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Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation
Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx
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Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation Xinhe Zhang a,b, Guannan Zhao c, Qian Liu d, Nan Zhao a, Minglu Jin a,⇑ a
School of Information and Communication Engineering, Dalian University of Technology, Dalian 116024, China School of Electronic and Information Engineering, University of Science and Technology Liaoning, Anshan 114051, China c School of Physics and Electronic Technology, Liaoning Normal University, Dalian 116029, China d School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China b
a r t i c l e
i n f o
Article history: Received 31 January 2016 Accepted 28 June 2016 Available online xxxx Keywords: Spatial modulation (SM) M-algorithm to maximum likelihood (MML) Computational complexity Multiple input multiple output (MIMO)
a b s t r a c t M-algorithm to maximum likelihood (MML) detector has been considered as an excellent choice for the detection of spatial modulation (SM) signals with lower computational complexity but satisfactory detection performance compared with the optimal maximum likelihood (ML) detector. However, MML algorithm requires a delicate tuning of the ‘‘M” values in order to achieve the optimal detection performance. Unfortunately, these values can only be retrieved via off-line experiments in general. Meanwhile, the detection order of MML algorithm is restricted to the ascending order of the receive antenna indices which neglects its huge impact on the bit-to-error-rate (BER) performance. In this paper, we propose three enhanced MML-based SM detectors with online ‘‘M” selection and strategic adjustment on the detection orders of MML algorithm: (1) hMML detector rearranges the detection order of MML algorithm with respect to the status of channel conditions; (2) yMML detector performs SM signal detection with a descending order of the received signal strength; (3) hyMML detector jointly considers the detection order of both hMML and yMML algorithms. Experimental results show that all proposed detectors outperform the MML detector in terms of reception BER performance. Ó 2016 Elsevier GmbH. All rights reserved.
1. Introduction Spatial modulation (SM) is a novel multi-antenna transmission scheme [1–3], which has been proposed recently to overcome the limits of conventional multiple input multiple output (MIMO) systems. In SM systems, the messages are transmitted with both constellation symbols and the transmit antenna index. For each channel, only a single transmit antenna is active for transmission. Therefore, the SM system only requires one mandatory radio frequency (RF) chain which can overcome the inter-channel interference (ICI), avoid the inter-antenna synchronization (IAS), and reduce the transmit power consumption. In addition, the spectrum efficiency of SM can be further enhanced by increasing the number of transmit antennas (in order to increase the size of the transmit antenna index set). It is obvious that SM holds promising potential for the next-generation large-scale MIMO communications.
⇑ Corresponding author. E-mail addresses: [email protected] (X. Zhang), [email protected] (G. Zhao), [email protected] (Q. Liu), [email protected] (N. Zhao), [email protected] (M. Jin).
For the detection/demodulation of SM signals, [4] proposed the optimal maximum likelihood (ML) detector with consideration of joint detection of both the transmit antenna index and the transmit symbol. However, the computational complexity of ML detector linearly increases with the number of transmit antennas (N T ), the size of the modulation scheme (N M ), and the number of receive antennas (N R ). Therefore, various detection algorithms have been developed in order to reduce the complexity of ML detection [5–10]. In particular, the M-algorithm to maximum likelihood (MML) detector was proposed in [11,12] which transformed the ML detection into a prioritized tree-search structure. In ML and MML detectors, the searching of the optimal solution can be viewed as a breadth-first search tree structure with N R layers, where the ith layer corresponds to the ith receive antenna. Intrinsically, ML detector traverses all the nodes in the tree, whereas MML only examines part of the nodes in each layer. Therefore, the MML detector has a lower computational complexity than ML detector. The MML detector has the same tree structure as the K-best sphere detector [13,14] where the K best nodes are selected on each layer as candidates to the next layer. The difference between the MML and the sphere detector is that the former has a breadth-first structure while the latter has a
http://dx.doi.org/10.1016/j.aeue.2016.06.015 1434-8411/Ó 2016 Elsevier GmbH. All rights reserved.
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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depth-first structure. In MML algorithm, the number of nodes kept at each layer can be different, as long as it follows a non-ascending order from the root layer to the bottom layer. It is noted that the detection performance of MML algorithm highly depends on the selection of M values, which unfortunately can only be obtained from off-line experiments [11]. Due to the complexity and rapid change of wireless environment, it is virtually impossible to carry out simulations for all communication scenarios resulting in a big compromise in the detection performance of MML algorithm. In addition, the detection order of MML detector is restricted to the ascending order of the receive antenna indices which neglects its huge impact on the BER performance. In order to overcome above shortcomings, we propose three enhanced MML-based detectors for SM detection with online M selection and intelligent adaptation of detection orders: (1) hMML detector rearranges the detection order of MML algorithm with respect to the status of channel conditions; (2) yMML detector performs SM signal detection with a descending order of the received signal strength; (3) hyMML detector jointly considers the detection order of both hMML and yMML algorithms. Experimental results show that all proposed detectors outperform the MML detector in terms of reception BER performance. The following notations are utilized throughout the entire paper. The boldface uppercase and lowercase letters represent matrices and vectors, respectively. kk2 denotes the two-norm of a vector. RðÞ and IðÞ denote the real and imaginary parts of a variable. ðÞ and ðÞT represent the conjugate and transpose of a vector, respectively. bc denotes the floor operation. 2. MML detector for SM systems Consider an SM system with N T transmit antennas and N R receive antennas. The input information bit stream is divided into frames with a length of ðlog2 N T þ log2 N M Þ bits, where N M is the modulation level. In each frame, the first log2 N T bits represent the index of the activated transmit antenna, while the other log2 N M bits are the transmitted symbol in the current channel. Assuming with quasi-static flat Rayleigh fading channels, the received signal y can be expressed as
y ¼ Hxj;s þ n ¼ hj s þ n; NR
ð1Þ NT
where y 2 C . The xj;s 2 C the following format
is the transmitted symbol vector with
xj;s ¼ ½0; . . . 0; s; 0; . . . ; 0T ; |fflfflffl{zfflfflffl} |fflfflfflffl{zfflfflfflffl}
ð2Þ
N T j
j1
where j 2 f1; . . . ; N T g is the index of the transmit antenna; s 2 S is a complex symbol from the signal set S, and the cardinality of set S is N M ; H ¼ h1 ; h2 ; . . . ; hNT 2 CNR NT is the channel matrix whose elements are drawn from a Gaussian distribution with zero mean and unit variance; n 2 CNR is a zero-mean Gaussian noise vector with variance of r2 . Assuming that the channel matrix is known at the receiver, the optimal ML detection becomes
( ) NR X 2 yi hi;j s2 ; ð^j; ^sÞ ¼ arg miny hj s2 ¼ arg min j;s
j;s
ð3Þ
i¼1
where ^j; ^s are the estimated transmit antenna index and the transmitted symbol, respectively; yi is the received signal at the ith receive antenna; hi;j is the channel coefficient between the jth transmit antenna and the ith receive antenna. For the solution of ML detection, we shall perform an exhausting search through all possible combinations to obtain the optimal pair of ð^j; ^sÞ.
