Low-Complexity Detection Algorithms for Spatial Modulation MIMO Systems

In this paper, the authors propose three low-complexity detection schemes for spatial modulation (SM) systems based on the modified beam search (MBS) detection. +e MBS detector, which splits the search tree into some subtrees, can reduce the computational complexity by decreasing the nodes retained in each layer. However, the MBS detector does not take into account the effect of subtree search order on computational complexity, and it does not consider the effect of layers search order on the biterror-rate (BER) performance.+e ost-MBS detector starts the search from the subtree where the optimal solution is most likely to be located, which can reduce total searches of nodes in the subsequent subtrees. +us, it can decrease the computational complexity. When the number of the retained nodes is fixed, which nodes are retained is very important. +at is, the different search orders of layers have a direct influence on BER. Based on this, we propose the oy-MBS detector. +e ost-oy-MBS detector combines the detection order of ost-MBS and oy-MBS together. +e algorithm analysis and experimental results show that the proposed detectors outstrip MBS with respect to the BER performance and the computational complexity.


Introduction
To meet the demand of wireless communication systems for higher data transmission rate, multiple-input multipleoutput (MIMO) technology has been adopted in mobile terminals.MIMO technology improves data throughput without increasing additional bandwidth and transmit power.Spatial modulation (SM) [1][2][3] is an emerging transmission scheme for MIMO systems.e main characteristic of SM is that only one transmit antenna is activated at one time slot, but simultaneously, the SM systems can use the original signal domain (signal constellation) and the transmit antenna (TA) indices (spatial constellation) to convey information.Compared to MIMO systems, SM systems can only equip one radio frequency (RF) chain, avoid interchannel interference (ICI) and interantenna synchronization (IAS), and also reduce the complexity of demodulation.
For the detection of SM signals, maximum ratio combining (MRC) algorithm was proposed in [4], in which the active-antenna index and the transmit symbol are separately estimated.
e MRC detector has a low computational complexity and only performs well on the constrained channels.is detector was improved in [5], and it further can be applied in conventional channel conditions.e optimum maximum likelihood (ML) detector which involves joint detection of the TA index and of the transmit symbol was proposed in [6].However, the computational complexity linearly grows as the number of TA (N T ), the number of receive antennas (N R ), and the size of the modulation scheme (N M ).In order to obtain the nearoptimal solution with a lower computational complexity, several low-complexity detectors have been put forward [7][8][9][10][11][12][13][14][15][16][17].In [7,8], two low-complexity hard-limiter-based ML (HL-ML) detectors which have the same BER performance as the ML detector were proposed for M-PSK and square-or rectangular-QAM modulation.e computational complexity has nothing to do with the constellation size.In [9][10][11], sphere-decoding (SD) algorithms were put forward for SM systems, which are capable of achieving near-optimal performance with a lower computational complexity on average.At worst, the computational complexity is equivalent to that of the ML detector.However, its detection performance depends mainly on the initial search radius and the transmit parameters.Compared with SD detectors, SD aided by the ordering strategy proposed in [12] can greatly reduce the