Research ArticleGeneralized Complex Quadrature Spatial Modulation

Manar Mohaisen

Department of EEC Engineering Korea Tech Cheonan Republic of Korea

Correspondence should be addressed to Manar Mohaisen manarsubhikoreatechackr

Received 7 September 2018 Revised 6 February 2019 Accepted 24 February 2019 Published 28 April 2019

Guest Editor Mohamad K A Rahim

Copyright copy 2019 Manar Mohaisen This 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

Spatial modulation (SM) is a multiple-input multiple-output (MIMO) system that achieves a MIMO high spectral efficiency whilemaintaining the transmitter computational complexity and requirements as low as those of the single-input systems The complexquadrature spatial modulation (CQSM) builds on the QSM scheme and improves the spectral efficiency by transmitting two signalsymbols at each channel use In this paper we propose two generalizations of CQSM namely generalized CQSM with uniquecombinations (GCQSM-UC) and with permuted combinations (GCQSM-PC)These two generalizations perform close to CQSMor outperform it depending on the system parameters Also the proposed schemes require much less transmit antennas to achievethe same spectral efficiency of CQSM for instance assuming 16-QAM GCQSM-PC and GCQSM-UC require 10 and 15 transmitantennas respectively to achieve the same spectral of CQSMwhich is equipped with 32 antennas

1 Introduction

Multiple-input multiple-output (MIMO) techniques arecapable of satisfying the continuous demand in high datarates in communication systems Index modulation (IM)is a group of MIMO techniques that use the indices of agiven resource(s) of the communication system to conveyadditional information besides that carried by the signalsymbols [1 2] These indices represent distinct antennas[3] spreading codes [4] polarities [5] subcarriers [6] andcombinations of angles [7] among others A virtual spatialmodulation (VSM) was also proposed in [8] where indexmodulation is performed on the parallel channels resultingfrom the singular value decomposition of the MIMO channelmatrix

Spatial modulation (SM) has attracted an increasinginterest from researchers due to its low-complexity and highdata rates In SM information is transmitted through the sig-nal symbols drawn from any conventional constellation setsand the index of the antenna from which the signal symbolis transmitted [3] To achieve a high spectral efficiency SMrequires relatively high number of transmit antennas A par-allel SM denoted by half-full transmit-diversity SM (HFTD-SM) is proposed recently where antennas are grouped into

two-antenna groupsThe conventional SM is then performedusing the same signal symbol in each group [9 10]

A conventional generalized SM (GSM) transmits thesame signal symbol from a combination of antennas ratherthan one as in SM leading to a reduction in the numberof transmit antennas required to achieve a given spectralefficiency [11] A different implementation of the GSM wasproposed in [12] referred to as MA-SM in which eachactivated antenna in a given combination transmits a differentsignal symbol This improves the spectral efficiency at thecost of additional hardware requirements and computationalcomplexity GSM was extended to the multiuser scenario in[13]

Quadrature spatial modulation (QSM) is another tech-nique that extends the spatial constellation into in-phase andquadrature dimensions that are used to transmit the real andimaginary part respectively of a single signal symbol [14] In[15] a generalized QSM (GQSM) that combines the benefitsof spatial multiplexing and QSMwas proposed Antennas aregrouped and independent QSM modulation is performed ineach group The hardware design of the generalized quadra-ture space shift keying modulation (GQSSK) and generalizedQSM (GQSM) using a single RF chain was investigated in[16]

HindawiWireless Communications and Mobile ComputingVolume 2019 Article ID 3137927 12 pageshttpsdoiorg10115520193137927

2 Wireless Communications and Mobile Computing

Building on the QSM a complex QSM (CQSM) improvesthe spectral efficiency by transmitting two signal symbols ateach channel use from antennas whose indices also carryinformation [17] When both signal symbols are transmittedfrom the same antenna the resulting combination is drawnfrom the Minkowski sum of the two sets from which thesymbols are drawn The second modulation set is designedas a rotated version of the first where the rotation angle isoptimized to minimize the average bit-error rate To avoidtransmitting the two signal symbols from the same antennathe transmitter is equipped with an additional antenna that isused to transmit the second symbol only when both symbolsare supposed to be transmitted from the same antennain CQSM The proposed modification is named improvedCQSM (ICQSM) in [18] The modulation set design of theICQSM was addressed in [19] Notice that the CQSM can beconceptualized based on the conventional SM or the GSMand not as an extension of the QSM While this approach iscorrect we built CQSM as an extension of the QSM whereinstead of transmitting the real and imaginary parts of asingle signal symbol from the in-phase and quadrature spatialdimensions two complex-valued signal symbols are trans-mitted through the two spatial dimensions To be consistentwith our previouswork we keep the samename that explicitlyincludes QSM

Contributions The contributions of this paper are summa-rized as follows

(i) We propose two generalized CQSM schemes Thefirst is GCQSMwith unique combinations (GCQSM-UC) where the spatial symbols are generated asin generalized GSM Each resulting combinationis then split into two subsets The first subset ofantennas is used to transmit the first symbol thesecond subset is used to transmit the second signalsymbol The second scheme is named GCQSM withpermuted combinations (GCQSM-PC) In GCQSM-PC the antenna combinations from which the firstsymbol is transmitted are generated as the full set ofcombinations of a given length For each combinationassociated with the first symbol the correspondinglist of combinations for the second symbol is gener-ated such that no overlap occurs between antennasused for transmitting the two symbolsThis algorithmreduces the number of transmit antennas required toachieve a given spectral efficiency

(ii) We also propose an analytical method to optimize therotation angle for CQSM and ICQSM systems Theobtained rotation angle minimizes the upper-boundof pairwise error probability

Based on the simulation results the proposed schemesperform very close to CQSM while requiring much smallernumber of antennas to achieve the same spectral efficiencyFor instance CQSM GCQSM-UC and GCQSM-PC require32 15 and 10 transmit antennas to achieve a spectral efficiencyof 14 bits per channel use (bpcu) assuming 2 bits per signalsymbol

Notations We denote the spectral efficiency achieved byspatial and signal symbols ES119904119901119886 and ES119904119894119892 respectively andthe total spectral efficiency ES = ES119904119894119892 + ES119904119901119886 The vector e119887119886is the 119886th column of the 119887times119887 identity matrix119860 = 1198861 119886119870is a set where the order of elements does not matter and119861 = (1198871 119887119870) is a tuple where order matters In thefollowing (sdot)119879 and (sdot)119867 denote the matrixvector transposeand Hermitian transpose respectively 119876(sdot) is the Gaussiantail function or simply the 119876-function A signal symbolis a complex element drawn from a quadrature amplitudemodulation (QAM) or phase shift keying (PSK) set Thenumber of bits carried by each signal symbol is equal to 119902where 2119902 is the cardinality of the modulation set A spatialsymbol is the index(es) of the antenna(s) from which a singleor several signal symbols are transmitted

The rest of this paper is organized as follows In Section 2we describe the system model and review related works InSection 3 we introduced the proposed generalized CQSMtechniques and analyze their error performance in Section 4In Section 5 we formulate the search of the optimal rotationangle for CQSM and ICQSM as an optimization problem thatreduced the asymptotic upper-bound on the error probabilityThe optimization of the rotation angle of the proposedgeneralized schemes is addressed in Section 6The simulationresults are given in Section 7 and conclusions are drawn inSection 8

2 System Model and Related Works

21 System Model Consider a MIMO system with 119899119879 trans-mit and 119899119877 receive antennas The system equation is given asfollows

y = Hs + n (1)

where H and n are the channel matrix and the noise vectorwhose elements are iid centered circularly symmetric com-plex Gaussian and have a variance of one and 1205902119899 respectivelyand s is the transmitted vector In SM techniques the vectors contains a few nonzero elements

22 Spatial Modulation In SM both a signal symbol andthe index of the antenna from which it is transmitted carryinformation As such SM achieves an improved spectralefficiency while keeping the transmitter as simple as that ofthe single-input communication systems Accordingly theSM system is modeled as follows

y = He119899119879119894 119904119896 + n = h119894119904119896 + n (2)

The spectral efficiency which is equivalent to the capacity athigh signal-to-noise ratio (SNR) is equal to 119902 + 119873 bits perchannel use (bpcu) with119873 = log2(119899119879)23 Generalized Spatial Modulation In SM the number oftransmit antennas increases exponentially in terms of thenumber of bits carried by each spatial symbol That is 119899119879 =2119873 where 119873 is the number of bits per spatial symbol GSM

Wireless Communications and Mobile Computing 3

reduces the number of transmit antennas by sending thesame signal symbol from a combination of two or moreantennas Let 119899119880 denote the number of active antennas ateach channel use then the number of spatial symbols alsoreferred to as combinations for a given 119899119879 is ( 119899119879119899119880 ) Since thenumber of spatial symbols should be a power of two only2ES119904119901119886 combinations can be used where ES119904119901119886 = lfloorlog2 ( 119899119879119899119880 )rfloorAssuming 119899119880 = 2 GSM requires 7 transmit antennas toachieve ES119904119901119886 = 4 bits per spatial symbol whereas SMrequires 16 antennas to achieve the same spectral efficiency

Let 119897 = 1198971 119897119899119880 isin L be a spatial symbol where L is theset of spatial symbols that can be used for transmission Thenthe received vector in the case of GSM is given by

y = 119904119896radic119899119880sum119894isin119897h119894 + n (3)

24 Multi-Active Spatial Modulation In contrast to GSMin which a single signal symbol is transmitted from acombination of antennas MA-SM transmits a different signalsymbol from each activated antenna Let 119899119880 be the number ofactivated antennas then the received vector is given by

y = sum119894isin119897

h119897119894119904119894 + n (4)

where 119897 = 1198971 119897119899119880 isin L and L is the list of spatialsymbols of MA-SM which is equivalent to that of GSMAdditionally a 3-dimensional constellation set is proposedin which each spatial symbol is associated with a uniquerotation angle Signal symbols transmitted from a givenspatial symbol are rotated before transmission Accordingto this description MA-SM benefits from the moderatecomputational complexity and low hardware requirements ofSM and the high multiplexing gain of the vertical-Bell Labslayered space-time (V-BLAST) systemThe spectral efficiencyof MA-SM is equal to (ES119904119901119886 + 119902 times 119899119880) where ES119904119901119886 is definedin Section 23

25 Complex Quadrature Spatial Modulation QSM expandsthe spatial constellation into in-phase and quadrature dimen-sions At each channel use a single signal symbol is transmit-ted the real part of the signal symbol is transmitted throughthe in-phase spatial dimension the imaginary through thequadrature dimension As such the spectral efficiency ofQSM is 119902+2log2(119899119879) where ES119904119901119886 = 2log2(119899119879) and ES119904119894119892 = 119902

CQSM transmits two signal symbols at each channel useleading to a spectral efficiency of 2119902+2log2(119899119879) where ES119904119901119886 =2log2(119899119879) and ES119904119894119892 = 2119902 The first signal symbol 1199041198961 isin Ω119886 istransmitted from the 1198941th transmit antenna and the secondsignal symbol 11990411989610158402 isin Ω119887 is transmitted from the 1198942th antennaTo make the distinction between 1199041198961 and 11990411989610158402 at the receiverside Ω119887 is obtained as follows

Ω119887 = 11990411989610158402 | 11990411989610158402 = 1199041198961119890119895120579 1199041198961 isin Ω119886 (5)

The rotation angle 120579 is optimized such that the probability oferror is reducedThe received vector of the CQSM is given by

y = Hs + n = h11989411199041198961 + h119894211990411989610158402 + n (6)

which is expanded to

y = h11989411199041198961 + h119894211990410158401198962 + n if 1198941 = 1198942h119894 (1199041198961 + 11990410158401198962) = h119894119904119896101584010158403 if 1198941 = 1198942 = 119894 (7)

where 119904119896101584010158403 isin Ω119888 and Ω119888 = Ω119886 oplus Ω119887 with oplus denoting theMinkowski sum [20]The full modulation setΩ119889 = Ω119886cupΩ119887cupΩ119888 is of size (|Ω119886|+ |Ω119887|+ |Ω119886|times |Ω119887|) For instance assuming|Ω119886| = |Ω119887| = 16 the number of symbols inΩ119889 becomes 288This high density of symbols reduces the Euclidean distanceamong signal symbols leading to degradation in the errorperformance

26 Improved Complex Quadrature Spatial Modulation InICQSM the transmitter is equipped with an additionalantenna indexed 119899119870 with 119899119870 = 119899119879 + 1 which is used onlywhen 1198941 = 1198942 In this case the first signal symbol is transmittedfrom its designated antenna and the second symbol istransmitted from the additional antenna Accordingly theICQSM system is modeled as follows

y = 1199041198961h1198941 + 11990411989610158402h1198942 + n if 1198941 = 11989421199041198961h1198941 + 11990411989610158402h119899119870 + n if 1198941 = 1198942 (8)

This strategy reduces the size of the total modulation set Ω119889from (|Ω119886| + |Ω119887| + |Ω119886| times |Ω119887|) in the case of CQSM toonly (|Ω119886| + |Ω119887|) in ICQSM This increases the Euclideandistance among the signal symbols and hence improves theerror performance

Based on the above descriptions CQSM activates eitherone or two antennas whereas ICQSM always has two activeantennas To perform a fair comparison between thesetwo systems and MA-SM we assume that 119899119880 = 2 Thatis the hardware requirements of the transmitter and thecomputational complexity of the detection algorithm areequivalent Since both systems transmit two signal symbolsat each channel use the comparison is conducted in terms ofthe spectral efficiency achieved by spatial symbols SM on theother hand transmits a single signal symbol at each channeluse Table 1 lists the number of transmit antennas required toachieve a given number of bits per spatial symbol (SE119904119901119886) byseveral systems For instance CQSM and ICQSM require 3and 2 antennas less than MA-SM to achieve the same SE119904119901119886 of4 bpcuThis gap increases for higher SE119904119901119886

3 Generalized Complex QuadratureSpatial Modulation

Both CQSM and ICQSM require the same number of RFchains and ICQSM requires one more physical transmitantenna compared to CQSM Transmitting each signal sym-bol from a combination of antennas instead of a singleantenna reduces the number of transmit antennas requiredto achieve a given SE119904119901119886 In the following subsections weintroduce two generalizations of CQSM namely generalizedCQSM with unique combinations (GCQSM-UC) and gen-eralized CQSM with permuted combinations (GCQSM-PC)

4 Wireless Communications and Mobile Computing

Table 1 The number of transmit antennas required by severalsystems to achieve the same number of bits per spatial symbol(SE119904119901119886)

SE119904119901119886 SM MA-SM ICQSM CQSM2 4 4 3 24 16 7 5 46 64 12 9 8

Table 2 An example of the spatial symbols (1198941 1198942) that can be usedfor transmission by GCQSM-UC assuming 119899119879 = 6 and 119899119880 = 21198941 11989421 2 3 41 2 3 51 2 3 61 2 4 51 2 4 61 2 5 61 3 4 51 3 4 61 3 5 61 4 5 62 3 4 52 3 4 62 3 5 62 4 5 63 4 5 6In the following we denote by 1198941 and 1198942 the set of antennasused for transmitting the first and second signal symbolsrespectively In the sequel the tuple 119860 = (119861119862) is composedof the two sets 119861 and119862 where the order of the sets in the tuplematters but the order of the elements in each set does not

31 Generalized CQSMwith Unique Combinations Let 119899119880 bethe number of antennas from which each of the two signalsymbols is transmitted then for a given 119899119879 the number ofcombinations that can be used for transmission is ( 1198991198792119899119880 )Assuming that 119894 = 1198941 1198942 is a spatial symbol ie acombination of antennas of length 2119899119880 then the first andsecond 119899119880 antennas are used to transmit the first and secondsignal symbols respectively Table 2 depicts an example ofthe spatial symbols for 119899119879 = 6 and 119899119880 = 2 According tothis description there will be no overlap between the sets ofantennas used to transmit the first and second signal symbolsSince the number of spatial symbols should be a power of twothen only 2SE119904119901119886 signal symbols can be used for transmissionwhere SE119904119901119886 = lfloorlog2 ( 1198991198792119899119880 )rfloor

Let 1199041198961 isin Ω119886 and 11990411989610158402 isin Ω119887 be the two signal symbols to betransmitted at a given channel use then the received vector isgiven by

y = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894 + n (9)

The receiver employs the maximum-likelihood principle torecover the two signal symbols and the spatial symbol asfollows (119894lowast 119904lowast1198961 119904lowast11989610158402) = arg min

1199041198961isinΩ11988611990411989610158402isinΩ119887

119894=(11989411198942)isinL

1003817100381710038171003817y minus g10038171003817100381710038172= arg min1199041198961isinΩ11988611990411989610158402

isinΩ119887119894=(11989411198942)isinL

1003817100381710038171003817g10038171003817100381710038172 minus 2R (y119867g) (10)

where

g = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894 (11)

