Differential Spatial Modulation with Gray Coded ...

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/LCOMM.2016.2557801 IEEE COMMUNICATIONS LETTERS, VOL XX, NO. XX, MONTH, 2016

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Differential Spatial Modulation with Gray Coded Antenna Activation Order Jun Li, Miaowen Wen, Member, IEEE, Xiang Cheng, Senior Member, IEEE, Yier Yan, Sangseob Song, Member, IEEE, Moon Ho Lee, Life Senior Member, IEEE

Abstract—In this letter, we propose a gray code order of antenna index permutations for differential spatial modulation (DSM). To facilitate the implementation, the well-known TrotterJohnson ranking and unranking algorithms are adopted, which results in similar computational complexity to the existing DSM that uses the lexicographic order. The signal-to-noise ratio gain achieved by the proposed gray code order over the lexicographic order is also analyzed and verified via simulations. Based on the Gray coding framework, we further propose a diversityenhancing scheme named intersected gray (I-gray) code order, where the permutations of active antenna indices are selected directly from the odd (or even) positions of the full permutations in the gray code order. From analysis and simulations, it is shown that the I-gray code order can harvest an additional transmit diversity order with respect to the gray code order. Index Terms—Gray coding, differential modulation, SNR gain, diversity order, spatial modulation (SM).

I. I NTRODUCTION HE optimal maximum-likelihood (ML) decoding in spatial modulation (SM) requires the knowledge of channel state information (CSI), which complicates the implementation [1]–[3]. To solve this problem, recently, differential (D-)SM is proposed, which dispenses with the CSI [4], [5]. In DSM, the antenna activation orders, which can be specified by the permutations of set {1, 2, . . . , NT } with NT denoting the number of transmit antennas, are used as an information carrying mechanism. Consequently, the difference between antenna activation orders plays an important role in the bit error rate (BER) performance of DSM. In our previous work [4], the antenna activation orders are set to follow the lexicographic order such that similar permutations may lead to huge bit difference between the corresponding information bit sequences. This, however, will degrade the BER performance of DSM as a detection error most probably occurs between similar permutations at high signal-to-noise ratio (SNR). In this letter, we resort to the idea of Gray coding [6] to improve the BER performance of DSM. In the proposed scheme,

T

This work was in part supported by MEST 2015R1A2A1A05000977, NRF, South Korea, the National Nature Science Foundation of China under Grants 61501190 and 61571020, the Major Project from Beijing Municipal Science and Technology Commission under Grant D151100000115004, and the Nature Science Foundation of Guangdong Province under Grant 2014A030310389. (Corresponding author: Miaowen Wen.) J. Li, S. Song and M. H. Lee are with the Department of Electronics and Information Chonbuk National University, Jeonju, South Korea (e-mail: {lijun52018, ssong, moonho}@jbnu.ac.kr). M. Wen is with School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China (e-mail: [email protected]). X. Cheng is with School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China (e-mail: [email protected]). Y. Yan is with School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510006, China (e-mail: [email protected]).

each information bit sequence has only one bit difference from the previous and next ones, while its corresponding antenna index permutation has only two antenna indices difference from the pervious and next ones. For the ease of implementation in the case of large NT , we apply the well-known Trotter-Johnson ranking and unranking algorithms to build the relationship between the information bit sequence and the corresponding antenna index permutation. From theoretical analysis, it is revealed that the proposed gray code order can achieve an SNR gain up to 1.2dB over the lexicographic coder with similar computational complexity. Based on the Gray coding framework, we also propose a new scheme called intersected gray (I-gray) code order to improve the diversity performance of DSM, which takes the odd (or even) positions of the full permutations in the gray code order only for information conveying purpose. We show that the I-gray code order achieves an additional transmit diversity order as opposed to the gray code order. Notations: (·)H stands for Hermitian transpose. The complex number field is represented by C. IM is an identity matrix of size M × M . R{·} represents the real component of the argument. T r{·}, rank{·} and mod(·, ·) denote the trace, rank and modulus operations, respectively. A(i, j) denotes the (i, j)th element of matrix A. ⌊·⌋ and ⌈·⌉ indicate the floor and ceil operations, respectively. II. S YSTEM M ODEL We consider a base-band NR × NT multi-input multioutput (MIMO) system, where NR represents the number of receive antennas. Specifically, DSM works as follows. At the transmitter, the information bits are partitioned into transmitted blocks of which each is composed of m = ⌊log2 (NT !)⌋ + NT log2 (M ) bits and are to be transmitted over NT time slots, where M denotes the cardinality of the constellation S. Note that both the number of transmit antennas and the length of one transmitted block equal to NT due to the concept of DSM. For each transmitted block, the proceeding m1 = ⌊log2 (NT !)⌋ bits are first mapped into an integer d ∈ {1, 2, . . . , 2m1 } T and then into a permutation of antenna indices {A(d, i)}N i=1 , while the remaining m2 = NT log2 (M ) bits are used to select T transmitted signals {si }N i=1 from S [5]. At the t-th transmission duration, the transmitted matrix St ∈ CNT ×NT is obtained as St = St−1 Xt ,

