Generalized Spatial Modulation for Large-Scale MIMO Systems: Analysis and Detection

Transcription

1 Generalized Spatial Modulation for Large-Scale MIMO Systems: Analysis and Detection T. Lakshmi Narasimhan, P. Raviteja, and A. Chockalingam Department of Electrical and Communication Engineering Indian Institute of Science, Bangalore, India Abstract Generalized Spatial modulation (GSM uses n t antenna elements but fewer radio frequency chains ( at the transmitter. Spatial modulation and spatial multiplexing are special cases of GSM with = 1 and = n t, respectively. In GSM, apart from conveying information bits through modulation symbols, information bits are also conveyed through the indices of the active transmit antennas. In this paper, we derive analytical bounds on the code-word and bit error probabilities of maximum likelihood detection in GSM. The bounds are shown to be tight at medium to high signal-tonoise ratios (SNR. We also present a low-complexity detection algorithm based oeactive tabu search (RTS for GSM in largescale MIMO systems. Simulatioesults show that the proposed algorithm performs well and scales well in complexity. Keywords Large-scale MIMO systems, generalized spatial modulation, performance analysis, detection, reactive tabu search. I. INTRODUCTION Multiple-input multiple-output (MIMO systems with a large number of antennas (tens to hundreds can provide several advantages like increased spectral and power efficiencies, which are key requirements in next generation of wireless communication systems. Key technological issues that need to be addressed in the practical realization of such large-scale MIMO systems include design and placement of compact antenna arrays, multiple radio frequency (RF chains, and large-dimension transmit/receive signal processing techniques and algorithms [1],[]. Spatial modulation [3], a relatively new modulation scheme for multi-antenna systems, can alleviate the need to have a large number of RF chains in large-scale MIMO systems. In spatial modulation (SM, the transmitter has multiple transmit antennas but only one transmit RF chain. This means that only one antenna can be active at a time and the remaining antennas have to remain silent. The choice of the active antenna at a given time is made based on information bits. If n t = m is the number of transmit antennas, then the index of the active antenna is chosen using log n t = m information bits. A conventional modulation (e.g., QAM symbol is sent on the chosen antenna. If A is the modulation alphabet used, then the number of bits conveyed in one channel use in SM is m+log A. It has been shown that SM outperforms conventional modulation in multiuser MIMO systems on the uplink [4],[5],[6]. This is because, for a given spectral efficiency, a reduced modulation alphabet size can be used in SM compared to that in conventional modulation. The advantages of SM can be further enhanced through generalized spatial modulation (GSM [1],[7]. In GSM, the transmitter is allowed to have more than one transmit RF chain. Let denote the number of RF chains at the transmitter. In GSM, 1 n t. Spatial modulation and spatial multiplexing are special cases of GSM with = 1 and = n t, respectively. In GSM, in each channel use, modulation symbols are transmitted from antennas out of then t available( antennas. The choice of out ofn t antennas conveys log nrf n t information bits. This is in addition to the information bits conveyed by the modulation symbols. It has been shown that for a given modulation alphabet and n t, there exits an optimum that maximizes the spectral efficiency, and that this optimum can be less than n t [7]. In this paper, we are interested in the performance analysis of GSM and detection of GSM signals in large-scale MIMO systems. Our new contributions in this paper can be summarized as follows. We first analytically characterize the code-word error probability (CEP and the bit error probability (BEP of the GSM system and derive closed-form expressions for the upper bounds on CEP and BEP for maximum likelihood (ML detection. The obtained bounds are tight in the moderate-to-high SNR regime. The analytical bounds and simulatioesults show that, for a given spectral efficiency, GSM