In order to reduce the computational complexity of ML algorithm, MML detection was developed in [11,12] which transformed the ML detection into a tree-search procedure with two key metrics: the branch metric and the accumulated metric. The detailed implementation procedure of MML detection is illustrated in Table 1, where Q i denotes the set containing possible transmit antenna indices and modulation symbols at the ith layer. For the sake of better understanding, we utilize a simple example (as shown in Fig. 1) to illustrate the key idea of MML detection. Suppose we have an SM system with 4 transmit antennas and 4 receive antennas. The transmit symbols are modulated with 4QAM. As a result, in MML detection, the root node has 4 branches representing the transmit antenna indices, while the tree has 4 levels representing the receive antenna indices. In addition, each transmit antenna has 4 branches corresponding to 4 symbols of the 4QAM constellation. In each layer, the identifier (i.e. hi;j ) on the left of each branch is the channel coefficient between the jth transmit antenna and the ith receive antenna. We define the branch metric of node (j; s) at the ith layer as the squared Euclidean distance between the received signal yi and the transmit symbol s, 2 which can be expressed as di ¼ yi hi;j s . The accumulated ðj;sÞ
metric of node (j; s) at the ith layer is defined as the summation of the branch metric at the ith layer and the accumulated metric at the ði 1Þth layer, which can be presented as Diðj;sÞ ¼ diðj;sÞ þ Di1 ðj;sÞ . MML detector first computes the branch metrics and the accumulated metrics in each layer. Then only M nodes with the smallest accumulated metric (the black nodes in Fig. 1) are retained as the candidates to the next layer, while all others (the white nodes in Fig. 1) are deleted. Intrinsically, ML detector traverses all the nodes in the tree, whereas MML only examine part of the nodes in each layer. Therefore, the MML algorithm has a lower computational complexity than ML detector. For each layer, M ¼ M 1 ; . . . ; M i ; . . . ; M NR 1 ; 1 nodes are retained where 1 6 M 1 6 N T N M , and 1 6 M iþ1 6 M i . 3. Proposed MML-based SM detectors It is noticed that the selection of M directly influences the detection performance. Unfortunately, the optimal M values can only be obtained through extensive off-line experiments. In this paper, we propose an online M selection strategy which achieves satisfactory BER performance when incorporating with the proposed detection order adaptation strategies. In particular, we adopt a simple geometric progression rule for the M-value selection. According to Fig. 1, we can observe that the number of layers in the tree is equal to the number of receive antennas. As a result, M is set to maxfN T ; N M g at the first layer, 1 at the last layer (i.e. NR k the N R -th layer), and ðmaxfN T ; N M gÞNR 1 at the kth layer (1 < k < N R ), following the geometric progression rule. It is apparent that we can obtain the M value for each layer at real-time with this strategy. Meanwhile, it is noticed that the ith layer corresponds to the ith receive antenna in the search tree of MML detector. The detection order of the MML detector is in the ascending order of the receive antenna indices. In fact, the essence of the MML detector is to compute (3) and choose the number of nodes properly. Since M i decreases gradually from the first layer to the last layer, the search space decreases gradually. In order to approximate the optimal BER performance, the optimal solution should be included in Q i as much as possible. When the number of the retained nodes is fixed, which nodes are retained is very important, that is, the calculation order of (3) has a direct influence on the BER performance of MML detector. Different calculation order corresponds to different
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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X. Zhang et al. / Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx Table 1 MML detector [12]. Initialization: Q 0 ¼ fðj; sÞjj 2 ½1; . . . ; N T ; s 2 Sg; M ¼ ½M 1 ; . . . ; M i ; . . . ; MNR 1 ; 1; 1 6 M 1 6 N T N M ; 1 6 Miþ1 6 Mi . Algorithm: 1: for i ¼ 1 : N R 2: 3: 4: 5: 6: 7: 8:
for ðj; sÞ 2 Q i1 2 di ¼ y hi;j s ðj;sÞ
i
Diðj;sÞ ¼ diðj;sÞ þ Di1 ðj;sÞ end
Q i ¼ argðj;sÞ M i smallest Dði j;sÞ end ^j; ^s ¼ Q NR
Fig. 1. The tree structure of 4 4 SM with 4QAM and M ¼ ½12; 6; 4; 1.