(1)

where Xt ∈ CNT ×NT is the information matrix, which is determined by the information bits. From above introduction, it is clear that Xt (A(d, i), i) = si , where i ∈ {1, 2, . . . , NT }.

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Example 1: Generating permutations of {1, 2, 3, 4} in the gray code order ← − 1234, 1243, 1423, 4123, 4 − → 4132 1432, 1342, 1324, 4 ← − 3124, 3142, 3412, 4312, 4 − → 4321, 3421, 3241,3214, 4 ← − 2314, 2341, 2431, 4231, 4 − → 4213, 2413, 2143, 2134, 4

TABLE I A LOOK - UP TABLE FOR NT = 3

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Bits in LO1 Permutations Bits in GCO2 Permutations 00 (1 2 3) 00 (1 2 3) 01 (1 3 2) 01 (1 3 2) 10 (2 1 3) 11 (3 1 2) 11 (2 3 1) 10 (3 2 1) LO=lexicographic order 2 GCO=gray code order

Let Ht ∈ CNR ×NT denote the channel matrix with covariance INT . Thus, the received signal matrix Yt ∈ CNR ×NT can be expressed as Yt = Ht St + Nt ,

(2)

where Nt ∈ CNR ×NT is the Gaussian noise matrix with zero mean and covariance σ 2 INR . Assuming quasi-static fading, i.e., Ht−1 = Ht , (2) can be thus expressed by the (t − 1)-th received signal matrix as Yt = Yt−1 Xt − Nt−1 Xt + Nt .

(3)

Accordingly, the optimal ML detection can be derived as [4] b t = arg max T r{R{YH Yt−1 Xt }}, X t Xt ∈G

(4)

where G stands for the set composed of all valid information matrices. Finally, the information bits are recovered by the demapping of the estimated antenna activation order and the b t. demodulation of the transmitted signals in X III. I NDEX - MAPPING IN G RAY C ODE O RDER In this section, we present the idea of encoding permutations of antenna indices in the gray code order for DSM, which is only related to the m1 information bits. A. Look-Up Table Table I shows the mapping between the information bits and the corresponding permutations in the lexicographic order and gray code order for NT = 3, respectively, where two out of in total six permutations have been discarded. Note that although not specified, it can be readily figured out that both manners give rise to the same result for NT = 2. From Table I, it is obvious that the discarded permutations in the gray code order are different from those in the lexicographic order. However, one can see surprisingly that the lexicographic order for NT = 3 also satisfies the requirement of the gray code order, i.e., any two permutations having only two indices difference differ from only one bit. This implies that both manners for NT = 3 will result in the same BER performance. For NT > 3, the lexicographic order cannot meet the above requirement any longer and both manners will lead to totally different performance. This will be verified in the sequel. B. Ranking and Unranking Methods The look-up table method necessitates a storage of all permutations at both the transmitter and receiver, which becomes impractical for large NT . For example, 3628800 and 2.4329 × 1018 permutations need to be stored for NT = 10 and 20, respectively. Obviously, it is impractical to store them

