Lowcomplexity softdecision aided detectors for coded spatial modulation MIMO systems
 Cong Li^{1},
 Jinlong Wang^{1},
 Yunpeng Cheng^{1}Email author and
 Yuzhen Huang^{1}
https://doi.org/10.1186/s1363801505099
© Li et al. 2016
Received: 29 July 2015
Accepted: 20 December 2015
Published: 3 February 2016
Abstract
In this paper, we present an efficient transmission scheme for multipleinput multipleoutput (MIMO) systems, i.e., coded spatial modulation (SM) systems with softdecision aided detector. To exploit the powerful error correction of channel coding, the key challenge of coded SM systems is on designing a reliable but lowcomplexity softoutput detector. Fighting against this problem, we first propose two softoutput detection algorithms by exploiting the features of Mphaseshift keying (PSK) and Mquadrature amplitude modulation (QAM) constellations, namely, PSKbased softoutput detector (PBSD) and QAMbased softoutput detector (QBSD). Furthermore, to further enhance the performance of the two algorithms, we propose another two softoutput detection algorithms taking into account of counterpart maximumlikelihood (ML) estimate, namely, improved PSKbased softoutput detector (IPBSD) and improved QAMbased softoutput detector (IQBSD). The findings of this paper demonstrate that: (1) The computational complexity of PBSD and QBSD algorithms are much lower than that of MaxLogLLR algorithm at the expense of error performance. (2) Both the IPBSD and IQBSD algorithms achieve the same performance as MaxLogLLR algorithm with reduced computational complexity. In addition, a comprehensive performance and computational complexity comparison between the proposed algorithms and the MaxLogLLR algorithm is provided to verify our proposed lowcomplexity softoutput detectors.
Keywords
Spatial modulation Multipleinput multipleoutput Maximum a posteriori probability MaxLogLLR detector1 Introduction
Spatial modulation (SM) has been identified as a new efficient multipleinput multipleoutput (MIMO) technique, which was first proposed in [1]. The main principle of SM is that it jointly uses antenna indices and a conventional signal set to convey information and activates only one antenna to transmit the traditional modulated symbol in each time slot. The advantages of SM, such as the relaxation of antenna synchronization, the avoidance of interchannel interference, and the reduction in transreceiver complexity, make it become a topic of recent research [2–4]. In order to achieve these potential benefits, an adequate harddecision detector for retrieving the information bits transmitted in the spatial and signal constellation diagrams is needed. A matched filter (MF) detection was first proposed in [1], which detects the antenna index and modulated symbol separately. In [5], the authors first investigated the optimum hardoutput maximumlikelihood (ML) detector, which jointly detects the transmit antenna index and the modulated symbol. To achieve a better tradeoff between the performance and computational complexity, various suboptimal hardoutput detectors based on ML and MF have been broadly investigated in [6–9], respectively.
Recently, to reduce the impact of channel fading and noise on bit error ratio (BER) performance, SMMIMO systems with powerful channel coding, such as turbo codes and lowdensity paritycheck (LDPC) codes, have gained rekindled interests [10–13]. A novel trellis coded spatial modulation (TCSM) scheme was proposed in [10], where the concept of trellis coded modulation was applied to the spatial constellation of SM systems. While it achieves better performance than that of uncoded SM system over correlated channels, it performs even worse in uncorrelated channels. To circumvent the problem, a novel MIMO transmission scheme was developed in [11], where a trellis encoder and a SM mapper are jointly designed to take advantage of the benefits of both. In [12], the authors designed a spectral efficiency transmission scheme, labeled as bitinterleaved coded spatial modulation (BICSM) with iterative demodulating/decoding, which provides substantial performance gains in all channel conditions. In [13], LDPC codes were optimized to match SMMIMO systems based on the extrinsic information transfer chart technique. Although these prior works have significantly improved the understanding of error performance of coded SMMIMO systems, the key limitation is that all of them employ the optimum maximum a posteriori probability (MAP) softoutput detector, the complexity of which is intractable when the number of transmit antennas is large or the modulation order is high. Hence, it is urgent to design lowcomplexity softoutput detectors for coded SMMIMO systems.