can outperform SM and spatial multiplexing. We then propose a algorithm for the detection of GSM signals in large-scale MIMO systems. The algorithm is based oeactive tabu search (RTS. An interesting aspect here is a neighborhood definition appropriate for GSM signal set. Simulatioesults show that the algorithm performs well in large-scale GSM-MIMO systems. The rest of the paper is organized as follows. The system model for GSM-MIMO is presented in Section II. The analysis of CEP and BEP of GSM-MIMO is presented in Section III. In Section IV, we present the detection algorithm for large-scale GSM-MIMO. Section V concludes the paper. II. GSM-MIMO SYSTEM MODEL Consider a GSM-MIMO system with n t antennas and RF chains at the transmitter, and antennas at the receiver. The transmitter uses GSM. The GSM transmitter is shown in Fig. 1. In each channel use, the transmitter selects out of n t antennas to transmit modulation symbols from /14/$ IEEE 171 Asilomar 14

2 antenna to the ith receive antenna, and w = [w 1 w w nr ] T is the noise vector whose entries are modeled as complex Gaussian with zero mean and variance σ. For this system model, the ML detectioule is given by ˆx = argmin where y Hx is the ML cost. y Hx, (4 Fig. 1. The GSM transmitter. a modulation alphabet A. This choice of antennas can be any one of the possible ( n t combinations. Thus, the number of information bits ( conveyed through indices of the chosen antennas is log nt. In addition to this, the number of information bits conveyed through the modulation symbols is log A. Therefore, the total number of bits conveyed in a channel use in GSM is given by ( nt η = log + log A bpcu. (1 Let S denote the GSM signal set, which is the set of all possible GSM signal vectors that can be transmitted. Out of the ( n t possible antenna activation patterns 1, only log ( n t activation patterns are needed for signaling. Let S denote this set of selected antenna activation patterns, where S = log ( n t n. Then, S rf is given by S = { x : x i A {}, x =, I(x S }, ( where x is the n t 1 transmit vector, x i is the ith entry of x, i = 1,,n t, x is the l -norm of the vector x, and I(x is a function that gives the activation pattern for x; for e.g., I(x = [ ] T = [1 1 1 ] T. Let us give an example of the GSM signal set. Let n t = 4, =, BPSK modulation, and S = {[1 1 ] T,[1 1 ] T,[1 1] T,[ 1 1 ] T }. The GSM signal set for this example is given by S 4,BPSK =,. The 1 received signal vector y = [y 1 y y nr ] T at the receiver can be written as y = Hx+w, (3 where x S n is the transmit vector, H t,a is the Cnr nt channel gain matrix, whose (i,jth entry h i,j CN(,1 denotes the complex channel gain from the jth transmit 1 An antenna activation pattern is a n t 1 vector consisting of 1 s and s, where a 1 in a coordinate indicates that the antenna corresponding to that coordinate is active and a indicates that the corresponding antenna is silent. III. CEP AND BEP ANALYSIS OF GSM-MIMO In this section, we analyze the CEP and BEP performance of ML detection in GSM-MIMO. Assume that all the transmit GSM signal vectors are equally likely. The ML detectioule in (4 can be written as ˆx = argmin n t y x k h k, (5 where x k is the kth element of x, and h k is the kth column of H. The pairwise error probability (PEP that x can be decoded as x S can be written as ( n t n t P(x x H=P y x k h k > y x k h k H ( n t n t = P y r x k h r,k > y r x k h r,k H, (6 whereh r,k is the(r,kth element ofh. LetA r = n t x kh r,k and Ãr = n t x kh r,k. Since x is the transmitted vector, y r = A r +w r, r = 1,,. Now, we can write ( P(x x H=P y r A r H > y r Ãr ( = P = P ( w r H > A r +w r Ãr H R((Ãr Arw r > A r, Ãr (7 where R((Ãr A r wr is a Gaussiaandom variable with mean zero and variance σ A r Ãr. Therefore, ( P(x x H = Q A r Ãr /σ ( n t = Q (x k x k h k /σ. (8 The argument in (8 is a central χ -distribution with degrees of freedom. Computation of the unconditional PEP requires the expectation of the Q(. function in (8 w.r.t. H, which can be obtained as follows [8]: PEP(x x=e H {P(x x H} r( nr +r =f(γ nr (1 f(γ r, (9 r r= 17