detection order, i.e., different search trees (or subtrees). Based on this important discovery, we proposed three enhanced MMLbased detectors by strategically adjusting the detection orders of MML algorithm. We shall demonstrate the proposed detectors in detail in this section. 3.1. hMML detector We notice that the channel gain has a direct effect on the receive signal. Given the noise power, the larger the channel gain is, the higher the power of the received signal is, which makes higher signal-to-noise ratio (SNR) and better performance of the signal detection. Based on this, we consider the detection according to the descending order of the channel gain. This idea is a bit similar to the detection in descending order of SNR or the receive signal strength adapted in successive decoding [15]. Assuming the symbol s sent by the jth transmit antenna, the received signal can be expressed as
yi ¼ hi;j s þ ni :
ð4Þ
At the receiver, the average SNR at the ith receive antenna is h 2 i 2 2 E hi;j s =E n2i , in which hi;j s ¼ hi;j jsj2 holds and E n2i is a con stant. It is to say, when sending specific symbol s, the larger hi:j will make larger hi:j jsj, thus will make higher SNR. Therefore, we propose hMML detector whose detection order is the descending order of hi:j for each transmit antenna. Thus, in the hMML detector, we first reorder each column of channel matrix according to the descending order of hi;j . Then obtain a new channel matrix e and the search tree is built based on H. e H The implementation procedure of hMML detector is described as follows.
T Step 1: Reorder hj ¼ h1;j ; . . . ; hi;j ; . . . ; hNR ;j ; j ¼ 1; ; N T , in h i descending order of hi;j , and obtain z j ¼ z1j ; ; zNj R , where q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 zij ¼ hi;j ¼ þ I hi;j . R hi;j Then, we sort the two-norm of channel gain in descending order and obtain
w1j ; . . . ; wNj R ¼ arg sort z j ;
ð5Þ
where sortðÞ defines the function of sorting in descending order; w1j and wNj R are indices of the maximal and minimal values in z j , respectively. The new channel matrix can be expressed as T h i ~ ¼ h j ;...;h j ;...;h j ~ ;...;h ~ N , where h ~1; . . . ; h e ¼ h , and H j j T w ;j w ;j w ;j 1
i
NR
e the search tree is built based on H. Step 2: Detect the transmit antenna index and transmitted symbol using MML detector. In the hMML detector, computing hi;j needs two real-valued multiplications. Therefore, the computational complexity is higher than the MML detector. To solve this problem, hi;j can be approximated by the sum of the absolute values of the real and imaginary parts of hi;j , that is, hi;j R hi;j þ I hi;j . 3.2. yMML detector For a single input multiple output (SIMO) system, the transmitted symbol arrives at the receive antennas through different channels. Therefore, the signal of each receive antenna is equal to the white Gaussian noise plus the product of channel gain and the transmitted symbol. Since the channel gain and noise of each channel are different, the channel and noise determine the strength of
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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the received signal. In general, the larger the magnitude of the received signal is , the better the BER performance is. Based on this observation, we propose the yMML detector by replacing the detection order of yMML detector with the descending order of the magnitude of receive signals. The new search tree can be obtained by exchanging the order of each layer according to the strength of the received signals. The implementation procedure of yMML detector is described as follows. Step 1: Rearrange the channel matrix and receive signal vector. ~ is generated by reordering the The new receive signal vector y ~1 j is entries of y according to the descending order of jyi j. Thus jy ~NR is the smallest. The new channel matrix can the largest, and y be obtained by reordering the rows of channel matrix in the same order with y. Thus, the received signal can be expressed as
Computer simulation analysis has verified that the detectors based on (8)–(10) have almost the same BER performance. Therefore these detectors can be referred as hyMML detector. In order to save space, only the simulation of hyMML detector based on (10) is given in the simulation section.
e j;s þ n ~: ~ ¼ Hx y
In this section, we assume that the channels are frequency-flat slow-varying Rayleigh fading channels. The perfect CSI is known at the receiver. The BER performance of the proposed detectors is shown in Fig. 2 with N T ¼ 16; N R ¼ 4; M ¼ ½16; 7; 3; 1 and 16QAM modulation. Fig. 3 illustrates the BER performance of the proposed detectors with N T ¼ 8; N R ¼ 4; M ¼ ½64; 16; 4; 1 and 64QAM modulation. We can observe from Figs. 2 and 3 that the proposed detectors have much better performance than MML. The performance of yMML is better than that of hMML at low SNRs, while the performance of hMML is better than that of yMML at high SNRs. The hyMML detector has the best BER performance at both low and high SNRs. The noise is relatively bigger at low SNRs, in general, the larger magnitude of the received signal is, the better is the BER performance. Thus, the BER performance of yMML is better than that of hMML at low SNRs. At high SNRs, the noise is relatively smaller. The channel gain has the main effect on the signal demodulation. In yMML detector, the order of the layers (receive antennas) for each column (transmit antennas) is changed simultaneously. However, the hMML detector finely tunes each of the columns separately, by changing the layers of a particular transmit antenna in a different manner than another transmit antenna. The every layer of the hMML detector in Fig. 1 will no longer correspond to the same receive antenna. Thus, the BER performance of hMML detector is better than that of yMML detector. We all know that a large hi;j cannot guarantee a large jyi j, while a large jyi j is not necessarily derived from a large hi;j . Since the
