Low-Complexity Detection Algorithms for Spatial Modulation ...

Research ArticleLow-Complexity Detection Algorithms for Spatial ModulationMIMO Systems

Xinhe Zhang 1 Yuehua Zhang1 Chang Liu2 and Hanzhong Jia3

1School of Electronic and Information Engineering University of Science and Technology Liaoning Anshan 114051 China2National Key Laboratory of Science and Technology on CommunicationsUniversity of Electronic Science and Technology of China Chengdu 611731 China3State Grid Liaoning Information and Communication Company Shenyang 110006 China

Correspondence should be addressed to Xinhe Zhang cdaszxhsinacom

Received 9 August 2018 Revised 11 October 2018 Accepted 25 October 2018 Published 15 November 2018

Academic Editor Jit S Mandeep

Copyright copy 2018 Xinhe Zhang et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

To meet the demand of wireless communication systems forhigher data transmission rate multiple-input multiple-output (MIMO) technology has been adopted in mobileterminals MIMO technology improves data throughputwithout increasing additional bandwidth and transmitpower Spatial modulation (SM) [1ndash3] is an emergingtransmission scheme for MIMO systems +e main char-acteristic of SM is that only one transmit antenna is activatedat one time slot but simultaneously the SM systems can usethe original signal domain (signal constellation) and thetransmit antenna (TA) indices (spatial constellation) toconvey information Compared to MIMO systems SMsystems can only equip one radio frequency (RF) chainavoid interchannel interference (ICI) and interantennasynchronization (IAS) and also reduce the complexity ofdemodulation

For the detection of SM signals maximum ratio com-bining (MRC) algorithm was proposed in [4] in which theactive-antenna index and the transmit symbol are separatelyestimated +e MRC detector has a low computationalcomplexity and only performs well on the constrainedchannels +is detector was improved in [5] and it furthercan be applied in conventional channel conditions +eoptimum maximum likelihood (ML) detector which in-volves joint detection of the TA index and of the transmitsymbol was proposed in [6] However the computationalcomplexity linearly grows as the number of TA (NT) thenumber of receive antennas (NR) and the size of themodulation scheme (NM) In order to obtain the near-optimal solution with a lower computational complexityseveral low-complexity detectors have been put forward[7ndash17] In [7 8] two low-complexity hard-limiter-based ML(HL-ML) detectors which have the same BER performanceas the ML detector were proposed forM-PSK and square- or

HindawiJournal of Electrical and Computer EngineeringVolume 2018 Article ID 4034625 7 pageshttpsdoiorg10115520184034625

rectangular-QAM modulation +e computational com-plexity has nothing to do with the constellation size In[9ndash11] sphere-decoding (SD) algorithms were put forwardfor SM systems which are capable of achieving near-optimalperformance with a lower computational complexity onaverage At worst the computational complexity is equiv-alent to that of the ML detector However its detectionperformance depends mainly on the initial search radius andthe transmit parameters Compared with SD detectors SDaided by the ordering strategy proposed in [12] can greatlyreduce the computational complexity Two matched filter-(MF-) based detectors were proposed in [13] In [14] Wanget al proposed a novel signal vector-based detection (SVD)scheme Tang et al [15] presented a distanced-based ordereddetection (DBD) algorithm to reduce the receiver com-plexity and achieve a near-maximum likelihood perfor-mance To reduce the detection complexity of ML detectionXu [16] presented simplified ML-based optimal detection(OD) and simplified multistage detection (MD) In thesimplified ML-based detection and multistage detectionschemes the signal set is firstly partitioned into four ldquolevel-one subsetsrdquo Each level-one subset is further partitionedinto four ldquolevel-two subsetsrdquo if each subset contains morethan four signals +e simple low-complexity detection(SLCD) and adaptive simple low-complexity detection(ASLCD) were proposed in [17]

In [18 19] the M-algorithm to maximum likelihood(MML) detector with prioritized tree-search structure waspresented +e detection is considered as a breadth-firstsearch tree with NTNM branches and NR layers in whichthe ith layer corresponds to the ith receive antenna (RA)+e MML detector only examines partial nodes in the treewhereas the ML detector traverses all nodes Comparedwith the ML detector the MML detector can achievea lower computational complexity In [20] a low-complexity symbol detection based on modified beamsearch (MBS) was proposed +e detection process of theMBS algorithm can be represented by constructing a treewith NT subtrees and 2NR layers where each subtree hasNM complete paths from the root node to the leaf nodesand each of the paths stands for a candidate solution +esolution is found by performing modified beam searchCompared with the MML algorithm the MBS algorithmreduces the computational complexity by discarding un-promising candidate solutions

