Signal Shaping for Generalized Spatial and Generalized Quadrature Spatial Modulation

Item Type Article

Authors Guo, Shuaishuai; Zhang, Haixia; Zhang, Peng; Dang, Shuping; Liang, Cong; Alouini, Mohamed-Slim

Citation Guo, S., Zhang, H., Zhang, P., Dang, S., Liang, C., & Alouini, M.- S. (2019). Signal Shaping for Generalized Spatial Modulation and Generalized Quadrature Spatial Modulation. IEEE Transactions on Wireless Communications, 18(8), 4047–4059. doi:10.1109/ twc.2019.2920822

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DOI 10.1109/TWC.2019.2920822

Publisher Institute of Electrical and Electronics Engineers (IEEE)

Journal IEEE Transactions on Wireless Communications

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Link to Item http://hdl.handle.net/10754/655513 1 Signal Shaping for Generalized Spatial Modulation and Generalized Quadrature Spatial Modulation Shuaishuai Guo, Member, IEEE, Haixia Zhang, Senior Member, IEEE, Peng Zhang, Member, IEEE, Shuping Dang, Member, IEEE, Cong Liang, and Mohamed-Slim Alouini, Fellow, IEEE

Abstract—This paper investigates generic signal shaping meth- multiple in-phase and quadrature (IQ) RF chains for multiple- ods for multiple-data-stream generalized spatial modulation data-steam transmission and additionally carry information by (GenSM) and generalized quadrature spatial modulation (Gen- the selection of transmit antenna combinations (TACs). By QSM). Three cases with different channel state information at the transmitter (CSIT) are considered, including no CSIT, varying the number of RF chains, they can achieve an attrac- statistical CSIT and perfect CSIT. A unified optimization problem tive trade-off between spectral efficiency (SE) and energy effi- is formulated to find the optimal transmit vector set under size, ciency (EE). For general (Nt,Nr,NRF , n) GenSM/GenQSM power and sparsity constraints. We propose an optimization- MIMO systems with Nt transmit antennas and Nr receive based signal shaping (OBSS) approach by solving the formulated antennas conveying a fixed length of n-bit stream via N RF problem directly and a codebook-based signal shaping (CBSS) RF approach by finding sub-optimal solutions in discrete space. chains, this paper investigates generic signal shaping methods n In the OBSS approach, we reformulate the original problem to find the optimal 2 transmit vectors. This is a rather to optimize the signal constellations used for each transmit intricate task because it couples the multi-dimensional signal antenna combination (TAC). Both the size and entry of all signal constellation optimization as well as the spatial constellation constellations are optimized. Specifically, we suggest the use of a optimization. In this paper, we aim to solve the problem based recursive design for size optimization. The entry optimization is formulated as a non-convex large-scale quadratically constrained on the criterion maximizing the minimum Euclidean distance quadratic programming (QCQP) problem and can be solved by (MMED). After that, we also discuss the designs based on existing optimization techniques with rather high complexity. To the criteria minimizing the symbol error rate (MSER) and reduce the complexity, we propose the CBSS approach using a maximizing the mutual information (MMI). codebook generated by quadrature (QAM) symbols and a low-complexity selection algorithm to choose the optimal transmit vector set. Simulation results show that the A. Prior Work OBSS approach exhibits the optimal performance in comparison with existing benchmarks. However, the OBSS approach is All existing signal constellation designs, spatial constella- impractical for large-size signal shaping and adaptive signal tion designs, precoding schemes (including phase rotation and shaping with instantaneous CSIT due to the demand of high power allocation schemes) as well as their combinations can computational complexity. As a low-complexity approach, CBSS be viewed as signal shaping methods, because all of them shows comparable performance and can be easily implemented in large-size systems. affect the transmit vectors. Based on the required information, we classify prior work into the following two categories: Index Terms—Multiple-input multiple-output, generalized s- patial modulation, generalized quadrature spatial modulation, 1) Signal Shaping without CSIT signal shaping, precoding, maximizing the minimum Euclidean Without CSIT, GenSM/GenQSM systems can benefit from distance, sparsity constraint the off-line design of signal and spatial constellation. Specif- ically, the analysis in [8] and [9] showed that the shape of I.I NTRODUCTION the signal constellation greatly affects the error performance, based on which [10]–[14] investigated the signal constellation ULTIPLE-DATA-STREAM generalized spatial mod- design for SM based on pre-defined constellation structures M ulation (GenSM) and generalized quadrature spatial or optimization techniques. However, the assumption that all modulation (GenQSM) have emerged as new techniques for TACs utilize the same signal constellation in [10]–[14] limits multiple-input multiple-output (MIMO) communications with the system performance improvement and the application reduced radio frequency (RF) chains and fast antenna switches. to SM systems with an arbitrary number of antennas. To As generalized forms of spatial modulation (SM) [1]–[4] and tackle this issue, the authors of [15] and [16] proposed an quadrature spatial modulation (QSM) [5]–[7], they employ optimization strategy jointly considering the signal and spatial S. Guo, S. Dang and M.-S. Alouini are with Computer Electrical constellations for SM. [16] showed that the joint optimization and Mathematical Sciences & Engineering (CEMSE) Division, King Ab- strategy achieves better performance than the sole signal dullah University of Science and Technology (KAUST),Thuwal 23955- constellation optimization strategies. On the other hand, the 6900, Kingdom of Saudi Arabia. (e-mail: {shuaishuai.guo; shuping.dang; slim.alouini}@kaust.edu.sa). gain is rather small yet costless, because the off-line design H. Zhang and C. Liang are with Shandong Provincial Key Laboratory of does not render any computational complexity for on-line Wireless Communication Technologies, Shandong University, Jinan 250061, data transmission. [16] extended the optimization strategy China, (e-mail: [email protected]; [email protected]). P. Zhang is with the School of Computer Engineering, Weifang Univer- regarding both signal and spatial constellations in GenSM sity, Weifang 261061, China (e-mail: [email protected]). systems using inter-channel interference (ICI)-free single-data- 2 stream transmission. However, the design for more spectral- the use of a recursive design presented in [16] for the efficient multiple-data-stream GenSM was left unconsidered. size optimization. The entry optimization is formulated Besides, little work is known on the constellation design as a large-scale non-convex quadratically constrained for GenQSM other than [17] proposed a heuristic signal quadratic programming (QCQP) problem, which can be constellation optimization for QSM and [18] studied a lattice- solved by existing optimization techniques with rather code-based constellation optimization strategy for GenQSM. high computational complexity. These are based on predefined codebooks and can be viewed • To reduce the computational complexity and also to facil- as an optimization strategy in the discrete space. Generic op- itate the implementation in realistic systems, we adopt a timization strategy in the continuous complex field regarding codebook generated by using quadrature amplitude mod- both signal and spatial constellations for GenQSM remains ulation (QAM) symbols to generate all feasible transmit unexplored. vectors. Then, we propose a low-complexity progressive 2) Signal Shaping with CSIT selection algorithm to choose the optimal transmit vector With instantaneous/statistical CSIT, GenSM/GenQSM sys- set. tems can benefit from adaptive signal shaping. For instance, • For extension, we discuss the designs relying on the adaptive signal constellation design was investigated for SM MSER and MMI criteria. Specifically, we formulate the in [19]; [20]–[22] studied antenna selection for SM systems, signal shaping problems to minimize the upper bound on which can be regarded as the adaptive spatial constellation the symbol error rate (SER) and to maximize the lower optimization; A large body of literature such as [23]–[38] bound on the mutual information, respectively. Analysis probed into precoding schemes as well as their combinations shows that both problems are reduced to the design based with the adaptive signal or spatial constellation optimization. on the MMED criterion in the high signal-to-noise ratio However, most literature considered SM or single-data-stream (SNR) regime. GenSM. To the best of our knowledge, few literature fo- • Numerical comparisons with existing designs are present- cused on the adaptive signal shaping for multiple-data-stream ed to verify the superiority of our designs. In particular, GenSM other than a recent work [38] that modifies a given we compare the proposed joint constellation design with multiple-dimensional signal constellation via a diagonal or the sole signal constellation design in open-loop systems full precoder for each TAC. However, such precoding-aided without CSIT and closed-loop systems with statistical C- signal shaping methods in [38] are suboptimal. The reasons SIT, where the well-recognized best signal constellations are twofolds: First, each TAC carries the same number of data (e.g., the binary phase shift keying, BPSK) are included symbol vectors in [38], while due to the random nature of in comparisons; we compare the proposed adaptive signal wireless channels, different TACs corresponding to separate shaping methods with precoding-aided signal shaping channels have distinct information-carrying capabilities; Sec- methods proposed in [38]. Comparison results show that ond, the data symbol vectors in the signal constellation of the proposed optimization-based signal shaping (OBSS) an activated TAC are modified by the same precoder, which approach considerably outperforms existing designs in cannot guarantee the global signal shaping optimality, because literature and the proposed codebook-based signal shap- the performance also highly depends on the previously given ing (CBSS) approach shows comparable performance signal constellation. In addition, few work was dedicated to with much lower complexity. Moreover, the performance the adaptive shaping for GenQSM, except [6] that adjusts the of the proposed designs with imperfect channel state precoding process for QSM systems with a single IQ RF chain. information (CSI) is also investigated by simulations. In summary, both constellation design and adaptive signal shaping have been extensively investigated for SM and single- C. Organization data-stream GenSM. A few of literature studied the signal The remainder of the paper is organized as follows. Section shaping for QSM systems with a single IQ RF chain. How- II describes the system model. Section III introduces the ever, the designs for multiple-data-stream GenSM/GenQSM unified problem formulation for the signal shaping of multiple- with/without CSIT call for a systematic investigation, which data-stream GenSM/GenQSM with/without CSIT based on motivates us to fill this gap with the contributions listed infra. the MMED criterion. In Sections IV and V, we introduce the OBSS and CBSS approaches, respectively. Sections VI B. Contributions discusses the extensions to the designs with MSER and MMI as optimization criteria. Numerical comparisons are presented • This paper formulates a unified signal shaping optimiza- in Section VII and conclusions are drawn in Section VIII. tion problem for multiple-data-stream GenSM/GenQSM with/without CSIT based on the MMED criterion, which aims to find the optimal transmit vector set under a size D. Notations constraint, a unit average power constraint and a special In this paper, a represents a scaler; a is a vector; A stands sparsity constraint. for a matrix. ||a||0 and ||a||2 stand for the l0 norm and the • To solve the formulated problem, we reformulate the l2 norm of a, receptively. diag (a) is a diagonal matrix whose original problem to find the optimal signal constellations diagonal entries are from vector a. a(i) denotes the ith entry for each TAC. We optimize both the size and entry of a. diag {A} stands for a vector formed by the diagonal of all signal constellations. In particular, we suggest elements of matrix A. det(A) represents the determination of 3