ð6Þ
Due to the reordering of the receive signals, the first layer of the search tree corresponds to the index of the receive antenna having the largest receive signal, while the last layer corresponds to the index of the receive antenna having the smallest receive signal. Step 2: Detect the transmit antenna index and transmitted symbol using MML detector. With analogy to the simplified hMML, we approximate jyi j by utilizing the sum of absolute values of the real and imaginary parts of yi (let jyi j jRðyi Þj þ jIðyi Þj) in order to reduce the computational complexity. 3.3. hyMML detector The hMML detector and yMML detector are based on the descending order of the channel gains and the receive signals, respectively. However, we discover that a large hi;j cannot guarantee a large jyi j, while a large jyi j is not necessarily derived from a large hi;j . As a result, we consider to combine the detection orders of hMML and yMML together. We propose hyMML detector whose detection order is based on the descending orders of two aspects: the sum of hi;j and jyi j, and the product of hi;j and jyi j. Under the condition of y; j; s, we should first select the calcula tion order of the receive antenna indices. By multiplying hi;j to each item in (4), we have
2 hi;j yi ¼ hi;j s þ hi;j ni :
ð7Þ
This way, hi;j yi is large when both the values of channel gain hi;j and the receive signal yi are large. Therefore, we perform detection at each transmit antenna based on the descending order of hi;j yi . We rearrange the channel gain of each transmit antenna according to the descending order of hi;j yi . As a result, the search tree of hyMML detector is different from that of hMML detector. hi;j yi can be expressed as
1=2 ¼ hi;j jyi j: hi;j yi ¼ yi hi;j hi;j y
ð8Þ
Computing (8) needs 5 real-valued multiplications. In order to simplify the calculation, (8) can be expressed approximately with
hi;j yi Rðhi;j Þ þ Iðhi;j Þ ðjRðyi Þj þ jIðyi ÞjÞ:
ð9Þ
In addition, we propose the (h+y)hyMML detector, which detects according to the descending order of hi;j þ jyi j. To simplify the calculation, hi;j þ jy j can be expressed approximately as i
hi;j þ jy j Rðhi;j Þ þ Iðhi;j Þ þ jRðy Þj þ jIðy Þj: i i i
ð10Þ
4. Simulation results and complexity analysis In this section, we demonstrate the performance of the proposed detectors, and compare it with MML and ML detectors. We also perform the complexity analysis of the proposed detectors in this section. 4.1. Simulation results
detection orders of hMML and yMML are combined in hyMML detector, the hyMML detector has the best BER performance at both low and high SNRs. 4.2. Complexity analysis In this section, we analyze the complexity of proposed detectors by utilizing the total number of the real-valued multiplications (divisions are also considered as multiplications) involved in an algorithm. We summarize the computational complexity of the ML [4], SVD [6,7], LSVD [8], DBD [9], LC-ML [10], MML [11,12] and the proposed detectors in Table 2. The computational complexity of the first six detectors can be directly obtained from the literatures shown in Table 2. Since hMML, yMML and hyMML detectors need no extra real-valued multiplications, the computational complexity of the proposed detectors are the same with MML detector. It noted that although LC-ML detector has relatively low complexity, it can only be applied to SM with PSK constellation. As a result, the proposed detectors show an excellent performance-complexity tradeoff among the compared detectors. In SM system with N T ¼ 16; N R ¼ 4; M ¼ ½16; 7; 3; 1 and 16QAM modulation, the complexity of ML, SVD, LSVD with L ¼ N2T , DBD with p ¼ N2T , the proposed detectors is 6144, 904, 3592, 1104, and 1692, respectively. In SM system with
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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10
10
10
−1
−2
−3
BER
10
0
10
10
10
10
−4
ML DBD SVD LSVD MML hMML yMML hyMML
−5
−6
−7
0
2
4
6
8
10
12 14 SNR, dB
16
18
20
22
24
26
Fig. 2. The BER performance of proposed detectors with N T ¼ 16; N R ¼ 4; M ¼ ½16; 7; 3; 1 and 16QAM modulation.
10
10
10
−1
−2
10
10
10
10
−2.24
−3
BER
10
0
−4
−5
10
10
−2.26
−2.28
−6
10 10
ML DBD SVD LSVD MML hMML yMML hyMML
−2.3
17.95 5
−7
0
18
18.05 18.1 10
15 SNR, dB
20
25
30
Fig. 3. The BER performance of proposed detectors with N T ¼ 8; N R ¼ 4; M ¼ ½64; 16; 4; 1 and 64QAM modulation.
Table 2 Comparison of the above-mentioned detectors. Detectors
Computational complexity
ML SVD LSVD DBD LC-ML MML
6N T N R N M [10] 8N T N R þ 2N R þ 6N R N M [8] 8N T N R þ 2N R þ 6LNR N M [8] N T ð6N R þ 4 þ 2N M Þ þ pð4N R þ 2Þ [5] N T ð6N R þ 9Þ [10] P R 1 6N T N M þ N i¼1 6M i [11] P R 1 6N T N M þ N i¼1 6M i
Proposed detectors
N T ¼ 8; N R ¼ 4; M ¼ ½64; 16; 4; 1 and 64QAM modulation, the complexity of ML, SVD, LSVD with L ¼ N2T , DBD with p ¼ N2T , the proposed detectors is 12288, 1736, 6344, 1320, and 3576, respectively. 5. Conclusion In this paper, we present hMML, yMML and hyMML detectors which enable online M value selection and efficient ordering strategies for the MML detector in SM systems. The proposed detectors achieve better BER performance, but comparable computational complexity with respect to MML algorithm.