In the MBS detector the detection sequence of differentsubtrees is confined to the ascending order of the subtreeindices whereas it ignores the influence of different searchorders on the computational complexity Moreover thedetection of all layers is confined to the ascending order ofthe layer indices whereas it ignores the influence of differentsearch orders on its bit-error-rate (BER) performance +atis to say the influence of different search orders on the BERperformance and the computational complexity is notconsidered in the MBS detector In recent years the sortingstrategy has attracted more and more attention To somedegree the sorting strategy can improve the algorithmdetection performance In [12 21] different orderingstrategies were proposed to improve the detection

performance In this paper we proposed three MBS-baseddetectors with novel ordering strategies (1) the ost-MBSdetector rearranges the search order of subtrees (2) the oy-MBS detector performs SM signal detection in a descendingorder of the received signal amplitude (3) the detectionorders of the abovementioned two detectors were jointlyconsidered in the ost-oy-MBS detector

+e rest of this paper is organized as follows In Section2 the system model of SM systems is introduced Section 3gives a brief overview of the MBS detector +e orderingstrategy is introduced to the MBS detector Section 4demonstrates ost-MBS oy-MBS and ost-oy-MBS detectorsSection 5 illustrates the simulation results Finally weconclude the paper with a summary in Section 5

Notations Boldface upperlower case symbols denote ma-trices and column vectors middot F is the Frobenius norm ofa vector or a matrix | middot | is the amplitude of a complexquantity or the cardinality of a setR(middot) andI(middot) are the realand imaginary parts of a complex-valued quantity (middot)H is theconjugate transpose of a vector or a matrix CN(μ σ2)denotes a complex Gaussian random variable with mean μand variance σ2

2 System Model

Consider an NT times NR SM system with constellationS s1 s2 sNM

1113966 1113967 In each time slot the incoming databits are rearranged into blocks of log2 NTNM bits in whichlog2 NT bits are used to select the activated TA and log2 NMbits are used to select the transmit symbol sm isin Sm isin 1 2 NM1113864 1113865 Hence the system model for SM sys-tems can be represented by

r G middot s + w (1)

where r isin CNR is the received signal vector s isin CNT is thetransmit symbol vector whose element is sm at the lthposition and zero at the other positions G isin CNRtimesNT andw isin CNR are the channel matrix and the noise vector whoseelements follow the circularly symmetric complex Gaussiandistributions with CN(0 1) and CN(0 σ2) respectively+e system model expressed in (1) can be reshaped as

y R(G) minusI(G)

I(G) R(G)1113890 1113891 middot

R(s)

I(s)1113890 1113891 +

R(w)

I(w)1113890 1113891 H middot x + n

(2)

where H isin R2NRtimes2NT x isin R2NT and n isin R2NR Since onlyone TA is activated in each time slot the system modelexpressed in (2) can be simplified as

y hl hl+NT1113960 1113961 middotR sm( 1113857

I sm( 11138571113890 1113891 + n (3)

where hl is the lth column of HIt follows from (3) that the optimal ML-based de-

modulator can be formulated as

2 Journal of Electrical and Computer Engineering

l sm( ) arg minlsm( )isinΛ

y minus hl hl+NT[ ] middotR sm( )I sm( )

[ ]

2

F

arg minlsmisinΛ

A lsm( )(4)

where Λ denotes the set containing all possible transmit an-tenna indices and complex constellation points Λ (l sm)|l isin 1 2 NT sm isin s1 s2 sNM

A(lsm) sum2NRk1 |yk minus hkl middotR(sm)minus hkl+NT

middotI(sm)|2 and hkl is the

(k l)th entry of matrix H

3 MBS Detector

According to Kim and Yi [20] the detection of the SM signalcan be regarded as a tree with NT subtrees and 2NR layerswhere each subtree has NM complete paths from the rootnode to the leaf nodes For ease of understanding we give anillustration (Figure 1) for the idea of the MBS detectorSuppose we have a 2 times 2 SM system with 4QAMmodulationthus the search tree has 4 layers and 8 branches 4 branchesof each subtree (TA) correspond to 4 symbols from the4QAM constellation We dene the branch metric of node(l sm) at the kth layer as the squared Euclidean distancebetween the received and the transmit signals which can bedenoted as B(lsm)k |yk minus hkl middotR(sm)minus hkl+NT

middotI(sm)|2eaccumulated metric of node (l sm) at the kth layer isthe summation of the branch metric at the kth layer andthe accumulated metric at (kminus 1)th layer which can beexpressed as A(lsm)k B(lsm)k + A(lsm)kminus 1

In the rst subtree theMk nodes in the kth layer with thesmallest accumulated metrics are kept as the candidatenodes for the next layer At the last layer the node with thesmallest accumulated metric is considered as the solution inthe rst subtree e smallest accumulated metric and itscorresponding TA index and the transmit symbol are rep-resented by ρΩ and Ω l sm respectively In the sub-sequent subtrees ρΩ andΩ are gradually updated In the kthlayer at most Mk nodes with the accumulated metricssmaller than the threshold value ρΩ are selected as thesurvival branches for the next layer If the accumulatedmetric is not less than ρΩ the search of the current branch isterminated Otherwise we should continue to search thenext branch If the accumulated metric is smaller than ρΩ atthe last layer ρΩ and Ω are updated Repeat the searchprocess until all subtrees are checked e cross symbol inFigure 1 shows that the branch with the accumulated metricnot less than the threshold value ρΩ is pruned

