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A New Low-Complexity Near-ML Detection Algorithm for Spatial Modulation Qian Tang, Yue Xiao, Ping Yang, Qiaoling Yu, and Shaoqian Li Abstract—In this letter, we propose a distance-based ordered detection (DBD) algorithm for spatial modulation (SM) to reduce the receiver complexity and achieve a near maximum likelihood (ML) performance. The proposed algorithm firstly compensates the channel attenuation and obtains the estimated symbols with transmit antenna indices, then orders the indices based on the distances between these symbols and their demodulation constellations. A searching method is developed to obtain the final decision through a trade-off between performance and complexity. The equivalence between DBD and ML algorithms is also proved by theoretical analysis. Furthermore, a new lowcomplexity soft output DBD (SODBD) algorithm is developed for coded SM systems. The simulation results show that DBD algorithm has a close performance to ML algorithm while effectively reducing the complexity compared to conventional near-ML algorithms. Index Terms—Spatial modulation, distance-based ordered detection algorithm, soft demodulator.
I. I NTRODUCTION
S
PATIAL modulation (SM) [1] is a special and promising MIMO [2] transmission technique by employing the active transmit antenna indices to convey information. As only one antenna is active in each time slot at the transmitter, the interchannel interference (ICI) is efficiently avoided [3]. SM also provides a general transceiver configuration for any number of transmit and receive antennas, especially for an unbalanced MIMO channel [3]. Several detection algorithms have been proposed for SM systems. Maximum ratio combination (MRC) which estimates the antenna index prior to detect the transmitted symbols was developed in [4], but it only operates under special channel assumptions [5]. Maximize likelihood (ML) algorithm was proposed in [6] to achieve an optimal performance. However, the high computational complexity remains a problem. To alleviate this problem, a class of simplified methods were proposed as SM sphere decoding (SM-SD) [7] algorithms. SM-SD provides a near-ML performance and reduces the complexity when the number of receive antennas NR is large. However, when NR is small, which is very common in the downlink communications, the complexity is still considerable. Recently, a new SM-SD algorithm was proposed in [10] with reduced complexity, but it cannot be used for soft demodulation. Another class of detection algorithms, called Manuscript received August 17, 2012. The associate editor coordinating the review of this letter and approving it for publication was I. Lee. This work is supported in part by the National Science Foundation of China under Grant number 60902026, the Fundamental Research Funds for the Central Universities (No. ZYGX2010J008), and the National High Tech R&D Program of China (863 Program: 2012AA011402). The authors are with the National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, 611731, Chengdu, Sichuan, China. Y. Xiao is the corresponding author (e-mail: [email protected]). Digital Object Identifier 10.1109/WCL.2012.120312.120601
b
x
1 0 . . .
0 s . . 0
Spaital mapping
Fig. 1.
1
1
2
2
......
h
NT
...... NR
b' Detection and Spaital demapping
1 0 . . .
Spatial modulation system model.
the MF-Based Detector [8][9], has been developed for Space Time Shifting Keying (STSK) systems [8], which can be used for SM systems as well. MF achieves an optimal performance with simplified complexity, but it is still with high complexity in the case of high order modulation. Yang proposed a lowcomplexity algorithm [11] which can be used for SM, but it only works for one receive antenna systems. Meanwhile, soft demodulation algorithm for coded SM systems was developed in [12]. However, it suffers from the problem of high complexity. In this case, simplified soft output demodulation algorithms are desired for SM systems [13],[14]. In this letter, we propose a low-complexity detection algorithm for SM systems called distance-based ordered detection (DBD) algorithm. The effect of DBD algorithm is proved by theoretical analysis and by the comparisons to current detection algorithms both in complexity and performance. We further propose a new low-complexity soft demodulation algorithm for coded SM systems, which provides a close performance to soft-output ML algorithm. II. P ROPOSED DBD H ARD D ETECTION A LGORITHM A. System Model The model of NT × NR SM systems is shown in Fig.1, where NT and NR are the number of transmit and receive antennas, respectively. Let L be the number of bits transmitted in each time slot. Then at the transmitter, the input bits are partitioned into two segments, containing log2 NT bits and log2 M (M is the constellation size) bits, respectively. The first segment is utilized to select a transmit antenna u for transmission, while the second segment is mapped to the modulated symbol s. Let h ∈ CNR ×NT denote the channel impulse response matrix, whose elements are assumed to be complex Gaussian random variables with zero mean and unit variance, so NR × 1 received signal is given by (1) y = hx + n = hu xu + n = hu s + n, where hu and xu are u-th column of h and u-th element of x, n is AWGN noise with zero-mean and variance of σ 2 . ML algorithm has an optimal performance for SM systems, however its computational complexity is high with exhaustive search of u and s, expressed as 2 (u , s ) = arg min y − hu st , (2) 1≤u≤NT ,st ∈S