1 1

TAC selection 2 2

(real part) . . . . . NRF-dim signal . . . n bits NRF I/Q

constellation . . .

TAC selection (imaginary part)

NRF Q/I Nt Nr x H y

Fig. 1. A typical GenSM/GenQSM MIMO system. matrix A. ⊗ stands for the Kronecker product. represents the where Hadamard product. (·)T , (·)† stand for the transpose and the " # " # <(yˆ) <(Hˆ ) −=(Hˆ ) conjugate, respectively. R is the real domain; Z represents the y = , H = , 2 ˆ ˆ integer domain; C stands for the complex domain. CN (µ, σ ) =(yˆ) =(H) <(H) " # " # (3) denotes the complex Gaussian distribution with mean µ and <(xˆ) <(nˆ) variance σ2. (·) and [·] represent the expectation op- x = , n = . EA EA =(xˆ) =(nˆ) eration with respect to A. A is a set and |A| represents the size of set A. [A]k,l represents the kth row lth column n We use XN = {x1, x2, ··· , xN } of size N = 2 to represent entry of A. <(·) and =(·) represent the real and imaginary = [<(ˆ )T , =(ˆ )T ]T ∈ n  the transmit vector set, where xi xi xi parts, respectively. b·c represents the floor operation. is 2Nt m R . It is assumed that the transmit vectors in XN are under a binomial coefficient. a unit average power constraint, which is expressed as

