of telecommunications with the Department of Electrical Engineering and Information Technology. His research activities include orthogonal/non-orthogonal multiple-access techniques, space-time processing for cooperative/cognitive communications, wireless systems optimization, and software-deﬁned networks. Prof. Verde has been involved in several technical program committees of major IEEE conferences in signal processing and wireless communications. He has served as Associate Editor for IEEE TRANSACTIONS ON COM- MUNICATIONS since 2017 and Senior Area Editor of the IEEE SIGNAL PROCESSING LETTERS since 2018. He was an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING (from 2010 to 2014) and IEEE SIGNAL PROCESSING LETTERS (from 2014 to 2018), as well as Guest Editor of the EURASIP Journal on Advances in Signal Processing in 2010 and SENSORS MDPI in 2018.

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