where S denotes the symbol alphabet. MRC is another conventional detection algorithm [1] for SM systems, it firstly
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estimates the transmit antenna index and then demodulates the data symbols. However, MRC was proved only for some special channel assumptions. B. MF Detection MF [8] was proposed as a low complexity detection algorithm for STSK systems and can be applied to SM H systems. the input Z = [z1, ..., zNT ] = h y, where i i For NT NT h = h / h , ..., h / h is the modified equivalent channels, MF detects u and s as u = arg max [2 hu {| (zu )| | (s t )| u,∀t (3) 2 + | (zu )| | (s t )|} −hu 2 |s t | , s = arg min |zu − hu st | = Q (zu / hu ) , st ∈{S}
(4)
where (·), (·) denote the real and imaginary part of the input, and Q(·) is the constellation demodulator as Q (a) = arg min |a − st | . (5) st ∈{S}
An improved MF detector (DMF) was proposed in [9] to further reduce the complexity. In general, MF simplifies the calculation compared to ML algorithm, but the complexity is still high by considering exhaustive (u, s) search for MF signals with high order modulation. C. Yang’s Detection For one receive antenna SM systems, Yang’s detection [11] first makes the candidate decisions as, 2 2 xˆm = arg min st − h∗m y/|hm | , m = 1, 2, ..., NT , (6) st ∈{S}
where * denotes the conjugate operator. Then the final decision is achieved as 2 u = arg min |y − hm x ˆm | , (7) m
(8) s = xu . Thus the complexity is reduced by avoiding exhaustive search. However, Yang’s detection does not consider the case of multiple receiver antennas. D. Proposed DBD Algorithm Here we present a detection algorithm for SM systems as distance-based ordered detection (DBD) algorithm, which achieves an ordered sequence of antenna indices based on the distances between the estimated symbols and their demodulation constellations. We introduce a parameter p to make a trade-off between performance and complexity. In DBD algorithm, the maximum number of searching points is reduced to NT compared to ML algorithm. For DBD algorithm, the received vector y is multiplied by h+ u as ru = h+ (9) u y (u = 1, 2, ..., NT ) , where (·)+ represents the pseudo inverse operation. Then the estimated symbol rˆu is obtained by the demodulator (10) rˆu = Q (ru ) . Let dn denote the distance between rˆu and ru as ru − ru | . (11) du = hu |ˆ
We preset a variable p (1 ≤ p ≤ NT ), then an ordered antenna index sequence could be obtained as U = [u1 , u2 , ..., up ], u1 and up denote the antenna index with minimal and maximal distance, respectively. According to U and the estimated symbols ˆr (ˆr = [ˆ r1 , rˆ2 , ..., rˆNT ]), DBD algorithm achieves the final decision as 2 u = arg min y − hu rˆu , (12) u∈U
s = rˆu .
(13)
Hence the estimated antenna index u and the corresponding modulated symbol s are obtained. And the detector proposed in [10] coincides with this DBD algorithm for p = NT . Additionally, if only one receive antenna is assumed as NR = 1, the proposed DBD algorithm can be further simplified as u = u1 = arg min (du ) = arg min |hu | |ru − rˆu | , (14) s = rˆu1 .
u
u
(15)
Thus the computational complexity is further reduced, without loss of system performance, which will be proved by theoretical analysis in the next section. E. Equivalence between DBD and ML Algorithms In this section, we prove the performance equivalence between DBD (p = NT ) and ML algorithms. For both algorithms, the calculation of Minimum Euclidean Distance (MED) [3] for antenna u is given as 2 M L : arg min y − hu s , (16) s∈S + 2 (17) DBD : arg min hu y − s ↔ Q h+ uy , s∈S
where y − hu s and |h+ u y − s| represent the distances for s in ML and DBD algorithms, respectively. Assuming s1 and s2 are any two constellation points, the corresponding distances for s1 and s2 determine which one is more closeful to the transmitted symbol. In ML algorithm, y − hu s1 < y − hu s2 denotes that s1 is more closeful to the transmitted + symbol, which holds true for |h+ u y − s1 | < |hu y − s2 | in DBD algorithm. Thus the performance equivalence between DBD and ML algorithms depends on the limitation of + y − hu s1 < y − hu s2 ⇔ |h+ u y − s1 | < |hu y − s2 |, expressed as 2 2 y − hu s1 − y − hu s2 (18) + 2 + 2 > 0. h u y − s 1 − hu y − s 2 The denominator and numerator of (18) can be calculated as
+ h y − s1 2 − h+ y − s2 2 u u
+ = (h+ u n + hu hu s − (s2 + s1 ))(s2 − s1 )
(19)
and y − hu s1 2 − y − hu s2 2
+ = hu 2 (h+ u n + hu hu s − (s2 + s1 ))(s2 − s1 ) . (20) After substituting equation (19) and (20) into (18), equation (18) can be simplified as 2 2 y − hu s1 − y − hu s2 2 (21) + 2 + 2 = hu > 0. hu y − s1 − hu y − s2
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Therefore, the performance equivalence between DBD and ML algorithms is proved. As DBD algorithm obtains the final result based on the MED criterion after the demodulation operation (17), the decision performance of DBD algorithm is the same as ML algorithm. Especially for the simplified DBD algorithm with NR = 1 in (14) and (15), as hu ∈ C1×1 , the detection result u1 and rˆu1 will be the same as the output of equation (12) and (13), since u DBD = arg min |hu | |ru − rˆu | u (22) ˆu = arg min |y − hu rˆu |. = arg min |hu | h+ uy −r u
u
In this case, the estimated antenna index is the same as the result in (12), thus the performance equivalence between DBD (NR = 1) and ML algorithms is also proved.