N 1 X P (X ) = (||x||2) = x T x ≤ 1. (4) II.S YSTEM MODEL N E N i i i=1 A. System Framework Besides, they are also under a sparsity constraint, because the number of data streams sent via RF chains should be less than In this paper, we consider an (Nt,Nr,NRF , n) GenS- or equal to the number of RF chains. Specifically, for GenSM, M/GenQSM MIMO system as illustrated in Fig. 1, where the sparsity constraint is given by Nt and Nr represent the numbers of transmit and receive √ R I antennas; NRF is the number of RF chains and n stands ||xi + jxi ||0 ≤ NRF , i = 1, 2, ··· , N, j = −1, (5) for the target transmission rate in bit per channel use (bpcu). Unlike the conventional GenSM and GenQSM that map data while for GenQSM, the sparsity constraint is given by bits separately to the signal and spatial constellation points, R I ||xi ||0 ≤ NRF , ||xi ||0 ≤ NRF , i = 1, 2, ··· ,N. (6) we map n data bits jointly to a transmit vector xˆ ∈ CNt , which can be viewed as a high-dimensional symbol. In such a R T Nt where xi , [xi(1), xi(2), ··· , xi(Nt)] = <(xˆ) ∈ R and system, the number of used TACs does not need to be a power I T Nt xi , [xi(Nt + 1), xi(Nt + 2), ··· , xi(2Nt)] = =(xˆ) ∈ R of two, and the signal constellations used for different TACs represent the vectors being composed of the first Nt entries are not required to be the same [15]. and the last Nt entries of vector xi, respectively. With xˆ being transmitted, the receive signal vector yˆ ∈ CNr Remark: The sparsity constraints for the transmit vectors can be written as of GenSM and GenQSM are different from their conventional √ ˆ counterparts, because the positions of non-zero elements in the yˆ = ρHxˆ + nˆ, (1) R I transmit vectors are constrained. That is, let Ii and Ii be the index sets of non-zero elements in xR and xI , respectively. ˆ Nr ×Nt i i where ρ denotes the average receive SNR; H ∈ C For GenSM, the sparsity constraint can be re-expressed as is the channel matrix and nˆ ∈ CNr represents the complex I R Gaussian noise vector with zero mean and unit variance, i.e., |Ii ∪ Ii | ≤ NRF , i = 1, 2, ··· ,N. (7) nˆ ∼ CN (0, INr ). The transmission can be re-expressed in the real domain as For GenQSM, it is √ I R y = ρHx + n, (2) |Ii | ≤ NRF , |Ii | ≤ NRF , i = 1, 2, ··· ,N. (8) 4

For comparison purposes, we also give the conventional spar- Therefore, the signal shaping optimization problems for GenS- sity constraint without the position limitation as follows: M and GenQSM of the aforementioned three cases can be formulated as ||xi||0 ≤ 2NRF , i = 1, 2, ··· ,N, (9) (P1) : Given : A,NRF ,N or Find : XN = {x1, x2, ··· , xN } I R |Ii | + |Ii | ≤ 2NRF , i = 1, 2, ··· ,N. (10) Maximize : dmin(XN , A)

Subject to : |XN | = N (17a)

B. Channel Model P (XN ) ≤ 1 (17b) R I In this paper, a specific transmit-correlated Rayleigh channel ||xi + jxi ||0 ≤ NRF (17c) model is adopted and its channel matrix can be written as [16] i = 1, 2, ··· ,N, ˆ ˆ 1/2 H = HwRtx , (11) and

ˆ Nr ×Nt where Hw ∈ C represents a complex Gaussian matrix (P2) : Given : A,NRF ,N ˆ Nt×Nt with [Hw]k,l ∼ CN (0, 1) and Rtx ∈ denotes the C Find : XN = {x1, x2, ··· , xN } transmit correlation matrix. The correlation weight matrix R ∈ Maximize : d (X , A) R2Nt×2Nt can be written in the real domain as min N Subject to : |X | = N (18a) " 1/2 1/2 # N <(Rtx ) −=(Rtx ) R = . P (XN ) ≤ 1 (18b) 1/2 1/2 (12) =(Rtx ) <(Rtx ) R I ||xi ||0 ≤ NRF , ||xi ||0 ≤ NRF (18c) Despite the specific channel model is used as an example i = 1, 2, ··· ,N, for illustration purposes, it should be noted that the proposed optimization strategies and obtained results are applicable to respectively. In (P1) and (P2), (17a), (18a) represent the size generalized channel models. constraints; (17b), (18b) are the unit average power constraints and (17c), (18c) are the special sparsity constraints. To replace the sparsity constraints in (17c) and (18c), we III.U NIFIED OPTIMIZATION PROBLEM FORMULATION k 2Nt×2NRF express xi as xi = Fksl , where Fk ∈ R is a matrix BASEDONTHE MMEDCRITERION k 2NRF that corresponds to the kth TAC and sl ∈ R is the lth data symbol vector when F is activated. For GenSM, F can In this section, we formulate a unified optimization problem k k be expressed as of the signal shaping for multiple-data-stream GenSM and GenQSM based on the MMED criterion in three cases that " # GenSM Cu 0 are without CSIT, and with statistical as well as instanta- Fk = , (19) 0 Cu neous CSIT, respectively. Without CSIT, XN is designed for maximizing the minimum Euclidean distance of the transmit where C ∈ Nt×NRF is an antenna selection matrix com- vectors by u R posed of NRF basis vectors of dimension Nt. Let FGenSM GenSM 0 denote the set of all feasible Fk . Since there are totally max dmin(XN , I2Nt ) = max min ||xi − xi ||2. (13)   xi6=x 0 ∈XN Nt GenSM i different Cu, the number of feasible F in NRF k  N   N  With statistical CSIT (i.e., R), XN is designed for maximizing t t FGenSM is N , corresponding to N feasible TACs. the minimum Euclidean distance of the correlation wighted RF RF For GenQSM, Fk can be given by transmit vectors by " # GenQSM Cu 0 max dmin(XN , R) = max min ||R(xi − xi0 )||2. (14) Fk = , (20) xi6=xi0 ∈XN 0 Cv