For coded SMMIMO systems with Graylabeled phaseshift keying (PSK), we first design a new calculation of loglikelihood ratio (LLR) method based on the harddecision detector in [6], denoted as PSKbased softoutput detector (PBSD) algorithm. The analytical results demonstrate that the PBSD algorithm significantly reduces the searched signal candidates from N _{ t } M to N _{ t }, and the computational complexity of calculating the LLR of each bit is independent of the constellation size.

For coded SMMIMO systems with Graylabeled quadrature amplitude modulation (QAM) modulation, we propose a lowcomplexity softoutput detector based on hardlimiterbased ML method proposed in [7], labeled as QAMbased softoutput detector (QBSD) algorithm. Compared to the MaxLogLLR algorithm, it has much lower computational complexity at the expense of performance loss.

Another contribution of this paper is that we further improve the above two softoutput detectors by adding a counterpart ML estimate to calculate the corresponding LLRs, namely, improved PSKbased softoutput detector (IPBSD) and improved QAMbased softoutput detector (IQBSD) algorithms, respectively. The analytical results, along with numerical analysis, reveal that both of them achieve the same performance as MaxLogLLR algorithm with reduced computational complexity.
The rest of the paper is organized as follows. Section 2 introduces the coded SMMIMO system model. Section 3 presents a brief overview of the existing detectors conceived for coded SMMIMO systems. In Section 4, the proposed lowcomplexity softoutput detectors are presented. In Section 5, a comprehensive analysis of computational complexity for all the detectors is provided. Simulation results are presented in Section 6, and we make a conclusion in Section 7.
Notations.
Upper/lower case bold symbols denote matrices/vectors. (·)^{ H }, (·)^{ T }, (·)^{∗}, and (·)^{ † } represent the hermitian transpose, transpose, complex conjugate and pseudo inverse of a vector, respectively. p(ab) denotes the probability density function (pdf) of random variable a conditioned on b, Pr(·) represents the probability of a variable. ⊕ denotes the bitwise XOR operation. ℜ(·) and I(·) denote the real and imaginary parts of a complexvalued quantity, respectively. round(a) is the operator that rounds the element a to its nearest integer. mod(a,n) denotes that a is computed modulus n.
2 Coded SM system model
which implies that the qth modulated symbol from the constellation \({\mathcal {M}}\) is transmitted from the jth transmit antenna. Taking into account of the code rate R, the total spectral efficiency is η=R(Q _{1}+Q _{2}) bits per channel use.
where \(\mathbf {H} = \left [\mathbf {h}_{1},\mathbf {h}_{2}, \ldots,\mathbf {h}_{N_{t}} \right ] \in \mathcal {C}^{N_{r}} \times N_{t}\) denotes the channel matrix, and h _{ j } is the jth column of H. Each entry in H, i.e., h _{ ij },i∈{1,…,N _{ r }},j∈{1,…,N _{ t }}, is zero mean and unit variance complex Gaussian random variable. n is a zeromean additive white Gaussian noise vector with \(\mathbb {E}\left [\mathbf {n}{\mathbf {n}}^{H} \right ] = {\sigma ^{2}}{\textbf {I}_{{N_{r}} \times {N_{r}}}}\).
At the receiver, to achieve the advantage of channel coding, a softoutput detector is employed, where the soft information for each coded bits is calculated based on the received signal. In coded SMMIMO systems, the soft information consists of two parts, i.e., the soft information of antenna index bits and the soft information of modulated symbol bits. For the LogMAP demapper, the soft information bits can be expressed in the form of the loglikelihood ratio (LLR). Without loss of generality, we define \({\Lambda _{p}^{A}}\) and \({\Lambda _{p}^{S}}\) as the a posterior LLRs of antenna index bits and modulated symbol bits, respectively.