3
antenna index. Although SOMLD is more effective than the hard-decision ML detection in coded SM systems, the computational complexity is also very high. B. Soft Output DBD Algorithm (p = NT ) In this section, we propose a new soft output DBD (SODBD) algorithm (p = NT ) for coded SM systems, so as to output LLR information for soft decoding. This SODBD algorithm firstly obtains the estimated symbols rˆu through equation (9) and (10) on different assumptions of antenna index u, then directly calculates the posterior LLR for the p-th antenna bit⎧ and q-th modulated bit as ⎫ ⎨ y−hu rˆu 2 y−hu rˆu 2 ⎬ −σ2 −σ2 L (up ) = log e / e ⎩ p ⎭ p u ∈u1
F. Complexity Analysis The computational complexity is evaluated by the number of float multiplication operations. For T = 4NR NT , L = log2 (NT · M ), the corresponding complexity per bit of ML, MF(DMF), Yang’s detection, and DBD algorithms is expressed as CML = (T M/τ + T M/2) /L, (23) CMF = ((T + NT M/4) /τ + T + 3NT M/4 + 2) /L, (24) CDMF = (T /τ + T + 2V NT + 6NR K) /L, C Y s = (T /τ + T + 6NT NR ) /L,
(25) (26)
C DBD = (T /τ + T + 3NT ) /L,
(27)
C DBD = (T /τ + T + 3NT + 6NR p )/L.
(28)
NR =1
NR =1
NR >1
For DMF algorithm in equation (25), V is the number of unit-norm vectors [8] related to the constellation size, and K is the number of probable antenna indices [9]. For all the above algorithms, the coherence interval τ is defined as the number of space-time blocks in which channel remains constant. In general, the complexity decreases with the raise of τ as the calculations related to h can be reused.
and L (sq ) = log
⎧ ⎨ ⎩
s ∈sq1
u ∈u0
e
y−hi s 2 −σ2
/
e
y−hi s 2 −σ2
s ∈sq1
(31)
⎫ ⎬ ⎭
,
(32)
where up1 , up0 are the vectors of the antenna indices with ’1’ and ’0’ at the p-th antenna bit, respectively given as {uq1 , uq0 } = {1, 2, ..., NT }, sq1 , sq0 denote the vectors of the data symbols with ’1’ and ’0’ at the q-th bit, respectively, and {sq1 , sq0 } = {ˆ r1 , rˆ2 , ..., rˆNT } in which i defines the corresponding antenna index with s . To simplify the calculation of LLR, SODBD chooses only NT best candidates of (u, s) based on the hard decision results, while conventional soft output ML algorithm considers all the candidates of (u, s). In this case, the accuracy of LLR in the proposed SODBD algorithm is slightly degraded. However, the simulation results in the next section will show that the performance loss is moderate. Meanwhile, SODBD effectively reduces the complexity, and the quantity of computation is associated with NT , while this of SOMLD is NT × M . IV. S IMULATION R ESULTS A. BER Performance
III. P ROPOSED S OFT O UTPUT DBD A LGORITHM A. Soft Output ML Demodulator In order to obtain better performance for coded SM systems, soft output ML demodulator for SM (SM-SOMLD) was proposed in [12]. SM-SOMLD calculates the posterior log-likelihood ratio (LLR) for the p-th antenna bit and q-th modulated bit ⎧ as ⎫ ⎪ ⎪ ⎪ ⎨ y−hu s 2 y−hu s 2 ⎪ ⎬ p 2 2 L (u ) = log e −σ / e −σ ⎪ ⎪ ⎪ ⎪ ⎩ u ∈up1 ⎭ u ∈up 0 s ∈S
and L (sq ) = log up1 ,
up0
⎧ ⎪ ⎪ ⎨
⎪ ⎪ ⎩ s ∈sq1 u ∈u
s ∈S
e
y−hu s 2 −σ2
/
s ∈sq0
u ∈u
e
y−hu
⎫ ⎪ ⎪ s 2 ⎬
(29)
⎪ ⎪ ⎭
, (30)
−σ2
where represent the vectors of the antenna indices with ’1’ and ’0’ at the p-th antenna bit, respectively, while sp1 , sp0 are the vectors of the data symbols with ’1’ and ’0’ at the q-th bit, respectively, and m denotes the candidate
We first perform the simulations of uncoded SM systems with 4 transmit antennas and 16QAM under Rayleigh fading channel. The bit error rate (BER) performance of the proposed DBD algorithm is shown in Fig.2 in comparison with four current detection algorithms as MRC, ML, MF(DMF,K = NT ) and Yang’s detection. When NR = 2, it is shown that for p = 4, the proposed DBD algorithm approaches ML in system performance, which proves the performance equivalence between DBD (p = NT ) and ML algorithms in Section II. We also show the performance of DBD algorithm degrades with the decreasing of p. In this case, p should be selected carefully to make a good tradeoff between system performance and computational complexity. For example, at a BER of 10−4 , in the case of p = 3, and p = 2, there is 0.1 and 1.9 dB penalty of SNR for DBD algorithms. Thus p = 3 could be a good choice for SM systems. Meanwhile, for both DBD and current algorithms such as ML, MF and Yang’s detection have similar system performance, which also proves the theoretical analysis for the performance equivalence between DBD and ML algorithms.
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100 10-1
10-3 10-4 10-5 10-6 0
MRC ML MF DBD p=1 DBD p=2 DBD p=3 DBD p=4 10 20 30 40 SNR(dB) (a) NR = 2
Complexity
BER
10-2
MRC ML MF Yang DBD 50
60
0
10
10
4
ML MF DMF DBD
ML MF DMF Yang DBD (p = 1)
10 3
p
20 30 40 SNR(dB) (b) NR = 1
50
60
100
10 14
NT , 3
5 6 7 8 Modulation order log2(M) / Number of antenna bits log2(NT) (a) NR = 2
Fig. 4.