With perfect CSIT, XN is designed for maximizing the min- Nt×NRF where Cu, Cv ∈ R are two independent antenna imum Euclidean distance of noise-free receive signal vectors selection matrices. Let F denote the set of all fea- GenQSM  by GenQSM Nt sible F . As there are different Cu and Cv, k NRF  2 max dmin(XN , H) = max min ||H(xi − xi0 )||2. (15) GenQSM Nt the number of feasible F in FGenQSM is , xi6=xi0 ∈XN k NRF  2 corresponding to Nt feasible TACs. According to the optimization problems formulated above, we NRF use a 2Nt-column weight matrix A to represent I2N , R or H, GenSM GenQSM t For unification, we use Fk to represent Fk or Fk and rewrite three objective functions in a unified form to be and F to represent FGenSM or FGenQSM. The kth signal k constellation Sk is defined as the set of sl when Fk is max dmin(XN , A) = max min ||A(xi − xi0 )||2. (16) xi6=xi0 ∈XN activated and the set of all signal constellations is represented 5

 by Z = S1, S2, ··· , S|F| . Based on these denotations, (P1) which can be regarded as an indicator of the signal constel- and (P2) can be expressed in a unified manner as follows: lation sizes {|Sk|} and the number of Fk in WN indicates the signal constellation size |S | for the kth TAC. With the (OP) : Given : A,N, F k definition of W , the extension of the recursive design in  N Find : Z = S1, S2, ··· , S|F| multiple-data-stream cases can be described as follows. Given Maximize : dmin(F, Z, A) WN−1, we can choose an Fk ∈ F to adjoin WN−1 generating |F| (21) |F| candidates of WN . For each candidate of WN , we perform X Subject to : |Sk| = N set size optimization and obtain the corresponding candidates k=1 of XN . Then, by comparing all the candidates of XN , we P (Z) ≤ 1, can obtain a suboptimal XN among all the candidates and the corresponding suboptimal WN . Based on this principle, where we use the optimal X2 and W2, which can be obtained by 0 k k exhaustive search, to find a suboptimal X3 and W3, then X4 dmin(F, Z, A) = min ||A(Fks − Fk0 s 0 )||2, (22) 0 l l F sk6=F sk k l k0 l0 and W4 and so on until the size constraint is satisfied. , ∈F Fk Fk0 0 sk∈S , sk ∈S l k l0 k0 S ,S ∈Z k k0 B. Set Entry Optimization and

|Z| |Sk| Given a fixed WN , the sizes of S1, S2, ··· , S|F| are de- 1 X X P (Z) = (sk)T sk. (23) termined and we now need to optimize the set entries in N l l k=1 l=1 each set to maximize dmin(F, Z, A) in (OP). To solve the k k problem, we define Sl , diag(sl ) for all k = 1, 2, ··· , |F|, IV.O PTIMIZATION-BASED SIGNAL SHAPING l = 1, 2, ··· , |Sk|, and a diagonal matrix Dq of dimension 2NNRF × 2NNRF as Since the problem (OP) for optimizing XN has been reformulated to search the optimal signal constellations S ,  1  1 S1 0 0 ··· 0 0 0 S2,... ,S|F| in the last section, the problem becomes a set  .   0 .. 0 ···· 0 0  optimization problem including the set size optimization and   the set entry optimization. We analyze both sub-problems with  1   0 0 S|S | ··· 0 · 0  details in the following subsections.  1   ......   ......  Dq ,  ......  , (25)   A. Set Size Optimization  · ··· |F|   0 0 S1 0 0    The set size optimization is a non-negative integer program-  ..  P|F|  0 0 ···· 0 . 0  ming satisfying k=1 |Sk| = N. There are a total number    N+|F|−1  0 0 0 ··· 0 0 S|F| of |F|−1 feasible solutions [39], which is rather large. |S|F|| Taking a MIMO system with Nt = 4, NRF = 2 and N = 16 4 2NNRF  as well as a vector ei ∈ as ei g ⊗ 12N where (i.e., n = 4 bpcu) as an example, |FGenSM| = 2 = 6 and R , i RF 2 4  gi is the ith N-dimensional vector basis with all zeros except |FGenQSM| = = 36. We can easily calculate that there   2 the ith entry being one. Based on these definitions, the square 16+6−1 ≈ 2×104 are 6−1 feasible size solutions for GenSM and of the pairwise Euclidean distances can be expressed as  16+36−1  12 36−1 ≈ 7.2×10 solutions for GenQSM, respectively. 2 2 ||Ax − Ax 0 || = ||AW D e − AW D e 0 || Therefore, exhaustive search for the optimal solution is infea- i i 2 N q i N q i 2 T T T T sible, because the set entry optimization is needed for each = (ei − ei0 ) Dq WN A AWN Dq(ei − ei0 ) feasible set size solution. In [16] that considers single-data-  T  = Tr D RAWDq∆Eii0 , stream transmission without CSIT, a greedy recursive method q (26) was proposed by finding the optimal set size solution for XN with size N according to XN−1 with size N −1. The recursive T T T = ∆ 0 = ( − 0 )( − 0 ) method shows comparable performance to the exhaustive where RAW WN A AWN and Eii ei ei ei ei . Tr( T ) = T ( ) search in the single-data-steam cases but demands much lower Adopting the equality DuUDvV u U V v, where = diag( ) = diag( ) complexity [16]. Thus, we also suggest the use of its extension Du u and Dv v , we rewrite (26) as in multiple-data-steam cases. To introduce the extension, we 2 T || − 0 || = 0 , 2Nt×2NNRF Axi Axi 2 q Qii q (27) define a constellation partition matrix WN ∈ R for XN with size N as 2NNRF T where q = diag{Dq} ∈ R and Qii0 = RAW ∆Eii0 ∈   2NNRF ×2NNRF |S1| |S2| |S|F|| R . As a consequence, the unit power constraint z }| { z }| { z }| { can be expressed as WN , F1, ··· , F1, F2, ··· , F2, ··· , ··· , F|F|, ··· , F|F| , 1   1 P (Z) = Tr D DT = qT q ≤ 1. (28) (24) N q q N 6