3 Conventional softdecision aided SM detectors
Before introducing our proposed lowcomplexity softdecision detectors, we first briefly summarize the conventional softoutput detectors for coded SMMIMO systems, which are studied in Section 5 and Section 6 as a benchmark for assessing the performance versus computational complexity tradeoff of the proposed softoutput detectors.
Note that the computational complexity of MaxLogLLR algorithm has been significantly reduced; however, the number of multiplications and comparisons remains high, which is the order of O(M N _{ t }).
4 The proposed lowcomplexity softdecision aided SM detectors
where \({\mathbf {h}}_{j}^{\dag } = \frac {{{\mathbf {h}}_{j}^{H}}} {{{\mathbf {h}}_{j}^{H}{{\mathbf {h}}_{j}}}}\) and j∈{1,2,⋯,N _{ t }}.
4.1 Lowcomplexity softoutput SM detectors with Graylabeled MPSK
where \(J = \sqrt { 1}\) and k∈{0,1,⋯,M−1}.
4.1.1 PBSD algorithm
4.1.2 IPBSD algorithm
As discussed in the above, when computing the LLRs of the modulated symbol bits, the proposed PBSD algorithm only finds the ML estimate of the symbol and ignores the counterpart ML candidate whose ith bit is contrary to the ML estimate. Therefore, this results in the performance gap between the proposed PBSD algorithm and the MaxLogLLR algorithm. In this subsection, we propose a new improved algorithm based on the PBSD algorithm, namely, IPBSD algorithm, by taking the counterpart ML estimate into account when computing the LLRs of the modulated symbol bits.
By careful inspection of Eq. (9), we find that the item \(\mathop {\min }\limits _{{s_{q}} \in {\mathcal {M}},j \in {\mathcal {S}}} {\left \ {{\mathbf {y}}  {{\mathbf {h}}_{j}}{s_{q}}} \right \^{2}}\), i.e., the squared Euclidean distance from the received signal vector y to the nearest constellation point \(s_{{\widehat {k}}^{j}}\) for the jth antenna, always appears in Eq. (9). It is equal to either \(\mathop {\min }\limits _{{s_{q}} \in {\mathcal {M}}_{i}^{0},j \in {\mathcal {S}}} {\left \ {{\mathbf {y}}  {{\mathbf {h}}_{j}}{s_{q}}} \right \^{2}}\) or \(\mathop {\min }\limits _{{s_{q}} \in {\mathcal {M}}_{i}^{1},j \in {\mathcal {S}}} {\left \ {{\mathbf {y}}  {{\mathbf {h}}_{j}}{s_{q}}} \right \^{2}}\), which depends on the ith bit of \(s_{{\widehat {k}}^{j}}\) being 0 or 1. Hence, our aim is to find another signal point \({s_{\overline {k}_{i}^{j}}}\) resulting in the other minimum term in Eq. (9), which means that the ith bit of \({s_{\overline {k}_{i}^{j}}}\) is opposite to the ith bit of \(s_{{\widehat {k}}^{j}}\). Moreover, as shown in Fig. 2, if the antenna index is j, the nearest signal point to the ZF output r _{ j } is s _{0}, the index of which is \({\widehat {k}^{j}}=0\). Hence, the first bit of s _{0} is 0, and the nearest signal point to r _{ j } with the first bit being 1 is s _{7}, that is, \(\overline {k}_{0}^{j}=7\). Similarly, for the second and third bits, we have \(\overline {k}_{1}^{j}=2\) and \(\overline {k}_{2}^{j}=1\), respectively.
Before introducing the detailed procedure of the proposed IPBSD algorithm, we first present the following lemmas for binaryreflected Graylabeled constellations [16].
Lemma 1.
Lemma 2.
where \(\overline b_{i}^{{{\widehat {k}}^{j}}} = 1  b_{i}^{{{\widehat {k}}^{j}}}\).
Proof: The proof can be found in [16].