GCQSM-UC can be easily extended to transmitting morethan two signal symbols per channel use at the cost ofrequiring more RF chains at the transmitter side This ispossible because the error performance of the system doesnot depend on the rotation angle 120579 This will be addressed inmore detail in subsequent sections

32 Generalized CQSM with Permuted Combinations Toreduce the number of transmit antennas required to achievea given spectral efficiency we use the structure of thetransmitted vector to increase the number of spatial symbolsLet 119894 = (1198941 1198942) be a spatial symbol where the first signalsymbol 1199041198961 is transmitted from the 1198941 combination of antennasand the second signal symbol 11990411989610158402 from the combination 1198942 Fora given channel realization the received noiseless vector g isgiven by

g = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894

= 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894

(12)

where 11990411989610158402 = 1199041198962119890119895120579 and 1199041198961 1199041198962 isin Ω119886 To obtain 1198941 and 1198942 weimpose the following conditions

(i) 1198941 cup 1198942 = 0 this implies that the set of antennas fromwhich the first signal symbol and that from whichthe second symbol are transmitted do not overlapThis condition reduces the detection ambiguity at thereceiver

(ii) (1198941 1198942) = (1198942 1198941) as described earlier the first andsecond symbols are drawn from different constella-tion sets Therefore for given signal symbolsrsquoindices 1198961 and 1198962 the following two received latticepointsmdashnoiseless received vectorsmdash are distinguish-able

g1 = 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894 (13)

and

g2 = 1199041198961radic119899119880sum119894isin1198942h119894 +1199041198962radic119899119880sum119894isin1198941119890119895120579h119894 (14)

Wireless Communications and Mobile Computing 5

Input A = 1 119899119879 119899119880 119899119879Output The set of spatial symbolsIC = COMB(A 119899119880)for 1198941 isin C do

B = A1198941

D = COMB(B 119899119880)for 1198942 isin D do

APPND(I (1198941 1198942))end

end

Algorithm 1 Generation of the spatial symbols assuming 119899119879transmit antennas and that each signal symbol is transmitted froma combination of 119899119880 antennasTable 3 An example of the spatial symbols (1198941 1198942) that can be usedfor transmission by GCQSM-PC assuming 119899119879 = 5 and 119899119880 = 21198941 11989421 2 3 4 3 5 4 51 3 2 4 2 5 4 51 4 2 3 2 5 3 51 5 2 3 2 4 3 42 3 1 4 1 5 4 52 4 1 3 1 5 3 52 5 1 3 1 4 3 43 4 1 2 1 5 2 53 5 1 2 1 4 2 44 5 1 2 1 3 2 3

This is rendered possible using the rotation angle 120579The optimization of the rotation angle is addressed ina following section

Algorithm 1 depicts a pseudocode of generating the set ofspatial symbolsI that can be used for transmission ThereinCOMB(A 119899119880) generates all combinations of length 119899119880 of theelements of the set A Also APPND(I (1198941 1198942)) appends thetuple (1198941 1198942) to the setI B = A

1198941is the difference of the two

sets119860 and 1198941 Accordingly the number of spatial symbols thatcan be used for transmission is given by

119862spa = (119899119879119899119880) times (119899119879 minus 119899119880119899119880 ) (15)

Table 3 depicts an example of the obtained spatial sym-bols assuming 119899119879 = 5 and 119899119880 = 2 In this case GCQSM-PC obtains 30 spatial symbols whereas GCQSM-UC obtainsonly five

Table 4 lists the number of transmit antennas required byeach system to achieve a given spectral efficiency per spatialsymbol For instance SM GSM (and MA-SM) CQSMGCQSM-UC andGCQSM-PC require 1024 46 32 15 and 10transmit antennas respectively to achieve 10 bits per spatialsymbol

4 Performance Analysis ofthe Generalized CQSM

Let

g119894 = 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894 (16)

and

g119896 = 1199041radic119899119880sum119894isin1h119894 +1199042radic119899119880sum119894isin2119890119895120579h119894 (17)

be two noiseless received codewords corresponding to thetransmitted symbol (1199041198961 1199041198962 1198941 1198942) and (1199041 1199042 1 2) where119904119896 is a signal symbol and 119894 = 1198971 119897119899119880 is a spatial symbolThe corresponding received vectors are denoted by y119894 and y119896respectively

The conditional pairwise error probability (PEP) of themaximum-likelihood (ML) receiver is given by [21 22]

Pr [g119894 997888rarr g119896 | H] = 119876(radic 1003817100381710038171003817y119894 minus y1198961003817100381710038171003817221205902119899 )

= 119876(radic 1003817100381710038171003817H (s119894 minus s119896)1003817100381710038171003817221205902119899 )(18)

where119876(sdot) is the Gaussian tail probability function or simplythe119876-function and 1198892119894119896 = y119894 minus y1198962 is the squared Euclideandistance between s119894 and s119896 at the receiver Similarly 1198892119894119896(tx) =s119894 minus s1198962 is the squared Euclidean distance between s119894 and s119896at the transmitter Note that

EH 1198892119894119895 = EH (s119894 minus s119896)119867H119867H (s119894 minus s119896)= (s119894 minus s119896)119867 (s119894 minus s119896) = 1198892119894119896(tx) (19)

The unconditional PEP (UPEP) assuming 119899119877 receivesantennas which is obtained by taking the expectation of (18)over the channel H is given by

Pr [g119894 997888rarr g119896] = 120583119899119877119894119895 119899119877minus1sum119897=0

(119899119877 minus 1 + 119897119897 ) [1 minus 120583119894119895]119897 (20)

where

120583119894119895 = 12 (1 minus radic 120574 sdot 1198892119894119896(tx)4 + 120574 sdot 1198892119894119896(tx)

) (21)

and 120574 = 1205902119904 1205902119899 is the signal-to-noise ratio (SNR) with 1205902119904 =E(119904lowast119904) the expected power of the signal symbol

The union bound on the pairwise error probability is thengiven by averaging all the pairwise probabilities as follows

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896] (22)

6 Wireless Communications and Mobile Computing

Table 4 Examples of the number of transmit antennas required to achieve a given SE119904119901119886 for several spatial modulation systems

SE119904119901119886 SM GSM CQSM QSM GCQSM-UC GCQSM-PC6 64 12 8 8 68 256 24 16 11 810 1024 46 32 15 1012 4096 92 64 20 13

Accordingly the average bit-error rate (BER) is upper-bounded by

119875119890 le 11198722119872 2119872sum119894=1

2119872sum119896=1

119863s119894s119896Pr [g119894 997888rarr g119896] (23)

where 119863s119894s119896 the hamming distance between s119894 and s119896 is thenumber of errors associated with the event [g119894 997888rarr g119896] and119872 is the spectral efficiency of the system Note that in eachtransmitted vector symbol s there are exactly (2times119899119880) nonzeroelements

5 Optimization of the Rotation Angles ofCQSMICQSM Revisited

In [17 18] the optimal rotation angles applied to the secondmodulation set of the CQSM and ICQSM respectively areobtained through extensive Monte Carlo simulations Inthis case the obtained rotation angle corresponds to theminimum bit-error rate (BER) averaged over a large numberof channel and noise realizations Alternatively the rotationangle can be obtained through minimizing the upper-boundof the unconditional pairwise error probability

Let

g119894 = Hs119894 = 1199041198961h1198941 + 11990410158401198962h1198942g119896 = Hs119896 = 1199041h1 + 1199041015840

2h2 (24)

where y119894 = g119894 + n and y119896 = g119896 + n are the correspondingreceived vectors for a given noise vector n

The PEP and UPEP of the CQSM and ICQSM can also beobtained using (18)ndash(21) The union bound on the pairwiseerror probability at high SNR values (aka asymptotic prob-ability of error) is then given by

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896]= (2119899119877 minus 1119899119877 ) 120574minus1198991198772119872 2

119872sum119894=1

2119872sum119896=1

(1198892119894119896(tx))minus119899119877= (2119899119877 minus 1119899119877 ) 120574minus1198991198771198712 119861sum

119894=1

119891119894Ω119894(25)

where the maximum value of 119861 equals 26 as shownin [19] The term Ω119894 is a function of the signal symbols(1199041198961 11990410158401198962 1199041 11990410158402) and 119899119877 and 119891119894 is the frequency of Ω119894 where

sum119861119894=1 119891119894 = 1198994119879 It is shown in [19] that 119861 = 7 in the case ofICQSM and it can be shown to be equal to 15 for CQSMThe rotation angle is then obtained by solving the followingoptimization problem

argmin0lt120579lt1205872

( 119861sum119894=1

119891119894Ω119894) (26)

where 119861 = 15 and 7 for CQSM and ICQSM respectivelyFigure 1 depicts the cost function of (26) for CQSM and

ICQSM and for several system configurations The optimalrotation angle corresponds to the minimum value of thecost function Assuming the case of CQSM and for thescenarios depicted in the figure the obtained optimal rotationangles are very close to those obtained in [17] Therefore thissmall and tolerable degradation in the error performancecomes at a high gain in the optimization time of the rotationangle On the other hand the obtained rotation angles forICQSM are identical to those obtained in [18] AssumingPSK modulation the optimal rotation angle for the ICQSMis equal to 120587119871 with 119871 denoting the cardinality of theconstellation set

6 Optimization of the Rotation Angles for theGeneralized CQSM

61 Rotation Angle of the GCQSM-UC As explained earlierand based on the design of the spatial symbols of theGCQSM-UC if (1198941 1198942) is a spatial symbol that can be used fortransmission then (1198942 1198941) is not a valid spatial symbol Thisimplies that the first symbol 1199041198961 and 11990410158401198962 are distinguishableusing the indices of the antennas fromwhich they are sent Assuch both symbols can be drawn from the same modulationset and hence the rotation angle will have no impact on theperformance of the system In the following 120579 is set to zerofor GCQSM-UC

62 Rotation Angle of the GCQSM-PC Opposed to theGCQSM-UC both (1198941 1198942) and (1198942 1198941) are valid spatial symbolsthat can be used for transmission in GCQSM-PC Thereforethe second symbol 11990410158401198962 should be rotated so that it can bedistinguished from the first signal symbol 1199041198961 The formula-tion of the upper-bound of the GCQSM-PC as in (22) is ahard problem Alternatively we obtained the rotation angleusing simulations Figure 2 depicts the BER of GCQSM-PCversus the rotation angle the upper subfigure assumes QPSKmodulation and the lower subfigure assumes 16-QAM Theoptimal rotation angle is about 1205874 for all the simulatedscenarios assuming QPSK modulation Also the optimal

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5004 8

0 20 40

400

600

800

10004 16

0 20 40

406080

100120

cost

0 20 4050

100

150

200

250

0 20 40100

200

300

400

0 20 40200

400

600

800

0 20 40

50

100

150

cost

8PSK

0 20 4050

100

150

200

16PSK

0 20 4050

100

150

200

16QAM

0 20 40

100

150

200

25032PSK

CQSM16QAM

CQSM8PSK

ICQSM

times times times times

4 2 4 4 4 8 4 16times times times times

4 4times

Figure 1 The cost function in (26) in dB scale versus the rotation angle for several system configurations The subfigures in the first andsecond row are obtained for CQSM and those in the bottom row are for ICQSM

rotation angle for the simulated scenarios using 16-QAMis approximately 23 degrees These values of the optimalrotation angle are affected by the sets from which symbolsare drawn and the values of 119899119879 and 119899119877 As a future work wewould like to analytically derive a generic formula to obtainthe optimal angle for arbitrary system configurations

7 Simulation Results

The following results are obtained assuming that the channelstate information (CSI) is known only at the receiver Tomake a fair comparison the rotation angles for the simulatedscenarios are optimized using simulations for both CQSMand the generalized CQSM The obtained rotation anglesusing simulations and the optimization method described inSection 5 for CQSM scheme have very similar values

Figure 3 depicts comparisons between the MA-SM andthe CQSM for several system configurations of (119899119879 119902) and afixed 119899119877 = 4 We make the following comparisons

(i) For the same 119899119879 and 119902 and different spectral effi-ciency (Figure 3(a)) in this case MA-SM outperformsCQSM by about 2 dB for 119899119879 = 8 and about 15dB for 119899119879 = 16 The gap is bridged between thetwo algorithms as 119899119879 becomes large This decreasein the performance gap is due to the decrease in theprobability of transmitting the two symbols from thesame transmit antenna in the CQSMThis probability

is equal to 1119899119879 The comparison here is not fairbecause the CQSM achieves a spectral efficiency 2bpcu higher than that of the MA-SM

(ii) For the spectral efficiency and 119902 and different 119899119879(Figure 3(b)) the performance of the two algorithmsis depicted for an equal spectral efficiency assuming119899119879 = 8 16 and 119899119879 = 12 24 for CQSM and MA-SMrespectively For these two scenarios CQSM requires4 and 8 transmit antennas less than MA-SM Thishuge reduction in the number of transmit antennascomes at a negligible degradation in the SNR for high119899119879 This implies that the CQSM is more attractive formassive MIMO systems

(iii) For the spectral efficiency and 119899119879 and different 119902(Figure 3(c)) finally the performance of the two algo-rithms is evaluated for the same spectral efficiencyand number of transmit antennas In this case 119902 is setto 2 and 3 in the CQSM and MA-SM respectively Ata target BER of 10minus4 CQSM outperforms MA-SM by2 and 24 dB respectively for the two depicted systemconfigurations

According to this analysis and results depicted inFigure 3 we conclude that CQSM can be used to reduce thenumber of transmit antennas at a marginal cost in the SNRfor high 119899119879 or it can achieve an improvement in the SNR ifthe same number of antennas is deployed by the two systems

8 Wireless Communications and Mobile Computing

0 10 20 30 40

8 times 4 =14 dB 10 times 4 = 17 dB13 times 4 = 17 dB

0 10 20 30 40

4 times 8 = 16 dB 8 times 4 = 22 dB 10 times 4 = 23 dB

10minus4

10minus2

10minus0

BER

10minus4

10minus2

10minus0

BER

Figure 2 BER of the GCQSM-PU versus the rotation angle applied to the second signal symbol using QPSK (first subfigure) and 16-QAM(second subfigure)

Since at high 119899119879 CQSM requires one antenna less thanICQSM to achieve the same spectral efficiency at the cost ofa moderate degradation in the BER performance CQSM isused in the following comparisons

Figure 4 depicts the BER performance of CQSMGCQSM-UC and GCQSM-PC for several system scenariosand 119899119880 = 2 The title of each subfigure is a tuple that includesthe number of transmit antennas required by CQSMGCQSM-UC and GCQSM-PC respectively to achievethe same spectral efficiency The spectral efficiency SE119904119901119886achieved assuming the systemrsquos parameters in Figure 4(a) is6 in 4(b) is 8 in 4(c) is 10 and in 4(d) is 12 bpcu For eachof the system configurations SE119904119894119892 is equal to 4 bpcu leadingto a total spectral efficiency SE = SE119904119901119886 + SE119904119894119892 of 10 12 14and 16 bpcu respectively The range over which the rotationangle is optimized is [0 1205874] The rotation angles appliedto the second signal symbol for the scenarios depictedin Figure 4 are listed in Table 5 The rotation angles areoptimized through Monte Carlo simulations Based on theresults depicted in Figure 4 we make the following remarksin terms of the BER performance the rotation angles andthe number of employed transmit antennas

(i) For 119899119877 = 2 the three techniques achieve the sameBER performance regardless of the number of trans-mit antennas

(ii) For a given value of 119899119877 the BER performanceis degraded as the number of transmit antennasincreases The degradation is relatively small as 119899119877increases This is an expected behavior of SM tech-niques because the size of hypothesis set over whichthe ML detector performs the search increases as 119899119879increases

(iii) In Figure 4(a) GCQSM-PC slightly outperforms theother two techniques for 119899119877 = 4 and 8 As 119899119879 getslarger the probability that the two signal symbols aretransmitted from the same antenna inCQSMreducesAccordingly the performance of CQSM and the twogeneralized algorithms almost coincide for 119899119877 = 4in Figures 4(b) 4(c) and 4(d) For 119899119877 = 8 CQSMslightly outperforms GCQSM-UC and GCQSM-PU

(iv) As shown in Figure 4(c) the generalized techniquesperform close to the CQSM for 119899119877 = 4 In thiscase GCQSM-UC and GCQSM-PC require 17 and 22

Wireless Communications and Mobile Computing 9

10 15 20SNR (dB)

BER

cqsm (8 2)ma-sm (8 2)cqsm (16 2)ma-sm (16 2)

10minus1

10minus2

10minus3

10minus4

(a)

10 15 20SNR (dB)

cqsm (8 2)ma-sm (12 2)cqsm (16 2)ma-sm (24 2)

BER

10minus1

10minus2

10minus3

10minus4

(b)

10 15 20 25SNR (dB)

cqsm (8 2)ma-sm (8 3)cqsm (16 2) ma-sm (16 3)

BER

10minus1

10minus2

10minus3

10minus4

(c)