4 Proposed Ordering MBS-Based Detectors

In MBS detector the detection of the SM signal is in theascending order of the subtree indices and the RA indicesEssentially the MBS detector is used to calculate (4) andselect partial reserved nodes When the number of the re-served nodes is a constant it is of great importance to choosewhich nodes In other words the computation order of (4)can directly aect the BER performance and the

computational complexity In this section in order to in-vestigate the inuence of dierent detection orders on theBER performance and computational complexity we pro-pose three ordering MBS-based detectors by adjusting thesearch orders

41Ost-MBSDetector e computational complexity of theMBS detector is reduced by pruning the branches whoseaccumulated metric is larger than or equal to ρΩ Howeverthe MBS algorithm does not take into account the inuenceof subtree search order on the computational complexity Ifthe optimal solution stands in the rst subtree only fewernodes are searched in the subsequent subtrees But if theoptimal solution exists in the last subtree we need to searchmore nodes in the top NT minus 1 subtrees In other wordssearching rstly from the subtree where the optimal solutionmost probably belongs to can reduce total searches of nodesin the subsequent subtrees us it can decrease the com-putational complexity at is the searching order of sub-trees directly aects the computational complexity Wepropose the ost-MBS detector which estimates the optimalsolution by rearranging the order of subtrees In the rststage we order the TA indices based on the suboptimalantenna detection algorithm In the second stage we rstsearch from the most probable TA index the estimatedsolution is obtained by searching all NT subtrees eproposed ost-MBS detector works as follows

Stage 1 Reorder the TA indices in descending order usingthe suboptimal modied maximum ratio combining(MMRC) algorithm proposed in [5] e MMRC lteroutputs are obtained by

tj hHj y∣∣∣∣∣

∣∣∣∣∣hjF j 1 NT (5)

e higher the value of tj the more likely it is that the jthTA was the one activated e TA indices are sorted indescending order by tj e set of ordered TA indices isdenoted by u Suppose T [t1 t2 tNT

]T the ordered TAindices can be obtained as

1st subtree 2nd subtree

1st layer

2nd layer

3th layer

4th layer

B1(lsm)

A1(lsm)

B2(lsm)

A2(lsm)

B3(lsm)

A3(lsm)

B4(lsm)

A4(lsm)

1

2 4

5

633

S1 S1S2 S2S3 S3S4 S4

3

Root

3

32

2

2

2

2

2

2

1

1

1

ρΩ = 6 Ω = 1s2

1

1

Figure 1 e tree structure of 2times2 SM with 4QAM andM [3 2 2 1]

Journal of Electrical and Computer Engineering 3

u u1 u2 uNT1113872 1113873 argsort(T) (6)

where sort(middot) denotes a descending order function and u1and uNT

are the indices of the maximum and minimumelements of T respectively In other words u1 and uNT

arethe most likely and the least likely estimates of the TA indexrespectively

Stage 2 Determine the TA index and the transmit symbolusing the MBS detector

42 Oy-MBSDetector Since only one TA is activated at onetime slot assuming that the lth antenna sends symbol sm thereceived signal can be expressed as

ri gli middot sm + wi i 1 NR (7)

Each signal at the receiver is related to the channel gainthe transmit symbol and the white Gaussian noise Due tothe difference of the channel gain and noise in each channelthe channel gain and noise together determine the amplitudeof the received signal Generally speaking a strong receivedsignal contributes to the demodulation In theMBS detectorthe TA index and the transmit symbol are estimated inascending order of the RA index from the root node to theleaf nodes However the effect of search order on the BER isnot taken into account For this reason we propose the oy-MBS detector which detects in descending order of theamplitude of the received signal +e oy-MBS detector isdescribed in detail as follows

Stage 1 Sort the RA indices in descending order by |ri| LetZ [z1 z2 zNR

]T where zi |ri| and the set of theordered RA indices can be obtained as

1113957v 1113957v1 1113957v2 1113957vNR1113872 1113873 argsort(Z) (8)

where 1113957v1 and 1113957vNRare the indices of the maximum and

minimum values in Z +e layer search order set v can beobtained as

v2iminus1 1113957vi

v2i NR + 1113957vi i 1 NR(9)

+e new search tree whose (2iminus 1)th and (2i)th layerscorrespond to the 1113957vth

i RA can be built by exchanging thelayers of Figure 1


43 Ost-Oy-MBS Detector In ost-MBS detector all NTsubtrees are searched in descending order of |hH

j y|||hj||F+at is to say we first detect the most probable subtree andthen detect the most impossible subtree at last To someextent the computational complexity can be reduced +eoy-MBS detector performs the MBS detection in descendingorder of the received signals amplitude which can improvethe BER performance In this subsection we combine thedetection orders of ost-MBS and oy-MBS detectors togetherWe propose the ost-oy-MBS detector whose detection order

is based on the subtrees and the received signals We detectall subtrees in descending order of |hH

j y|hjF and detecteach layer of subtrees in descending order of |ri|

+e detection process of the proposed ordering MBS-based detectors is summarized in Algorithm 1 In Algorithm1 lines 2ndash6 and 7ndash11 correspond to subtree-ordering andreceiver-ordering strategies respectively whereas lines12ndash28 describe the detection process of the MBS detector