4
5 6 7 8 Modulation order log2(M) / Number of antenna bits log2(NT) (b) NR = 1
Complexity comparison with NR = 2 (a) and NR = 1 (b).
V. C ONCLUSION
10-1
BER
10-2 10-3
10-5 0
5
10 2
Fig. 2. BER performance comparison in a NT = 4, 16QAM uncoded SM systems with NR = 2 (a) and NR = 1 (b).
10-4
10
Hard-MLD Soft-MLD Soft-DBD 5
SNR(dB)
10
15
Fig. 3. BER performance of the proposed SODBD algorithm and the conventional SOMLD in a 8×2 QPSK-modulated coded SM systems.
Furthermore, we show the BER performance of an 8×2 coded SM systems with QPSK modulation. A rate 1/2 convolutional encoder is considered with the length 9. The code generator matrix is with the constraint [g1 = (561)o ; g2 = (753)o ] where ’o’ denotes octal number. As shown in Fig.3, combined with coding, soft demodulators outperform the original harddecision ML algorithm. It is shown that at BER= 10−4 , there is a 0.4dB performance gap between SODBD (p = NT ) and SOMLD, owing to the simplifying calculation of LLR in which SODBD utilizes less candidates compared to SOMLD. However, the complexity of SODBD algorithm is much lower. B. Comparison of Complexity In this section, the complexity for proposed and current detection algorithms are analyzed to show the advantage of the proposed DBD algorithm. Since in Fig.2 the detection algorithms including ML, MF, Yang’s detection, DBD can achieve an optimal performance, Fig.4 shows the computational complexity of each bit for these algorithms for NR = 2 (a) and NR = 1 (b) respectively, in which the coherence interval τ = 100, and log2 (M ) = log2 (NT ). In Fig.4, we show that the complexity of all algorithms increases with the raise of the modulation order or number of antenna bits in SM systems. Among these algorithms, DBD algorithm shows its advantage in complexity especially in the case of high order modulation. And the complexity of DBD algorithm further decreases with the reduction of p. As a result, DBD algorithm can make a flexible tradeoff between complexity and performance by carefully choosing the value of p, which is not available for current SM detection algorithms.
In this letter, a low-complexity near-ML detection algorithm, called DBD algorithm, has been proposed for SM systems to achieve a better tradeoff between system performance and computational complexity. Furthermore, a soft output DBD algorithm is also developed for coded SM systems. Theoretical analysis and simulation results show that the proposed DBD algorithm can achieve a near-ML performance for SM systems while effectively reducing the complexity compared to current detection algorithms. R EFERENCES [1] R. Mesleh, H. Haas, C. W. Ahn, and S. Yun, “Spatial modulation—a new low complexity spectral effciency enhancing technique,” in Proc. 2006 IEEE CHINACOM, pp. 1–5. [2] G. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Techn. J., vol. 2, pp. 41–59, 1996. [3] J. Jeganathan, A. Ghrayeb, and L. Szczecinski, “Spatial modulation: optimal detection and performance analysis,” IEEE Commun. Lett., vol. 12, no. 8, pp. 545–547, Aug. 2008. [4] R. Mesleh, H. Hass, S. Sinanovic, C. W. Ahn, and S. Yun, “Spatial modulation,” IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 2228–2241, July 2008. [5] N. R. Naidoo, H. J. Xu, and T. A. Quazi, “Spatial modulation: optimal detector asymptotic performance and multiple-stage detection,” IET Commun., vol. 5, no. 10, pp. 1368–1376, July 2011. [6] M. di Renzo and H. Hass, “Performance analysis of spatial modulation,” in Proc. 2010 IEEE CHINACOM, pp. 1–7. [7] A. Younis, R. Mesleh, H. Hass, and P. Grant, “Reduced complexity sphere decoder for spatial modulation detection receivers,” in Proc. 2010 IEEE GLOBECOM, pp. 1–5. [8] S. Sugiura, C. Xu, S. X. Ng, and L. Hanzo, “Reduced-complexity coherent versus non-coherent QAM-aided space-time shift keying,” IEEE Trans. Commun., vol. 59, no. 11, pp. 3090–3101, Nov. 2011. [9] P. Yang, Y. Xiao, L. Li, Q. Tang, and S. Li, “An improved matched-filter based detection algorithm for space-time shift keying systems,” IEEE Signal Process. Lett., vol. 19, no. 5, pp. 271–274, May 2012. [10] R. Rajashekar and K. V. S. Hari, “Low complexity maximum likelihood detection in spatial modulation systems.” Available: arXiv:1206.6190v1, June 2012. [11] Y. Yang and B. Jiao, “Information-guided channel-hopping for high data rate wireless communication,” IEEE Commun. Lett., vol. 12, no. 4, pp. 225–227, Apr. 2008. [12] S. U. Hwang, S. Jeon, S. Lee, and J. Seo, “Soft-output ML detector for spatial modulation OFDM systems,” IEICE Elect. Expr., vol. 6, no. 19, pp. 1426–1431, Oct. 2009. [13] S. Sugiura, S. Chen, and L. Hanzo, “Coherent and differential spacetime shift keying: a dispersion matrix approach,” IEEE Trans. Commun., vol. 58, no. 11, pp. 3219–3230, Nov. 2010. [14] S. Sugiura, C. Xu, S. X. Ng, and L. Hanzo, “Reduced-complexity iterative-detection aided generalized space-time shift keying,” IEEE Trans. Veh. Technol., vol. 61, no. 8, pp. 3656–3664, Oct. 2012.