Figure 3 A comparison between CQSM andMA-SM (a) for the same 119899119879 and different spectral efficiency (b) for the same spectral efficiencyand 119902 and different 119899119879 and finally (c) for the same spectral efficiency and 119899119879 and different 119902

Table 5 Range of optimal rotation angles used to obtain the BER results depicted in Figure 4 assuming QPSK modulation

SE119904119901119886 119899119877 120579CQSM 120579GCQSMminusPC

62 ge25 ge304 ge35 ge418 ge38 ge43

82 ge20 ge214 ge34 ge358 ge38 ge42

102 ge9 ge124 ge30 ge328 ge38 ge39

122 ge9 ge124 ge25 ge308 ge38 ge39

transmit antennas less than CQSM while achievingthe same spectral efficiency

Finally we compare the BER performance of SM GSMQSM CQSM and the proposed generalized CQSM tech-niques We assume that all the schemes achieve the sameES119904119901119886 and ES119904119894119892 leading to the same total spectral efficiencyof ES = (ES119904119901119886 + ES119904119894119892) Also the three generalized schemesuse a combination of two antennas to transmit each signalsymbol Figures 5(a) and 5(b) depict the BER performancefor a spectral efficiency of 16 and 18 bpcu respectively Thenumber of transmit antennas and bits per signal symbol isrepresented as a tuple of the form (119899119879 119902) for each of the

systems The rotation angles applied to CQSM and GCQSM-PC are 20 and 23 degrees respectively to obtain the results inthe two subfigures We make the following two remarks

(i) CQSM GCQSM-PC and GCQSM-UC apply 16-QAM modulation whereas SM GSM and QSM use256-QAM to achieve an ES119904119894119892 of 8 bpcu The numberof transmit antennas required to achieve the spectralefficiency of 16 and 18 bpcu is given in the secondand third row respectively in Table 4 GCQSM-PCrequires the least number of transmit antennas of8 and 10 for the results in Figures 5(a) and 5(b)respectively On the other hand SM requires 256 and

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

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Wireless Communications and Mobile Computing 3

reduces the number of transmit antennas by sending thesame signal symbol from a combination of two or moreantennas Let 119899119880 denote the number of active antennas ateach channel use then the number of spatial symbols alsoreferred to as combinations for a given 119899119879 is ( 119899119879119899119880 ) Since thenumber of spatial symbols should be a power of two only2ES119904119901119886 combinations can be used where ES119904119901119886 = lfloorlog2 ( 119899119879119899119880 )rfloorAssuming 119899119880 = 2 GSM requires 7 transmit antennas toachieve ES119904119901119886 = 4 bits per spatial symbol whereas SMrequires 16 antennas to achieve the same spectral efficiency

Let 119897 = 1198971 119897119899119880 isin L be a spatial symbol where L is theset of spatial symbols that can be used for transmission Thenthe received vector in the case of GSM is given by

y = 119904119896radic119899119880sum119894isin119897h119894 + n (3)

24 Multi-Active Spatial Modulation In contrast to GSMin which a single signal symbol is transmitted from acombination of antennas MA-SM transmits a different signalsymbol from each activated antenna Let 119899119880 be the number ofactivated antennas then the received vector is given by

y = sum119894isin119897

h119897119894119904119894 + n (4)

where 119897 = 1198971 119897119899119880 isin L and L is the list of spatialsymbols of MA-SM which is equivalent to that of GSMAdditionally a 3-dimensional constellation set is proposedin which each spatial symbol is associated with a uniquerotation angle Signal symbols transmitted from a givenspatial symbol are rotated before transmission Accordingto this description MA-SM benefits from the moderatecomputational complexity and low hardware requirements ofSM and the high multiplexing gain of the vertical-Bell Labslayered space-time (V-BLAST) systemThe spectral efficiencyof MA-SM is equal to (ES119904119901119886 + 119902 times 119899119880) where ES119904119901119886 is definedin Section 23

25 Complex Quadrature Spatial Modulation QSM expandsthe spatial constellation into in-phase and quadrature dimen-sions At each channel use a single signal symbol is transmit-ted the real part of the signal symbol is transmitted throughthe in-phase spatial dimension the imaginary through thequadrature dimension As such the spectral efficiency ofQSM is 119902+2log2(119899119879) where ES119904119901119886 = 2log2(119899119879) and ES119904119894119892 = 119902

CQSM transmits two signal symbols at each channel useleading to a spectral efficiency of 2119902+2log2(119899119879) where ES119904119901119886 =2log2(119899119879) and ES119904119894119892 = 2119902 The first signal symbol 1199041198961 isin Ω119886 istransmitted from the 1198941th transmit antenna and the secondsignal symbol 11990411989610158402 isin Ω119887 is transmitted from the 1198942th antennaTo make the distinction between 1199041198961 and 11990411989610158402 at the receiverside Ω119887 is obtained as follows

Ω119887 = 11990411989610158402 | 11990411989610158402 = 1199041198961119890119895120579 1199041198961 isin Ω119886 (5)

The rotation angle 120579 is optimized such that the probability oferror is reducedThe received vector of the CQSM is given by

y = Hs + n = h11989411199041198961 + h119894211990411989610158402 + n (6)

which is expanded to

y = h11989411199041198961 + h119894211990410158401198962 + n if 1198941 = 1198942h119894 (1199041198961 + 11990410158401198962) = h119894119904119896101584010158403 if 1198941 = 1198942 = 119894 (7)

where 119904119896101584010158403 isin Ω119888 and Ω119888 = Ω119886 oplus Ω119887 with oplus denoting theMinkowski sum [20]The full modulation setΩ119889 = Ω119886cupΩ119887cupΩ119888 is of size (|Ω119886|+ |Ω119887|+ |Ω119886|times |Ω119887|) For instance assuming|Ω119886| = |Ω119887| = 16 the number of symbols inΩ119889 becomes 288This high density of symbols reduces the Euclidean distanceamong signal symbols leading to degradation in the errorperformance

26 Improved Complex Quadrature Spatial Modulation InICQSM the transmitter is equipped with an additionalantenna indexed 119899119870 with 119899119870 = 119899119879 + 1 which is used onlywhen 1198941 = 1198942 In this case the first signal symbol is transmittedfrom its designated antenna and the second symbol istransmitted from the additional antenna Accordingly theICQSM system is modeled as follows

y = 1199041198961h1198941 + 11990411989610158402h1198942 + n if 1198941 = 11989421199041198961h1198941 + 11990411989610158402h119899119870 + n if 1198941 = 1198942 (8)

This strategy reduces the size of the total modulation set Ω119889from (|Ω119886| + |Ω119887| + |Ω119886| times |Ω119887|) in the case of CQSM toonly (|Ω119886| + |Ω119887|) in ICQSM This increases the Euclideandistance among the signal symbols and hence improves theerror performance

Based on the above descriptions CQSM activates eitherone or two antennas whereas ICQSM always has two activeantennas To perform a fair comparison between thesetwo systems and MA-SM we assume that 119899119880 = 2 Thatis the hardware requirements of the transmitter and thecomputational complexity of the detection algorithm areequivalent Since both systems transmit two signal symbolsat each channel use the comparison is conducted in terms ofthe spectral efficiency achieved by spatial symbols SM on theother hand transmits a single signal symbol at each channeluse Table 1 lists the number of transmit antennas required toachieve a given number of bits per spatial symbol (SE119904119901119886) byseveral systems For instance CQSM and ICQSM require 3and 2 antennas less than MA-SM to achieve the same SE119904119901119886 of4 bpcuThis gap increases for higher SE119904119901119886

3 Generalized Complex QuadratureSpatial Modulation

Both CQSM and ICQSM require the same number of RFchains and ICQSM requires one more physical transmitantenna compared to CQSM Transmitting each signal sym-bol from a combination of antennas instead of a singleantenna reduces the number of transmit antennas requiredto achieve a given SE119904119901119886 In the following subsections weintroduce two generalizations of CQSM namely generalizedCQSM with unique combinations (GCQSM-UC) and gen-eralized CQSM with permuted combinations (GCQSM-PC)

4 Wireless Communications and Mobile Computing

Table 1 The number of transmit antennas required by severalsystems to achieve the same number of bits per spatial symbol(SE119904119901119886)

SE119904119901119886 SM MA-SM ICQSM CQSM2 4 4 3 24 16 7 5 46 64 12 9 8

Table 2 An example of the spatial symbols (1198941 1198942) that can be usedfor transmission by GCQSM-UC assuming 119899119879 = 6 and 119899119880 = 21198941 11989421 2 3 41 2 3 51 2 3 61 2 4 51 2 4 61 2 5 61 3 4 51 3 4 61 3 5 61 4 5 62 3 4 52 3 4 62 3 5 62 4 5 63 4 5 6In the following we denote by 1198941 and 1198942 the set of antennasused for transmitting the first and second signal symbolsrespectively In the sequel the tuple 119860 = (119861119862) is composedof the two sets 119861 and119862 where the order of the sets in the tuplematters but the order of the elements in each set does not

31 Generalized CQSMwith Unique Combinations Let 119899119880 bethe number of antennas from which each of the two signalsymbols is transmitted then for a given 119899119879 the number ofcombinations that can be used for transmission is ( 1198991198792119899119880 )Assuming that 119894 = 1198941 1198942 is a spatial symbol ie acombination of antennas of length 2119899119880 then the first andsecond 119899119880 antennas are used to transmit the first and secondsignal symbols respectively Table 2 depicts an example ofthe spatial symbols for 119899119879 = 6 and 119899119880 = 2 According tothis description there will be no overlap between the sets ofantennas used to transmit the first and second signal symbolsSince the number of spatial symbols should be a power of twothen only 2SE119904119901119886 signal symbols can be used for transmissionwhere SE119904119901119886 = lfloorlog2 ( 1198991198792119899119880 )rfloor

Let 1199041198961 isin Ω119886 and 11990411989610158402 isin Ω119887 be the two signal symbols to betransmitted at a given channel use then the received vector isgiven by

y = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894 + n (9)

The receiver employs the maximum-likelihood principle torecover the two signal symbols and the spatial symbol asfollows (119894lowast 119904lowast1198961 119904lowast11989610158402) = arg min

1199041198961isinΩ11988611990411989610158402isinΩ119887

119894=(11989411198942)isinL

1003817100381710038171003817y minus g10038171003817100381710038172= arg min1199041198961isinΩ11988611990411989610158402

isinΩ119887119894=(11989411198942)isinL

1003817100381710038171003817g10038171003817100381710038172 minus 2R (y119867g) (10)

where

g = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894 (11)

GCQSM-UC can be easily extended to transmitting morethan two signal symbols per channel use at the cost ofrequiring more RF chains at the transmitter side This ispossible because the error performance of the system doesnot depend on the rotation angle 120579 This will be addressed inmore detail in subsequent sections

32 Generalized CQSM with Permuted Combinations Toreduce the number of transmit antennas required to achievea given spectral efficiency we use the structure of thetransmitted vector to increase the number of spatial symbolsLet 119894 = (1198941 1198942) be a spatial symbol where the first signalsymbol 1199041198961 is transmitted from the 1198941 combination of antennasand the second signal symbol 11990411989610158402 from the combination 1198942 Fora given channel realization the received noiseless vector g isgiven by

g = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894

= 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894

(12)

where 11990411989610158402 = 1199041198962119890119895120579 and 1199041198961 1199041198962 isin Ω119886 To obtain 1198941 and 1198942 weimpose the following conditions

(i) 1198941 cup 1198942 = 0 this implies that the set of antennas fromwhich the first signal symbol and that from whichthe second symbol are transmitted do not overlapThis condition reduces the detection ambiguity at thereceiver

(ii) (1198941 1198942) = (1198942 1198941) as described earlier the first andsecond symbols are drawn from different constella-tion sets Therefore for given signal symbolsrsquoindices 1198961 and 1198962 the following two received latticepointsmdashnoiseless received vectorsmdash are distinguish-able

g1 = 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894 (13)

and

g2 = 1199041198961radic119899119880sum119894isin1198942h119894 +1199041198962radic119899119880sum119894isin1198941119890119895120579h119894 (14)

Wireless Communications and Mobile Computing 5

Input A = 1 119899119879 119899119880 119899119879Output The set of spatial symbolsIC = COMB(A 119899119880)for 1198941 isin C do

B = A1198941

D = COMB(B 119899119880)for 1198942 isin D do

APPND(I (1198941 1198942))end

end

Algorithm 1 Generation of the spatial symbols assuming 119899119879transmit antennas and that each signal symbol is transmitted froma combination of 119899119880 antennasTable 3 An example of the spatial symbols (1198941 1198942) that can be usedfor transmission by GCQSM-PC assuming 119899119879 = 5 and 119899119880 = 21198941 11989421 2 3 4 3 5 4 51 3 2 4 2 5 4 51 4 2 3 2 5 3 51 5 2 3 2 4 3 42 3 1 4 1 5 4 52 4 1 3 1 5 3 52 5 1 3 1 4 3 43 4 1 2 1 5 2 53 5 1 2 1 4 2 44 5 1 2 1 3 2 3

This is rendered possible using the rotation angle 120579The optimization of the rotation angle is addressed ina following section

Algorithm 1 depicts a pseudocode of generating the set ofspatial symbolsI that can be used for transmission ThereinCOMB(A 119899119880) generates all combinations of length 119899119880 of theelements of the set A Also APPND(I (1198941 1198942)) appends thetuple (1198941 1198942) to the setI B = A

1198941is the difference of the two

sets119860 and 1198941 Accordingly the number of spatial symbols thatcan be used for transmission is given by

119862spa = (119899119879119899119880) times (119899119879 minus 119899119880119899119880 ) (15)

Table 3 depicts an example of the obtained spatial sym-bols assuming 119899119879 = 5 and 119899119880 = 2 In this case GCQSM-PC obtains 30 spatial symbols whereas GCQSM-UC obtainsonly five

Table 4 lists the number of transmit antennas required byeach system to achieve a given spectral efficiency per spatialsymbol For instance SM GSM (and MA-SM) CQSMGCQSM-UC andGCQSM-PC require 1024 46 32 15 and 10transmit antennas respectively to achieve 10 bits per spatialsymbol

4 Performance Analysis ofthe Generalized CQSM

Let

g119894 = 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894 (16)

and

g119896 = 1199041radic119899119880sum119894isin1h119894 +1199042radic119899119880sum119894isin2119890119895120579h119894 (17)

be two noiseless received codewords corresponding to thetransmitted symbol (1199041198961 1199041198962 1198941 1198942) and (1199041 1199042 1 2) where119904119896 is a signal symbol and 119894 = 1198971 119897119899119880 is a spatial symbolThe corresponding received vectors are denoted by y119894 and y119896respectively

The conditional pairwise error probability (PEP) of themaximum-likelihood (ML) receiver is given by [21 22]

Pr [g119894 997888rarr g119896 | H] = 119876(radic 1003817100381710038171003817y119894 minus y1198961003817100381710038171003817221205902119899 )

= 119876(radic 1003817100381710038171003817H (s119894 minus s119896)1003817100381710038171003817221205902119899 )(18)

where119876(sdot) is the Gaussian tail probability function or simplythe119876-function and 1198892119894119896 = y119894 minus y1198962 is the squared Euclideandistance between s119894 and s119896 at the receiver Similarly 1198892119894119896(tx) =s119894 minus s1198962 is the squared Euclidean distance between s119894 and s119896at the transmitter Note that

EH 1198892119894119895 = EH (s119894 minus s119896)119867H119867H (s119894 minus s119896)= (s119894 minus s119896)119867 (s119894 minus s119896) = 1198892119894119896(tx) (19)

The unconditional PEP (UPEP) assuming 119899119877 receivesantennas which is obtained by taking the expectation of (18)over the channel H is given by

Pr [g119894 997888rarr g119896] = 120583119899119877119894119895 119899119877minus1sum119897=0

(119899119877 minus 1 + 119897119897 ) [1 minus 120583119894119895]119897 (20)

where

120583119894119895 = 12 (1 minus radic 120574 sdot 1198892119894119896(tx)4 + 120574 sdot 1198892119894119896(tx)

) (21)

and 120574 = 1205902119904 1205902119899 is the signal-to-noise ratio (SNR) with 1205902119904 =E(119904lowast119904) the expected power of the signal symbol

The union bound on the pairwise error probability is thengiven by averaging all the pairwise probabilities as follows

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896] (22)

6 Wireless Communications and Mobile Computing

Table 4 Examples of the number of transmit antennas required to achieve a given SE119904119901119886 for several spatial modulation systems

SE119904119901119886 SM GSM CQSM QSM GCQSM-UC GCQSM-PC6 64 12 8 8 68 256 24 16 11 810 1024 46 32 15 1012 4096 92 64 20 13

Accordingly the average bit-error rate (BER) is upper-bounded by

119875119890 le 11198722119872 2119872sum119894=1

2119872sum119896=1

119863s119894s119896Pr [g119894 997888rarr g119896] (23)

where 119863s119894s119896 the hamming distance between s119894 and s119896 is thenumber of errors associated with the event [g119894 997888rarr g119896] and119872 is the spectral efficiency of the system Note that in eachtransmitted vector symbol s there are exactly (2times119899119880) nonzeroelements