5 Simulation Results

In this section the computational complexity and the BERperformance of the proposed detectors and the MML MBSML simplfied OD simplified MD SLCD and ASLCDdetectors are compared+e label (N1 N2) OD denotes thesimplified OD detector with N1 level-one subsets and N2level-two subsets +e label (N N1 and N2) MD stands forthe simplified MD detector with N estimated transmit an-tennas N1 level-one subsets and N2 level-two subsets +elabel (N) SLCD denotes the simplified low-complexitydetection with N most probable estimates +e label(N α) ASLCD stands for the adaptive low-complexity de-tection with N most probable estimates and threshold co-efficient α +e ideal channel state information (CSI) isassumed available at the receiver In the simulation thesignal-to-noise ratio (SNR) is the ratio of the signal power tothe noise power ie ρ (1113936

NMm1s

2mNM)σ2

To validate the BER performance of the abovementioneddetectors the theoretical bound [17] is drawn in BERsimulation figures Figures 2ndash3 compare the BER perfor-mance and the computational complexity of the proposeddetectors and existing detectors for 4 times 4 64QAM SM sys-tems with M [64 26 26 8 8 2 2 1] in MBS and theproposed detectors and M [256 104 104 32 32 8 8 1] inthe MML detector +e BER performance and computa-tional complexity of the proposed detectors are shown inFigures 4ndash5 with M [16 10 8 4 4 2 1 1] in MBS and theproposed detectors and M [128 80 64 32 32 16 8 1] inthe MML detector for 8 times 4 16QAM SM systems Since eachlevel-two subset must contain more than four signals(ie the modulation order NM gt 16) in OD and MD de-tectors the simulation curves of OD and MD detectors arenot listed in Figures 4ndash5

To estimate the computational complexity of an algo-rithm we define the computational complexity as the totalnumber of the real-valuedmultiplicationsdivisions requiredin the detection process

In the MML algorithm we need to compute the accu-mulated metrics of all NTNM nodes in the first layer andcompute the accumulated metrics of Mkminus1 nodes in the kth(2le kle 2NR) layer Since computing the accumulatedmetrics of one node needs 3 real multiplications thecomputational complexity of the MML detector is3NTNM + 1113936

2NRminus1i1 3Mi

Since the ML detector computes the accumulatedmetrics of all nodes in the tree the computational com-plexity of the ML detector is 6NTNRNM

According to Section 32 and 33 in [16] we can obtainthe computational complexity of OD and MD +e


computational complexity of the (N1 N2) OD detector and(NN1 N2) MD detector is 10NR(4NT + 4N1 +NMN216)and 10NR(NT + 4N + 4N1 +NMN216) respectively

According to Section 35 and 36 in [17] the computationalcomplexity of SLCD and ASLCD depends on the parameterN and the size of estimated transmit symbol set Since the sizeof estimated transmit symbol set is not constant the com-putational complexity can only be obtained by simulation

0 5 10 15 20 2510minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

BER

eoryML(2)SLCD(205)ASLCD

MML(44)OD(245)MDMBS

OstminusMBSOyminusMBSOstminusoyminusMBS

Figure 2 e BER performance of the proposed detectors withNT 4 NR 4 and 64QAM modulation

0 5 10 15 20 25

2500

800

1000

1200

1400

1600

1800

2000

2200

2400

SNR (dB)

Com

puta

tiona

l com

plex

ity

(2)SLCD(205)ASLCDMML

(44)OD(245)MDMBS


Figure 3 e computational complexity of the proposed detectorswith NT 4 NR 4 and 64QAM modulation

(1) Initialization Λl (l sm)|sm isin s1 middot middot middot sNM for each l isin 1 middot middot middot NT

(2) if subtree-ordering is used then(3) u (u1 uNT

) argsort(|hHj y|||hj||F)(4) else(5) u (1 middot middot middot NT)(6) end if(7) if receiver-ordering is used then(8) v (v1 vNR

) argsort(|ri|) obtain layer search order v by Equation (9)(9) else(10) v (1 middot middot middot NR) obtain layer search order v by Equation (9)(11) end if(12) ρΩ 0Ω ϕ(13) for i 1NT(14) Ψ Λui [(p q) value] search subtree(Ψv)(15) Ω (p q) and ρΩ value if (p q) is not null and valuelt ρΩ(16) end for(17) End the algorithm by returning Ω corresponding to ρΩ(18) function search subtree(Ψ v)(19) For each (p q) isin Ψ A(pq) 0(20) for i 12NR(21) k vi for each (p q isin Ψ) A(pq) A(pq) + B(pq)k (22) while ilt 2NR and |Ψ|gtMi(23) Ψ Ψminus (p q) where (p q) argmax