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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Short communication
Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation Xinhe Zhang a,b, Guannan Zhao c, Qian Liu d, Nan Zhao a, Minglu Jin a,⇑ a
School of Information and Communication Engineering, Dalian University of Technology, Dalian 116024, China School of Electronic and Information Engineering, University of Science and Technology Liaoning, Anshan 114051, China c School of Physics and Electronic Technology, Liaoning Normal University, Dalian 116029, China d School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China b
a r t i c l e
i n f o
Article history: Received 31 January 2016 Accepted 28 June 2016 Available online xxxx Keywords: Spatial modulation (SM) M-algorithm to maximum likelihood (MML) Computational complexity Multiple input multiple output (MIMO)
a b s t r a c t M-algorithm to maximum likelihood (MML) detector has been considered as an excellent choice for the detection of spatial modulation (SM) signals with lower computational complexity but satisfactory detection performance compared with the optimal maximum likelihood (ML) detector. However, MML algorithm requires a delicate tuning of the ‘‘M” values in order to achieve the optimal detection performance. Unfortunately, these values can only be retrieved via off-line experiments in general. Meanwhile, the detection order of MML algorithm is restricted to the ascending order of the receive antenna indices which neglects its huge impact on the bit-to-error-rate (BER) performance. In this paper, we propose three enhanced MML-based SM detectors with online ‘‘M” selection and strategic adjustment on the detection orders of MML algorithm: (1) hMML detector rearranges the detection order of MML algorithm with respect to the status of channel conditions; (2) yMML detector performs SM signal detection with a descending order of the received signal strength; (3) hyMML detector jointly considers the detection order of both hMML and yMML algorithms. Experimental results show that all proposed detectors outperform the MML detector in terms of reception BER performance. Ó 2016 Elsevier GmbH. All rights reserved.
1. Introduction Spatial modulation (SM) is a novel multi-antenna transmission scheme [1–3], which has been proposed recently to overcome the limits of conventional multiple input multiple output (MIMO) systems. In SM systems, the messages are transmitted with both constellation symbols and the transmit antenna index. For each channel, only a single transmit antenna is active for transmission. Therefore, the SM system only requires one mandatory radio frequency (RF) chain which can overcome the inter-channel interference (ICI), avoid the inter-antenna synchronization (IAS), and reduce the transmit power consumption. In addition, the spectrum efficiency of SM can be further enhanced by increasing the number of transmit antennas (in order to increase the size of the transmit antenna index set). It is obvious that SM holds promising potential for the next-generation large-scale MIMO communications.
⇑ Corresponding author. E-mail addresses: [email protected] (X. Zhang), [email protected] (G. Zhao), [email protected] (Q. Liu), [email protected] (N. Zhao), [email protected] (M. Jin).
For the detection/demodulation of SM signals, [4] proposed the optimal maximum likelihood (ML) detector with consideration of joint detection of both the transmit antenna index and the transmit symbol. However, the computational complexity of ML detector linearly increases with the number of transmit antennas (N T ), the size of the modulation scheme (N M ), and the number of receive antennas (N R ). Therefore, various detection algorithms have been developed in order to reduce the complexity of ML detection [5–10]. In particular, the M-algorithm to maximum likelihood (MML) detector was proposed in [11,12] which transformed the ML detection into a prioritized tree-search structure. In ML and MML detectors, the searching of the optimal solution can be viewed as a breadth-first search tree structure with N R layers, where the ith layer corresponds to the ith receive antenna. Intrinsically, ML detector traverses all the nodes in the tree, whereas MML only examines part of the nodes in each layer. Therefore, the MML detector has a lower computational complexity than ML detector. The MML detector has the same tree structure as the K-best sphere detector [13,14] where the K best nodes are selected on each layer as candidates to the next layer. The difference between the MML and the sphere detector is that the former has a breadth-first structure while the latter has a
http://dx.doi.org/10.1016/j.aeue.2016.06.015 1434-8411/Ó 2016 Elsevier GmbH. All rights reserved.
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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depth-first structure. In MML algorithm, the number of nodes kept at each layer can be different, as long as it follows a non-ascending order from the root layer to the bottom layer. It is noted that the detection performance of MML algorithm highly depends on the selection of M values, which unfortunately can only be obtained from off-line experiments [11]. Due to the complexity and rapid change of wireless environment, it is virtually impossible to carry out simulations for all communication scenarios resulting in a big compromise in the detection performance of MML algorithm. In addition, the detection order of MML detector is restricted to the ascending order of the receive antenna indices which neglects its huge impact on the BER performance. In order to overcome above shortcomings, we propose three enhanced MML-based detectors for SM detection with online M selection and intelligent adaptation of detection orders: (1) hMML detector rearranges the detection order of MML algorithm with respect to the status of channel conditions; (2) yMML detector performs SM signal detection with a descending order of the received signal strength; (3) hyMML detector jointly considers the detection order of both hMML and yMML algorithms. Experimental results show that all proposed detectors outperform the MML detector in terms of reception BER performance. The following notations are utilized throughout the entire paper. The boldface uppercase and lowercase letters represent matrices and vectors, respectively. kk2 denotes the two-norm of a vector. RðÞ and IðÞ denote the real and imaginary parts of a variable. ðÞ and ðÞT represent the conjugate and transpose of a vector, respectively. bc denotes the floor operation. 2. MML detector for SM systems Consider an SM system with N T transmit antennas and N R receive antennas. The input information bit stream is divided into frames with a length of ðlog2 N T þ log2 N M Þ bits, where N M is the modulation level. In each frame, the first log2 N T bits represent the index of the activated transmit antenna, while the other log2 N M bits are the transmitted symbol in the current channel. Assuming with quasi-static flat Rayleigh fading channels, the received signal y can be expressed as
y ¼ Hxj;s þ n ¼ hj s þ n; NR
ð1Þ NT
where y 2 C . The xj;s 2 C the following format
is the transmitted symbol vector with
xj;s ¼ ½0; . . . 0; s; 0; . . . ; 0T ; |fflfflffl{zfflfflffl} |fflfflfflffl{zfflfflfflffl}
ð2Þ
N T j
j1
where j 2 f1; . . . ; N T g is the index of the transmit antenna; s 2 S is a complex symbol from the signal set S, and the cardinality of set S is N M ; H ¼ h1 ; h2 ; . . . ; hNT 2 CNR NT is the channel matrix whose elements are drawn from a Gaussian distribution with zero mean and unit variance; n 2 CNR is a zero-mean Gaussian noise vector with variance of r2 . Assuming that the channel matrix is known at the receiver, the optimal ML detection becomes
( ) NR X 2 yi hi;j s2 ; ð^j; ^sÞ ¼ arg miny hj s2 ¼ arg min j;s
j;s
ð3Þ
i¼1
where ^j; ^s are the estimated transmit antenna index and the transmitted symbol, respectively; yi is the received signal at the ith receive antenna; hi;j is the channel coefficient between the jth transmit antenna and the ith receive antenna. For the solution of ML detection, we shall perform an exhausting search through all possible combinations to obtain the optimal pair of ð^j; ^sÞ.