computational complexity.Two matched filter-(MF-) based detectors were proposed in [13].In [14], Wang et al. proposed a novel signal vector-based detection (SVD) scheme.Tang et al. [15] presented a distanced-based ordered detection (DBD) algorithm to reduce the receiver complexity and achieve a near-maximum likelihood performance.To reduce the detection complexity of ML detection, Xu [16] presented simplified ML-based optimal detection (OD) and simplified multistage detection (MD).In the simplified ML-based detection and multistage detection schemes, the signal set is firstly partitioned into four "levelone subsets".Each level-one subset is further partitioned into four "level-two subsets" if each subset contains more than four signals.
In [18,19], the M-algorithm to maximum likelihood (MML) detector with prioritized tree-search structure was presented.
e detection is considered as a breadth-first search tree with N T N M branches and N R layers, in which the ith layer corresponds to the ith receive antenna (RA).
e MML detector only examines partial nodes in the tree, whereas the ML detector traverses all nodes.Compared with the ML detector, the MML detector can achieve a lower computational complexity.In [20], a lowcomplexity symbol detection based on modified beam search (MBS) was proposed.e detection process of the MBS algorithm can be represented by constructing a tree with N T subtrees and 2N R layers, where each subtree has N M complete paths from the root node to the leaf nodes, and each of the paths stands for a candidate solution.e solution is found by performing modified beam search.Compared with the MML algorithm, the MBS algorithm reduces the computational complexity by discarding unpromising candidate solutions.
In the MBS detector, the detection sequence of different subtrees is confined to the ascending order of the subtree indices, whereas it ignores the influence of different search orders on the computational complexity.Moreover, the detection of all layers is confined to the ascending order of the layer indices, whereas it ignores the influence of different search orders on its bit-error-rate (BER) performance.at is to say, the influence of different search orders on the BER performance and the computational complexity is not considered in the MBS detector.In recent years, the sorting strategy has attracted more and more attention.To some degree, the sorting strategy can improve the algorithm detection performance.In [12,21], different ordering strategies were proposed to improve the detection performance.In this paper, we proposed three MBS-based detectors with novel ordering strategies: (1) the ost-MBS detector rearranges the search order of subtrees; (2) the oy-MBS detector performs SM signal detection in a descending order of the received signal amplitude; (3) the detection orders of the abovementioned two detectors were jointly considered in the ost-oy-MBS detector.e rest of this paper is organized as follows.In Section 2, the system model of SM systems is introduced.Section 3 gives a brief overview of the MBS detector.e ordering strategy is introduced to the MBS detector.Section 4 demonstrates ost-MBS, oy-MBS, and ost-oy-MBS detectors.Section 5 illustrates the simulation results.Finally, we conclude the paper with a summary in Section 5.
Notations.Boldface upper/lower case symbols denote matrices and column vectors; ‖ • ‖ F is the Frobenius norm of a vector or a matrix; | • | is the amplitude of a complex quantity or the cardinality of a set; R(•) and I(•) are the real and imaginary parts of a complex-valued quantity; (•) H is the conjugate transpose of a vector or a matrix; CN(μ, σ 2 ) denotes a complex Gaussian random variable with mean μ and variance σ 2 .