(pq)isinΨA(pq)

(24) end while(25) For each (p q) isin Ψ Ψ Ψminus (p q) if A(pq) ge ρΩ(26) end for(27) return [(p q) value] argmin

(pq)isinΨA(pq) if Ψ not empty otherwise return null

(28) end function

ALGORITHM 1 Ordering-aided MBS-based detectors


From the above analysis we can conclude that thecomputational complexity of MML OD and MD detectorsis lower than that of the ML detector but with a BERperformance loss Meanwhile the complexity of MML ODMD SLCD and ASLCD depends on the preset parameters

e computational complexity of the proposed MBS-based detectors includes the number of real-valued multi-plications of computing |hHj y|||hj||F and |ri| in stage 1 andthe number of real-valued multiplications of MBS detectorin stage 2 e computational complexity of stage 1 can beeasily obtained by calculation e computational com-plexity of stage 2 depends on the number of retained nodes

In MBS and the proposed MBS-based detectors the pa-rameterMk is at most the number of retained nodes in eachlayer at is the number of retained nodes is not xederefore the computational complexity of proposed MBS-based detectors can only be obtained by simulation

From the above simulation curves we can draw thefollowing conclusions

(1) e computational complexity of MBS OD MDSLCD ASLCD MML and the proposed detectors islower than that of theMLdetector Since the number ofretained nodes ofMMLOD andMDdetectors is xedunder dierent SNRs the computational complexitydoes not change with the SNR e computationalcomplexity of SLCD ASLCDMBS ost-MBS oy-MBSand ost-oy-MBS detectors changes with the SNR

(2) e BER performance of ost-MBS detector is thesame as that of the MBS detector and the complexityis lower than that of the MBS detector e ost-MBSdetector only changes the search order of subtreesand does not aect the detection performanceerefore the ost-MBS and MBS detectors have thesame BER performance e ost-MBS detectorsearches the subtrees in the descending order of|hHj y|hjF which increases the probability of theoptimal solution in the rst subtree Since the ac-cumulated metric of the optimal solution is minimalthe number of retained nodes can be reduced in thesubsequent subtrees thus reducing the total com-putational complexity

(3) e BER performance of the oy-MBS detector issuperior to that of the MBS detector e oy-MBSdetector estimates the solution in the descendingorder of the received signals amplitude To somedegree the strong received signal contributes to thedemodulation erefore compared with the MBSdetector the oy-MBS detector has better BER per-formance Meanwhile we also notice that the sortingstrategy of oy-MBS also reduces the computationalcomplexity

(4) e ost-oy-MBS detector and oy-MBS detector havethe same BER performance which is superior to theMBS detector e ost-oy-MBS detector has theadvantages of both ost-MBS and oy-MBS detectorsat is the ost-oy-MBS detector has the best BERperformance and the lowest computational com-plexity among the proposed MBS-based detectors

(5) Under the current simulation conditions the BERperformance of OD MD SLCD ASLCD MML andost-oy-MBS detectors is almost the same as the MLdetector Compared to OD and MD detectors MBS-based detectors have a lower computational com-plexity in moderate-to-high SNRs

6 Conclusion

In this paper novel ordering MBS-based detectors for SMsystems are proposed to improve the BER performance and

0 5 10 15 20 2510ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

SNR (dB)

BER

eoryML(3)SLCD

(305)ASLCDMMLMBS



0 5 10 15 20 25600

700

800

900

1000

1100

1200

1300

1400

1500

SNR (dB)

Com

puta

tiona

l com

plex

ity


MBSOstminusMBS

OyminusMBSOstminusoyminusMBS



reduce the computational complexity +e ost-MBS detectorfirst searches each subtree from the most probable TACompared to the MBS detector it has the lower compu-tational complexity +e oy-MBS algorithm detects eachsubtree in the descending order of the received signal am-plitude +e BER performance of the oy-MBS detector issuperior to that of the MBS detector +e ost-oy-MBS de-tector combined the orders of ost-MBS and oy-MBS de-tectors Among all proposed MBS-based methods the ost-oy-MBS detector has the best BER performance and thelowest computational complexity Meanwhile we notice thatthe computational complexity and the BER performance ofOD MD SCLD ASLCD MML and MBS-based detectorsdepend on the preset parameters Regrettably how to selectthe parameters in the proposed MBS-based detectors canonly be obtained by simulation Next we will study how toselect parameters and try to give a theoretical derivation

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] R Mesleh H Haas C W Ahn and S Yun ldquoSpatialmodulation-a new low complexity spectral efficiency en-hancing techniquerdquo in Proceedings of 2006 First InternationalConference on Communications and Networking in Chinapp 1ndash5 Beijing China October 2006