Based on the above reformulations, the set entry optimization 10 becomes 0 8 (S-OP) : Given : Qii0 , ∀i 6= i ∈ {1, 2, ··· ,N} Find : q 6 T (29) Maximize : min q Qijq 4 Subject to : qT q ≤ N 2 By introducing an auxiliary variable τ, the optimization prob- lem can be equivalently transformed to be 0 N=4 N=8 N=16 0 (S-OP-a) : Given : Qii0 , ∀i 6= i ∈ {1, 2, ··· ,N} Find : q, τ Fig. 2. Average number of iterations that the algorithm proposed in [38] takes to converge for solving (S-OP-b). Maximize : τ T 0 Subject to : q Qijq ≥ τ, ∀i 6= i ∈ {1, 2, ··· ,N} Algorithm 1 OBSS Procedure T q q ≤ N Input: A, Nt, NRF , F and N (30) Output: XN %% Exhaustive Search for X2 and W2 Problem (S-OP-a) is a non-convex large-scale QCQP problem 2  N  Generate |F| feasible candidates of W2. with 2NNRF variables and constraints, and can be 2 For each W2 candidate, compute q2 by solving (S-OP). solved by the iterative algorithm developed in [30] with Combining all W2 with the corresponding q2, we obtain complexity about O(N 4N 2 ) in each iteration1. Moreover, 2 RF the |F| candidates of X2. because the minimum Euclidean distance is monotonically Compare all the candidates of X2 in terms of minimum increasing with the increase of the average power, problem Euclidean distance to find the optimal one and the corre- (S-OP-a) that maximizes the minimum distance subject to sponding W2. an average power constraint can be reformulated to an op- %% Initialization timization problem minimizing the average power for a target Initialize t = 3. minimum distance, which can be expressed as %% Recursive Optimization for XN and WN 0 repeat (S-OP-b) : Given : Qii0 , ∀i 6= i ∈ {1, 2, ··· ,N} Generate |F| feasible W based on W . Find : q t t−1 For each Wt candidate, compute qt by solving (S-OP). Minimize : qT q Combining all Wt with the corresponding qt, we obtain T 0 Subject to : q Qijq ≥ d, ∀i 6= i ∈ {1, 2, ··· ,N}, the |F| candidates of Xt. (31) Compare all the candidates of Xt in terms of minimum Euclidean distance to find the suboptimal one and the where d is the target minimum distance. The problem (S-OP- related W . b) can be solved by the Lagrangian method developed in [38]. t Update t ← t + 1. The complexity is also around O(N 4N 2 ) in each iteration1, RF until t > N. but fortunately the algorithm in [38] converges faster than that Output the optimized X . in [30]. Additionally, it should be noted that problem (S-OP- N b) is formulated without any power constraint and one should further scale the optimized transmit vectors to meet the unit average power constraint. determine the number of variables and N solely determines In the paper, we use the advanced algorithm developed in the number of constraints. [38] to solve the formulated QCQP problem (S-OP-b). The number of iterations that the algorithm in [38] takes to con- verge is one of the key factors dominating the computational C. Complexity Analysis complexity. Furthermore, we demonstrate how the parameters N and NRF affect the key factor in Fig. 2. In the simulations For clearly viewing the detailed design procedure of OBSS, with N = 4 and N = 4, the number of iterations that the t r we list it in Algorithm 1, where qt, Wt and Xt are the algorithm in [38] takes to converge is obtained by averaging temporary variables in the tth iteration. The complexity of over 100 realizations. Observing the results demonstrated in Algorithm 1 is determined by the exhaustive search for X2 and Fig. 2, we find that the average number of iterations that the the recursive optimization for XN . In the exhaustive search for 2 algorithm [38] takes to converge for solving (S-OP-b) slightly X2, we need to solve (S-OP) for |F| times. The aggregate increases as NRF and N increase, because both of them 2 4 2 complexity is O(|F| Niter2 2 NRF ), where Niter2 denotes the number of iterations for solving (S-OP). It is obvious that 1We omit the other terms in the complexity analysis in [30] and [38] for simplicity, since N is much larger than other components constituting the the complexity is rather low and thereby negligible. In the complexity. recursive optimization for XN , we need to solve (S-OP) for 7

TABLE I COMPARISONS AMONG DIFFERENT SIGNAL SHAPING APPROACHES

Approaches Number of variables Number of constraints Complexity Applications  N  5 2 OBSS 2NNRF 2 O(Nitert |F|N NRF ) GenSM/GenQSM  N  D 2 2 2 Diagonal precoding [38] 2|Fs|NRF 2 O(Niter|Fs| NRF N ) GenSM 2  N  F 2 4 2 Full precoding [38] 2|Fs|NRF 2 O(Niter|Fs| NRF N ) GenSM 2 CBSS - - O(Nc ) GenSM/GenQSM