System Model
Consider an N T × N R SM system with constellation S � s 1 , s 2 , . . ., s N M  .In each time slot, the incoming data bits are rearranged into blocks of log 2 N T N M bits, in which log 2 N T bits are used to select the activated TA and log 2 N M bits are used to select the transmit symbol s m ∈ S, m ∈ 1, 2, . . ., N M  .Hence, the system model for SM systems can be represented by where r ∈ C N R is the received signal vector; s ∈ C N T is the transmit symbol vector, whose element is s m at the lth position and zero at the other positions; G ∈ C N R ×N T and w ∈ C N R are the channel matrix and the noise vector, whose elements follow the circularly symmetric complex Gaussian distributions with CN(0, 1) and CN(0, σ 2 ), respectively.e system model expressed in (1) can be reshaped as where Since only one TA is activated in each time slot, the system model expressed in (2) can be simplified as where h l is the lth column of H.It follows from (3) that the optimal ML-based demodulator can be formulated as 2 Journal of Electrical and Computer Engineering where Λ denotes the set containing all possible transmit antenna indices and complex constellation points, , and h k,l is the (k, l) th entry of matrix H.

MBS Detector
According to Kim and Yi [20], the detection of the SM signal can be regarded as a tree with N T subtrees and 2N R layers, where each subtree has N M complete paths from the root node to the leaf nodes.For ease of understanding, we give an illustration (Figure 1) for the idea of the MBS detector.Suppose we have a 2 × 2 SM system with 4QAM modulation; thus the search tree has 4 layers and 8 branches.4 branches of each subtree (TA) correspond to 4 symbols from the 4QAM constellation.We de ne the branch metric of node (l, s m ) at the kth layer as the squared Euclidean distance between the received and the transmit signals, which can be denoted as e accumulated metric of node (l, s m ) at the kth layer is the summation of the branch metric at the kth layer and the accumulated metric at (k − 1)th layer, which can be expressed as In the rst subtree, the M k nodes in the kth layer with the smallest accumulated metrics are kept as the candidate nodes for the next layer.At the last layer, the node with the smallest accumulated metric is considered as the solution in the rst subtree.e smallest accumulated metric and its corresponding TA index and the transmit symbol are represented by ρ Ω and Ω l, s m , respectively.In the subsequent subtrees, ρ Ω and Ω are gradually updated.In the kth layer, at most M k nodes with the accumulated metrics smaller than the threshold value, ρ Ω are selected as the survival branches for the next layer.If the accumulated metric is not less than ρ Ω , the search of the current branch is terminated.Otherwise, we should continue to search the next branch.If the accumulated metric is smaller than ρ Ω at the last layer, ρ Ω and Ω are updated.Repeat the search process until all subtrees are checked.e cross symbol in Figure 1 shows that the branch with the accumulated metric not less than the threshold value ρ Ω is pruned.

Proposed Ordering MBS-Based Detectors
In MBS detector, the detection of the SM signal is in the ascending order of the subtree indices and the RA indices.Essentially, the MBS detector is used to calculate (4) and select partial reserved nodes.When the number of the reserved nodes is a constant, it is of great importance to choose which nodes.In other words, the computation order of (4) can directly a ect the BER performance and the computational complexity.In this section, in order to investigate the in uence of di erent detection orders on the BER performance and computational complexity, we propose three ordering MBS-based detectors by adjusting the search orders.

Ost-MBS Detector.
e computational complexity of the MBS detector is reduced by pruning the branches, whose accumulated metric is larger than or equal to ρ Ω .However, the MBS algorithm does not take into account the in uence of subtree search order on the computational complexity.If the optimal solution stands in the rst subtree, only fewer nodes are searched in the subsequent subtrees.But, if the optimal solution exists in the last subtree, we need to search more nodes in the top N T − 1 subtrees.In other words, searching rstly from the subtree where the optimal solution most probably belongs to can reduce total searches of nodes in the subsequent subtrees.us, it can decrease the computational complexity.at is, the searching order of subtrees directly a ects the computational complexity.We propose the ost-MBS detector, which estimates the optimal solution by rearranging the order of subtrees.In the rst stage, we order the TA indices based on the suboptimal antenna detection algorithm.In the second stage, we rst search from the most probable TA index, the estimated solution is obtained by searching all N T subtrees.e proposed ost-MBS detector works as follows.
Stage 1. Reorder the TA indices in descending order using the suboptimal modi ed maximum ratio combining (MMRC) algorithm proposed in [5].
e MMRC lter outputs are obtained by e higher the value of t j , the more likely it is that the jth TA was the one activated.
e TA indices are sorted in descending order by t j .e set of ordered TA indices is denoted by u.Suppose T [t 1 , t 2 , . . ., t N T ] T , the ordered TA indices can be obtained as e tree structure of 2×2 SM with 4QAM and where sort(•) denotes a descending order function and u 1 and u N T are the indices of the maximum and minimum elements of T, respectively.In other words, u 1 and u N T are the most likely and the least likely estimates of the TA index, respectively.
Stage 2. Determine the TA index and the transmit symbol using the MBS detector.

Oy-MBS Detector.
Since only one TA is activated at one time slot, assuming that the lth antenna sends symbol s m , the received signal can be expressed as Each signal at the receiver is related to the channel gain, the transmit symbol, and the white Gaussian noise.Due to the difference of the channel gain and noise in each channel, the channel gain and noise together determine the amplitude of the received signal.Generally speaking, a strong received signal contributes to the demodulation.In the MBS detector, the TA index and the transmit symbol are estimated in ascending order of the RA index from the root node to the leaf nodes.However, the effect of search order on the BER is not taken into account.For this reason, we propose the oy-MBS detector, which detects in descending order of the amplitude of the received signal.e oy-MBS detector is described in detail as follows.
where  v 1 and  v N R are the indices of the maximum and minimum values in Z. e layer search order set v can be obtained as e new search tree, whose (2i − 1) th and (2i) th layers correspond to the  v th i RA, can be built by exchanging the layers of Figure 1.
Stage 2. Determine the TA index and the transmit symbol using the MBS detector.