for current computers. Therefore, it is advisable to create an easy-to-implement one-to-one mapping from the information bits to the corresponding permutations, called unranking, and inverse mapping from the permutations to the corresponding information bits, called ranking. To begin with, we have to first figure out how to generate permutations in the gray code order. Aiming at this, we resort to the idea presented in [7]. Example 1 gives a solution for NT = 4. From this example, we can expect some important properties of gray code order for a general NT as follows. For ease of exposition, let us define left and right moving ←− −→ directions for number NT as NT and NT , respectively. Also, define a directional indicator for number NT as INT , where ←− INT = 1 stands for the situation of NT and INT = 0 for the −→ situation of NT . If the whole swaps between number NT and the adjacent element in the set {1, 2, . . . , NT } for the same direction is said to be one round for number NT , one will see that the total number of rounds of number NT , denoted by LR , is (NT − 1)!. In addition, it can be concluded that INT = 1 if lNT is odd and INT = 0 if lNT is even, where lNT is a variable indicating that number NT is in the lNT th round. Details on the generation rule can be referred to the Trotter-Johnson ranking and unranking algorithms [7]. In what follows, we summarize them in our notations and mainly focus on their application to DSM. 1) Trotter-Johnson ranking algorithm: Assume that we have a permutation a = {a1 , a2 , . . . , aNT } where u, au ∈ {1, 2, . . . , NT }. Define Pau as the position of the element au in a[au ] , where a[au ] indicates the sub-permutation of a, which discards the elements greater than au . Initially, we will determine whether a is generated from the permutations {1, 2} or {2, 1} for obtaining l3 . It is obvious that a is generated from {1, 2} if P2 = 2, which indicates l3 = 1 (I3 = 1), while a is generated from {2, 1} if P2 = 1, which indicates l3 = 2 T (I3 = 0). For the case of NT > 3, {lj }N j=4 can be calculated recursively by lj = (lj−1−1) · (j−1)+ I¯j−1 · Pj−1 +Ij−1 · (j−Pj−1 ), (5) where I¯j−1 = 1 − Ij−1 . Finally, the integer d ∈ {1, . . . , 2m1 } corresponding to a is calculated by d = (lNT −1) · NT + I¯NT · PNT +INT · (NT +1−PNT ). (6) Then, the information bit sequence in the gray code order can be directly obtained from d. 2) Trotter-Johnson unranking algorithm: The unranking process is the inversion of the ranking process. Firstly, the information bit sequence in the gray code order is converted



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into an integer d. Then, lNT and PNT can be derived as lNT = ⌈d/NT ⌉ and PNT = mod(d, NT ) + 1, respectively. For the case of NT > 3, lk and Pk with k = NT − 1 can be calculated by lk = ⌈lk+1 /k⌉ and Pk = mod(lk+1 , k) + 1, respectively. Then, we set k := k − 1 and iteratively obtain lk and Pk , respectively. The process continues until k = 3, NT NT T when {lk }N k=3 and {Pk }k=3 are all ready. Note that {Ik }k=3 NT can be determined by {lk }k=3 . Finally, we can rebuild the permutation from the ranking algorithm.

TABLE II I- GRAY CODE ORDER FOR NT = 3

Candidates Permutations in GCO in GCO (1 2 3) (1 3 2) (1 2 3) (3 1 2) (1 3 2) (3 2 1) (3 1 2) (2 3 1) (3 2 1) (2 1 3) 1 I-GCO=I-gray code order

C. SNR Gain Analysis We now analyze the performance of the gray code order by comparing it with that of the lexicographic order. To isolate the effect of Gray coding, we consider the special case of DSM, T i.e., differential space shift keying (DSSK), in which {si }N i=1 are set to 1s for all time. An upper bound on the average bit error probability (ABEP) can be derived according to the union bound technique as 2 ∑1 2∑1 1 b q ), (7) b q ) Pr(Xp → X N (Xp → X m1 · 2m1 p=1 q=1 m