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A New Low-Complexity Near-ML Detection Algorithm for Spatial Modulation Qian Tang, Yue Xiao, Ping Yang, Qiaoling Yu, and Shaoqian Li Abstract—In this letter, we propose a distance-based ordered detection (DBD) algorithm for spatial modulation (SM) to reduce the receiver complexity and achieve a near maximum likelihood (ML) performance. The proposed algorithm firstly compensates the channel attenuation and obtains the estimated symbols with transmit antenna indices, then orders the indices based on the distances between these symbols and their demodulation constellations. A searching method is developed to obtain the final decision through a trade-off between performance and complexity. The equivalence between DBD and ML algorithms is also proved by theoretical analysis. Furthermore, a new lowcomplexity soft output DBD (SODBD) algorithm is developed for coded SM systems. The simulation results show that DBD algorithm has a close performance to ML algorithm while effectively reducing the complexity compared to conventional near-ML algorithms. Index Terms—Spatial modulation, distance-based ordered detection algorithm, soft demodulator.
I. I NTRODUCTION
S
PATIAL modulation (SM) [1] is a special and promising MIMO [2] transmission technique by employing the active transmit antenna indices to convey information. As only one antenna is active in each time slot at the transmitter, the interchannel interference (ICI) is efficiently avoided [3]. SM also provides a general transceiver configuration for any number of transmit and receive antennas, especially for an unbalanced MIMO channel [3]. Several detection algorithms have been proposed for SM systems. Maximum ratio combination (MRC) which estimates the antenna index prior to detect the transmitted symbols was developed in [4], but it only operates under special channel assumptions [5]. Maximize likelihood (ML) algorithm was proposed in [6] to achieve an optimal performance. However, the high computational complexity remains a problem. To alleviate this problem, a class of simplified methods were proposed as SM sphere decoding (SM-SD) [7] algorithms. SM-SD provides a near-ML performance and reduces the complexity when the number of receive antennas NR is large. However, when NR is small, which is very common in the downlink communications, the complexity is still considerable. Recently, a new SM-SD algorithm was proposed in [10] with reduced complexity, but it cannot be used for soft demodulation. Another class of detection algorithms, called Manuscript received August 17, 2012. The associate editor coordinating the review of this letter and approving it for publication was I. Lee. This work is supported in part by the National Science Foundation of China under Grant number 60902026, the Fundamental Research Funds for the Central Universities (No. ZYGX2010J008), and the National High Tech R&D Program of China (863 Program: 2012AA011402). The authors are with the National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, 611731, Chengdu, Sichuan, China. Y. Xiao is the corresponding author (e-mail: [email protected]). Digital Object Identifier 10.1109/WCL.2012.120312.120601
b
x
1 0 . . .
0 s . . 0
Spaital mapping
Fig. 1.
1
1
2
2
......
h
NT
...... NR
b' Detection and Spaital demapping
1 0 . . .
Spatial modulation system model.
the MF-Based Detector [8][9], has been developed for Space Time Shifting Keying (STSK) systems [8], which can be used for SM systems as well. MF achieves an optimal performance with simplified complexity, but it is still with high complexity in the case of high order modulation. Yang proposed a lowcomplexity algorithm [11] which can be used for SM, but it only works for one receive antenna systems. Meanwhile, soft demodulation algorithm for coded SM systems was developed in [12]. However, it suffers from the problem of high complexity. In this case, simplified soft output demodulation algorithms are desired for SM systems [13],[14]. In this letter, we propose a low-complexity detection algorithm for SM systems called distance-based ordered detection (DBD) algorithm. The effect of DBD algorithm is proved by theoretical analysis and by the comparisons to current detection algorithms both in complexity and performance. We further propose a new low-complexity soft demodulation algorithm for coded SM systems, which provides a close performance to soft-output ML algorithm. II. P ROPOSED DBD H ARD D ETECTION A LGORITHM A. System Model The model of NT × NR SM systems is shown in Fig.1, where NT and NR are the number of transmit and receive antennas, respectively. Let L be the number of bits transmitted in each time slot. Then at the transmitter, the input bits are partitioned into two segments, containing log2 NT bits and log2 M (M is the constellation size) bits, respectively. The first segment is utilized to select a transmit antenna u for transmission, while the second segment is mapped to the modulated symbol s. Let h ∈ CNR ×NT denote the channel impulse response matrix, whose elements are assumed to be complex Gaussian random variables with zero mean and unit variance, so NR × 1 received signal is given by (1) y = hx + n = hu xu + n = hu s + n, where hu and xu are u-th column of h and u-th element of x, n is AWGN noise with zero-mean and variance of σ 2 . ML algorithm has an optimal performance for SM systems, however its computational complexity is high with exhaustive search of u and s, expressed as 2 (u , s ) = arg min y − hu st , (2) 1≤u≤NT ,st ∈S