Step 1: The first step is to compute the ZF filtered signal r _{ j } for each transmit antenna based on Eq. (10).

Step 2: According to Eq. (16) and Eq. (17), the second step is to find the index of the modulated signal \({\widehat k^{j}}\) and the estimated modulated symbol \({s_{{{\widehat {k}}^{j}}}}\). Afterwards, the binary representation \({{\mathbf {b}}^{{{\widehat {k}}^{j}}}}\) of \({\widehat k^{j}}\) can be achieved, and thus, we can get the Gray labeling representation \({{\mathbf {g}}^{{{\widehat {k}}^{j}}}}\) from Eq. (21).

Step 3: After performing Step 2, we have obtained the estimated signal symbol, which results in both minimum terms in Eq. (8), while only one of the minimum term in Eq. (9). Hence, in this step, we need to find the signal points that result in another minimum term of Eq. (9) based on Lemma 2.

Step 4: Finally, the a posteriori LLR of the modulated symbol bits can be derived as$$ \begin{aligned} {\Lambda_{p}^{S}}\left({{c_{S,i}}} \right) = \frac{1} {{2{\sigma^{2}}}}\left[\mathop {\min }\limits_{s \in \mathcal{W}_{i}^{1}} \left({{{\left\ {{{\mathbf{h}}_{j}}} \right\}^{2}}\left({1  2\Re \left({{r_{j}}{s^ * }} \right)} \right)} \right)\right. \\ \left. \mathop {\min }\limits_{s \in \mathcal{W}_{i}^{0}} \left({{{\left\ {{{\mathbf{h}}_{j}}} \right\}^{2}}\left({1  2\Re \left({{r_{j}}{s^ * }} \right)} \right)} \right) \right] \end{aligned} $$(23)
while the a posteriori LLR of the antenna index bits is the same as the proposed PBSD algorithm in Eq. (19).
4.2 Lowcomplexity softoutput SM detector with Graylabeled MQAM
4.2.1 QBSD algorithm
Noting from Eqs. (30) and (31), we find that the length of the candidate list of the proposed QBSD detector is reduced from N _{ t } M to N _{ t }.
4.2.2 IQBSD algorithm
As the above discussion, the QBSD algorithm only calculates the ML estimated symbol, which results in one of the minimum of MaxLogLLR algorithm in Eq. (9). Hence, the error performance of the QBSD algorithm is inferior to that of MaxLogLLR algorithm. Against this observation, the aim of IQBSD algorithm is to eliminate the performance gap by searching the counterpart ML estimate.
As illustrated in Fig. 3, the ML estimate of the modulated signal is the point z _{2} in the 4PAM constellation, in which the first and second bits are both “1.” Therefore, the constellation point with the first bit being 0 and nearest to the real part of r _{ j } is the signal point z _{1} in the 4PAM constellation. Moreover, the constellation point with the second bit being 0 and nearest to the real part of r _{ j } is the signal point z _{3}. Before describing the procedure of the IQBSD algorithm in detail, we first present the following lemma.
Lemma 3.
where m _{1}=log2(N _{1}) and i∈{0,1,⋯,m _{1}−1}.
Note that, if the bit belong to the imaginary part of the modulated symbol, we can only substitute m _{1} with m _{2} in Eq. (32), where m _{2}=log2(N _{2}).

Step 1: The first step of IQBSD algorithm is the same as the IPBSD algorithm, i.e., calculating the filter output r _{ j } for each antenna based on Eq. (10).

Step 2 After obtaining r _{ j }, the symbol \(\Re \left ({{{\,\widehat s}^{\,\,j}}} \right)\) in the N _{1}PAM constellation nearest to ℜ(r _{ j }) can be calculated by Eq. (27). \({\widehat k}^{j}\) is the index of the symbol \(\Re \left ({{{\widehat {\,s}}^{\,\,j}}} \right)\) in N _{1}PAM constellation, and the binary representation of the index \({\widehat {k}}^{j}\) is \({{\mathbf {b}}^{{{\widehat {k}}^{j}}}} = \left ({b_{0}^{{{\widehat {k}}^{j}}},b_{1}^{{{\widehat {k}}^{j}}}, \cdots,b_{{m_{1}}  1}^{{{\widehat {k}}^{j}}}} \right)\). Then, the binaryreflected Gray labeling \({{\mathbf {g}}^{{{\widehat {k}}^{j}}}}\) can be achieved by Lemma 1.

Step 3: The objective of this step is to find the counterpart ML estimate \({z_{\overline {k}_{i}^{j}}}\) for the ith bit, where the corresponding index is denoted as \(\overline {k}_{i}^{j}\). Considering the symmetric structure of Graylabeled PAM, we can resort to Lemma 3 to find the signal \({z_{\overline {k}_{i}^{j}}}\), which only requires the binary representation of index \({\widehat k}^{j}\) and addition operation, instead of computing the Euclidean distance between all the signals with the ith bit being opposite to that of \({z_{{{\widehat k}^{j}}}}\) and ℜ(r _{ j }).