|F| times for designing Xt and the aggregate complexity is Algorithm 2 Progressive Selection Algorithm N Input: A, Nc, N and Xc X 4 2  5 2 Output: X C = O Nitert |F|t NRF ≈ O(Nitert |F|N NRF ), t=3 Compute the powers of ∀xi ∈ Xc and ||A(xi − xi0 )||, where (32) xi, 6= xi0 ∈ Xc. Save them for the computation of the CFMs. where Nitert represents the number of iterations for solving %% Exhaustive Search for X   2 Nc (S-OP) to attain Xt and we assume {Nitert } are of the same Generate 2 feasible candidates of X2. order for any t. Compare all candidates in terms of CFM to find the optimal one. D. Remarks %% Initialization The proposed OBSS approach aims to directly optimize the Initialize t = 3. transmit vectors, which is different from the precoding-aided %% Progressive Selection to Design XN signal shaping proposed in [38] that modifies a given signal repeat constellation of size M NRF via a diagonal preceding matrix or Select a vector xi ∈ Xc \Xt−1 and add it into Xt−1 as a full precoding matrix for each TAC, where they assume M- the candidates of Xt. ary modulation is adopted for each data stream transmission. Compare all candidates of Xt in terms of CFM and find Even though the diagonal precoder and the full precoder can be the optimal one. designed by solving a similar QCQP problem as (S-OP), there Update t ← t + 1. exists a difference in the number of optimization variables. until t > N. For the diagonal precoder optimization, the total number of Output the selected XN . variables is 2|Fs|NRF , while for the full precoder optimiza- 2 tion, the total number of variables is 2|Fs|NRF , where Fs is a selected subset of F and |F | = 2blog2 |F|c. Compared to s approach. We use a codebook generated by Mc-ary QAM 2 2|Fs|NRF and 2|Fs|NRF , the number of variables 2NNRF in modulation symbols, which is inspired by the work presented NRF (S-OP) is much larger, because N = |Fs|M . Additionally, NRF in [18]. With the codebook, there are Mc feasible signal the number of constraints in (S-OP) and that in the precoding- NRF constellation points and a total number of Nc = |F|Mc aided signal shaping in [38] are the same. That is, the scale of feasible transmit vectors in the candidate set Xc. Then, we the QCQP problem in the proposed OBSS approach is larger need to select a subset of size N. To reduce the selection than those formulated in precoding-aided shaping methods complexity, we progressively select vectors to maximize the proposed in [38]. As a result, the computational complexity constellation figure merit (CFM), which is equivalent to the of the proposed OBSS approach is much higher than the MMED criterion under normalized power constraint and de- precoding-aided shaping. Anyway, the proposed OBSS ap- fined by [40] proach can be treated as a generalized design and hence can 2 dmin(XN , A) yield better performance. Also, the global optimality of the CFM(XN ) , . (33) proposed OBSS approach cannot be guaranteed, because of P (XN ) the greedy set size optimization and the non-convexity of (S- This selection procedure is illustrated in Algorithm 2. The OP) in the set entry optimization. complexity of Algorithm 2 is dominated by the calculations 2 In summary, the OBSS approach is proposed for the sake of pairwise Euclidean distances, which is of the order O(Nc ). of performance enhancement, whereas its complexity is rather For clarity, comprehensive comparisons among the proposed high. It is affordable for off-line designs without CSIT or on- approach and other existing approaches are given in Table line designs with long-term invariant statistical CSIT. How- I. From Table I, we find that its complexity is much lower ever, the complexity is a heavy burden for large-size signal than existing approaches since it does not need to solve the shaping designs or adaptive designs with instantaneous CSIT. large-scale QCQP problems, which enables its extensions to For adaptive designs, the complexity needs to be greatly large-size signal shaping designs or adaptive designs with reduced before implementing these designs in practice. instantaneous CSIT.

V.C ODEBOOK-BASED SIGNAL SHAPING VI.E XTENSIONSTO MSER AND MMIDESIGNS To alleviate the heavy computation burden of the OBSS In this section, we investigate the extensions to the signal approach, we propose an alternative low-complexity CBSS shaping with the MSER and MMI as the design criteria, where 8 perfect CSI is assumed to be available at the transmitter and C. Remarks the receiver. By replacing (S-OP-b) with (SEP-OP) and (MI-OP), the OBSS approach can be extended to the designs based on A. MSER Signal Shaping MSER and MMI criteria, respectively. This problems can With a maximum likelihood (ML) employed at also be solved by the existing algorithms, e.g., the algorithm the receiver, the SER of GenSM/GenQSM systems is upper proposed in [33] and the complexity are also of the same order, bounded by [24] because the objective functions of (SEP-OP) and (MI-OP) are N N the functions of {qQii0 q} and the computation of {qQii0 q} 1 X X  ρ 2 Ps(XN ) = exp − ||H(xi − xi0 )|| . (34) dominates the computational complexity. Similarly, the CBSS 2N 4 2 i=1 i0=1, approach can also be extended by using the SER and MI as i06=i selection metrics. We remark that the MSER and MMI metrics Based on the re-formulation in Section IV-B with a fixed W , N are reduced to the MMED metric in the high SNR regime, we have 2 T because the SER upper bound in (36) and the MI lower bound ||H(x − x 0 )|| = q Q 0 q, (35) i i 2 ii in (40) are dominated by the minimum Euclidean distance term and the upper bound can be re-expressed as with the help of the exponential operator. Moreover, it should N N be noted that compared to the MMED design, the MSER and 1 X X  ρ T  P (q) = exp − q Q 0 q . (36) MMI designs are SNR-dependent, which means that they need s 2N 4 ii i=1 i0=1, to be re-designed as SNR varies. i06=i Thus, the optimization problem to minimize the upper bound VII.S IMULATIONS AND DISCUSSIONS on SER can be formulated as In the simulations, we investigate the performance of the 0 (SER-OP) : Given : Qii0 , ∀i 6= i ∈ {1, 2, ··· ,N}, ρ proposed OBSS and CBSS approaches in variously configured Find : q (Nt,Nr,NRF , n) GenSM and GenQSM MIMO systems. The (37) transmit-correlated Rayleigh channel model is adopted as Minimize : Ps(q) T described in Section II-B and the correlation matrix entries Subject to : q q ≤ N are defined by [16] ( B. MMI Signal Shaping δk−l, k ≤ l [Rtx]k,l = k, l = 1, ··· ,Nt (42) According to the similar analysis in [26], the mutual infor- (δl−k)†, l > k mation (MI) of GenSM/GenQSM systems given X as inputs where δ represents the transmit correlation coefficient. More- can be expressed as over, it is assumed that the transmitter and receiver both know I(x; y|H) = log2 N − · · · δ. We compare different signal shaping methods not only in the N ( N ) minimum Euclidean distance but also in the SER, where the 1 X X  2 2 2  En log2 exp −ρ(||H(xi − xi0 + n )||2 − ||n||2) .ML detector is employed. Over transmit-correlated Rayleigh N 0 i=1 i =1 model, the asymptotic upper bound on the SER given XN can (38) be expressed as [42] The expression is lower bounded by [41] N N X X 0 −2Nr ILB(x; y|H) = log2 N + Nr(1 − log2 e) Ps1 = c ||R(xi − xi)||2 , (43) i=1 0 0 N N  2  i =1,i6=i 1 X X ρ||H(xi − xi0 ||2 − log exp − . −Nr   N 2 2 where c = ρ 2Nr −1 . With CSIT, the upper bound on i=1 i0=1 N Nr (39) the SER is given be 2 T P = P (X ) . (44) Similarly, based on the expression ||H(xi − xi0 )||2 = q Qii0 q s2 EH s N 0 when i 6= i , we can obtain For clarity, we divide the section into three subsections. In