where S denotes the symbol alphabet. MRC is another conventional detection algorithm [1] for SM systems, it firstly
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estimates the transmit antenna index and then demodulates the data symbols. However, MRC was proved only for some special channel assumptions. B. MF Detection MF [8] was proposed as a low complexity detection algorithm for STSK systems and can be applied to SM H systems. the input Z = [z1, ..., zNT ] = h y, where i i For NT NT h = h / h , ..., h / h is the modified equivalent channels, MF detects u and s as u = arg max [2 hu {| (zu )| | (s t )| u,∀t (3) 2 + | (zu )| | (s t )|} −hu 2 |s t | , s = arg min |zu − hu st | = Q (zu / hu ) , st ∈{S}
(4)
where (·), (·) denote the real and imaginary part of the input, and Q(·) is the constellation demodulator as Q (a) = arg min |a − st | . (5) st ∈{S}
An improved MF detector (DMF) was proposed in [9] to further reduce the complexity. In general, MF simplifies the calculation compared to ML algorithm, but the complexity is still high by considering exhaustive (u, s) search for MF signals with high order modulation. C. Yang’s Detection For one receive antenna SM systems, Yang’s detection [11] first makes the candidate decisions as, 2 2 xˆm = arg min st − h∗m y/|hm | , m = 1, 2, ..., NT , (6) st ∈{S}
where * denotes the conjugate operator. Then the final decision is achieved as 2 u = arg min |y − hm x ˆm | , (7) m
(8) s = xu . Thus the complexity is reduced by avoiding exhaustive search. However, Yang’s detection does not consider the case of multiple receiver antennas. D. Proposed DBD Algorithm Here we present a detection algorithm for SM systems as distance-based ordered detection (DBD) algorithm, which achieves an ordered sequence of antenna indices based on the distances between the estimated symbols and their demodulation constellations. We introduce a parameter p to make a trade-off between performance and complexity. In DBD algorithm, the maximum number of searching points is reduced to NT compared to ML algorithm. For DBD algorithm, the received vector y is multiplied by h+ u as ru = h+ (9) u y (u = 1, 2, ..., NT ) , where (·)+ represents the pseudo inverse operation. Then the estimated symbol rˆu is obtained by the demodulator (10) rˆu = Q (ru ) . Let dn denote the distance between rˆu and ru as ru − ru | . (11) du = hu |ˆ
We preset a variable p (1 ≤ p ≤ NT ), then an ordered antenna index sequence could be obtained as U = [u1 , u2 , ..., up ], u1 and up denote the antenna index with minimal and maximal distance, respectively. According to U and the estimated symbols ˆr (ˆr = [ˆ r1 , rˆ2 , ..., rˆNT ]), DBD algorithm achieves the final decision as 2 u = arg min y − hu rˆu , (12) u∈U
s = rˆu .
(13)
Hence the estimated antenna index u and the corresponding modulated symbol s are obtained. And the detector proposed in [10] coincides with this DBD algorithm for p = NT . Additionally, if only one receive antenna is assumed as NR = 1, the proposed DBD algorithm can be further simplified as u = u1 = arg min (du ) = arg min |hu | |ru − rˆu | , (14) s = rˆu1 .
u
u
(15)
Thus the computational complexity is further reduced, without loss of system performance, which will be proved by theoretical analysis in the next section. E. Equivalence between DBD and ML Algorithms In this section, we prove the performance equivalence between DBD (p = NT ) and ML algorithms. For both algorithms, the calculation of Minimum Euclidean Distance (MED) [3] for antenna u is given as 2 M L : arg min y − hu s , (16) s∈S + 2 (17) DBD : arg min hu y − s ↔ Q h+ uy , s∈S
where y − hu s and |h+ u y − s| represent the distances for s in ML and DBD algorithms, respectively. Assuming s1 and s2 are any two constellation points, the corresponding distances for s1 and s2 determine which one is more closeful to the transmitted symbol. In ML algorithm, y − hu s1 < y − hu s2 denotes that s1 is more closeful to the transmitted + symbol, which holds true for |h+ u y − s1 | < |hu y − s2 | in DBD algorithm. Thus the performance equivalence between DBD and ML algorithms depends on the limitation of + y − hu s1 < y − hu s2 ⇔ |h+ u y − s1 | < |hu y − s2 |, expressed as 2 2 y − hu s1 − y − hu s2 (18) + 2 + 2 > 0. h u y − s 1 − hu y − s 2 The denominator and numerator of (18) can be calculated as
+ h y − s1 2 − h+ y − s2 2 u u
+ = (h+ u n + hu hu s − (s2 + s1 ))(s2 − s1 )
(19)
and y − hu s1 2 − y − hu s2 2
+ = hu 2 (h+ u n + hu hu s − (s2 + s1 ))(s2 − s1 ) . (20) After substituting equation (19) and (20) into (18), equation (18) can be simplified as 2 2 y − hu s1 − y − hu s2 2 (21) + 2 + 2 = hu > 0. hu y − s1 − hu y − s2
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Therefore, the performance equivalence between DBD and ML algorithms is proved. As DBD algorithm obtains the final result based on the MED criterion after the demodulation operation (17), the decision performance of DBD algorithm is the same as ML algorithm. Especially for the simplified DBD algorithm with NR = 1 in (14) and (15), as hu ∈ C1×1 , the detection result u1 and rˆu1 will be the same as the output of equation (12) and (13), since u DBD = arg min |hu | |ru − rˆu | u (22) ˆu = arg min |y − hu rˆu |. = arg min |hu | h+ uy −r u
u
In this case, the estimated antenna index is the same as the result in (12), thus the performance equivalence between DBD (NR = 1) and ML algorithms is also proved.