Step 4: Finally, the a posteriori LLR of the antenna index bits can be derived as Eq. (30), and the a posteriori LLR of the symbol bits is obtained as$$ \begin{aligned} {}{\Lambda_{p}^{S}}\left({{c_{S,i}}} \right) = \frac{1} {{2{\sigma^{2}}}}\left[\mathop {\min }\limits_{s' \in \mathcal{V}_{i}^{1}} \left(s{{{\left\ {{{\mathbf{h}}_{j}}} \right\}^{2}}\left({{{\left {{r_{j}}  s'} \right}^{2}}  {{\left {{r_{j}}} \right}^{2}}} \right)} \right)\right.\\ \left.  \frac{1} {{2{\sigma^{2}}}}\mathop {\min }\limits_{s' \in \mathcal{V}_{i}^{0}} \left({{{\left\ {{{\mathbf{h}}_{j}}} \right\}^{2}}\left({{{\left {{r_{j}}  s'} \right}^{2}}  {{\left {{r_{j}}} \right}^{2}}} \right)} \right) \right], \end{aligned} $$(35)
where j is the antenna index corresponding to the modulated symbol s ^{′} selected from the set \({\mathcal {V}}_{i}\) in Eq. (33).
5 Complexity analysis
In this section, the computational complexity of the proposed softdecision aided detectors and the MaxLogLLR detector will be discussed in detail. Without loss of generality, the computational complexity is evaluated in terms of the number of real multiplications, real additions, and comparisons, respectively.
5.1 MaxLogLLR algorithm
5.2 PBSD algorithm
5.3 IPBSD algorithm
5.4 QBSD algorithm
5.5 IQBSD
Computational complexity comparison
Detectors  Multiplications  Additions  Comparisons 

MaxLogLLR  12,296  12,040  2048 
PBSD  904  808  128 
IPBSD  1224  1000  192 
QBSD  952  888  128 
IQBSD  1272  1208  192 
Remark 1.
Different from other existing detectors, the length of the candidate list for PBSD and QBSD algorithms does not grow linearly with the modulation order M. Thus, it is suitable for coded SM systems with high spectral efficiency, i.e., high modulation order.
Remark 2.
Compared with MaxLogLLR algorithm, the proposed IPBSD and IQBSD algorithms further reduce the number of multiplications, additions, and comparisons without loss of any performance. In addition, both achieve better performance than the PBSD and QBSD algorithms, respectively, with a neglected complexity overhead.
6 Simulation results
In this section, representative numerical simulations are provided to verify the proposed detectors under Rayleigh fading channels to verify the proposed detectors described in the previous sections. Unless otherwise stated, the following set of parameters are used: N _{ t }=4 and N _{ r }=2. The code length of LDPC is 360, and the code rate is \(R = \frac {1}{2}\). In addition, the signal to noise ratio is defined as \({\text {SNR}} = 10\lg \left ({\frac {{{E_{b}}}}{{{N_{0}}}}} \right) = 10\lg \left ({\frac {{{E_{s}}}}{{\eta {N_{0}}}}} \right) = 10\lg \left ({\frac {{{E_{s}}}}{{{N_{0}}R{{\log }_{2}}\left ({{N_{t}}M} \right)}}} \right)\), where E _{ s } is the average energy of the transmit signal.
7 Conclusions
In this paper, by exploiting the features of MPSK and MQAM constellations, we first have designed two novel lowcomplexity softoutput algorithms based on the existing harddecision detection algorithms. The computational complexity of the proposed two detections is much lower than that of MaxLogLLR algorithm with a acceptable performance degradation. Motivated by this, to further enhance the performance of the two softdecision aided detectors, another two improved algorithms were proposed, i.e., IPBSD and IQBSD detectors. The key idea of the two detectors is finding the counterpart ML estimate for each bit of the modulated symbol. The theoretical analysis and simulation results have shown that the proposed IPBSD and IQBSD detectors achieve exactly the same performance as that of MaxLogLLR detector with reduced computational complexity.
Declarations
Acknowledgments
This work was supported by the National Science Foundation of China under Grant No. 61501507, and the Jiangsu Provincial Natural Science Foundation of China under Grant No. BK20150719.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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