In order to reduce the computational complexity of ML algorithm, MML detection was developed in [11,12] which transformed the ML detection into a tree-search procedure with two key metrics: the branch metric and the accumulated metric. The detailed implementation procedure of MML detection is illustrated in Table 1, where Q i denotes the set containing possible transmit antenna indices and modulation symbols at the ith layer. For the sake of better understanding, we utilize a simple example (as shown in Fig. 1) to illustrate the key idea of MML detection. Suppose we have an SM system with 4 transmit antennas and 4 receive antennas. The transmit symbols are modulated with 4QAM. As a result, in MML detection, the root node has 4 branches representing the transmit antenna indices, while the tree has 4 levels representing the receive antenna indices. In addition, each transmit antenna has 4 branches corresponding to 4 symbols of the 4QAM constellation. In each layer, the identifier (i.e. hi;j ) on the left of each branch is the channel coefficient between the jth transmit antenna and the ith receive antenna. We define the branch metric of node (j; s) at the ith layer as the squared Euclidean distance between the received signal yi and the transmit symbol s, 2 which can be expressed as di ¼ yi hi;j s . The accumulated ðj;sÞ
metric of node (j; s) at the ith layer is defined as the summation of the branch metric at the ith layer and the accumulated metric at the ði 1Þth layer, which can be presented as Diðj;sÞ ¼ diðj;sÞ þ Di1 ðj;sÞ . MML detector first computes the branch metrics and the accumulated metrics in each layer. Then only M nodes with the smallest accumulated metric (the black nodes in Fig. 1) are retained as the candidates to the next layer, while all others (the white nodes in Fig. 1) are deleted. Intrinsically, ML detector traverses all the nodes in the tree, whereas MML only examine part of the nodes in each layer. Therefore, the MML algorithm has a lower computational complexity than ML detector. For each layer, M ¼ M 1 ; . . . ; M i ; . . . ; M NR 1 ; 1 nodes are retained where 1 6 M 1 6 N T N M , and 1 6 M iþ1 6 M i . 3. Proposed MML-based SM detectors It is noticed that the selection of M directly influences the detection performance. Unfortunately, the optimal M values can only be obtained through extensive off-line experiments. In this paper, we propose an online M selection strategy which achieves satisfactory BER performance when incorporating with the proposed detection order adaptation strategies. In particular, we adopt a simple geometric progression rule for the M-value selection. According to Fig. 1, we can observe that the number of layers in the tree is equal to the number of receive antennas. As a result, M is set to maxfN T ; N M g at the first layer, 1 at the last layer (i.e. NR k the N R -th layer), and ðmaxfN T ; N M gÞNR 1 at the kth layer (1 < k < N R ), following the geometric progression rule. It is apparent that we can obtain the M value for each layer at real-time with this strategy. Meanwhile, it is noticed that the ith layer corresponds to the ith receive antenna in the search tree of MML detector. The detection order of the MML detector is in the ascending order of the receive antenna indices. In fact, the essence of the MML detector is to compute (3) and choose the number of nodes properly. Since M i decreases gradually from the first layer to the last layer, the search space decreases gradually. In order to approximate the optimal BER performance, the optimal solution should be included in Q i as much as possible. When the number of the retained nodes is fixed, which nodes are retained is very important, that is, the calculation order of (3) has a direct influence on the BER performance of MML detector. Different calculation order corresponds to different
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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X. Zhang et al. / Int. J. Electron. Commun. (AEÜ) xxx (2016) xxx–xxx Table 1 MML detector [12]. Initialization: Q 0 ¼ fðj; sÞjj 2 ½1; . . . ; N T ; s 2 Sg; M ¼ ½M 1 ; . . . ; M i ; . . . ; MNR 1 ; 1; 1 6 M 1 6 N T N M ; 1 6 Miþ1 6 Mi . Algorithm: 1: for i ¼ 1 : N R 2: 3: 4: 5: 6: 7: 8:
for ðj; sÞ 2 Q i1 2 di ¼ y hi;j s ðj;sÞ
i
Diðj;sÞ ¼ diðj;sÞ þ Di1 ðj;sÞ end
Q i ¼ argðj;sÞ M i smallest Dði j;sÞ end ^j; ^s ¼ Q NR
Fig. 1. The tree structure of 4 4 SM with 4QAM and M ¼ ½12; 6; 4; 1.