5 Optimization of the Rotation Angles ofCQSMICQSM Revisited

In [17 18] the optimal rotation angles applied to the secondmodulation set of the CQSM and ICQSM respectively areobtained through extensive Monte Carlo simulations Inthis case the obtained rotation angle corresponds to theminimum bit-error rate (BER) averaged over a large numberof channel and noise realizations Alternatively the rotationangle can be obtained through minimizing the upper-boundof the unconditional pairwise error probability

Let

g119894 = Hs119894 = 1199041198961h1198941 + 11990410158401198962h1198942g119896 = Hs119896 = 1199041h1 + 1199041015840

2h2 (24)

where y119894 = g119894 + n and y119896 = g119896 + n are the correspondingreceived vectors for a given noise vector n

The PEP and UPEP of the CQSM and ICQSM can also beobtained using (18)ndash(21) The union bound on the pairwiseerror probability at high SNR values (aka asymptotic prob-ability of error) is then given by

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896]= (2119899119877 minus 1119899119877 ) 120574minus1198991198772119872 2

119872sum119894=1

2119872sum119896=1

(1198892119894119896(tx))minus119899119877= (2119899119877 minus 1119899119877 ) 120574minus1198991198771198712 119861sum

119894=1

119891119894Ω119894(25)

where the maximum value of 119861 equals 26 as shownin [19] The term Ω119894 is a function of the signal symbols(1199041198961 11990410158401198962 1199041 11990410158402) and 119899119877 and 119891119894 is the frequency of Ω119894 where

sum119861119894=1 119891119894 = 1198994119879 It is shown in [19] that 119861 = 7 in the case ofICQSM and it can be shown to be equal to 15 for CQSMThe rotation angle is then obtained by solving the followingoptimization problem

argmin0lt120579lt1205872

( 119861sum119894=1

119891119894Ω119894) (26)

where 119861 = 15 and 7 for CQSM and ICQSM respectivelyFigure 1 depicts the cost function of (26) for CQSM and

ICQSM and for several system configurations The optimalrotation angle corresponds to the minimum value of thecost function Assuming the case of CQSM and for thescenarios depicted in the figure the obtained optimal rotationangles are very close to those obtained in [17] Therefore thissmall and tolerable degradation in the error performancecomes at a high gain in the optimization time of the rotationangle On the other hand the obtained rotation angles forICQSM are identical to those obtained in [18] AssumingPSK modulation the optimal rotation angle for the ICQSMis equal to 120587119871 with 119871 denoting the cardinality of theconstellation set

6 Optimization of the Rotation Angles for theGeneralized CQSM

61 Rotation Angle of the GCQSM-UC As explained earlierand based on the design of the spatial symbols of theGCQSM-UC if (1198941 1198942) is a spatial symbol that can be used fortransmission then (1198942 1198941) is not a valid spatial symbol Thisimplies that the first symbol 1199041198961 and 11990410158401198962 are distinguishableusing the indices of the antennas fromwhich they are sent Assuch both symbols can be drawn from the same modulationset and hence the rotation angle will have no impact on theperformance of the system In the following 120579 is set to zerofor GCQSM-UC

62 Rotation Angle of the GCQSM-PC Opposed to theGCQSM-UC both (1198941 1198942) and (1198942 1198941) are valid spatial symbolsthat can be used for transmission in GCQSM-PC Thereforethe second symbol 11990410158401198962 should be rotated so that it can bedistinguished from the first signal symbol 1199041198961 The formula-tion of the upper-bound of the GCQSM-PC as in (22) is ahard problem Alternatively we obtained the rotation angleusing simulations Figure 2 depicts the BER of GCQSM-PCversus the rotation angle the upper subfigure assumes QPSKmodulation and the lower subfigure assumes 16-QAM Theoptimal rotation angle is about 1205874 for all the simulatedscenarios assuming QPSK modulation Also the optimal

Wireless Communications and Mobile Computing 7

0 20 4040

60

80

100

120

140co

st

4 2

0 20 40

100

150

200

2504 4

0 20 40

200

300

400

5004 8

0 20 40

400

600

800

10004 16

0 20 40

406080

100120

cost

0 20 4050

100

150

200

250

0 20 40100

200

300

400

0 20 40200

400

600

800

0 20 40

50

100

150

cost

8PSK

0 20 4050

100

150

200

16PSK

0 20 4050

100

150

200

16QAM

0 20 40

100

150

200

25032PSK

CQSM16QAM

CQSM8PSK

ICQSM

times times times times

4 2 4 4 4 8 4 16times times times times

4 4times

Figure 1 The cost function in (26) in dB scale versus the rotation angle for several system configurations The subfigures in the first andsecond row are obtained for CQSM and those in the bottom row are for ICQSM

rotation angle for the simulated scenarios using 16-QAMis approximately 23 degrees These values of the optimalrotation angle are affected by the sets from which symbolsare drawn and the values of 119899119879 and 119899119877 As a future work wewould like to analytically derive a generic formula to obtainthe optimal angle for arbitrary system configurations

7 Simulation Results

The following results are obtained assuming that the channelstate information (CSI) is known only at the receiver Tomake a fair comparison the rotation angles for the simulatedscenarios are optimized using simulations for both CQSMand the generalized CQSM The obtained rotation anglesusing simulations and the optimization method described inSection 5 for CQSM scheme have very similar values

Figure 3 depicts comparisons between the MA-SM andthe CQSM for several system configurations of (119899119879 119902) and afixed 119899119877 = 4 We make the following comparisons

(i) For the same 119899119879 and 119902 and different spectral effi-ciency (Figure 3(a)) in this case MA-SM outperformsCQSM by about 2 dB for 119899119879 = 8 and about 15dB for 119899119879 = 16 The gap is bridged between thetwo algorithms as 119899119879 becomes large This decreasein the performance gap is due to the decrease in theprobability of transmitting the two symbols from thesame transmit antenna in the CQSMThis probability

is equal to 1119899119879 The comparison here is not fairbecause the CQSM achieves a spectral efficiency 2bpcu higher than that of the MA-SM

(ii) For the spectral efficiency and 119902 and different 119899119879(Figure 3(b)) the performance of the two algorithmsis depicted for an equal spectral efficiency assuming119899119879 = 8 16 and 119899119879 = 12 24 for CQSM and MA-SMrespectively For these two scenarios CQSM requires4 and 8 transmit antennas less than MA-SM Thishuge reduction in the number of transmit antennascomes at a negligible degradation in the SNR for high119899119879 This implies that the CQSM is more attractive formassive MIMO systems

(iii) For the spectral efficiency and 119899119879 and different 119902(Figure 3(c)) finally the performance of the two algo-rithms is evaluated for the same spectral efficiencyand number of transmit antennas In this case 119902 is setto 2 and 3 in the CQSM and MA-SM respectively Ata target BER of 10minus4 CQSM outperforms MA-SM by2 and 24 dB respectively for the two depicted systemconfigurations

According to this analysis and results depicted inFigure 3 we conclude that CQSM can be used to reduce thenumber of transmit antennas at a marginal cost in the SNRfor high 119899119879 or it can achieve an improvement in the SNR ifthe same number of antennas is deployed by the two systems

8 Wireless Communications and Mobile Computing

0 10 20 30 40

8 times 4 =14 dB 10 times 4 = 17 dB13 times 4 = 17 dB

0 10 20 30 40

4 times 8 = 16 dB 8 times 4 = 22 dB 10 times 4 = 23 dB

10minus4

10minus2

10minus0

BER

10minus4

10minus2

10minus0

BER

Figure 2 BER of the GCQSM-PU versus the rotation angle applied to the second signal symbol using QPSK (first subfigure) and 16-QAM(second subfigure)

Since at high 119899119879 CQSM requires one antenna less thanICQSM to achieve the same spectral efficiency at the cost ofa moderate degradation in the BER performance CQSM isused in the following comparisons

Figure 4 depicts the BER performance of CQSMGCQSM-UC and GCQSM-PC for several system scenariosand 119899119880 = 2 The title of each subfigure is a tuple that includesthe number of transmit antennas required by CQSMGCQSM-UC and GCQSM-PC respectively to achievethe same spectral efficiency The spectral efficiency SE119904119901119886achieved assuming the systemrsquos parameters in Figure 4(a) is6 in 4(b) is 8 in 4(c) is 10 and in 4(d) is 12 bpcu For eachof the system configurations SE119904119894119892 is equal to 4 bpcu leadingto a total spectral efficiency SE = SE119904119901119886 + SE119904119894119892 of 10 12 14and 16 bpcu respectively The range over which the rotationangle is optimized is [0 1205874] The rotation angles appliedto the second signal symbol for the scenarios depictedin Figure 4 are listed in Table 5 The rotation angles areoptimized through Monte Carlo simulations Based on theresults depicted in Figure 4 we make the following remarksin terms of the BER performance the rotation angles andthe number of employed transmit antennas

(i) For 119899119877 = 2 the three techniques achieve the sameBER performance regardless of the number of trans-mit antennas

(ii) For a given value of 119899119877 the BER performanceis degraded as the number of transmit antennasincreases The degradation is relatively small as 119899119877increases This is an expected behavior of SM tech-niques because the size of hypothesis set over whichthe ML detector performs the search increases as 119899119879increases

(iii) In Figure 4(a) GCQSM-PC slightly outperforms theother two techniques for 119899119877 = 4 and 8 As 119899119879 getslarger the probability that the two signal symbols aretransmitted from the same antenna inCQSMreducesAccordingly the performance of CQSM and the twogeneralized algorithms almost coincide for 119899119877 = 4in Figures 4(b) 4(c) and 4(d) For 119899119877 = 8 CQSMslightly outperforms GCQSM-UC and GCQSM-PU

(iv) As shown in Figure 4(c) the generalized techniquesperform close to the CQSM for 119899119877 = 4 In thiscase GCQSM-UC and GCQSM-PC require 17 and 22

Wireless Communications and Mobile Computing 9

10 15 20SNR (dB)

BER

cqsm (8 2)ma-sm (8 2)cqsm (16 2)ma-sm (16 2)

10minus1

10minus2

10minus3

10minus4

(a)

10 15 20SNR (dB)

cqsm (8 2)ma-sm (12 2)cqsm (16 2)ma-sm (24 2)

BER

10minus1

10minus2

10minus3

10minus4

(b)

10 15 20 25SNR (dB)

cqsm (8 2)ma-sm (8 3)cqsm (16 2) ma-sm (16 3)

BER

10minus1

10minus2

10minus3

10minus4

(c)

Figure 3 A comparison between CQSM andMA-SM (a) for the same 119899119879 and different spectral efficiency (b) for the same spectral efficiencyand 119902 and different 119899119879 and finally (c) for the same spectral efficiency and 119899119879 and different 119902

Table 5 Range of optimal rotation angles used to obtain the BER results depicted in Figure 4 assuming QPSK modulation

SE119904119901119886 119899119877 120579CQSM 120579GCQSMminusPC

62 ge25 ge304 ge35 ge418 ge38 ge43

82 ge20 ge214 ge34 ge358 ge38 ge42

102 ge9 ge124 ge30 ge328 ge38 ge39

122 ge9 ge124 ge25 ge308 ge38 ge39

transmit antennas less than CQSM while achievingthe same spectral efficiency

Finally we compare the BER performance of SM GSMQSM CQSM and the proposed generalized CQSM tech-niques We assume that all the schemes achieve the sameES119904119901119886 and ES119904119894119892 leading to the same total spectral efficiencyof ES = (ES119904119901119886 + ES119904119894119892) Also the three generalized schemesuse a combination of two antennas to transmit each signalsymbol Figures 5(a) and 5(b) depict the BER performancefor a spectral efficiency of 16 and 18 bpcu respectively Thenumber of transmit antennas and bits per signal symbol isrepresented as a tuple of the form (119899119879 119902) for each of the

systems The rotation angles applied to CQSM and GCQSM-PC are 20 and 23 degrees respectively to obtain the results inthe two subfigures We make the following two remarks

(i) CQSM GCQSM-PC and GCQSM-UC apply 16-QAM modulation whereas SM GSM and QSM use256-QAM to achieve an ES119904119894119892 of 8 bpcu The numberof transmit antennas required to achieve the spectralefficiency of 16 and 18 bpcu is given in the secondand third row respectively in Table 4 GCQSM-PCrequires the least number of transmit antennas of8 and 10 for the results in Figures 5(a) and 5(b)respectively On the other hand SM requires 256 and

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

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4 Wireless Communications and Mobile Computing

Table 1 The number of transmit antennas required by severalsystems to achieve the same number of bits per spatial symbol(SE119904119901119886)

SE119904119901119886 SM MA-SM ICQSM CQSM2 4 4 3 24 16 7 5 46 64 12 9 8

Table 2 An example of the spatial symbols (1198941 1198942) that can be usedfor transmission by GCQSM-UC assuming 119899119879 = 6 and 119899119880 = 21198941 11989421 2 3 41 2 3 51 2 3 61 2 4 51 2 4 61 2 5 61 3 4 51 3 4 61 3 5 61 4 5 62 3 4 52 3 4 62 3 5 62 4 5 63 4 5 6In the following we denote by 1198941 and 1198942 the set of antennasused for transmitting the first and second signal symbolsrespectively In the sequel the tuple 119860 = (119861119862) is composedof the two sets 119861 and119862 where the order of the sets in the tuplematters but the order of the elements in each set does not

31 Generalized CQSMwith Unique Combinations Let 119899119880 bethe number of antennas from which each of the two signalsymbols is transmitted then for a given 119899119879 the number ofcombinations that can be used for transmission is ( 1198991198792119899119880 )Assuming that 119894 = 1198941 1198942 is a spatial symbol ie acombination of antennas of length 2119899119880 then the first andsecond 119899119880 antennas are used to transmit the first and secondsignal symbols respectively Table 2 depicts an example ofthe spatial symbols for 119899119879 = 6 and 119899119880 = 2 According tothis description there will be no overlap between the sets ofantennas used to transmit the first and second signal symbolsSince the number of spatial symbols should be a power of twothen only 2SE119904119901119886 signal symbols can be used for transmissionwhere SE119904119901119886 = lfloorlog2 ( 1198991198792119899119880 )rfloor

Let 1199041198961 isin Ω119886 and 11990411989610158402 isin Ω119887 be the two signal symbols to betransmitted at a given channel use then the received vector isgiven by

y = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894 + n (9)

The receiver employs the maximum-likelihood principle torecover the two signal symbols and the spatial symbol asfollows (119894lowast 119904lowast1198961 119904lowast11989610158402) = arg min

1199041198961isinΩ11988611990411989610158402isinΩ119887

119894=(11989411198942)isinL

1003817100381710038171003817y minus g10038171003817100381710038172= arg min1199041198961isinΩ11988611990411989610158402

isinΩ119887119894=(11989411198942)isinL

1003817100381710038171003817g10038171003817100381710038172 minus 2R (y119867g) (10)

where

g = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894 (11)

GCQSM-UC can be easily extended to transmitting morethan two signal symbols per channel use at the cost ofrequiring more RF chains at the transmitter side This ispossible because the error performance of the system doesnot depend on the rotation angle 120579 This will be addressed inmore detail in subsequent sections

32 Generalized CQSM with Permuted Combinations Toreduce the number of transmit antennas required to achievea given spectral efficiency we use the structure of thetransmitted vector to increase the number of spatial symbolsLet 119894 = (1198941 1198942) be a spatial symbol where the first signalsymbol 1199041198961 is transmitted from the 1198941 combination of antennasand the second signal symbol 11990411989610158402 from the combination 1198942 Fora given channel realization the received noiseless vector g isgiven by

g = 1199041198961radic119899119880sum119894isin1198941h119894 +11990411989610158402radic119899119880sum119894isin1198942h119894

= 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894

(12)

where 11990411989610158402 = 1199041198962119890119895120579 and 1199041198961 1199041198962 isin Ω119886 To obtain 1198941 and 1198942 weimpose the following conditions

(i) 1198941 cup 1198942 = 0 this implies that the set of antennas fromwhich the first signal symbol and that from whichthe second symbol are transmitted do not overlapThis condition reduces the detection ambiguity at thereceiver

(ii) (1198941 1198942) = (1198942 1198941) as described earlier the first andsecond symbols are drawn from different constella-tion sets Therefore for given signal symbolsrsquoindices 1198961 and 1198962 the following two received latticepointsmdashnoiseless received vectorsmdash are distinguish-able

g1 = 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894 (13)

and

g2 = 1199041198961radic119899119880sum119894isin1198942h119894 +1199041198962radic119899119880sum119894isin1198941119890119895120579h119894 (14)

Wireless Communications and Mobile Computing 5

Input A = 1 119899119879 119899119880 119899119879Output The set of spatial symbolsIC = COMB(A 119899119880)for 1198941 isin C do