Ost-Oy-MBS Detector.
In ost-MBS detector, all N T subtrees are searched in descending order of |h H j y|/||h j || F . at is to say, we first detect the most probable subtree and then detect the most impossible subtree at last.To some extent, the computational complexity can be reduced.e oy-MBS detector performs the MBS detection in descending order of the received signals amplitude, which can improve the BER performance.In this subsection, we combine the detection orders of ost-MBS and oy-MBS detectors together.We propose the ost-oy-MBS detector whose detection order is based on the subtrees and the received signals.We detect all subtrees in descending order of |h H j y|/‖h j ‖ F and detect each layer of subtrees in descending order of |r i |.
e detection process of the proposed ordering MBSbased detectors is summarized in Algorithm 1.In Algorithm 1, lines 2-6 and 7-11 correspond to subtree-ordering and receiver-ordering strategies, respectively, whereas lines 12-28 describe the detection process of the MBS detector.

Simulation Results
In this section, the computational complexity and the BER performance of the proposed detectors and the MML, MBS, ML, simplfied OD, simplified MD, SLCD, and ASLCD detectors are compared.e label, (N 1 , N 2 ) OD, denotes the simplified OD detector with N 1 level-one subsets and N 2 level-two subsets.e label, (N, N 1 , and N 2 ) MD, stands for the simplified MD detector with N estimated transmit antennas, N 1 level-one subsets, and N 2 level-two subsets.e label, (N) SLCD, denotes the simplified low-complexity detection with N most probable estimates.
e label, (N, α) ASLCD, stands for the adaptive low-complexity detection with N most probable estimates and threshold coefficient α.
e ideal channel state information (CSI) is assumed available at the receiver.In the simulation, the signal-to-noise ratio (SNR) is the ratio of the signal power to the noise power, i.e., ρ � ( To validate the BER performance of the abovementioned detectors, the theoretical bound [17] is drawn in BER simulation figures.e BER performance and computational complexity of the proposed detectors are shown in Figures 4-5 with M � [16, 10, 8, 4, 4, 2, 1, 1] in MBS and the proposed detectors and M � [128, 80, 64, 32, 32, 16, 8, 1] in the MML detector for 8 × 4 16QAM SM systems.Since each level-two subset must contain more than four signals (i.e., the modulation order N M > 16) in OD and MD detectors, the simulation curves of OD and MD detectors are not listed in Figures 4-5.
To estimate the computational complexity of an algorithm, we define the computational complexity as the total number of the real-valued multiplications/divisions required in the detection process.
In the MML algorithm, we need to compute the accumulated metrics of all N T N M nodes in the first layer and compute the accumulated metrics of M k−1 nodes in the kth (2 ≤ k ≤ 2N R ) layer.Since computing the accumulated metrics of one node needs 3 real multiplications, the computational complexity of the MML detector is Since the ML detector computes the accumulated metrics of all nodes in the tree, the computational complexity of the ML detector is 6N T N R N M .
According to Section 3.2 and 3.3 in [16], we can obtain the computational complexity of OD and MD. e 4 Journal of Electrical and Computer Engineering computational complexity of the (N 1 , N 2 ) OD detector and (N, ), respectively.