[2] M D Renzo H Haas A Ghrayeb S Sugiura and L HanzoldquoSpatail modulation for generalized MIMO challenges op-portunities and implementationrdquo Proceedings of the IEEEvol 102 no 1 pp 56ndash103 2014

[3] P Yang M D Renzo Y Xiao S Li and L Hanzo ldquoDesignguidelines for spatial modulationrdquo IEEE CommunicationsSurveys and Tutorials vol 17 no 1 pp 6ndash26 2015

[4] R Mesleh H Haas S Sinanovic C W Ahn and S YunldquoSpatial modulationrdquo IEEE Transactions on Vehicular Tech-nology vol 57 no 4 pp 2228ndash2241 2008

[5] N R Naidoo H Xu and T Quazi ldquoSpatial modulationoptimal detector asymptotic performance and multiple-stagedetectionrdquo IET Communications vol 5 no 10 pp 1368ndash1376 2011

[6] J Jeganathan A Ghrayeb and L Szczecinski ldquoSpatialmodulation optimal detection and performance analysisrdquoIEEE Communications Letters vol 12 no 8 pp 545ndash5472008

[7] R Rajashekar K V S Hari and L Hanzo ldquoReduced-complexity ML detection and capacity-optimized trainingfor spatial modulation systemsrdquo IEEE Transactions onCommunications vol 62 no 1 pp 112ndash125 2014

[8] H Men and M Jin ldquoA low-complexity ML detection algo-rithm for spatial modulation system with MPSK constella-tionrdquo IEEE Communications Letters vol 18 no 8pp 1375ndash1378 2014

[9] A Younis R Mesleh H Haas and P Grant ldquoReducedcomplexity sphere decoder for spatial modulation detection

receiversrdquo in Proceedings of 2010 IEEE Global Telecommu-nications Conference GLOBECOM 2010 pp 1ndash5 Miami FLUSA December 2010

[10] A Younis M D Renzo R Mesleh and H Hass ldquoSpheredecoding for spatial modulationrdquo in Proceedings of 2011 IEEEInternational Conference on Communications (ICC) pp 1ndash6Kyoto Japan June 2011

[11] A Younis S Sinanovic M D Renzo R Mesleh and H HaasldquoGeneralised sphere decoding for spatial modulationrdquo IEEETransactions on Communications vol 61 no 7 pp 2805ndash2815 2013

[12] K Lee ldquoDoubly ordered sphere decoding for spatial modu-lationrdquo IIEEE Communications Letters vol 19 no 5pp 795ndash798 2015

[13] S Sugiura C Xu S X Ng and L Hanzo ldquoReduced-complexity coherent versus non-coherent QAM-aidedspace-time shift keyingrdquo IEEE Transactions on Communi-cations vol 59 no 11 pp 3090ndash3101 2011

[14] J Wang S Jia and J Song ldquoSignal vector based detectionscheme for spatial modulationrdquo IEEE Communications Let-ters vol 16 no 1 pp 19ndash21 2012

[15] Q Tang Y Xiao P Yang Q Yu and S Li ldquoA new low-complexity near-ML detection algorithm for spatial modu-lationrdquo IEEE Wireless Communications Letters vol 2 no 1pp 90ndash93 2013

[16] H Xu ldquoSimplified maximum likelihood-based detectionschemes for M-ary quadrature amplitude modulation spatialmodulationrdquo IET Communications vol 6 no 11 pp 1356ndash1363 2012

[17] H Xu ldquoSimple low-complexity detection schemes for M-aryquadrature amplitude modulation spatial modulationrdquo IETCommunications vol 6 no 17 pp 2840ndash2847 2012

[18] J Zheng X Yang and Z Li ldquoLow-complexity detectionmethod for spatial modulation based on M-algorithmrdquoElectronics Letters vol 52 no 21 pp 1552ndash1554 2014

[19] Z Tian J Yang and Z Li ldquoM-algorithm-based optimaldetectors for spatial modulationrdquo Journal of Communicationsvol 10 no 4 pp 245ndash251 2015

[20] T Kim and K Yi ldquoLow-complexity symbol detection basedon modified beam search for spatial modulation MIMOsystemsrdquo Electronics Letters vol 51 no 19 pp 1546ndash15482015

[21] X Zhang G Zhao Q Liu N Zhao and M Jin ldquoEnhancedM-algorithm-based maximum likelihood detectors for spatialmodulationrdquo AEU - International Journal of Electronics andCommunications vol 70 no 9 pp 1361ndash1366 2016