ILB(q) = log2 N + Nr(1 − log2 e) the first subsection, we compare the proposed OBSS solution   with the sole signal constellation optimized results in open- N N  T  1 X X ρq Q 0 q loop systems without CSIT and closed-loop systems with − log 1 + exp − ii  . N 2  2  statistical CSIT, where the optimal signal constellation has i=1 i0=1 i06=i already been known and adopted for comparison. In the second (40) subsection, we investigate the proposed OBSS and CBSS Thus, the optimization problem to maximize the lower bound optimization strategies in closed-loop systems with instanta- on MI can be formulated to be neous CSIT and compare them with the precoding-aided signal 0 shaping methods in [38]. Also, we investigate the proposed (MI-OP) : Given : Qii0 , ∀i 6= i ∈ {1, 2, ··· ,N}, ρ CBSS approach in large-scale systems and compare it with Find : q (41) the sole spatial constellation design. In the last subsection, Maximize : ILB(q) we investigate the impact of channel uncertainty on the error Subject to : qT q ≤ N performance. 9

TABLE II MINIMUM EUCLIDEAN DISTANCE COMPARISONS IN (3, 2, 2, 3) GENSM 100 MIMOSYSTEMS

- δ = 0 δ = 0.1 δ = 0.3

BPSK 1 0.7007 0.4850 -1 OBSS 1.4768 1.5738 1.6508 10

0 10 10-2 SER

10-1 10-3

10-2

-4

SER 10 0 2 4 6 8 10 12 14 16 18 20 10-3 SNR (dB)

Fig. 4. SER comparisons in (4, 2, 2, 4) GenSM MIMO systems with no 10-4 CSIT (δ = 0)/with statistical CSIT (δ = 0.1, 0.3).

10-1 0 2 4 6 8 10 12 14 16 18 20 (3,2,2,3) GenSM MIMO, OBSS SNR (dB) (3,2,2,3) GenSM MIMO, OBSS, UB (43) (4,2,2,4) GenSM MIMO, OBSS (4,2,2,4) GenSM MIMO, OBSS, UB (43) Fig. 3. SER comparisons in (3, 2, 2, 3) GenSM MIMO systems with no CSIT (δ = 0)/with statistical CSIT (δ = 0.1, 0.3). 10-2

A. Superiority of the Proposed Design in Open-Loop Systems SER without CSIT and Closed-Loop Systems with Statistical

CSIT 10-3 Firstly, we compare the proposed OBSS design with the well-recognized optimal signal constellation in (3, 2, 2, 3) GenSM MIMO systems under different channel conditions.

In (3, 2, 2, 3) GenSM systems, two TACs selected from all 10-4 3  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 = 3 ones can provide a rate of 1 bpcu, subtracting which we can derive that binary modulation is used for 2-data- stream carrying 2 bpcu. The BPSK as the optimal binary signal Fig. 5. SER performance at an SNR of 14 dB with respect to the correlation constellation is adopted for comparison. We compare the coefficient δ ranging from 0 to 1. optimized signal shaping with BPSK-based shaping in terms of the minimum Euclidean distances dmin(X8, R) as listed in Table II. It is found that the proposed OBSS optimization and 0.3, respectively. All above comparisons show that the strategy has a much larger dmin(X8, R) for δ = 0, δ = 0.1 OBSS approach can be used to combat transmit correlation and δ = 0.3 and as δ increases, dmin(X8, R) also increases. and can even benefit from the transmit correlation. Also, we The SER comparisons are illustrated in Fig. 3, where the present the SER performance at an SNR of 14 dB in highly- analytical upper bounds presented in (43) are also included. correlated systems, where the transmit correlation coefficient It demonstrates that OBSS outperforms (3, 2, 2, 3) GenSM δ is set to range from 0 to 1 as depicted in Fig. 5. The results with BPSK by 1 dB over independent Rayleigh channels (i.e., show that the SER performance of OBSS gets better as δ δ = 0). The gain is achieved at no expense. Over the transmit- increases under the given system setups. The reason behind correlated channels, results show that the OBSS approach can this is that even through the correlation increases the overlap bring a higher gain. Specifically, more than 3.5 dB and 8 between the column subspaces, it enlarges these subspaces. For −2 dB are achieved at an SER of 10 when δ = 0.1 and 0.3 general (Nt,Nr,NRF , n) GenSM MIMO systems with OBSS, respectively owing to the joint optimization of spatial and whether the correlation is always good is still an open question, signal constellation. because the exact relationship between dmin(XN , R) (which To show more results, we also make the comparisons determines the SER performance in the high SNR regime) and regarding SER in (4, 2, 2, 4) GenSM MIMO systems where R is mathematically unknown for general (Nt,Nr,NRF , n) BPSK is also employed for comparison. Fig. 4 demonstrates GenSM MIMO systems. But for some specific systems, the similar trends that more than 1 dB, 3 dB and 6 dB are correlation can be proved to be good, e.g., the (2, 2, 1, 1) SM achieved compared to GenSM with BPSK when δ = 0, 0.1 MIMO system with the square root of correlation matrix given 10

TABLE III MINIMUM EUCLIDEAN DISTANCE COMPARISONS IN (3, 2, 2, 4) GENQSM 100 MIMOSYSTEMS

- δ = 0 δ = 0.1 δ = 0.3 π 10-1 4 -BPSK 0.7071 0.4954 0.3430 OBSS 1.2852 1.2754 1.2614