Journal of Electrical and Computer Engineering
From the above analysis, we can conclude that the computational complexity of MML, OD, and MD detectors is lower than that of the ML detector but with a BER performance loss.Meanwhile, the complexity of MML, OD, MD, SLCD, and ASLCD depends on the preset parameters.e computational complexity of the proposed MBSbased detectors includes the number of real-valued multiplications of computing |h H j y|/||h j || F and |r i | in stage 1, and the number of real-valued multiplications of MBS detector in stage 2. e computational complexity of stage 1 can be easily obtained by calculation.e computational complexity of stage 2 depends on the number of retained nodes.
In MBS and the proposed MBS-based detectors, the parameter M k is at most the number of retained nodes in each layer.at is, the number of retained nodes is not xed.erefore, the computational complexity of proposed MBSbased detectors can only be obtained by simulation.
From the above simulation curves, we can draw the following conclusions: (1) e computational complexity of MBS, OD, MD, SLCD, ASLCD, MML, and the proposed detectors is lower than that of the ML detector.Since the number of retained nodes of MML, OD, and MD detectors is xed under di erent SNRs, the computational complexity does not change with the SNR.e computational complexity of SLCD, ASLCD, MBS, ost-MBS, oy-MBS, and ost-oy-MBS detectors changes with the SNR.
(2) e BER performance of ost-MBS detector is the same as that of the MBS detector, and the complexity is lower than that of the MBS detector.e ost-MBS detector only changes the search order of subtrees and does not a ect the detection performance.erefore, the ost-MBS and MBS detectors have the same BER performance.
e ost-MBS detector searches the subtrees in the descending order of |h H j y|/ h j F , which increases the probability of the optimal solution in the rst subtree.Since the accumulated metric of the optimal solution is minimal, the number of retained nodes can be reduced in the subsequent subtrees, thus reducing the total computational complexity.
(3) e BER performance of the oy-MBS detector is superior to that of the MBS detector.e oy-MBS detector estimates the solution in the descending order of the received signals amplitude.To some degree, the strong received signal contributes to the demodulation.erefore, compared with the MBS detector, the oy-MBS detector has better BER performance.Meanwhile, we also notice that the sorting strategy of oy-MBS also reduces the computational complexity.(4) e ost-oy-MBS detector and oy-MBS detector have the same BER performance, which is superior to the MBS detector.e ost-oy-MBS detector has the advantages of both ost-MBS and oy-MBS detectors.
at is, the ost-oy-MBS detector has the best BER performance and the lowest computational complexity among the proposed MBS-based detectors.
(5) Under the current simulation conditions, the BER performance of OD, MD, SLCD, ASLCD, MML, and ost-oy-MBS detectors is almost the same as the ML detector.Compared to OD and MD detectors, MBSbased detectors have a lower computational complexity in moderate-to-high SNRs.

Conclusion
In this paper, novel ordering MBS-based detectors for SM systems are proposed to improve the BER performance and    Journal of Electrical and Computer Engineering reduce the computational complexity.e ost-MBS detector first searches each subtree from the most probable TA.Compared to the MBS detector, it has the lower computational complexity.e oy-MBS algorithm detects each subtree in the descending order of the received signal amplitude.e BER performance of the oy-MBS detector is superior to that of the MBS detector.e ost-oy-MBS detector combined the orders of ost-MBS and oy-MBS detectors.Among all proposed MBS-based methods, the ostoy-MBS detector has the best BER performance and the lowest computational complexity.Meanwhile, we notice that the computational complexity and the BER performance of OD, MD, SCLD, ASLCD, MML, and MBS-based detectors depend on the preset parameters.Regrettably, how to select the parameters in the proposed MBS-based detectors can only be obtained by simulation.Next, we will study how to select parameters and try to give a theoretical derivation.

Figure 2 :
Figure 2: e BER performance of the proposed detectors with N T 4, N R 4, and 64QAM modulation.

Figure 3 :
Figure 3: e computational complexity of the proposed detectors with N T 4, N R 4, and 64QAM modulation.

Figure 4 :
Figure 4: e BER performance of the proposed detectors with N T 8, N R 4, and 16QAM modulation.

Figure 5 :
Figure 5: e computational complexity of the proposed detectors with N T 8, N R 4, and 16QAM modulation.