detection order, i.e., different search trees (or subtrees). Based on this important discovery, we proposed three enhanced MMLbased detectors by strategically adjusting the detection orders of MML algorithm. We shall demonstrate the proposed detectors in detail in this section. 3.1. hMML detector We notice that the channel gain has a direct effect on the receive signal. Given the noise power, the larger the channel gain is, the higher the power of the received signal is, which makes higher signal-to-noise ratio (SNR) and better performance of the signal detection. Based on this, we consider the detection according to the descending order of the channel gain. This idea is a bit similar to the detection in descending order of SNR or the receive signal strength adapted in successive decoding [15]. Assuming the symbol s sent by the jth transmit antenna, the received signal can be expressed as
yi ¼ hi;j s þ ni :
ð4Þ
At the receiver, the average SNR at the ith receive antenna is h 2 i 2 2 E hi;j s =E n2i , in which hi;j s ¼ hi;j jsj2 holds and E n2i is a con stant. It is to say, when sending specific symbol s, the larger hi:j will make larger hi:j jsj, thus will make higher SNR. Therefore, we propose hMML detector whose detection order is the descending order of hi:j for each transmit antenna. Thus, in the hMML detector, we first reorder each column of channel matrix according to the descending order of hi;j . Then obtain a new channel matrix e and the search tree is built based on H. e H The implementation procedure of hMML detector is described as follows.
T Step 1: Reorder hj ¼ h1;j ; . . . ; hi;j ; . . . ; hNR ;j ; j ¼ 1; ; N T , in h i descending order of hi;j , and obtain z j ¼ z1j ; ; zNj R , where q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 zij ¼ hi;j ¼ þ I hi;j . R hi;j Then, we sort the two-norm of channel gain in descending order and obtain
w1j ; . . . ; wNj R ¼ arg sort z j ;
ð5Þ
where sortðÞ defines the function of sorting in descending order; w1j and wNj R are indices of the maximal and minimal values in z j , respectively. The new channel matrix can be expressed as T h i ~ ¼ h j ;...;h j ;...;h j ~ ;...;h ~ N , where h ~1; . . . ; h e ¼ h , and H j j T w ;j w ;j w ;j 1
i
NR
e the search tree is built based on H. Step 2: Detect the transmit antenna index and transmitted symbol using MML detector. In the hMML detector, computing hi;j needs two real-valued multiplications. Therefore, the computational complexity is higher than the MML detector. To solve this problem, hi;j can be approximated by the sum of the absolute values of the real and imaginary parts of hi;j , that is, hi;j R hi;j þ I hi;j . 3.2. yMML detector For a single input multiple output (SIMO) system, the transmitted symbol arrives at the receive antennas through different channels. Therefore, the signal of each receive antenna is equal to the white Gaussian noise plus the product of channel gain and the transmitted symbol. Since the channel gain and noise of each channel are different, the channel and noise determine the strength of
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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the received signal. In general, the larger the magnitude of the received signal is , the better the BER performance is. Based on this observation, we propose the yMML detector by replacing the detection order of yMML detector with the descending order of the magnitude of receive signals. The new search tree can be obtained by exchanging the order of each layer according to the strength of the received signals. The implementation procedure of yMML detector is described as follows. Step 1: Rearrange the channel matrix and receive signal vector. ~ is generated by reordering the The new receive signal vector y ~1 j is entries of y according to the descending order of jyi j. Thus jy ~NR is the smallest. The new channel matrix can the largest, and y be obtained by reordering the rows of channel matrix in the same order with y. Thus, the received signal can be expressed as
Computer simulation analysis has verified that the detectors based on (8)–(10) have almost the same BER performance. Therefore these detectors can be referred as hyMML detector. In order to save space, only the simulation of hyMML detector based on (10) is given in the simulation section.
e j;s þ n ~: ~ ¼ Hx y
In this section, we assume that the channels are frequency-flat slow-varying Rayleigh fading channels. The perfect CSI is known at the receiver. The BER performance of the proposed detectors is shown in Fig. 2 with N T ¼ 16; N R ¼ 4; M ¼ ½16; 7; 3; 1 and 16QAM modulation. Fig. 3 illustrates the BER performance of the proposed detectors with N T ¼ 8; N R ¼ 4; M ¼ ½64; 16; 4; 1 and 64QAM modulation. We can observe from Figs. 2 and 3 that the proposed detectors have much better performance than MML. The performance of yMML is better than that of hMML at low SNRs, while the performance of hMML is better than that of yMML at high SNRs. The hyMML detector has the best BER performance at both low and high SNRs. The noise is relatively bigger at low SNRs, in general, the larger magnitude of the received signal is, the better is the BER performance. Thus, the BER performance of yMML is better than that of hMML at low SNRs. At high SNRs, the noise is relatively smaller. The channel gain has the main effect on the signal demodulation. In yMML detector, the order of the layers (receive antennas) for each column (transmit antennas) is changed simultaneously. However, the hMML detector finely tunes each of the columns separately, by changing the layers of a particular transmit antenna in a different manner than another transmit antenna. The every layer of the hMML detector in Fig. 1 will no longer correspond to the same receive antenna. Thus, the BER performance of hMML detector is better than that of yMML detector. We all know that a large hi;j cannot guarantee a large jyi j, while a large jyi j is not necessarily derived from a large hi;j . Since the