10-2 by " √ # SER 1/2 1 δ -3 Rˆ = √ , (45) 10 tx δ 1

-4 in which one of the exact optimal complex vector sets Xˆ2 can 10 be obtained as (" # " #) 1 −1 Xˆ2 = {xˆ1, xˆ2} = , . (46) 0 2 4 6 8 10 12 14 16 18 20 0 0 SNR (dB) Based on these, we can calculate Fig. 6. SER comparisons in (3, 2, 2, 4) GenQSM MIMO systems with no √ ˆ ˆ 1/2 CSIT (δ = 0)/with statistical CSIT (δ = 0.1, 0.3). dmin(X2, R) = dmin(X2, Rtx ) = 2 1 + δ, (47) which increases as δ increases. Additionally, we note that the price paid to achieve such substantial gains in the transmit 1 correlated systems is just the knowledge of δ. Knowing δ 0.9 is sometimes costless as δ can be simply determined by the 0.8 antenna deployment. Secondly, to validate the performance superiority of the 0.7 proposed optimization strategies in GenQSM systems, we 0.6 make the minimum Euclidean distance and SER comparisons 0.5 in (3, 2, 2, 4) GenQSM MIMO systems. Since the original CCDF 0.4 BPSK cannot be directly applied to GenQSM MIMO systems π 0.3 because of the zeros in the imaginary parts, we use a 4 phase- rotated BPSK as the symbol modulation for GenQSM and 0.2 the rotation does not change its optimality. The comparison 0.1 results in terms of minimum Euclidean distance and SER are 0 presented in Table III and Fig. 6, respectively. It is observed 0 0.5 1 1.5 2 2.5 3 3.5 4 from Table III that the proposed OBSS approach can also bring considerable performance improvement in maximizing the minimum Euclidean distances when applied to GenQSM Fig. 7. CCDF comparisons in adaptive (3, 2, 2, 3) GenSM MIMO systems systems. For δ = 0, 0.1 and 0.3, the optimized minimum with perfect CSIT. Euclidean distances are almost the same. SER comparisons in Fig. 6 demonstrate that when δ = 0.3 the systems BPSK and the diagonal precoding proposed in [38], GenSM achieves a lower SER. It is inconsistent with the minimum with BPSK with the full precoding given in [38], GenSM with Euclidean distance comparisons. The reason is that the SER CBSS and 16QAM as the codebook, GenSM with MMED is determined not only by the minimum Euclidean distance OBSS, and GenSM with MSER OBSS (at an SNR of 10 dB) but also by the other pairwise Euclidean distances (c.f., (43)). is 0.1, 0.43, 0.5, 0.8, 0.86 and 0.858, respectively. From these Compared with GenQSM using π -BPSK, the proposed OBSS 4 results, MMED OBSS is the best and the open-loop BPSK- approaches bring substantial performance improvements by based shaping is the worst. The SER comparisons among these about 2 dB, 5 dB and 9 dB for δ = 0, 0.1 and 0.3, respectively. schemes are illustrated in Fig. 8, from which we observe that This validates the performance superiority of our design in the proposed OBSS approaches outperforms BPSK with the GenQSM systems. full precoding and the one with diagonal precoding by around 1 dB and 4 dB at an SER of 10−2, respectively. The proposed B. Superiority of the Proposed Designs in Closed-Loop Sys- CBSS approaches outperforms BPSK with diagonal precoding tems with Instantaneous CSIT by more than 2 dB at an SER of 10−2. Moreover, as shown With perfect CSIT, we investigate the proposed designs in in Table I, the CBSS approach is of very low computational (3, 2, 2, 3) GenSM MIMO systems, we plot the complemen- complexity compared to other schemes, and it would suit tary cumulative distribution function (CCDF) of the minimum practical implementation in realistic MIMO systems with a Euclidean distance dmin(X8, H) in Fig. 7. The probability limited processing capability. As expected, MSER OBSS s- Pr {dmin(X8, H) > 1.5} for GenSM with BPSK, GenSM with lightly outperforms MMED OBSS in terms of SER. Moreover, 11

100 100

10-1

10-1 10-2

10-3 10-2 SER SER 10-4

10-5 -3 10 BPSK and 32 fixed TACs, no CSIT BPSK and 32 adaptive TACs, with CSIT 10-6 4-QAM and 8 fixed TACs, no CSIT 4-QAM and 8 adative TACs, with CSIT CBSS with 4-QAM, with CSIT 10-7 10-4 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 SNR (dB) SNR (dB)

Fig. 8. SER comparisons in adaptive (3, 2, 2, 3) GenSM MIMO systems Fig. 9. SER comparisons in adaptive (10, 5, 2, 7) GenSM MIMO systems with perfect CSIT. with perfect CSIT. we find from the numerical and simulation results in Fig. 8 100 that the OBSS and CBSS approaches can increase the diversity order. That is, the SER performance of OBSS and CBSS with 10-1 CSIT decays faster as SNR increases than the shaping without CSIT. However, it should be mentioned that the exact diversity order is unknown, because the signal set XN with CSIT is 10-2 obtained according to H by optimization techniques. As a EH[ln Ps(XN )] SER result, the diversity order D = limρ→+∞ is not ln ρ 10-3 mathematically tractable. To show the performance of the proposed signal shaping in large-scale systems, we investigate the adaptive signal shaping 10-4 in (10, 5, 2, 7) GenSM MIMO systems. The computational complexity burden of the proposed OBSS approach and that of the precoding-based signal shaping proposed in [38] are over- 0 2 4 6 8 10 12 whelming and are infeasible for implementing the adaptive SNR (dB) shaping in the given systems. Therefore, we only investigate the low-complexity CBSS approach with 4-QAM as the code- Fig. 10. SER comparisons in adaptive (3, 2, 2, 3) GenSM MIMO systems with perfect and imperfect CSIT. book. For comparison, the SER performance of (10, 5, 2, 7) GSM MIMO systems with BPSK, 4-QAM employed for 2- data-stream and fixed/adaptive spatial constellation designs In the presence of channel estimation errors, the performance is also included. Results show that the CBSS approach that of all schemes degrades by a certain level. Despite this, the jointly optimizes the multi-dimensional signal constellation proposed OBSS approach maintains its championship. Clearly, and spatial constellation outperforms the schemes that solely the performance gains brought by our designs are robust to optimize the spatial constellation. channel estimation errors.

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