International Journal of

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rectangular-QAM modulation +e computational com-plexity has nothing to do with the constellation size In[9ndash11] sphere-decoding (SD) algorithms were put forwardfor SM systems which are capable of achieving near-optimalperformance with a lower computational complexity onaverage At worst the computational complexity is equiv-alent to that of the ML detector However its detectionperformance depends mainly on the initial search radius andthe transmit parameters Compared with SD detectors SDaided by the ordering strategy proposed in [12] can greatlyreduce the computational complexity Two matched filter-(MF-) based detectors were proposed in [13] In [14] Wanget al proposed a novel signal vector-based detection (SVD)scheme Tang et al [15] presented a distanced-based ordereddetection (DBD) algorithm to reduce the receiver com-plexity and achieve a near-maximum likelihood perfor-mance To reduce the detection complexity of ML detectionXu [16] presented simplified ML-based optimal detection(OD) and simplified multistage detection (MD) In thesimplified ML-based detection and multistage detectionschemes the signal set is firstly partitioned into four ldquolevel-one subsetsrdquo Each level-one subset is further partitionedinto four ldquolevel-two subsetsrdquo if each subset contains morethan four signals +e simple low-complexity detection(SLCD) and adaptive simple low-complexity detection(ASLCD) were proposed in [17]

In [18 19] the M-algorithm to maximum likelihood(MML) detector with prioritized tree-search structure waspresented +e detection is considered as a breadth-firstsearch tree with NTNM branches and NR layers in whichthe ith layer corresponds to the ith receive antenna (RA)+e MML detector only examines partial nodes in the treewhereas the ML detector traverses all nodes Comparedwith the ML detector the MML detector can achievea lower computational complexity In [20] a low-complexity symbol detection based on modified beamsearch (MBS) was proposed +e detection process of theMBS algorithm can be represented by constructing a treewith NT subtrees and 2NR layers where each subtree hasNM complete paths from the root node to the leaf nodesand each of the paths stands for a candidate solution +esolution is found by performing modified beam searchCompared with the MML algorithm the MBS algorithmreduces the computational complexity by discarding un-promising candidate solutions

In the MBS detector the detection sequence of differentsubtrees is confined to the ascending order of the subtreeindices whereas it ignores the influence of different searchorders on the computational complexity Moreover thedetection of all layers is confined to the ascending order ofthe layer indices whereas it ignores the influence of differentsearch orders on its bit-error-rate (BER) performance +atis to say the influence of different search orders on the BERperformance and the computational complexity is notconsidered in the MBS detector In recent years the sortingstrategy has attracted more and more attention To somedegree the sorting strategy can improve the algorithmdetection performance In [12 21] different orderingstrategies were proposed to improve the detection

performance In this paper we proposed three MBS-baseddetectors with novel ordering strategies (1) the ost-MBSdetector rearranges the search order of subtrees (2) the oy-MBS detector performs SM signal detection in a descendingorder of the received signal amplitude (3) the detectionorders of the abovementioned two detectors were jointlyconsidered in the ost-oy-MBS detector

+e rest of this paper is organized as follows In Section2 the system model of SM systems is introduced Section 3gives a brief overview of the MBS detector +e orderingstrategy is introduced to the MBS detector Section 4demonstrates ost-MBS oy-MBS and ost-oy-MBS detectorsSection 5 illustrates the simulation results Finally weconclude the paper with a summary in Section 5

Notations Boldface upperlower case symbols denote ma-trices and column vectors middot F is the Frobenius norm ofa vector or a matrix | middot | is the amplitude of a complexquantity or the cardinality of a setR(middot) andI(middot) are the realand imaginary parts of a complex-valued quantity (middot)H is theconjugate transpose of a vector or a matrix CN(μ σ2)denotes a complex Gaussian random variable with mean μand variance σ2

2 System Model

Consider an NT times NR SM system with constellationS s1 s2 sNM

1113966 1113967 In each time slot the incoming databits are rearranged into blocks of log2 NTNM bits in whichlog2 NT bits are used to select the activated TA and log2 NMbits are used to select the transmit symbol sm isin Sm isin 1 2 NM1113864 1113865 Hence the system model for SM sys-tems can be represented by

r G middot s + w (1)

where r isin CNR is the received signal vector s isin CNT is thetransmit symbol vector whose element is sm at the lthposition and zero at the other positions G isin CNRtimesNT andw isin CNR are the channel matrix and the noise vector whoseelements follow the circularly symmetric complex Gaussiandistributions with CN(0 1) and CN(0 σ2) respectively+e system model expressed in (1) can be reshaped as

y R(G) minusI(G)

I(G) R(G)1113890 1113891 middot

R(s)

I(s)1113890 1113891 +

R(w)

I(w)1113890 1113891 H middot x + n

(2)

where H isin R2NRtimes2NT x isin R2NT and n isin R2NR Since onlyone TA is activated in each time slot the system modelexpressed in (2) can be simplified as

y hl hl+NT1113960 1113961 middotR sm( 1113857

I sm( 11138571113890 1113891 + n (3)

where hl is the lth column of HIt follows from (3) that the optimal ML-based de-

modulator can be formulated as




[ ]

2

F

arg minlsmisinΛ

A lsm( )(4)





3 MBS Detector










∣∣∣∣∣hjF j 1 NT (5)




1st layer

2nd layer

3th layer

4th layer

B1(lsm)

A1(lsm)

B2(lsm)

A2(lsm)

B3(lsm)

A3(lsm)

B4(lsm)