3 ( where f(γ 1 1 γ 1+γ, γ α 4σ, α = nt θ k, and θ k x k x k. Now, an upper bound on the average CEP can be obtained as P CEP 1 η \x PEP(x x. (1 From (1, the average BEP can be upper bounded as P BEP 1 d(x, x η PEP(x x, (11 η \x where d(x, x is the number of bits in which x differs from x. The total number of PEPs that are to be calculated is ( η ( η. Therefore, the complexity of the computation of the above bounds on CEP and BEP will increase exponentially for large values of n t,. In the following subsection, we propose a simplification that reduces this computational complexity. A. Computation of the upper bounds for large n t, The CEP expression in (1 can be written in the form: P CEP 1 S S η j=1 x:i(x=s i S x:i( x=s j S, x x PEP(x x. (1 For a given pair of activation patterns s i and s j, i,j {1,, S }, the total number of PEPs are A when i j, and A ( A when i = j. Complexity reduction 1: For a pair of activation patterns s i and s j, let A ij denote the set of active antennas that are common to both s i and s j. Define β ij = A ij. Note that β ij {,1,,min(,n t }. Also, note that for any i,j for which β ij = q, the value of the summation PEP(x x in (1 will be the same, x:i(x=s i x:i( x=s j, x x and so it is enough to compute this summation only once for each q. With this simplification, (1 can be written as P CEP 1 min(,n t η φ(q q= x:i(x=s i x:i( x=s j β ij=q PEP(x x, (13 where φ(q is the number of (s i,s j pairs for which β ij = q, which can be computed easily. Complexity reduction : For each value of q, we need to compute A PEPs. We propose to reduce this complexity as follows. The parameter α in (9 is the summation of n t terms. Out of these n t terms, n t ( + q terms will be zero for a given value of q. Of the ( +q non-zero terms, q terms will take values from J { c : c A}, and q terms will take values from L { c c : c, c A}. Let J = {j 1,j,,j m } and L = {l 1,l,,l n }, where j 1 < j < < j m, l 1 < l < < l n, m = J, and n = L. We write α as α = α 1 +α, where α 1 is the sum of q terms from J and α is the sum of q terms from L. Note that α 1 can take values in the range qj 1 to qj m. For a given value of α 1, the following equations must be satisfied: m m j i v i = α 1, v i = q, (14 where v i is an integer such that v i {,1,, (α 1 m k=i j kv k /j i }. Similarly, α can take values in the range ( ql 1 to ( ql n, and, for a given value of α, the following equations must be satisfied: l i u i = α, u i = q, (15 where u i is an integer such that u i {,1,, (α n k=i l ku k /l i }. Since α = α 1 +α, α lies in the range qj 1 +( ql 1 to qj m +( ql n. The choice of v i s and u i s to attain a particular α is not unique, i.e., there exist multiple pairs of x and x that correspond to different values of v i s and u i s but the same value of α. Thus, we need to evaluate (9 only once for a given value of α and count the number of possible combinations of v i s and u i s that correspond to that α. Remark: The above complexity reduction schemes significantly simplify the computation of (1, because without these simplifications the sum PEP(x x x:i(x=s i x:i( x=s j, x x needs to be computed for all i,j, which is prohibitive for large n t,. The following examples illustrate the achieved complexity reduction. Example 1: For n t =, = 16, we have S = 16. A direct computation of (1 which involves a double summation from 1 to S is prohibitive. Whereas for these parameters, q {,1,,6}. Hence (13 can be easily computed in much fewer computations. This illustrates complexity reduction 1. Example : For n t = 4, = 3, A = { j,+j,1 j,1+ j}, we have J = {}, L = {,4,8}. For a particular value of q, say q = 1, the summation in (13 requires computation of the PEPs for 64 different pairs of GSM signal vectors. But since α lies in the range 4 to, we need to compute only 17 PEPs. This illustrates complexity reduction. B. Results and discussion In this subsection, we present numerical results of the CEP and BEP performance of GSM-MIMO. We compare the analytical upper bounds with the simulatioesults. We use the notation (n t, -GSM to refer to a GSM-MIMO system with n t transmit antennas and transmit RF chains. In Fig., we compare the simulated CEP and BEP with the analytical upper bounds for the (4,3-GSM system with = 4 and 4-QAM, at a spectral efficiency of 8 bits per channel use (bpcu. From Fig., we see that the analytical upper bound is quite tight in the medium-to-high SNR regime. In Figs. 3 and 4, we present a comparison between the performance of GSM-MIMO with those of SM-MIMO and V- BLAST (spatial multiplexing MIMO. Recall that SM-MIMO and V-BLAST MIMO are special cases of GSM-MIMO with = 1 and = n t, respectively. Figure 3 shows the CEP 173