B = A1198941

D = COMB(B 119899119880)for 1198942 isin D do

APPND(I (1198941 1198942))end

end

Algorithm 1 Generation of the spatial symbols assuming 119899119879transmit antennas and that each signal symbol is transmitted froma combination of 119899119880 antennasTable 3 An example of the spatial symbols (1198941 1198942) that can be usedfor transmission by GCQSM-PC assuming 119899119879 = 5 and 119899119880 = 21198941 11989421 2 3 4 3 5 4 51 3 2 4 2 5 4 51 4 2 3 2 5 3 51 5 2 3 2 4 3 42 3 1 4 1 5 4 52 4 1 3 1 5 3 52 5 1 3 1 4 3 43 4 1 2 1 5 2 53 5 1 2 1 4 2 44 5 1 2 1 3 2 3

This is rendered possible using the rotation angle 120579The optimization of the rotation angle is addressed ina following section

Algorithm 1 depicts a pseudocode of generating the set ofspatial symbolsI that can be used for transmission ThereinCOMB(A 119899119880) generates all combinations of length 119899119880 of theelements of the set A Also APPND(I (1198941 1198942)) appends thetuple (1198941 1198942) to the setI B = A

1198941is the difference of the two

sets119860 and 1198941 Accordingly the number of spatial symbols thatcan be used for transmission is given by

119862spa = (119899119879119899119880) times (119899119879 minus 119899119880119899119880 ) (15)

Table 3 depicts an example of the obtained spatial sym-bols assuming 119899119879 = 5 and 119899119880 = 2 In this case GCQSM-PC obtains 30 spatial symbols whereas GCQSM-UC obtainsonly five

Table 4 lists the number of transmit antennas required byeach system to achieve a given spectral efficiency per spatialsymbol For instance SM GSM (and MA-SM) CQSMGCQSM-UC andGCQSM-PC require 1024 46 32 15 and 10transmit antennas respectively to achieve 10 bits per spatialsymbol

4 Performance Analysis ofthe Generalized CQSM

Let

g119894 = 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894 (16)

and

g119896 = 1199041radic119899119880sum119894isin1h119894 +1199042radic119899119880sum119894isin2119890119895120579h119894 (17)

be two noiseless received codewords corresponding to thetransmitted symbol (1199041198961 1199041198962 1198941 1198942) and (1199041 1199042 1 2) where119904119896 is a signal symbol and 119894 = 1198971 119897119899119880 is a spatial symbolThe corresponding received vectors are denoted by y119894 and y119896respectively

The conditional pairwise error probability (PEP) of themaximum-likelihood (ML) receiver is given by [21 22]

Pr [g119894 997888rarr g119896 | H] = 119876(radic 1003817100381710038171003817y119894 minus y1198961003817100381710038171003817221205902119899 )

= 119876(radic 1003817100381710038171003817H (s119894 minus s119896)1003817100381710038171003817221205902119899 )(18)

where119876(sdot) is the Gaussian tail probability function or simplythe119876-function and 1198892119894119896 = y119894 minus y1198962 is the squared Euclideandistance between s119894 and s119896 at the receiver Similarly 1198892119894119896(tx) =s119894 minus s1198962 is the squared Euclidean distance between s119894 and s119896at the transmitter Note that

EH 1198892119894119895 = EH (s119894 minus s119896)119867H119867H (s119894 minus s119896)= (s119894 minus s119896)119867 (s119894 minus s119896) = 1198892119894119896(tx) (19)

The unconditional PEP (UPEP) assuming 119899119877 receivesantennas which is obtained by taking the expectation of (18)over the channel H is given by

Pr [g119894 997888rarr g119896] = 120583119899119877119894119895 119899119877minus1sum119897=0

(119899119877 minus 1 + 119897119897 ) [1 minus 120583119894119895]119897 (20)

where

120583119894119895 = 12 (1 minus radic 120574 sdot 1198892119894119896(tx)4 + 120574 sdot 1198892119894119896(tx)

) (21)

and 120574 = 1205902119904 1205902119899 is the signal-to-noise ratio (SNR) with 1205902119904 =E(119904lowast119904) the expected power of the signal symbol

The union bound on the pairwise error probability is thengiven by averaging all the pairwise probabilities as follows

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896] (22)

6 Wireless Communications and Mobile Computing

Table 4 Examples of the number of transmit antennas required to achieve a given SE119904119901119886 for several spatial modulation systems

SE119904119901119886 SM GSM CQSM QSM GCQSM-UC GCQSM-PC6 64 12 8 8 68 256 24 16 11 810 1024 46 32 15 1012 4096 92 64 20 13

Accordingly the average bit-error rate (BER) is upper-bounded by

119875119890 le 11198722119872 2119872sum119894=1

2119872sum119896=1

119863s119894s119896Pr [g119894 997888rarr g119896] (23)

where 119863s119894s119896 the hamming distance between s119894 and s119896 is thenumber of errors associated with the event [g119894 997888rarr g119896] and119872 is the spectral efficiency of the system Note that in eachtransmitted vector symbol s there are exactly (2times119899119880) nonzeroelements

5 Optimization of the Rotation Angles ofCQSMICQSM Revisited

In [17 18] the optimal rotation angles applied to the secondmodulation set of the CQSM and ICQSM respectively areobtained through extensive Monte Carlo simulations Inthis case the obtained rotation angle corresponds to theminimum bit-error rate (BER) averaged over a large numberof channel and noise realizations Alternatively the rotationangle can be obtained through minimizing the upper-boundof the unconditional pairwise error probability

Let

g119894 = Hs119894 = 1199041198961h1198941 + 11990410158401198962h1198942g119896 = Hs119896 = 1199041h1 + 1199041015840

2h2 (24)

where y119894 = g119894 + n and y119896 = g119896 + n are the correspondingreceived vectors for a given noise vector n

The PEP and UPEP of the CQSM and ICQSM can also beobtained using (18)ndash(21) The union bound on the pairwiseerror probability at high SNR values (aka asymptotic prob-ability of error) is then given by

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896]= (2119899119877 minus 1119899119877 ) 120574minus1198991198772119872 2

119872sum119894=1

2119872sum119896=1

(1198892119894119896(tx))minus119899119877= (2119899119877 minus 1119899119877 ) 120574minus1198991198771198712 119861sum

119894=1

119891119894Ω119894(25)

where the maximum value of 119861 equals 26 as shownin [19] The term Ω119894 is a function of the signal symbols(1199041198961 11990410158401198962 1199041 11990410158402) and 119899119877 and 119891119894 is the frequency of Ω119894 where

sum119861119894=1 119891119894 = 1198994119879 It is shown in [19] that 119861 = 7 in the case ofICQSM and it can be shown to be equal to 15 for CQSMThe rotation angle is then obtained by solving the followingoptimization problem

argmin0lt120579lt1205872

( 119861sum119894=1

119891119894Ω119894) (26)

where 119861 = 15 and 7 for CQSM and ICQSM respectivelyFigure 1 depicts the cost function of (26) for CQSM and

ICQSM and for several system configurations The optimalrotation angle corresponds to the minimum value of thecost function Assuming the case of CQSM and for thescenarios depicted in the figure the obtained optimal rotationangles are very close to those obtained in [17] Therefore thissmall and tolerable degradation in the error performancecomes at a high gain in the optimization time of the rotationangle On the other hand the obtained rotation angles forICQSM are identical to those obtained in [18] AssumingPSK modulation the optimal rotation angle for the ICQSMis equal to 120587119871 with 119871 denoting the cardinality of theconstellation set

6 Optimization of the Rotation Angles for theGeneralized CQSM

61 Rotation Angle of the GCQSM-UC As explained earlierand based on the design of the spatial symbols of theGCQSM-UC if (1198941 1198942) is a spatial symbol that can be used fortransmission then (1198942 1198941) is not a valid spatial symbol Thisimplies that the first symbol 1199041198961 and 11990410158401198962 are distinguishableusing the indices of the antennas fromwhich they are sent Assuch both symbols can be drawn from the same modulationset and hence the rotation angle will have no impact on theperformance of the system In the following 120579 is set to zerofor GCQSM-UC

62 Rotation Angle of the GCQSM-PC Opposed to theGCQSM-UC both (1198941 1198942) and (1198942 1198941) are valid spatial symbolsthat can be used for transmission in GCQSM-PC Thereforethe second symbol 11990410158401198962 should be rotated so that it can bedistinguished from the first signal symbol 1199041198961 The formula-tion of the upper-bound of the GCQSM-PC as in (22) is ahard problem Alternatively we obtained the rotation angleusing simulations Figure 2 depicts the BER of GCQSM-PCversus the rotation angle the upper subfigure assumes QPSKmodulation and the lower subfigure assumes 16-QAM Theoptimal rotation angle is about 1205874 for all the simulatedscenarios assuming QPSK modulation Also the optimal

Wireless Communications and Mobile Computing 7

0 20 4040

60

80

100

120

140co

st

4 2

0 20 40

100

150

200

2504 4

0 20 40

200

300

400

5004 8

0 20 40

400

600

800

10004 16

0 20 40

406080

100120

cost

0 20 4050

100

150

200

250

0 20 40100

200

300

400

0 20 40200

400

600

800

0 20 40

50

100

150

cost

8PSK

0 20 4050

100

150

200

16PSK

0 20 4050

100

150

200

16QAM

0 20 40

100

150

200

25032PSK

CQSM16QAM

CQSM8PSK

ICQSM

times times times times

4 2 4 4 4 8 4 16times times times times

4 4times

Figure 1 The cost function in (26) in dB scale versus the rotation angle for several system configurations The subfigures in the first andsecond row are obtained for CQSM and those in the bottom row are for ICQSM

rotation angle for the simulated scenarios using 16-QAMis approximately 23 degrees These values of the optimalrotation angle are affected by the sets from which symbolsare drawn and the values of 119899119879 and 119899119877 As a future work wewould like to analytically derive a generic formula to obtainthe optimal angle for arbitrary system configurations

7 Simulation Results

The following results are obtained assuming that the channelstate information (CSI) is known only at the receiver Tomake a fair comparison the rotation angles for the simulatedscenarios are optimized using simulations for both CQSMand the generalized CQSM The obtained rotation anglesusing simulations and the optimization method described inSection 5 for CQSM scheme have very similar values

Figure 3 depicts comparisons between the MA-SM andthe CQSM for several system configurations of (119899119879 119902) and afixed 119899119877 = 4 We make the following comparisons

(i) For the same 119899119879 and 119902 and different spectral effi-ciency (Figure 3(a)) in this case MA-SM outperformsCQSM by about 2 dB for 119899119879 = 8 and about 15dB for 119899119879 = 16 The gap is bridged between thetwo algorithms as 119899119879 becomes large This decreasein the performance gap is due to the decrease in theprobability of transmitting the two symbols from thesame transmit antenna in the CQSMThis probability

is equal to 1119899119879 The comparison here is not fairbecause the CQSM achieves a spectral efficiency 2bpcu higher than that of the MA-SM

(ii) For the spectral efficiency and 119902 and different 119899119879(Figure 3(b)) the performance of the two algorithmsis depicted for an equal spectral efficiency assuming119899119879 = 8 16 and 119899119879 = 12 24 for CQSM and MA-SMrespectively For these two scenarios CQSM requires4 and 8 transmit antennas less than MA-SM Thishuge reduction in the number of transmit antennascomes at a negligible degradation in the SNR for high119899119879 This implies that the CQSM is more attractive formassive MIMO systems

(iii) For the spectral efficiency and 119899119879 and different 119902(Figure 3(c)) finally the performance of the two algo-rithms is evaluated for the same spectral efficiencyand number of transmit antennas In this case 119902 is setto 2 and 3 in the CQSM and MA-SM respectively Ata target BER of 10minus4 CQSM outperforms MA-SM by2 and 24 dB respectively for the two depicted systemconfigurations

According to this analysis and results depicted inFigure 3 we conclude that CQSM can be used to reduce thenumber of transmit antennas at a marginal cost in the SNRfor high 119899119879 or it can achieve an improvement in the SNR ifthe same number of antennas is deployed by the two systems

8 Wireless Communications and Mobile Computing

0 10 20 30 40

8 times 4 =14 dB 10 times 4 = 17 dB13 times 4 = 17 dB

0 10 20 30 40

4 times 8 = 16 dB 8 times 4 = 22 dB 10 times 4 = 23 dB

10minus4

10minus2

10minus0

BER

10minus4

10minus2

10minus0

BER

Figure 2 BER of the GCQSM-PU versus the rotation angle applied to the second signal symbol using QPSK (first subfigure) and 16-QAM(second subfigure)

Since at high 119899119879 CQSM requires one antenna less thanICQSM to achieve the same spectral efficiency at the cost ofa moderate degradation in the BER performance CQSM isused in the following comparisons

Figure 4 depicts the BER performance of CQSMGCQSM-UC and GCQSM-PC for several system scenariosand 119899119880 = 2 The title of each subfigure is a tuple that includesthe number of transmit antennas required by CQSMGCQSM-UC and GCQSM-PC respectively to achievethe same spectral efficiency The spectral efficiency SE119904119901119886achieved assuming the systemrsquos parameters in Figure 4(a) is6 in 4(b) is 8 in 4(c) is 10 and in 4(d) is 12 bpcu For eachof the system configurations SE119904119894119892 is equal to 4 bpcu leadingto a total spectral efficiency SE = SE119904119901119886 + SE119904119894119892 of 10 12 14and 16 bpcu respectively The range over which the rotationangle is optimized is [0 1205874] The rotation angles appliedto the second signal symbol for the scenarios depictedin Figure 4 are listed in Table 5 The rotation angles areoptimized through Monte Carlo simulations Based on theresults depicted in Figure 4 we make the following remarksin terms of the BER performance the rotation angles andthe number of employed transmit antennas

(i) For 119899119877 = 2 the three techniques achieve the sameBER performance regardless of the number of trans-mit antennas

(ii) For a given value of 119899119877 the BER performanceis degraded as the number of transmit antennasincreases The degradation is relatively small as 119899119877increases This is an expected behavior of SM tech-niques because the size of hypothesis set over whichthe ML detector performs the search increases as 119899119879increases

(iii) In Figure 4(a) GCQSM-PC slightly outperforms theother two techniques for 119899119877 = 4 and 8 As 119899119879 getslarger the probability that the two signal symbols aretransmitted from the same antenna inCQSMreducesAccordingly the performance of CQSM and the twogeneralized algorithms almost coincide for 119899119877 = 4in Figures 4(b) 4(c) and 4(d) For 119899119877 = 8 CQSMslightly outperforms GCQSM-UC and GCQSM-PU

(iv) As shown in Figure 4(c) the generalized techniquesperform close to the CQSM for 119899119877 = 4 In thiscase GCQSM-UC and GCQSM-PC require 17 and 22

Wireless Communications and Mobile Computing 9

10 15 20SNR (dB)

BER

cqsm (8 2)ma-sm (8 2)cqsm (16 2)ma-sm (16 2)

10minus1

10minus2

10minus3

10minus4

(a)

10 15 20SNR (dB)

cqsm (8 2)ma-sm (12 2)cqsm (16 2)ma-sm (24 2)

BER

10minus1

10minus2

10minus3

10minus4

(b)

10 15 20 25SNR (dB)

cqsm (8 2)ma-sm (8 3)cqsm (16 2) ma-sm (16 3)

BER

10minus1

10minus2

10minus3

10minus4

(c)

Figure 3 A comparison between CQSM andMA-SM (a) for the same 119899119879 and different spectral efficiency (b) for the same spectral efficiencyand 119902 and different 119899119879 and finally (c) for the same spectral efficiency and 119899119879 and different 119902

Table 5 Range of optimal rotation angles used to obtain the BER results depicted in Figure 4 assuming QPSK modulation

SE119904119901119886 119899119877 120579CQSM 120579GCQSMminusPC

62 ge25 ge304 ge35 ge418 ge38 ge43

82 ge20 ge214 ge34 ge358 ge38 ge42

102 ge9 ge124 ge30 ge328 ge38 ge39

122 ge9 ge124 ge25 ge308 ge38 ge39

transmit antennas less than CQSM while achievingthe same spectral efficiency

Finally we compare the BER performance of SM GSMQSM CQSM and the proposed generalized CQSM tech-niques We assume that all the schemes achieve the sameES119904119901119886 and ES119904119894119892 leading to the same total spectral efficiencyof ES = (ES119904119901119886 + ES119904119894119892) Also the three generalized schemesuse a combination of two antennas to transmit each signalsymbol Figures 5(a) and 5(b) depict the BER performancefor a spectral efficiency of 16 and 18 bpcu respectively Thenumber of transmit antennas and bits per signal symbol isrepresented as a tuple of the form (119899119879 119902) for each of the

systems The rotation angles applied to CQSM and GCQSM-PC are 20 and 23 degrees respectively to obtain the results inthe two subfigures We make the following two remarks