A4(lsm)

1

2 4

5

633


3

Root

3

32

2

2

2

2

2

2

1

1

1

ρΩ = 6 Ω = 1s2

1

1



u u1 u2 uNT1113872 1113873 argsort(T) (6)










1113957v 1113957v1 1113957v2 1113957vNR1113872 1113873 argsort(Z) (8)



v2iminus1 1113957vi

v2i NR + 1113957vi i 1 NR(9)











NMm1s

2mNM)σ2




2NRminus1i1 3Mi






0 5 10 15 20 2510minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

BER


MML(44)OD(245)MDMBS



0 5 10 15 20 25

2500

800

1000

1200

1400

1600

1800

2000

2200

2400

SNR (dB)

Com

puta

tiona

l com

plex

ity


(44)OD(245)MDMBS







(pq)isinΨA(pq)



(28) end function












6 Conclusion


0 5 10 15 20 2510ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

SNR (dB)

BER

eoryML(3)SLCD

(305)ASLCDMMLMBS



0 5 10 15 20 25600

700

800

900

1000

1100

1200

1300

1400

1500

SNR (dB)

Com

puta

tiona

l com

plex

ity


MBSOstminusMBS





Data Availability




References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




[ ]

2

F

arg minlsmisinΛ

A lsm( )(4)





3 MBS Detector










∣∣∣∣∣hjF j 1 NT (5)




1st layer

2nd layer

3th layer

4th layer

B1(lsm)

A1(lsm)

B2(lsm)

A2(lsm)

B3(lsm)

A3(lsm)

B4(lsm)

A4(lsm)

1

2 4

5

633


3

Root

3

32

2

2

2

2

2

2

1

1

1

ρΩ = 6 Ω = 1s2

1

1



u u1 u2 uNT1113872 1113873 argsort(T) (6)










1113957v 1113957v1 1113957v2 1113957vNR1113872 1113873 argsort(Z) (8)



v2iminus1 1113957vi

v2i NR + 1113957vi i 1 NR(9)











NMm1s

2mNM)σ2




2NRminus1i1 3Mi






0 5 10 15 20 2510minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

BER


MML(44)OD(245)MDMBS



0 5 10 15 20 25

2500

800

1000

1200

1400

1600

1800

2000

2200

2400

SNR (dB)

Com

puta

tiona

l com

plex

ity


(44)OD(245)MDMBS







(pq)isinΨA(pq)



(28) end function












6 Conclusion


0 5 10 15 20 2510ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

SNR (dB)

BER

eoryML(3)SLCD

(305)ASLCDMMLMBS



0 5 10 15 20 25600

700

800

900

1000

1100

1200

1300

1400

1500

SNR (dB)

Com

puta

tiona

l com

plex

ity


MBSOstminusMBS





Data Availability




References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


u u1 u2 uNT1113872 1113873 argsort(T) (6)










1113957v 1113957v1 1113957v2 1113957vNR1113872 1113873 argsort(Z) (8)



v2iminus1 1113957vi

v2i NR + 1113957vi i 1 NR(9)











NMm1s

2mNM)σ2




2NRminus1i1 3Mi






0 5 10 15 20 2510minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

BER


MML(44)OD(245)MDMBS



0 5 10 15 20 25

2500

800

1000

1200

1400

1600

1800

2000

2200

2400

SNR (dB)

Com

puta

tiona

l com

plex

ity


(44)OD(245)MDMBS







(pq)isinΨA(pq)



(28) end function












6 Conclusion


0 5 10 15 20 2510ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

SNR (dB)

BER

eoryML(3)SLCD

(305)ASLCDMMLMBS



0 5 10 15 20 25600

700

800

900

1000

1100

1200

1300

1400

1500

SNR (dB)

Com

puta

tiona

l com

plex

ity


MBSOstminusMBS





Data Availability




References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




0 5 10 15 20 2510minus5

10minus4

10minus3

10minus2

10minus1

100

SNR (dB)

BER


MML(44)OD(245)MDMBS



0 5 10 15 20 25

2500

800

1000

1200

1400

1600

1800

2000

2200

2400

SNR (dB)

Com

puta

tiona

l com

plex

ity


(44)OD(245)MDMBS







(pq)isinΨA(pq)



(28) end function












6 Conclusion


0 5 10 15 20 2510ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

SNR (dB)

BER

eoryML(3)SLCD

(305)ASLCDMMLMBS



0 5 10 15 20 25600

700

800

900

1000

1100

1200

1300

1400

1500

SNR (dB)

Com

puta

tiona

l com

plex

ity


MBSOstminusMBS





Data Availability




References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











6 Conclusion


0 5 10 15 20 2510ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

SNR (dB)

BER

eoryML(3)SLCD

(305)ASLCDMMLMBS



0 5 10 15 20 25600

700

800

900

1000

1100

1200

1300

1400

1500

SNR (dB)

Com

puta

tiona

l com

plex

ity


MBSOstminusMBS





Data Availability




References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Data Availability




References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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