4 1 1 CEP (Sim. CEP (Ana. BEP (Ana. BEP (Sim. 1 1 CEP, BEP 1 n t = 4, = 3, = 4, 4 QAM spectral efficiency = 8 bpcu BEP 1 = spectral efficiency = 6 bpcu Fig.. BEP and CEP performance of (4,3-GSM system with = 4, 4-QAM, 8 bpcu. CEP =, spectral efficiency = 6 bpcu 1 3 (4, GSM, 4 QAM, (Ana. (4, GSM, 4 QAM, (Sim. (4,1 SM, 16 QAM, (Ana. (4,1 SM, 16 QAM, (Sim. (, VBLAST, 8 QAM, (Ana. (, VBLAST, 8 QAM, (Sim Fig. 3. CEP comparison between i (4,-GSM system with 4-QAM, ii (4,1-GSM system (i.e., SM-MIMO system with 16-QAM, and iii (,- GSM system (i.e., V-BLAST system with 8-QAM. =, 6 bpcu. comparison s between i (4,-GSM system with 4-QAM, ii (4,1-GSM system (i.e., SM system with 16-QAM, and iii (,-GSM system (i.e., V-BLAST system with 8-QAM, with =. Note that all the three systems have the same spectral efficiency of 6 bpcu. Figure 4 shows the corresponding BEP plots. From Figs. 3 and 4, we can observe that i the upper bounds are tight at medium-to-high SNRs, and ii the GSM- MIMO system outperforms both SM-MIMO and V-BLAST MIMO systems. IV. DETECTION IN LARGE-SCALE GSM-MIMO The complexity of ML detection in GSM-MIMO increases exponentially with increase in n t,. In this section, we propose a low complexity algorithm for GSM-MIMO signal detection. The algorithm is based oeactive tabu search with random restarts (R3TS. Details of the R3TS algorithm for detection in V-BLAST MIMO systems are available in [9],[1]. For adapting this algorithm for detection of GSM- MIMO signals, we need to define appropriate neighborhood for the GSM signal set. We define the neighborhood as follows. Neighborhood definition for GSM signal set: We define the neighborhood N(x for a GSM signal vector x S n as t,a the set of all possible signal vectors which differ from x in 1 3 (4, GSM, 4 QAM, (Ana. (4, GSM, 4 QAM, (Sim. (4,1 SM, 16 QAM, (Ana. (4,1 SM, 16 QAM, (Sim. (, VBLAST, 8 QAM, (Ana. (, VBLAST, 8 QAM, (Sim Fig. 4. BEP comparison between i (4,-GSM system with 4-QAM, ii (4,1-GSM system (i.e., SM-MIMO system with 16-QAM, and iii (,- GSM system (i.e., V-BLAST system with 8-QAM. =, 6 bpcu. either one modulation symbol or in one active antenna index. That is, N(x = N 1 (x N (x, where N 1 (x={z : z k = x k, k except for some k 1 ; I(z = I(x, z k1 A\x k1 }, (16 N (x={z : β ij = 1, where I(x = s i,i(z = s j ; z k = x k, k except for some k 1,k s.t. x k1 =,z k1 A,z k = }. (17 So, a transmitted vector x will have ( A 1 + (n t A neighbors. For e.g., for n t = 3, = and BPSK, N 1 =,. (18 N =,,. (19 Tabu matrix: For the above neighborhood definition, the tabu matrix T is of size (n t + ( n t A A, where the first n t A rows correspond to N 1 (x and the next ( n t A rows correspond to N (x. For a solution vector x, if z N 1 (x then it corresponds to ((k 1 1 A + t,t th position in the tabu matrix, where x k1 = a t, z k1 = a t, and a t,a t A. If z N (x, then it corresponds to ( n t ( k 1(n t k 1 ( k k 1 t,t th position in the tabu matrix T, where k 1 = min(k 1,k, k = max(k 1,k, x k1 =,z k1 = a t, x k = a t, and a t,a t A. R3TS-GSM detection algorithm: The algorithm starts with an initial solution vector x ( as the current solution. For example, x ( can be the MMSE solution vector x MMSE. All the entries of the tabu matrix are initially set to zero. Let m denote the iteration index, P the tabu period, g (m the vector with the least ML cost till the mth iteration, l rep the average number of iterations between two successive occurrences of the same solution vector. Initialize P = P, g ( = x (, and l rep =. In each iteration (e.g., mth iteration, perform the following steps. 174