Pe ≤

m

b q ) is the pairwise error probability accountwhere Pr(Xp → X bq ing for the probability of detecting the information matrix X b q ) is the number when Xp is transmitted, and N (Xp → X b q . Assuming a rich of bits in difference between Xp and X scattering environment, at high SNR the upper bound in (7) can be further approximated as [8] c · SNR−rmin NR ∑ b q |Rp,q = rmin ), (8) Pe ≤ N (Xp → X m1 · 2m1 p,q where c is a constant, N (·|·) is the conditional number of b q } and rmin = min Rp,q . bits in error, Rp,q = rank{Xp − X It can be readily figured out that for both the gray code order and lexicographic order we have rmin = 1, which implies that DSSK achieves unit transmit diversity order regardless of which encoding manner is employed. Therefore, the performance of the gray code order and lexicographic order differ from the SNR gain only. To evaluate the value, let us define the total number of bits in difference with Rp,q = 1 ∑ G b q |Rp,q = 1) for both manners as Nerror = p,q N G (Xp → X ∑ L L b q |Rp,q = 1), where the suband Nerror = p,q N (Xp → X scripts G and L refer to the gray code order and lexicographic order, respectively. Then, from (8) the SNR gain achieved by the gray code order over lexicographic order in dB, which is defined based on the BER criterion, can be thus calculated by [9] L G γ = 10 log10 (Nerror /Nerror )/NR .

IV. A

(9)

DIVERSITY- ENHANCING SCHEME

From above analysis, we see that the transmit diversity order achieved by DSM systems depends on rmin . Since in the gray code order two adjacent permutations have two elements difference, the minimum rank equals one and in return the transmit diversity order remains unit. To improve the diversity performance of DSM, in this section we propose a novel scheme, i.e., I-gray code order, based on the gray code order.

Selected Permutations

Permutations in I-GCO1

(1 2 3) (3 1 2) (2 3 1)

(1 2 3) (3 1 2)

A. Application to DSSK In the I-gray code order, the new permutations consist of the ones which locate the odd (or even) positions of the full permutations in the gray code order. The example for NT = 3 is given in Table II, where only the odd positions, i.e., 1st, 3rd and 5th, of the full permutations in the gray code order are selected for the purpose. To modulate the information bits, however, only 2⌊log2 (NT !/2)⌋ = 2 permutations are permitted to be used, which obtains the legitimate permutations as (1 2 3) and (3 1 2). It is clear that two adjacent permutations in the I-gray code order will have three elements difference, which improves an additional transmit diversity order in DSSK. B. Application to DSM The direct application of the I-gray code order to conventional DSM fails to achieve a transmit diversity order of two since the independent symbol-by-symbol transmission scheme will limit rmin to be unit. To overcome this problem, it is advisable to introduce correlation between two adjacent modulated symbols in a DSM block under the framework of the I-gray code order. To this end, we extend the idea of coordinate interleaving design (CID) [10], [11] to our design. Specifically, in a DSM block, for a pair of modulated symbols drawn from a phase rotated constellation with angle θ, the real and imaginary parts of one modulated symbol is combined with the imaginary and real parts of the other modulated symbol. For example, the information matrix for NT = 2, 3 with antenna activation orders (2 1) and (3 1 2) can be expressed by ] [ I 0 sR 2 + js3 , (10) Xt = R s1 + jsI2 0 and



0 0 Xt =  sR + jsI2 1

I sR 2 + js3 0 0

 0 I sR 3 + js1 , 0

(11)

I 4 θ θ respectively, where {si = sR i + jsi }i=1 ∈ S and S denotes the rotated constellation S with angle θ. The information matrix for NT > 3 can be constructed by combining (10) and (11). Note that the value of θ affects the BER performance of DSM and the optimization of θ is considered as our further research.

V. S IMULATION R ESULTS AND A NALYSIS In this section, we conduct simulations to evaluate the BER performance of DSM/DSSK with the proposed schemes,



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Fig. 1. BER performance between the gray code order and lexicographic order for different configurations in DSSK.

Fig. 3. BER performance between the gray code order and I-gray code order with NT = 4, NR = 2 in DSSK and DSM.

gray code order with 4QAM outperforms the gray code order with either 4QAM or BPSK in DSM at high SNR. The Igray code order with 4QAM performs worse than the gray code order with BPSK at low SNR while better at high SNR. This is because the coding effect dominates the BER at low SNR while the diversity effect takes place at high SNR. In DSSK, I-gray code order still obtains an additional transmit diversity order at the price of one information bit loss for each transmission.