(i) CQSM GCQSM-PC and GCQSM-UC apply 16-QAM modulation whereas SM GSM and QSM use256-QAM to achieve an ES119904119894119892 of 8 bpcu The numberof transmit antennas required to achieve the spectralefficiency of 16 and 18 bpcu is given in the secondand third row respectively in Table 4 GCQSM-PCrequires the least number of transmit antennas of8 and 10 for the results in Figures 5(a) and 5(b)respectively On the other hand SM requires 256 and

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

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Wireless Communications and Mobile Computing 5

Input A = 1 119899119879 119899119880 119899119879Output The set of spatial symbolsIC = COMB(A 119899119880)for 1198941 isin C do

B = A1198941

D = COMB(B 119899119880)for 1198942 isin D do

APPND(I (1198941 1198942))end

end

Algorithm 1 Generation of the spatial symbols assuming 119899119879transmit antennas and that each signal symbol is transmitted froma combination of 119899119880 antennasTable 3 An example of the spatial symbols (1198941 1198942) that can be usedfor transmission by GCQSM-PC assuming 119899119879 = 5 and 119899119880 = 21198941 11989421 2 3 4 3 5 4 51 3 2 4 2 5 4 51 4 2 3 2 5 3 51 5 2 3 2 4 3 42 3 1 4 1 5 4 52 4 1 3 1 5 3 52 5 1 3 1 4 3 43 4 1 2 1 5 2 53 5 1 2 1 4 2 44 5 1 2 1 3 2 3

This is rendered possible using the rotation angle 120579The optimization of the rotation angle is addressed ina following section

Algorithm 1 depicts a pseudocode of generating the set ofspatial symbolsI that can be used for transmission ThereinCOMB(A 119899119880) generates all combinations of length 119899119880 of theelements of the set A Also APPND(I (1198941 1198942)) appends thetuple (1198941 1198942) to the setI B = A

1198941is the difference of the two

sets119860 and 1198941 Accordingly the number of spatial symbols thatcan be used for transmission is given by

119862spa = (119899119879119899119880) times (119899119879 minus 119899119880119899119880 ) (15)

Table 3 depicts an example of the obtained spatial sym-bols assuming 119899119879 = 5 and 119899119880 = 2 In this case GCQSM-PC obtains 30 spatial symbols whereas GCQSM-UC obtainsonly five

Table 4 lists the number of transmit antennas required byeach system to achieve a given spectral efficiency per spatialsymbol For instance SM GSM (and MA-SM) CQSMGCQSM-UC andGCQSM-PC require 1024 46 32 15 and 10transmit antennas respectively to achieve 10 bits per spatialsymbol

4 Performance Analysis ofthe Generalized CQSM

Let

g119894 = 1199041198961radic119899119880sum119894isin1198941h119894 +1199041198962radic119899119880sum119894isin1198942119890119895120579h119894 (16)

and

g119896 = 1199041radic119899119880sum119894isin1h119894 +1199042radic119899119880sum119894isin2119890119895120579h119894 (17)

be two noiseless received codewords corresponding to thetransmitted symbol (1199041198961 1199041198962 1198941 1198942) and (1199041 1199042 1 2) where119904119896 is a signal symbol and 119894 = 1198971 119897119899119880 is a spatial symbolThe corresponding received vectors are denoted by y119894 and y119896respectively

The conditional pairwise error probability (PEP) of themaximum-likelihood (ML) receiver is given by [21 22]

Pr [g119894 997888rarr g119896 | H] = 119876(radic 1003817100381710038171003817y119894 minus y1198961003817100381710038171003817221205902119899 )

= 119876(radic 1003817100381710038171003817H (s119894 minus s119896)1003817100381710038171003817221205902119899 )(18)

where119876(sdot) is the Gaussian tail probability function or simplythe119876-function and 1198892119894119896 = y119894 minus y1198962 is the squared Euclideandistance between s119894 and s119896 at the receiver Similarly 1198892119894119896(tx) =s119894 minus s1198962 is the squared Euclidean distance between s119894 and s119896at the transmitter Note that

EH 1198892119894119895 = EH (s119894 minus s119896)119867H119867H (s119894 minus s119896)= (s119894 minus s119896)119867 (s119894 minus s119896) = 1198892119894119896(tx) (19)

The unconditional PEP (UPEP) assuming 119899119877 receivesantennas which is obtained by taking the expectation of (18)over the channel H is given by

Pr [g119894 997888rarr g119896] = 120583119899119877119894119895 119899119877minus1sum119897=0

(119899119877 minus 1 + 119897119897 ) [1 minus 120583119894119895]119897 (20)

where

120583119894119895 = 12 (1 minus radic 120574 sdot 1198892119894119896(tx)4 + 120574 sdot 1198892119894119896(tx)

) (21)

and 120574 = 1205902119904 1205902119899 is the signal-to-noise ratio (SNR) with 1205902119904 =E(119904lowast119904) the expected power of the signal symbol

The union bound on the pairwise error probability is thengiven by averaging all the pairwise probabilities as follows

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896] (22)

6 Wireless Communications and Mobile Computing

Table 4 Examples of the number of transmit antennas required to achieve a given SE119904119901119886 for several spatial modulation systems

SE119904119901119886 SM GSM CQSM QSM GCQSM-UC GCQSM-PC6 64 12 8 8 68 256 24 16 11 810 1024 46 32 15 1012 4096 92 64 20 13

Accordingly the average bit-error rate (BER) is upper-bounded by

119875119890 le 11198722119872 2119872sum119894=1

2119872sum119896=1

119863s119894s119896Pr [g119894 997888rarr g119896] (23)

where 119863s119894s119896 the hamming distance between s119894 and s119896 is thenumber of errors associated with the event [g119894 997888rarr g119896] and119872 is the spectral efficiency of the system Note that in eachtransmitted vector symbol s there are exactly (2times119899119880) nonzeroelements

5 Optimization of the Rotation Angles ofCQSMICQSM Revisited

In [17 18] the optimal rotation angles applied to the secondmodulation set of the CQSM and ICQSM respectively areobtained through extensive Monte Carlo simulations Inthis case the obtained rotation angle corresponds to theminimum bit-error rate (BER) averaged over a large numberof channel and noise realizations Alternatively the rotationangle can be obtained through minimizing the upper-boundof the unconditional pairwise error probability

Let

g119894 = Hs119894 = 1199041198961h1198941 + 11990410158401198962h1198942g119896 = Hs119896 = 1199041h1 + 1199041015840

2h2 (24)

where y119894 = g119894 + n and y119896 = g119896 + n are the correspondingreceived vectors for a given noise vector n

The PEP and UPEP of the CQSM and ICQSM can also beobtained using (18)ndash(21) The union bound on the pairwiseerror probability at high SNR values (aka asymptotic prob-ability of error) is then given by

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896]= (2119899119877 minus 1119899119877 ) 120574minus1198991198772119872 2

119872sum119894=1

2119872sum119896=1

(1198892119894119896(tx))minus119899119877= (2119899119877 minus 1119899119877 ) 120574minus1198991198771198712 119861sum

119894=1

119891119894Ω119894(25)

where the maximum value of 119861 equals 26 as shownin [19] The term Ω119894 is a function of the signal symbols(1199041198961 11990410158401198962 1199041 11990410158402) and 119899119877 and 119891119894 is the frequency of Ω119894 where

sum119861119894=1 119891119894 = 1198994119879 It is shown in [19] that 119861 = 7 in the case ofICQSM and it can be shown to be equal to 15 for CQSMThe rotation angle is then obtained by solving the followingoptimization problem

argmin0lt120579lt1205872

( 119861sum119894=1

119891119894Ω119894) (26)

where 119861 = 15 and 7 for CQSM and ICQSM respectivelyFigure 1 depicts the cost function of (26) for CQSM and

ICQSM and for several system configurations The optimalrotation angle corresponds to the minimum value of thecost function Assuming the case of CQSM and for thescenarios depicted in the figure the obtained optimal rotationangles are very close to those obtained in [17] Therefore thissmall and tolerable degradation in the error performancecomes at a high gain in the optimization time of the rotationangle On the other hand the obtained rotation angles forICQSM are identical to those obtained in [18] AssumingPSK modulation the optimal rotation angle for the ICQSMis equal to 120587119871 with 119871 denoting the cardinality of theconstellation set

6 Optimization of the Rotation Angles for theGeneralized CQSM

61 Rotation Angle of the GCQSM-UC As explained earlierand based on the design of the spatial symbols of theGCQSM-UC if (1198941 1198942) is a spatial symbol that can be used fortransmission then (1198942 1198941) is not a valid spatial symbol Thisimplies that the first symbol 1199041198961 and 11990410158401198962 are distinguishableusing the indices of the antennas fromwhich they are sent Assuch both symbols can be drawn from the same modulationset and hence the rotation angle will have no impact on theperformance of the system In the following 120579 is set to zerofor GCQSM-UC

62 Rotation Angle of the GCQSM-PC Opposed to theGCQSM-UC both (1198941 1198942) and (1198942 1198941) are valid spatial symbolsthat can be used for transmission in GCQSM-PC Thereforethe second symbol 11990410158401198962 should be rotated so that it can bedistinguished from the first signal symbol 1199041198961 The formula-tion of the upper-bound of the GCQSM-PC as in (22) is ahard problem Alternatively we obtained the rotation angleusing simulations Figure 2 depicts the BER of GCQSM-PCversus the rotation angle the upper subfigure assumes QPSKmodulation and the lower subfigure assumes 16-QAM Theoptimal rotation angle is about 1205874 for all the simulatedscenarios assuming QPSK modulation Also the optimal

Wireless Communications and Mobile Computing 7

0 20 4040

60

80

100

120

140co

st

4 2

0 20 40

100

150

200

2504 4

0 20 40

200

300

400

5004 8

0 20 40

400

600

800

10004 16

0 20 40

406080

100120

cost

0 20 4050

100

150

200

250

0 20 40100

200

300

400

0 20 40200

400

600

800

0 20 40

50

100

150

cost

8PSK

0 20 4050

100

150

200

16PSK

0 20 4050

100

150

200

16QAM

0 20 40

100

150

200

25032PSK

CQSM16QAM

CQSM8PSK

ICQSM

times times times times

4 2 4 4 4 8 4 16times times times times

4 4times

Figure 1 The cost function in (26) in dB scale versus the rotation angle for several system configurations The subfigures in the first andsecond row are obtained for CQSM and those in the bottom row are for ICQSM

rotation angle for the simulated scenarios using 16-QAMis approximately 23 degrees These values of the optimalrotation angle are affected by the sets from which symbolsare drawn and the values of 119899119879 and 119899119877 As a future work wewould like to analytically derive a generic formula to obtainthe optimal angle for arbitrary system configurations

7 Simulation Results

The following results are obtained assuming that the channelstate information (CSI) is known only at the receiver Tomake a fair comparison the rotation angles for the simulatedscenarios are optimized using simulations for both CQSMand the generalized CQSM The obtained rotation anglesusing simulations and the optimization method described inSection 5 for CQSM scheme have very similar values

Figure 3 depicts comparisons between the MA-SM andthe CQSM for several system configurations of (119899119879 119902) and afixed 119899119877 = 4 We make the following comparisons

(i) For the same 119899119879 and 119902 and different spectral effi-ciency (Figure 3(a)) in this case MA-SM outperformsCQSM by about 2 dB for 119899119879 = 8 and about 15dB for 119899119879 = 16 The gap is bridged between thetwo algorithms as 119899119879 becomes large This decreasein the performance gap is due to the decrease in theprobability of transmitting the two symbols from thesame transmit antenna in the CQSMThis probability

is equal to 1119899119879 The comparison here is not fairbecause the CQSM achieves a spectral efficiency 2bpcu higher than that of the MA-SM

(ii) For the spectral efficiency and 119902 and different 119899119879(Figure 3(b)) the performance of the two algorithmsis depicted for an equal spectral efficiency assuming119899119879 = 8 16 and 119899119879 = 12 24 for CQSM and MA-SMrespectively For these two scenarios CQSM requires4 and 8 transmit antennas less than MA-SM Thishuge reduction in the number of transmit antennascomes at a negligible degradation in the SNR for high119899119879 This implies that the CQSM is more attractive formassive MIMO systems

(iii) For the spectral efficiency and 119899119879 and different 119902(Figure 3(c)) finally the performance of the two algo-rithms is evaluated for the same spectral efficiencyand number of transmit antennas In this case 119902 is setto 2 and 3 in the CQSM and MA-SM respectively Ata target BER of 10minus4 CQSM outperforms MA-SM by2 and 24 dB respectively for the two depicted systemconfigurations

According to this analysis and results depicted inFigure 3 we conclude that CQSM can be used to reduce thenumber of transmit antennas at a marginal cost in the SNRfor high 119899119879 or it can achieve an improvement in the SNR ifthe same number of antennas is deployed by the two systems

8 Wireless Communications and Mobile Computing

0 10 20 30 40

8 times 4 =14 dB 10 times 4 = 17 dB13 times 4 = 17 dB

0 10 20 30 40

4 times 8 = 16 dB 8 times 4 = 22 dB 10 times 4 = 23 dB

10minus4

10minus2

10minus0

BER

10minus4

10minus2

10minus0

BER

Figure 2 BER of the GCQSM-PU versus the rotation angle applied to the second signal symbol using QPSK (first subfigure) and 16-QAM(second subfigure)

Since at high 119899119879 CQSM requires one antenna less thanICQSM to achieve the same spectral efficiency at the cost ofa moderate degradation in the BER performance CQSM isused in the following comparisons

Figure 4 depicts the BER performance of CQSMGCQSM-UC and GCQSM-PC for several system scenariosand 119899119880 = 2 The title of each subfigure is a tuple that includesthe number of transmit antennas required by CQSMGCQSM-UC and GCQSM-PC respectively to achievethe same spectral efficiency The spectral efficiency SE119904119901119886achieved assuming the systemrsquos parameters in Figure 4(a) is6 in 4(b) is 8 in 4(c) is 10 and in 4(d) is 12 bpcu For eachof the system configurations SE119904119894119892 is equal to 4 bpcu leadingto a total spectral efficiency SE = SE119904119901119886 + SE119904119894119892 of 10 12 14and 16 bpcu respectively The range over which the rotationangle is optimized is [0 1205874] The rotation angles appliedto the second signal symbol for the scenarios depictedin Figure 4 are listed in Table 5 The rotation angles areoptimized through Monte Carlo simulations Based on theresults depicted in Figure 4 we make the following remarksin terms of the BER performance the rotation angles andthe number of employed transmit antennas

(i) For 119899119877 = 2 the three techniques achieve the sameBER performance regardless of the number of trans-mit antennas

(ii) For a given value of 119899119877 the BER performanceis degraded as the number of transmit antennasincreases The degradation is relatively small as 119899119877increases This is an expected behavior of SM tech-niques because the size of hypothesis set over whichthe ML detector performs the search increases as 119899119879increases

(iii) In Figure 4(a) GCQSM-PC slightly outperforms theother two techniques for 119899119877 = 4 and 8 As 119899119879 getslarger the probability that the two signal symbols aretransmitted from the same antenna inCQSMreducesAccordingly the performance of CQSM and the twogeneralized algorithms almost coincide for 119899119877 = 4in Figures 4(b) 4(c) and 4(d) For 119899119877 = 8 CQSMslightly outperforms GCQSM-UC and GCQSM-PU

(iv) As shown in Figure 4(c) the generalized techniquesperform close to the CQSM for 119899119877 = 4 In thiscase GCQSM-UC and GCQSM-PC require 17 and 22

Wireless Communications and Mobile Computing 9

10 15 20SNR (dB)

BER

cqsm (8 2)ma-sm (8 2)cqsm (16 2)ma-sm (16 2)

10minus1

10minus2

10minus3

10minus4

(a)

10 15 20SNR (dB)

cqsm (8 2)ma-sm (12 2)cqsm (16 2)ma-sm (24 2)

BER

10minus1

10minus2

10minus3

10minus4

(b)

10 15 20 25SNR (dB)

cqsm (8 2)ma-sm (8 3)cqsm (16 2) ma-sm (16 3)

BER

10minus1

10minus2

10minus3

10minus4

(c)

Figure 3 A comparison between CQSM andMA-SM (a) for the same 119899119879 and different spectral efficiency (b) for the same spectral efficiencyand 119902 and different 119899119879 and finally (c) for the same spectral efficiency and 119899119879 and different 119902

Table 5 Range of optimal rotation angles used to obtain the BER results depicted in Figure 4 assuming QPSK modulation