3
antenna index. Although SOMLD is more effective than the hard-decision ML detection in coded SM systems, the computational complexity is also very high. B. Soft Output DBD Algorithm (p = NT ) In this section, we propose a new soft output DBD (SODBD) algorithm (p = NT ) for coded SM systems, so as to output LLR information for soft decoding. This SODBD algorithm firstly obtains the estimated symbols rˆu through equation (9) and (10) on different assumptions of antenna index u, then directly calculates the posterior LLR for the p-th antenna bit⎧ and q-th modulated bit as ⎫ ⎨ y−hu rˆu 2 y−hu rˆu 2 ⎬ −σ2 −σ2 L (up ) = log e / e ⎩ p ⎭ p u ∈u1
F. Complexity Analysis The computational complexity is evaluated by the number of float multiplication operations. For T = 4NR NT , L = log2 (NT · M ), the corresponding complexity per bit of ML, MF(DMF), Yang’s detection, and DBD algorithms is expressed as CML = (T M/τ + T M/2) /L, (23) CMF = ((T + NT M/4) /τ + T + 3NT M/4 + 2) /L, (24) CDMF = (T /τ + T + 2V NT + 6NR K) /L, C Y s = (T /τ + T + 6NT NR ) /L,
(25) (26)
C DBD = (T /τ + T + 3NT ) /L,
(27)
C DBD = (T /τ + T + 3NT + 6NR p )/L.
(28)
NR =1
NR =1
NR >1
For DMF algorithm in equation (25), V is the number of unit-norm vectors [8] related to the constellation size, and K is the number of probable antenna indices [9]. For all the above algorithms, the coherence interval τ is defined as the number of space-time blocks in which channel remains constant. In general, the complexity decreases with the raise of τ as the calculations related to h can be reused.
and L (sq ) = log
⎧ ⎨ ⎩
s ∈sq1
u ∈u0
e
y−hi s 2 −σ2
/
e
y−hi s 2 −σ2
s ∈sq1
(31)
⎫ ⎬ ⎭
,
(32)
where up1 , up0 are the vectors of the antenna indices with ’1’ and ’0’ at the p-th antenna bit, respectively given as {uq1 , uq0 } = {1, 2, ..., NT }, sq1 , sq0 denote the vectors of the data symbols with ’1’ and ’0’ at the q-th bit, respectively, and {sq1 , sq0 } = {ˆ r1 , rˆ2 , ..., rˆNT } in which i defines the corresponding antenna index with s . To simplify the calculation of LLR, SODBD chooses only NT best candidates of (u, s) based on the hard decision results, while conventional soft output ML algorithm considers all the candidates of (u, s). In this case, the accuracy of LLR in the proposed SODBD algorithm is slightly degraded. However, the simulation results in the next section will show that the performance loss is moderate. Meanwhile, SODBD effectively reduces the complexity, and the quantity of computation is associated with NT , while this of SOMLD is NT × M . IV. S IMULATION R ESULTS A. BER Performance
III. P ROPOSED S OFT O UTPUT DBD A LGORITHM A. Soft Output ML Demodulator In order to obtain better performance for coded SM systems, soft output ML demodulator for SM (SM-SOMLD) was proposed in [12]. SM-SOMLD calculates the posterior log-likelihood ratio (LLR) for the p-th antenna bit and q-th modulated bit ⎧ as ⎫ ⎪ ⎪ ⎪ ⎨ y−hu s 2 y−hu s 2 ⎪ ⎬ p 2 2 L (u ) = log e −σ / e −σ ⎪ ⎪ ⎪ ⎪ ⎩ u ∈up1 ⎭ u ∈up 0 s ∈S
and L (sq ) = log up1 ,
up0
⎧ ⎪ ⎪ ⎨
⎪ ⎪ ⎩ s ∈sq1 u ∈u
s ∈S
e
y−hu s 2 −σ2
/
s ∈sq0
u ∈u
e
y−hu
⎫ ⎪ ⎪ s 2 ⎬
(29)
⎪ ⎪ ⎭
, (30)
−σ2
where represent the vectors of the antenna indices with ’1’ and ’0’ at the p-th antenna bit, respectively, while sp1 , sp0 are the vectors of the data symbols with ’1’ and ’0’ at the q-th bit, respectively, and m denotes the candidate
We first perform the simulations of uncoded SM systems with 4 transmit antennas and 16QAM under Rayleigh fading channel. The bit error rate (BER) performance of the proposed DBD algorithm is shown in Fig.2 in comparison with four current detection algorithms as MRC, ML, MF(DMF,K = NT ) and Yang’s detection. When NR = 2, it is shown that for p = 4, the proposed DBD algorithm approaches ML in system performance, which proves the performance equivalence between DBD (p = NT ) and ML algorithms in Section II. We also show the performance of DBD algorithm degrades with the decreasing of p. In this case, p should be selected carefully to make a good tradeoff between system performance and computational complexity. For example, at a BER of 10−4 , in the case of p = 3, and p = 2, there is 0.1 and 1.9 dB penalty of SNR for DBD algorithms. Thus p = 3 could be a good choice for SM systems. Meanwhile, for both DBD and current algorithms such as ML, MF and Yang’s detection have similar system performance, which also proves the theoretical analysis for the performance equivalence between DBD and ML algorithms.
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IEEE WIRELESS COMMUNICATIONS LETTERS, ACCEPTED
100 10-1
10-3 10-4 10-5 10-6 0
MRC ML MF DBD p=1 DBD p=2 DBD p=3 DBD p=4 10 20 30 40 SNR(dB) (a) NR = 2
Complexity
BER
10-2
MRC ML MF Yang DBD 50
60
0
10
10
4
ML MF DMF DBD
ML MF DMF Yang DBD (p = 1)
10 3
p
20 30 40 SNR(dB) (b) NR = 1
50
60
100
10 14
NT , 3
5 6 7 8 Modulation order log2(M) / Number of antenna bits log2(NT) (a) NR = 2
Fig. 4.