1.4 1.2 coding gain ratio(dB)

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Fig. 2. The SNR gain ratio between the gray code order and lexicographic order for NT = {3, 4, 5, 6} and NR = 1 in DSSK.

where slow-varying Rayleigh flat fading channels are assumed. Fig. 1 shows the comparison results between the BER performances of the gray code order and lexicographic order for NT = 4, 6 and NR = 1, 2, 3, 4 in DSSK under the spectral efficiencies of 1bps/Hz and 1.5bps/Hz, respectively. At BER = 10−3 , it can be seen that in the case of NR = 1 the proposed gray code order achieves about 1.2dB and 0.3dB SNR gains over the lexicographic order for NT = 4 and NT = 6, respectively. Note that the performance gain in DSM is still impressive though it becomes a little smaller than that in DSSK. Due to page limit, we have not added the results for DSM. In addition, from the figures, it is notable that the SNR gain brought by coding decrease as NR increases for a given NT . Fig. 2 shows the calculated SNR gains achieved by the gray code order over the lexicographic order for NT = {3, 4, 5, 6} and NR = 1 from (9) in DSSK. It is expected that no SNR gain is available for NT = 3 and it achieves almost 1.2dB SNR L G gain for NT = 4, in which Nerror = 136 and Nerror = 104. On the other hand, we see that the theoretical results match their simulation counterparts in Fig. 1. Fig. 3 shows the BER performances of the I-gray code order and gray code order with NT = 4, NR = 2 in DSSK and DSM, respectively. Similar to [10], we set θ = 15◦ . We see that due to the improvement of diversity performance, the I-

VI. C ONCLUSION In this letter, we presented the gray code order to improve the performance of DSM. Further, the I-gray code order for DSM was also proposed to achieve an additional transmit diversity order compared with the gray code order. R EFERENCES [1] M. Di Renzo, H. Haas, and P. M. Grant, “Spatial modulation for multipleantenna wireless systems: a survey,” IEEE Commun. Mag., vol. 49, no. 12, pp. 182-191, Dec. 2011. [2] P. Yang, M. Di Renzo, Y. Xiao, S. Li, and L. Hanzo, “Design guidelines for spatial modulation,” IEEE Commun. Surveys & Tuts., vol. 17, no. 1, pp. 6-26, First Quarter 2015. [3] M. Di Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo, “Spatial modulation for generalized MIMO: challenges, opportunities and implementation,” Proc. of the IEEE, vol. 102, no. 1, pp. 56-103, Jan. 2014. [4] Y. Bian, X. Cheng, M. Wen, L. Yang, H. V. Poor, and B. Jiao, “Differential spatial modulation,” IEEE Trans. Veh. Technol., vol. 64, no. 7, pp. 32623268, July 2015. [5] M. Wen, X. Cheng, Y. Bian, and H. V. Poor, “A low-complexity near-ML differential spatial modulation detector,” IEEE Signal Process. Lett., vol. 22, no. 11, pp. 1834-1838, Nov. 2015. [6] M. Cohn and S. Even, “A gray code counter,” IEEE Trans. Computers, vol. C-18, no. 7, pp. 662-664, July 1969. [7] D. L. Kreher and D. R. Stinson, Combinatorial Algorithms: Generation, Enumeration, and Search (Discrete Mathematics and Its Applications), 1st ed., CRC Press, 1998. [8] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communications: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, no. 2, pp. 744-765, Mar. 1998. [9] Z. Wang and G. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun., vol. 51, no. 8, pp. 1389-1398, Aug. 2003. [10] E. Basar, “OFDM with index modulation using coordinate interleaving,” IEEE Wireless Commun. Lett., vol. 4, no. 4, pp. 381-384, Aug. 2015. [11] R. Rajashekar and K. V. S. Hari, “Modulation diversity for spatial modulation using complex interleaved orthogonal design,” in Proc. of IEEE TENCON 2012, Cebu, Philippines, Nov. 2012, pp. 1-6.