SE119904119901119886 119899119877 120579CQSM 120579GCQSMminusPC

62 ge25 ge304 ge35 ge418 ge38 ge43

82 ge20 ge214 ge34 ge358 ge38 ge42

102 ge9 ge124 ge30 ge328 ge38 ge39

122 ge9 ge124 ge25 ge308 ge38 ge39

transmit antennas less than CQSM while achievingthe same spectral efficiency

Finally we compare the BER performance of SM GSMQSM CQSM and the proposed generalized CQSM tech-niques We assume that all the schemes achieve the sameES119904119901119886 and ES119904119894119892 leading to the same total spectral efficiencyof ES = (ES119904119901119886 + ES119904119894119892) Also the three generalized schemesuse a combination of two antennas to transmit each signalsymbol Figures 5(a) and 5(b) depict the BER performancefor a spectral efficiency of 16 and 18 bpcu respectively Thenumber of transmit antennas and bits per signal symbol isrepresented as a tuple of the form (119899119879 119902) for each of the

systems The rotation angles applied to CQSM and GCQSM-PC are 20 and 23 degrees respectively to obtain the results inthe two subfigures We make the following two remarks

(i) CQSM GCQSM-PC and GCQSM-UC apply 16-QAM modulation whereas SM GSM and QSM use256-QAM to achieve an ES119904119894119892 of 8 bpcu The numberof transmit antennas required to achieve the spectralefficiency of 16 and 18 bpcu is given in the secondand third row respectively in Table 4 GCQSM-PCrequires the least number of transmit antennas of8 and 10 for the results in Figures 5(a) and 5(b)respectively On the other hand SM requires 256 and

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

6 Wireless Communications and Mobile Computing

Table 4 Examples of the number of transmit antennas required to achieve a given SE119904119901119886 for several spatial modulation systems

SE119904119901119886 SM GSM CQSM QSM GCQSM-UC GCQSM-PC6 64 12 8 8 68 256 24 16 11 810 1024 46 32 15 1012 4096 92 64 20 13

Accordingly the average bit-error rate (BER) is upper-bounded by

119875119890 le 11198722119872 2119872sum119894=1

2119872sum119896=1

119863s119894s119896Pr [g119894 997888rarr g119896] (23)

where 119863s119894s119896 the hamming distance between s119894 and s119896 is thenumber of errors associated with the event [g119894 997888rarr g119896] and119872 is the spectral efficiency of the system Note that in eachtransmitted vector symbol s there are exactly (2times119899119880) nonzeroelements

5 Optimization of the Rotation Angles ofCQSMICQSM Revisited

In [17 18] the optimal rotation angles applied to the secondmodulation set of the CQSM and ICQSM respectively areobtained through extensive Monte Carlo simulations Inthis case the obtained rotation angle corresponds to theminimum bit-error rate (BER) averaged over a large numberof channel and noise realizations Alternatively the rotationangle can be obtained through minimizing the upper-boundof the unconditional pairwise error probability

Let

g119894 = Hs119894 = 1199041198961h1198941 + 11990410158401198962h1198942g119896 = Hs119896 = 1199041h1 + 1199041015840

2h2 (24)

where y119894 = g119894 + n and y119896 = g119896 + n are the correspondingreceived vectors for a given noise vector n

The PEP and UPEP of the CQSM and ICQSM can also beobtained using (18)ndash(21) The union bound on the pairwiseerror probability at high SNR values (aka asymptotic prob-ability of error) is then given by

Pr [119890] le 12119872 2119872sum119894=1

2119872sum119896=1

Pr [g119894 997888rarr g119896]= (2119899119877 minus 1119899119877 ) 120574minus1198991198772119872 2

119872sum119894=1

2119872sum119896=1

(1198892119894119896(tx))minus119899119877= (2119899119877 minus 1119899119877 ) 120574minus1198991198771198712 119861sum

119894=1

119891119894Ω119894(25)

where the maximum value of 119861 equals 26 as shownin [19] The term Ω119894 is a function of the signal symbols(1199041198961 11990410158401198962 1199041 11990410158402) and 119899119877 and 119891119894 is the frequency of Ω119894 where

sum119861119894=1 119891119894 = 1198994119879 It is shown in [19] that 119861 = 7 in the case ofICQSM and it can be shown to be equal to 15 for CQSMThe rotation angle is then obtained by solving the followingoptimization problem

argmin0lt120579lt1205872

( 119861sum119894=1

119891119894Ω119894) (26)

where 119861 = 15 and 7 for CQSM and ICQSM respectivelyFigure 1 depicts the cost function of (26) for CQSM and

ICQSM and for several system configurations The optimalrotation angle corresponds to the minimum value of thecost function Assuming the case of CQSM and for thescenarios depicted in the figure the obtained optimal rotationangles are very close to those obtained in [17] Therefore thissmall and tolerable degradation in the error performancecomes at a high gain in the optimization time of the rotationangle On the other hand the obtained rotation angles forICQSM are identical to those obtained in [18] AssumingPSK modulation the optimal rotation angle for the ICQSMis equal to 120587119871 with 119871 denoting the cardinality of theconstellation set

6 Optimization of the Rotation Angles for theGeneralized CQSM

61 Rotation Angle of the GCQSM-UC As explained earlierand based on the design of the spatial symbols of theGCQSM-UC if (1198941 1198942) is a spatial symbol that can be used fortransmission then (1198942 1198941) is not a valid spatial symbol Thisimplies that the first symbol 1199041198961 and 11990410158401198962 are distinguishableusing the indices of the antennas fromwhich they are sent Assuch both symbols can be drawn from the same modulationset and hence the rotation angle will have no impact on theperformance of the system In the following 120579 is set to zerofor GCQSM-UC

62 Rotation Angle of the GCQSM-PC Opposed to theGCQSM-UC both (1198941 1198942) and (1198942 1198941) are valid spatial symbolsthat can be used for transmission in GCQSM-PC Thereforethe second symbol 11990410158401198962 should be rotated so that it can bedistinguished from the first signal symbol 1199041198961 The formula-tion of the upper-bound of the GCQSM-PC as in (22) is ahard problem Alternatively we obtained the rotation angleusing simulations Figure 2 depicts the BER of GCQSM-PCversus the rotation angle the upper subfigure assumes QPSKmodulation and the lower subfigure assumes 16-QAM Theoptimal rotation angle is about 1205874 for all the simulatedscenarios assuming QPSK modulation Also the optimal

Wireless Communications and Mobile Computing 7

0 20 4040

60

80

100

120

140co

st

4 2

0 20 40

100

150

200

2504 4

0 20 40

200

300

400

5004 8

0 20 40

400

600

800

10004 16

0 20 40

406080

100120

cost

0 20 4050

100

150

200

250

0 20 40100

200

300

400

0 20 40200

400

600

800

0 20 40

50

100

150

cost

8PSK

0 20 4050

100

150

200

16PSK

0 20 4050

100

150

200

16QAM

0 20 40

100

150

200

25032PSK

CQSM16QAM

CQSM8PSK

ICQSM

times times times times

4 2 4 4 4 8 4 16times times times times

4 4times

Figure 1 The cost function in (26) in dB scale versus the rotation angle for several system configurations The subfigures in the first andsecond row are obtained for CQSM and those in the bottom row are for ICQSM

rotation angle for the simulated scenarios using 16-QAMis approximately 23 degrees These values of the optimalrotation angle are affected by the sets from which symbolsare drawn and the values of 119899119879 and 119899119877 As a future work wewould like to analytically derive a generic formula to obtainthe optimal angle for arbitrary system configurations

7 Simulation Results

The following results are obtained assuming that the channelstate information (CSI) is known only at the receiver Tomake a fair comparison the rotation angles for the simulatedscenarios are optimized using simulations for both CQSMand the generalized CQSM The obtained rotation anglesusing simulations and the optimization method described inSection 5 for CQSM scheme have very similar values

Figure 3 depicts comparisons between the MA-SM andthe CQSM for several system configurations of (119899119879 119902) and afixed 119899119877 = 4 We make the following comparisons

(i) For the same 119899119879 and 119902 and different spectral effi-ciency (Figure 3(a)) in this case MA-SM outperformsCQSM by about 2 dB for 119899119879 = 8 and about 15dB for 119899119879 = 16 The gap is bridged between thetwo algorithms as 119899119879 becomes large This decreasein the performance gap is due to the decrease in theprobability of transmitting the two symbols from thesame transmit antenna in the CQSMThis probability

is equal to 1119899119879 The comparison here is not fairbecause the CQSM achieves a spectral efficiency 2bpcu higher than that of the MA-SM

(ii) For the spectral efficiency and 119902 and different 119899119879(Figure 3(b)) the performance of the two algorithmsis depicted for an equal spectral efficiency assuming119899119879 = 8 16 and 119899119879 = 12 24 for CQSM and MA-SMrespectively For these two scenarios CQSM requires4 and 8 transmit antennas less than MA-SM Thishuge reduction in the number of transmit antennascomes at a negligible degradation in the SNR for high119899119879 This implies that the CQSM is more attractive formassive MIMO systems

(iii) For the spectral efficiency and 119899119879 and different 119902(Figure 3(c)) finally the performance of the two algo-rithms is evaluated for the same spectral efficiencyand number of transmit antennas In this case 119902 is setto 2 and 3 in the CQSM and MA-SM respectively Ata target BER of 10minus4 CQSM outperforms MA-SM by2 and 24 dB respectively for the two depicted systemconfigurations

According to this analysis and results depicted inFigure 3 we conclude that CQSM can be used to reduce thenumber of transmit antennas at a marginal cost in the SNRfor high 119899119879 or it can achieve an improvement in the SNR ifthe same number of antennas is deployed by the two systems

8 Wireless Communications and Mobile Computing

0 10 20 30 40

8 times 4 =14 dB 10 times 4 = 17 dB13 times 4 = 17 dB

0 10 20 30 40

4 times 8 = 16 dB 8 times 4 = 22 dB 10 times 4 = 23 dB

10minus4

10minus2

10minus0

BER

10minus4

10minus2

10minus0

BER

Figure 2 BER of the GCQSM-PU versus the rotation angle applied to the second signal symbol using QPSK (first subfigure) and 16-QAM(second subfigure)

Since at high 119899119879 CQSM requires one antenna less thanICQSM to achieve the same spectral efficiency at the cost ofa moderate degradation in the BER performance CQSM isused in the following comparisons

Figure 4 depicts the BER performance of CQSMGCQSM-UC and GCQSM-PC for several system scenariosand 119899119880 = 2 The title of each subfigure is a tuple that includesthe number of transmit antennas required by CQSMGCQSM-UC and GCQSM-PC respectively to achievethe same spectral efficiency The spectral efficiency SE119904119901119886achieved assuming the systemrsquos parameters in Figure 4(a) is6 in 4(b) is 8 in 4(c) is 10 and in 4(d) is 12 bpcu For eachof the system configurations SE119904119894119892 is equal to 4 bpcu leadingto a total spectral efficiency SE = SE119904119901119886 + SE119904119894119892 of 10 12 14and 16 bpcu respectively The range over which the rotationangle is optimized is [0 1205874] The rotation angles appliedto the second signal symbol for the scenarios depictedin Figure 4 are listed in Table 5 The rotation angles areoptimized through Monte Carlo simulations Based on theresults depicted in Figure 4 we make the following remarksin terms of the BER performance the rotation angles andthe number of employed transmit antennas

(i) For 119899119877 = 2 the three techniques achieve the sameBER performance regardless of the number of trans-mit antennas

(ii) For a given value of 119899119877 the BER performanceis degraded as the number of transmit antennasincreases The degradation is relatively small as 119899119877increases This is an expected behavior of SM tech-niques because the size of hypothesis set over whichthe ML detector performs the search increases as 119899119879increases

(iii) In Figure 4(a) GCQSM-PC slightly outperforms theother two techniques for 119899119877 = 4 and 8 As 119899119879 getslarger the probability that the two signal symbols aretransmitted from the same antenna inCQSMreducesAccordingly the performance of CQSM and the twogeneralized algorithms almost coincide for 119899119877 = 4in Figures 4(b) 4(c) and 4(d) For 119899119877 = 8 CQSMslightly outperforms GCQSM-UC and GCQSM-PU

(iv) As shown in Figure 4(c) the generalized techniquesperform close to the CQSM for 119899119877 = 4 In thiscase GCQSM-UC and GCQSM-PC require 17 and 22

Wireless Communications and Mobile Computing 9

10 15 20SNR (dB)

BER

cqsm (8 2)ma-sm (8 2)cqsm (16 2)ma-sm (16 2)

10minus1

10minus2

10minus3

10minus4

(a)

10 15 20SNR (dB)

cqsm (8 2)ma-sm (12 2)cqsm (16 2)ma-sm (24 2)

BER

10minus1

10minus2

10minus3

10minus4

(b)

10 15 20 25SNR (dB)

cqsm (8 2)ma-sm (8 3)cqsm (16 2) ma-sm (16 3)

BER

10minus1

10minus2

10minus3

10minus4

(c)

Figure 3 A comparison between CQSM andMA-SM (a) for the same 119899119879 and different spectral efficiency (b) for the same spectral efficiencyand 119902 and different 119899119879 and finally (c) for the same spectral efficiency and 119899119879 and different 119902

Table 5 Range of optimal rotation angles used to obtain the BER results depicted in Figure 4 assuming QPSK modulation

SE119904119901119886 119899119877 120579CQSM 120579GCQSMminusPC

62 ge25 ge304 ge35 ge418 ge38 ge43

82 ge20 ge214 ge34 ge358 ge38 ge42

102 ge9 ge124 ge30 ge328 ge38 ge39

122 ge9 ge124 ge25 ge308 ge38 ge39

transmit antennas less than CQSM while achievingthe same spectral efficiency

Finally we compare the BER performance of SM GSMQSM CQSM and the proposed generalized CQSM tech-niques We assume that all the schemes achieve the sameES119904119901119886 and ES119904119894119892 leading to the same total spectral efficiencyof ES = (ES119904119901119886 + ES119904119894119892) Also the three generalized schemesuse a combination of two antennas to transmit each signalsymbol Figures 5(a) and 5(b) depict the BER performancefor a spectral efficiency of 16 and 18 bpcu respectively Thenumber of transmit antennas and bits per signal symbol isrepresented as a tuple of the form (119899119879 119902) for each of the

systems The rotation angles applied to CQSM and GCQSM-PC are 20 and 23 degrees respectively to obtain the results inthe two subfigures We make the following two remarks

(i) CQSM GCQSM-PC and GCQSM-UC apply 16-QAM modulation whereas SM GSM and QSM use256-QAM to achieve an ES119904119894119892 of 8 bpcu The numberof transmit antennas required to achieve the spectralefficiency of 16 and 18 bpcu is given in the secondand third row respectively in Table 4 GCQSM-PCrequires the least number of transmit antennas of8 and 10 for the results in Figures 5(a) and 5(b)respectively On the other hand SM requires 256 and

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Wireless Communications and Mobile Computing 7

0 20 4040

60

80

100

120

140co

st

4 2

0 20 40

100

150

200

2504 4

0 20 40

200

300

400

5004 8

0 20 40

400

600

800

10004 16

0 20 40

406080

100120

cost

0 20 4050

100

150

200

250

0 20 40100

200

300

400

0 20 40200

400

600

800

0 20 40

50

100

150

cost

8PSK

0 20 4050

100

150

200

16PSK

0 20 4050

100

150

200

16QAM

0 20 40

100

150

200

25032PSK

CQSM16QAM

CQSM8PSK

ICQSM

times times times times

4 2 4 4 4 8 4 16times times times times

4 4times

Figure 1 The cost function in (26) in dB scale versus the rotation angle for several system configurations The subfigures in the first andsecond row are obtained for CQSM and those in the bottom row are for ICQSM

rotation angle for the simulated scenarios using 16-QAMis approximately 23 degrees These values of the optimalrotation angle are affected by the sets from which symbolsare drawn and the values of 119899119879 and 119899119877 As a future work wewould like to analytically derive a generic formula to obtainthe optimal angle for arbitrary system configurations

7 Simulation Results

The following results are obtained assuming that the channelstate information (CSI) is known only at the receiver Tomake a fair comparison the rotation angles for the simulatedscenarios are optimized using simulations for both CQSMand the generalized CQSM The obtained rotation anglesusing simulations and the optimization method described inSection 5 for CQSM scheme have very similar values

Figure 3 depicts comparisons between the MA-SM andthe CQSM for several system configurations of (119899119879 119902) and afixed 119899119877 = 4 We make the following comparisons

(i) For the same 119899119879 and 119902 and different spectral effi-ciency (Figure 3(a)) in this case MA-SM outperformsCQSM by about 2 dB for 119899119879 = 8 and about 15dB for 119899119879 = 16 The gap is bridged between thetwo algorithms as 119899119879 becomes large This decreasein the performance gap is due to the decrease in theprobability of transmitting the two symbols from thesame transmit antenna in the CQSMThis probability

is equal to 1119899119879 The comparison here is not fairbecause the CQSM achieves a spectral efficiency 2bpcu higher than that of the MA-SM

(ii) For the spectral efficiency and 119902 and different 119899119879(Figure 3(b)) the performance of the two algorithmsis depicted for an equal spectral efficiency assuming119899119879 = 8 16 and 119899119879 = 12 24 for CQSM and MA-SMrespectively For these two scenarios CQSM requires4 and 8 transmit antennas less than MA-SM Thishuge reduction in the number of transmit antennascomes at a negligible degradation in the SNR for high119899119879 This implies that the CQSM is more attractive formassive MIMO systems