5 Step 1: Find z best1 = argmin y Hz. The move from z N(x m x m to z best1 is accepted if any one of the following conditions is satisfied: (i y Hz best1 < y Hx m, (ii If z best1 N 1 (x m, then T((k 1 t,t =. If z best1 N (x m, then T ( n t A + ( k 1(n t k 1 A + ( k k 1 1 A + t,t =. If a move is accepted, then x (m = z best1. If a move is not accepted, then find z best = argmin z N(x m \z best 1 y Hz, and check the above conditions for z best. If this is also not accepted, theepeat the procedure for z best3, and so on. If all the neighbors are tabu, then all the entries in the T are decremented by the minimum value in T. Theepeat the procedure from z best1 to find x m. Step : After step 1, the new solution x m is checked for repetition. Repetition can be checked by comparing ML costs of all the solutions in the previous iterations. If there is a repetition, then l rep is updated and P P + 1. If y Hx m < y Hx m, then do if x m N 1 (x m then T((k 1 t,t =, if x m N (x m then T ( n t ( k 1(n t k 1 ( k k 1 t,t =, if I(x m S, then g (m = x m else if x m N 1 (x m, then T((k 1 t,t = P, if x m N (x m, then T ( n t ( k 1(n t k 1 ( k k 1 t,t = P, g (m = g (m. Step 3: Update the entries of the tabu matrix as T(r,s = max(t(r,s,. The algorithm can be stopped after a maximum number of iterationsmax iter or when thel rep value exceeds a threshold max rep. The performance of the algorithm can be improved by using multiple restarts, where, in each restart, we start with a different initial solution. The algorithm is stopped after a particular number of maximum restarts max rest or if the ML cost of the solution vector obtained so far is below the σ + nr σ 4. The best solution with least ML cost is declared as the final output solution. A. Results and discussions In Fig. 5, we show the performance (,16-GSM system with 4-QAM and (16,16-GSM system (i.e., V-BLAST system with 8-QAM, both at 48 bpcu and = 16. For V- BLAST detection, we used sphere decoding (i.e., ML detection. For GSM detection, we used the R3TS algorithm with the following parameters: max iter = 1, max rep = 3, max rest = 5. We have plotted CEP upper bounds as well as simulated CEP and BEP. The following observations can be made from Fig. 5: i the proposed complexity reduction techniques allow us to compute the CEP bounds for ML detection for large n t (=16, and (=16, and these bounds are tight at moderate-to-high SNRs (e.g., for n t = = 16, ii at CEP, BEP = 16 Spectral efficiency = 48 bpcu 1 3 (16,16 VBLAST, 8 QAM, ML det, CEP (Ana. (16,16 VBLAST, 8 QAM, ML det, CEP (Sim. (,16 GSM, 4 QAM, ML det, CEP (Ana. (,16 GSM, 4 QAM, R3TS det, CEP (Sim. (16,16 VBLAST, 8 QAM, ML det, BEP (Sim. (,16 GSM, 4 QAM, R3TS det, BEP (Sim Fig. 5. CEP and BEP of (,16-GSM system with 4-QAM and (16,16-GSM system (i.e., V-BLAST system with 8-QAM. = 16, 48 bpcu. high SNRs, the simulated CEP of R3TS detection (which a low complexity suboptimum detection is close to the CEP upper bound of ML detection for (,16-GSM system, and iii at moderate-to-high SNRs, (,16-GSM system with R3TS detection outperforms V-BLAST system with sphere decoding. V. CONCLUSIONS We studied large-scale generalized spatial modulation MIMO (GSM-MIMO systems. We first derived analytical upper bounds on the CEP and BEP performance of GSM- MIMO and showed that the bounds are tight at medium to high SNRs. We also proposed complexity reduction schemes that allowed the computation of the bounds for large n t,. We then presented a reactive tabu search (RTS based algorithm for the detection of large-scale GSM-MIMO signals. Our analytical and simulatioesults show that, for a given spectral efficiency, GSM-MIMO system can achieve better performance compared to SM-MIMO and V-BLAST systems. REFERENCES [1] A. Chockalingam and B. Sundar Rajan, Large MIMO Systems, Cambridge Univ. Press, Feb. 14. [] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L.Marzetta, O. Edfors, F. Tufvesson, Scaling up MIMO: opportunities and challenges with very large arrays, IEEE Sig. Proc. Mag., vol. 3, no. 1, pp. 4-6, Jan. 13. [3] M. Di Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo, Spatial modulation for generalized MIMO: challenges, opportunities and implementation, Proceedings of the IEEE, vol. 1, no. 1, Jan. 14. [4] N. Serafimovski1, S. Sinanovic, M. Di Renzo, and H. Haas, Multiple access spatial modulation, EURASIP J. Wireless Commun. and Networking 1, 1:99. [5] T. Lakshmi Narasimhan, P. Raviteja, and A. Chockalingam, Large-scale multiuser SM-MIMO versus massive MIMO, Proc. ITA 14, Feb. 14. [6] P. 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