ð6Þ
Due to the reordering of the receive signals, the first layer of the search tree corresponds to the index of the receive antenna having the largest receive signal, while the last layer corresponds to the index of the receive antenna having the smallest receive signal. Step 2: Detect the transmit antenna index and transmitted symbol using MML detector. With analogy to the simplified hMML, we approximate jyi j by utilizing the sum of absolute values of the real and imaginary parts of yi (let jyi j jRðyi Þj þ jIðyi Þj) in order to reduce the computational complexity. 3.3. hyMML detector The hMML detector and yMML detector are based on the descending order of the channel gains and the receive signals, respectively. However, we discover that a large hi;j cannot guarantee a large jyi j, while a large jyi j is not necessarily derived from a large hi;j . As a result, we consider to combine the detection orders of hMML and yMML together. We propose hyMML detector whose detection order is based on the descending orders of two aspects: the sum of hi;j and jyi j, and the product of hi;j and jyi j. Under the condition of y; j; s, we should first select the calcula tion order of the receive antenna indices. By multiplying hi;j to each item in (4), we have
2 hi;j yi ¼ hi;j s þ hi;j ni :
ð7Þ
This way, hi;j yi is large when both the values of channel gain hi;j and the receive signal yi are large. Therefore, we perform detection at each transmit antenna based on the descending order of hi;j yi . We rearrange the channel gain of each transmit antenna according to the descending order of hi;j yi . As a result, the search tree of hyMML detector is different from that of hMML detector. hi;j yi can be expressed as
1=2 ¼ hi;j jyi j: hi;j yi ¼ yi hi;j hi;j y
ð8Þ
Computing (8) needs 5 real-valued multiplications. In order to simplify the calculation, (8) can be expressed approximately with
hi;j yi Rðhi;j Þ þ Iðhi;j Þ ðjRðyi Þj þ jIðyi ÞjÞ:
ð9Þ
In addition, we propose the (h+y)hyMML detector, which detects according to the descending order of hi;j þ jyi j. To simplify the calculation, hi;j þ jy j can be expressed approximately as i
hi;j þ jy j Rðhi;j Þ þ Iðhi;j Þ þ jRðy Þj þ jIðy Þj: i i i
ð10Þ
4. Simulation results and complexity analysis In this section, we demonstrate the performance of the proposed detectors, and compare it with MML and ML detectors. We also perform the complexity analysis of the proposed detectors in this section. 4.1. Simulation results
detection orders of hMML and yMML are combined in hyMML detector, the hyMML detector has the best BER performance at both low and high SNRs. 4.2. Complexity analysis In this section, we analyze the complexity of proposed detectors by utilizing the total number of the real-valued multiplications (divisions are also considered as multiplications) involved in an algorithm. We summarize the computational complexity of the ML [4], SVD [6,7], LSVD [8], DBD [9], LC-ML [10], MML [11,12] and the proposed detectors in Table 2. The computational complexity of the first six detectors can be directly obtained from the literatures shown in Table 2. Since hMML, yMML and hyMML detectors need no extra real-valued multiplications, the computational complexity of the proposed detectors are the same with MML detector. It noted that although LC-ML detector has relatively low complexity, it can only be applied to SM with PSK constellation. As a result, the proposed detectors show an excellent performance-complexity tradeoff among the compared detectors. In SM system with N T ¼ 16; N R ¼ 4; M ¼ ½16; 7; 3; 1 and 16QAM modulation, the complexity of ML, SVD, LSVD with L ¼ N2T , DBD with p ¼ N2T , the proposed detectors is 6144, 904, 3592, 1104, and 1692, respectively. In SM system with
Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015
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10
10
10
−1
−2
−3
BER
10
0
10
10
10
10
−4
ML DBD SVD LSVD MML hMML yMML hyMML
−5
−6
−7
0
2
4
6
8
10
12 14 SNR, dB
16
18
20
22
24
26
Fig. 2. The BER performance of proposed detectors with N T ¼ 16; N R ¼ 4; M ¼ ½16; 7; 3; 1 and 16QAM modulation.
10
10
10
−1
−2
10
10
10
10
−2.24
−3
BER
10
0
−4
−5
10
10
−2.26
−2.28
−6
10 10
ML DBD SVD LSVD MML hMML yMML hyMML
−2.3
17.95 5
−7
0
18
18.05 18.1 10
15 SNR, dB
20
25
30
Fig. 3. The BER performance of proposed detectors with N T ¼ 8; N R ¼ 4; M ¼ ½64; 16; 4; 1 and 64QAM modulation.
Table 2 Comparison of the above-mentioned detectors. Detectors
Computational complexity
ML SVD LSVD DBD LC-ML MML
6N T N R N M [10] 8N T N R þ 2N R þ 6N R N M [8] 8N T N R þ 2N R þ 6LNR N M [8] N T ð6N R þ 4 þ 2N M Þ þ pð4N R þ 2Þ [5] N T ð6N R þ 9Þ [10] P R 1 6N T N M þ N i¼1 6M i [11] P R 1 6N T N M þ N i¼1 6M i
Proposed detectors
N T ¼ 8; N R ¼ 4; M ¼ ½64; 16; 4; 1 and 64QAM modulation, the complexity of ML, SVD, LSVD with L ¼ N2T , DBD with p ¼ N2T , the proposed detectors is 12288, 1736, 6344, 1320, and 3576, respectively. 5. Conclusion In this paper, we present hMML, yMML and hyMML detectors which enable online M value selection and efficient ordering strategies for the MML detector in SM systems. The proposed detectors achieve better BER performance, but comparable computational complexity with respect to MML algorithm.
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Please cite this article in press as: Zhang X et al. Enhanced M-algorithm-based maximum likelihood detectors for spatial modulation. Int J Electron Commun (AEÜ) (2016), http://dx.doi.org/10.1016/j.aeue.2016.06.015