4
5 6 7 8 Modulation order log2(M) / Number of antenna bits log2(NT) (b) NR = 1
Complexity comparison with NR = 2 (a) and NR = 1 (b).
V. C ONCLUSION
10-1
BER
10-2 10-3
10-5 0
5
10 2
Fig. 2. BER performance comparison in a NT = 4, 16QAM uncoded SM systems with NR = 2 (a) and NR = 1 (b).
10-4
10
Hard-MLD Soft-MLD Soft-DBD 5
SNR(dB)
10
15
Fig. 3. BER performance of the proposed SODBD algorithm and the conventional SOMLD in a 8×2 QPSK-modulated coded SM systems.
Furthermore, we show the BER performance of an 8×2 coded SM systems with QPSK modulation. A rate 1/2 convolutional encoder is considered with the length 9. The code generator matrix is with the constraint [g1 = (561)o ; g2 = (753)o ] where ’o’ denotes octal number. As shown in Fig.3, combined with coding, soft demodulators outperform the original harddecision ML algorithm. It is shown that at BER= 10−4 , there is a 0.4dB performance gap between SODBD (p = NT ) and SOMLD, owing to the simplifying calculation of LLR in which SODBD utilizes less candidates compared to SOMLD. However, the complexity of SODBD algorithm is much lower. B. Comparison of Complexity In this section, the complexity for proposed and current detection algorithms are analyzed to show the advantage of the proposed DBD algorithm. Since in Fig.2 the detection algorithms including ML, MF, Yang’s detection, DBD can achieve an optimal performance, Fig.4 shows the computational complexity of each bit for these algorithms for NR = 2 (a) and NR = 1 (b) respectively, in which the coherence interval τ = 100, and log2 (M ) = log2 (NT ). In Fig.4, we show that the complexity of all algorithms increases with the raise of the modulation order or number of antenna bits in SM systems. Among these algorithms, DBD algorithm shows its advantage in complexity especially in the case of high order modulation. And the complexity of DBD algorithm further decreases with the reduction of p. As a result, DBD algorithm can make a flexible tradeoff between complexity and performance by carefully choosing the value of p, which is not available for current SM detection algorithms.
In this letter, a low-complexity near-ML detection algorithm, called DBD algorithm, has been proposed for SM systems to achieve a better tradeoff between system performance and computational complexity. Furthermore, a soft output DBD algorithm is also developed for coded SM systems. Theoretical analysis and simulation results show that the proposed DBD algorithm can achieve a near-ML performance for SM systems while effectively reducing the complexity compared to current detection algorithms. R EFERENCES [1] R. Mesleh, H. Haas, C. W. Ahn, and S. Yun, “Spatial modulation—a new low complexity spectral effciency enhancing technique,” in Proc. 2006 IEEE CHINACOM, pp. 1–5. [2] G. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Techn. J., vol. 2, pp. 41–59, 1996. [3] J. Jeganathan, A. Ghrayeb, and L. Szczecinski, “Spatial modulation: optimal detection and performance analysis,” IEEE Commun. Lett., vol. 12, no. 8, pp. 545–547, Aug. 2008. [4] R. Mesleh, H. Hass, S. Sinanovic, C. W. Ahn, and S. Yun, “Spatial modulation,” IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 2228–2241, July 2008. [5] N. R. Naidoo, H. J. Xu, and T. A. Quazi, “Spatial modulation: optimal detector asymptotic performance and multiple-stage detection,” IET Commun., vol. 5, no. 10, pp. 1368–1376, July 2011. [6] M. di Renzo and H. Hass, “Performance analysis of spatial modulation,” in Proc. 2010 IEEE CHINACOM, pp. 1–7. [7] A. Younis, R. Mesleh, H. Hass, and P. Grant, “Reduced complexity sphere decoder for spatial modulation detection receivers,” in Proc. 2010 IEEE GLOBECOM, pp. 1–5. [8] S. Sugiura, C. Xu, S. X. Ng, and L. Hanzo, “Reduced-complexity coherent versus non-coherent QAM-aided space-time shift keying,” IEEE Trans. Commun., vol. 59, no. 11, pp. 3090–3101, Nov. 2011. [9] P. Yang, Y. Xiao, L. Li, Q. Tang, and S. Li, “An improved matched-filter based detection algorithm for space-time shift keying systems,” IEEE Signal Process. Lett., vol. 19, no. 5, pp. 271–274, May 2012. [10] R. Rajashekar and K. V. S. Hari, “Low complexity maximum likelihood detection in spatial modulation systems.” Available: arXiv:1206.6190v1, June 2012. [11] Y. Yang and B. Jiao, “Information-guided channel-hopping for high data rate wireless communication,” IEEE Commun. Lett., vol. 12, no. 4, pp. 225–227, Apr. 2008. [12] S. U. Hwang, S. Jeon, S. Lee, and J. Seo, “Soft-output ML detector for spatial modulation OFDM systems,” IEICE Elect. Expr., vol. 6, no. 19, pp. 1426–1431, Oct. 2009. [13] S. Sugiura, S. Chen, and L. Hanzo, “Coherent and differential spacetime shift keying: a dispersion matrix approach,” IEEE Trans. Commun., vol. 58, no. 11, pp. 3219–3230, Nov. 2010. [14] S. Sugiura, C. Xu, S. X. Ng, and L. Hanzo, “Reduced-complexity iterative-detection aided generalized space-time shift keying,” IEEE Trans. Veh. Technol., vol. 61, no. 8, pp. 3656–3664, Oct. 2012.