(iii) For the spectral efficiency and 119899119879 and different 119902(Figure 3(c)) finally the performance of the two algo-rithms is evaluated for the same spectral efficiencyand number of transmit antennas In this case 119902 is setto 2 and 3 in the CQSM and MA-SM respectively Ata target BER of 10minus4 CQSM outperforms MA-SM by2 and 24 dB respectively for the two depicted systemconfigurations

According to this analysis and results depicted inFigure 3 we conclude that CQSM can be used to reduce thenumber of transmit antennas at a marginal cost in the SNRfor high 119899119879 or it can achieve an improvement in the SNR ifthe same number of antennas is deployed by the two systems

8 Wireless Communications and Mobile Computing

0 10 20 30 40

8 times 4 =14 dB 10 times 4 = 17 dB13 times 4 = 17 dB

0 10 20 30 40

4 times 8 = 16 dB 8 times 4 = 22 dB 10 times 4 = 23 dB

10minus4

10minus2

10minus0

BER

10minus4

10minus2

10minus0

BER

Figure 2 BER of the GCQSM-PU versus the rotation angle applied to the second signal symbol using QPSK (first subfigure) and 16-QAM(second subfigure)

Since at high 119899119879 CQSM requires one antenna less thanICQSM to achieve the same spectral efficiency at the cost ofa moderate degradation in the BER performance CQSM isused in the following comparisons

Figure 4 depicts the BER performance of CQSMGCQSM-UC and GCQSM-PC for several system scenariosand 119899119880 = 2 The title of each subfigure is a tuple that includesthe number of transmit antennas required by CQSMGCQSM-UC and GCQSM-PC respectively to achievethe same spectral efficiency The spectral efficiency SE119904119901119886achieved assuming the systemrsquos parameters in Figure 4(a) is6 in 4(b) is 8 in 4(c) is 10 and in 4(d) is 12 bpcu For eachof the system configurations SE119904119894119892 is equal to 4 bpcu leadingto a total spectral efficiency SE = SE119904119901119886 + SE119904119894119892 of 10 12 14and 16 bpcu respectively The range over which the rotationangle is optimized is [0 1205874] The rotation angles appliedto the second signal symbol for the scenarios depictedin Figure 4 are listed in Table 5 The rotation angles areoptimized through Monte Carlo simulations Based on theresults depicted in Figure 4 we make the following remarksin terms of the BER performance the rotation angles andthe number of employed transmit antennas

(i) For 119899119877 = 2 the three techniques achieve the sameBER performance regardless of the number of trans-mit antennas

(ii) For a given value of 119899119877 the BER performanceis degraded as the number of transmit antennasincreases The degradation is relatively small as 119899119877increases This is an expected behavior of SM tech-niques because the size of hypothesis set over whichthe ML detector performs the search increases as 119899119879increases

(iii) In Figure 4(a) GCQSM-PC slightly outperforms theother two techniques for 119899119877 = 4 and 8 As 119899119879 getslarger the probability that the two signal symbols aretransmitted from the same antenna inCQSMreducesAccordingly the performance of CQSM and the twogeneralized algorithms almost coincide for 119899119877 = 4in Figures 4(b) 4(c) and 4(d) For 119899119877 = 8 CQSMslightly outperforms GCQSM-UC and GCQSM-PU

(iv) As shown in Figure 4(c) the generalized techniquesperform close to the CQSM for 119899119877 = 4 In thiscase GCQSM-UC and GCQSM-PC require 17 and 22

Wireless Communications and Mobile Computing 9

10 15 20SNR (dB)

BER

cqsm (8 2)ma-sm (8 2)cqsm (16 2)ma-sm (16 2)

10minus1

10minus2

10minus3

10minus4

(a)

10 15 20SNR (dB)

cqsm (8 2)ma-sm (12 2)cqsm (16 2)ma-sm (24 2)

BER

10minus1

10minus2

10minus3

10minus4

(b)

10 15 20 25SNR (dB)

cqsm (8 2)ma-sm (8 3)cqsm (16 2) ma-sm (16 3)

BER

10minus1

10minus2

10minus3

10minus4

(c)

Figure 3 A comparison between CQSM andMA-SM (a) for the same 119899119879 and different spectral efficiency (b) for the same spectral efficiencyand 119902 and different 119899119879 and finally (c) for the same spectral efficiency and 119899119879 and different 119902

Table 5 Range of optimal rotation angles used to obtain the BER results depicted in Figure 4 assuming QPSK modulation

SE119904119901119886 119899119877 120579CQSM 120579GCQSMminusPC

62 ge25 ge304 ge35 ge418 ge38 ge43

82 ge20 ge214 ge34 ge358 ge38 ge42

102 ge9 ge124 ge30 ge328 ge38 ge39

122 ge9 ge124 ge25 ge308 ge38 ge39

transmit antennas less than CQSM while achievingthe same spectral efficiency

Finally we compare the BER performance of SM GSMQSM CQSM and the proposed generalized CQSM tech-niques We assume that all the schemes achieve the sameES119904119901119886 and ES119904119894119892 leading to the same total spectral efficiencyof ES = (ES119904119901119886 + ES119904119894119892) Also the three generalized schemesuse a combination of two antennas to transmit each signalsymbol Figures 5(a) and 5(b) depict the BER performancefor a spectral efficiency of 16 and 18 bpcu respectively Thenumber of transmit antennas and bits per signal symbol isrepresented as a tuple of the form (119899119879 119902) for each of the

systems The rotation angles applied to CQSM and GCQSM-PC are 20 and 23 degrees respectively to obtain the results inthe two subfigures We make the following two remarks

(i) CQSM GCQSM-PC and GCQSM-UC apply 16-QAM modulation whereas SM GSM and QSM use256-QAM to achieve an ES119904119894119892 of 8 bpcu The numberof transmit antennas required to achieve the spectralefficiency of 16 and 18 bpcu is given in the secondand third row respectively in Table 4 GCQSM-PCrequires the least number of transmit antennas of8 and 10 for the results in Figures 5(a) and 5(b)respectively On the other hand SM requires 256 and

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

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VLSI Design

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Hindawiwwwhindawicom Volume 2018

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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

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Advances in

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Submit your manuscripts atwwwhindawicom

8 Wireless Communications and Mobile Computing

0 10 20 30 40

8 times 4 =14 dB 10 times 4 = 17 dB13 times 4 = 17 dB

0 10 20 30 40

4 times 8 = 16 dB 8 times 4 = 22 dB 10 times 4 = 23 dB

10minus4

10minus2

10minus0

BER

10minus4

10minus2

10minus0

BER

Figure 2 BER of the GCQSM-PU versus the rotation angle applied to the second signal symbol using QPSK (first subfigure) and 16-QAM(second subfigure)

Since at high 119899119879 CQSM requires one antenna less thanICQSM to achieve the same spectral efficiency at the cost ofa moderate degradation in the BER performance CQSM isused in the following comparisons

Figure 4 depicts the BER performance of CQSMGCQSM-UC and GCQSM-PC for several system scenariosand 119899119880 = 2 The title of each subfigure is a tuple that includesthe number of transmit antennas required by CQSMGCQSM-UC and GCQSM-PC respectively to achievethe same spectral efficiency The spectral efficiency SE119904119901119886achieved assuming the systemrsquos parameters in Figure 4(a) is6 in 4(b) is 8 in 4(c) is 10 and in 4(d) is 12 bpcu For eachof the system configurations SE119904119894119892 is equal to 4 bpcu leadingto a total spectral efficiency SE = SE119904119901119886 + SE119904119894119892 of 10 12 14and 16 bpcu respectively The range over which the rotationangle is optimized is [0 1205874] The rotation angles appliedto the second signal symbol for the scenarios depictedin Figure 4 are listed in Table 5 The rotation angles areoptimized through Monte Carlo simulations Based on theresults depicted in Figure 4 we make the following remarksin terms of the BER performance the rotation angles andthe number of employed transmit antennas

(i) For 119899119877 = 2 the three techniques achieve the sameBER performance regardless of the number of trans-mit antennas

(ii) For a given value of 119899119877 the BER performanceis degraded as the number of transmit antennasincreases The degradation is relatively small as 119899119877increases This is an expected behavior of SM tech-niques because the size of hypothesis set over whichthe ML detector performs the search increases as 119899119879increases

(iii) In Figure 4(a) GCQSM-PC slightly outperforms theother two techniques for 119899119877 = 4 and 8 As 119899119879 getslarger the probability that the two signal symbols aretransmitted from the same antenna inCQSMreducesAccordingly the performance of CQSM and the twogeneralized algorithms almost coincide for 119899119877 = 4in Figures 4(b) 4(c) and 4(d) For 119899119877 = 8 CQSMslightly outperforms GCQSM-UC and GCQSM-PU

(iv) As shown in Figure 4(c) the generalized techniquesperform close to the CQSM for 119899119877 = 4 In thiscase GCQSM-UC and GCQSM-PC require 17 and 22

Wireless Communications and Mobile Computing 9

10 15 20SNR (dB)

BER

cqsm (8 2)ma-sm (8 2)cqsm (16 2)ma-sm (16 2)

10minus1

10minus2

10minus3

10minus4

(a)

10 15 20SNR (dB)

cqsm (8 2)ma-sm (12 2)cqsm (16 2)ma-sm (24 2)

BER

10minus1

10minus2

10minus3

10minus4

(b)

10 15 20 25SNR (dB)

cqsm (8 2)ma-sm (8 3)cqsm (16 2) ma-sm (16 3)

BER

10minus1

10minus2

10minus3

10minus4

(c)

Figure 3 A comparison between CQSM andMA-SM (a) for the same 119899119879 and different spectral efficiency (b) for the same spectral efficiencyand 119902 and different 119899119879 and finally (c) for the same spectral efficiency and 119899119879 and different 119902

Table 5 Range of optimal rotation angles used to obtain the BER results depicted in Figure 4 assuming QPSK modulation

SE119904119901119886 119899119877 120579CQSM 120579GCQSMminusPC

62 ge25 ge304 ge35 ge418 ge38 ge43

82 ge20 ge214 ge34 ge358 ge38 ge42

102 ge9 ge124 ge30 ge328 ge38 ge39

122 ge9 ge124 ge25 ge308 ge38 ge39

transmit antennas less than CQSM while achievingthe same spectral efficiency

Finally we compare the BER performance of SM GSMQSM CQSM and the proposed generalized CQSM tech-niques We assume that all the schemes achieve the sameES119904119901119886 and ES119904119894119892 leading to the same total spectral efficiencyof ES = (ES119904119901119886 + ES119904119894119892) Also the three generalized schemesuse a combination of two antennas to transmit each signalsymbol Figures 5(a) and 5(b) depict the BER performancefor a spectral efficiency of 16 and 18 bpcu respectively Thenumber of transmit antennas and bits per signal symbol isrepresented as a tuple of the form (119899119879 119902) for each of the

systems The rotation angles applied to CQSM and GCQSM-PC are 20 and 23 degrees respectively to obtain the results inthe two subfigures We make the following two remarks

(i) CQSM GCQSM-PC and GCQSM-UC apply 16-QAM modulation whereas SM GSM and QSM use256-QAM to achieve an ES119904119894119892 of 8 bpcu The numberof transmit antennas required to achieve the spectralefficiency of 16 and 18 bpcu is given in the secondand third row respectively in Table 4 GCQSM-PCrequires the least number of transmit antennas of8 and 10 for the results in Figures 5(a) and 5(b)respectively On the other hand SM requires 256 and

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Wireless Communications and Mobile Computing 9

10 15 20SNR (dB)

BER

cqsm (8 2)ma-sm (8 2)cqsm (16 2)ma-sm (16 2)

10minus1

10minus2

10minus3

10minus4

(a)

10 15 20SNR (dB)

cqsm (8 2)ma-sm (12 2)cqsm (16 2)ma-sm (24 2)

BER

10minus1

10minus2

10minus3

10minus4

(b)

10 15 20 25SNR (dB)

cqsm (8 2)ma-sm (8 3)cqsm (16 2) ma-sm (16 3)

BER

10minus1

10minus2

10minus3

10minus4

(c)

Figure 3 A comparison between CQSM andMA-SM (a) for the same 119899119879 and different spectral efficiency (b) for the same spectral efficiencyand 119902 and different 119899119879 and finally (c) for the same spectral efficiency and 119899119879 and different 119902

Table 5 Range of optimal rotation angles used to obtain the BER results depicted in Figure 4 assuming QPSK modulation

SE119904119901119886 119899119877 120579CQSM 120579GCQSMminusPC

62 ge25 ge304 ge35 ge418 ge38 ge43

82 ge20 ge214 ge34 ge358 ge38 ge42

102 ge9 ge124 ge30 ge328 ge38 ge39

122 ge9 ge124 ge25 ge308 ge38 ge39

transmit antennas less than CQSM while achievingthe same spectral efficiency

Finally we compare the BER performance of SM GSMQSM CQSM and the proposed generalized CQSM tech-niques We assume that all the schemes achieve the sameES119904119901119886 and ES119904119894119892 leading to the same total spectral efficiencyof ES = (ES119904119901119886 + ES119904119894119892) Also the three generalized schemesuse a combination of two antennas to transmit each signalsymbol Figures 5(a) and 5(b) depict the BER performancefor a spectral efficiency of 16 and 18 bpcu respectively Thenumber of transmit antennas and bits per signal symbol isrepresented as a tuple of the form (119899119879 119902) for each of the

systems The rotation angles applied to CQSM and GCQSM-PC are 20 and 23 degrees respectively to obtain the results inthe two subfigures We make the following two remarks

(i) CQSM GCQSM-PC and GCQSM-UC apply 16-QAM modulation whereas SM GSM and QSM use256-QAM to achieve an ES119904119894119892 of 8 bpcu The numberof transmit antennas required to achieve the spectralefficiency of 16 and 18 bpcu is given in the secondand third row respectively in Table 4 GCQSM-PCrequires the least number of transmit antennas of8 and 10 for the results in Figures 5(a) and 5(b)respectively On the other hand SM requires 256 and

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Wireless Communications and Mobile Computing

0 10 20 30SNR (dB)

BER

CQSM

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (8 8 6)

Ｈ２=2

Ｈ２=4

Ｈ２=8

GCQSM-UCGCQSM-PC

(a)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (16 11 8)

CQSMGCQSM-UCGCQSM-PC

(b)

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (32 15 10)

CQSMGCQSM-UCGCQSM-PC

(c)

CQSM

0 10 20 30SNR (dB)

BER

100

10minus1

10minus2

10minus3

10minus4

Ｈ４ = (64 20 13)

GCQSM-UCGCQSM-PC

(d)

Figure 4 BER performance of CQSM GCQSM-UC and GCQSM-PC using QPSK modulation where the spectral efficiency is equal to(a) 10 (b) 12 (c) 14 and (d) 16 bpcu The title of each subfigure is a tuple of the number of antennas required by CQSM GCQSM-UC andGCQSM-PC respectively

1024 antennas respectively to achieve the same SECompared to CQSM GCQSM-PC requires 8 and 22less antennas to achieve the same SE119904119901119886 of 8 and 10bpcu respectively

(ii) For both scenarios GCQSM-UC achieves the bestperformance followed by GCQSM-PC and CQSM InFigure 5(b) GCQSM-UC outperforms GCQSM-PCand CQSM by 1 and 16 dB respectively However

GCQSM-PC requires 5 less transmit antennas thanGCQSM-UCTherefore this slight degradation in theSNR is tolerable in the case of GCQSM-PC given thehigh reduction in the number of transmit antennasused to achieve the same spectral efficiency

Future Work We would like to investigate the case ofgeneralized CQSM where the number of transmit antennasused for transmitting each of the two signal symbols can be

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Wireless Communications and Mobile Computing 11

0 5 10 15 20 25 30SNR (dB)

BER

SM (256 8)GSM (24 8)QSM (16 8)

GCQSM-PC (8 4)GCQSM-UC (11 4)CQSM (16 4)

SE = 16 bpcu

100

10minus1

10minus2

10minus3

10minus4

(a)

0 5 10 15 20 25 30SNR (dB)

BER

SM (1024 8)GSM (46 8)QSM (32 8)

GCQSM-PC (10 4)GCQSM-UC (15 4)CQSM (32 4)

SE = 18 bpcu

100

10minus1

10minus2

10minus3

10minus4

(b)

Figure 5 BER comparison between the generalized CQSM schemes CQSM SM GSM and QSM for spectral efficiency of (a) 16 and (b) 18bpcu CQSM GCQSM-UC and GCQSM-PC use 16-QAM and the remaining schemes use 256-QAM

variable Accordingly the total number of spatial symbols willbe increased leading to higher spectral efficiency

8 Conclusions

In this paper we introduced two generalizations of thecomplex quadrature spatial modulation namely GCQSMwith unique combinations (GCQSM-UC) andwith permutedcombinations (GCQSM-PC) While the former generaliza-tion uses the conventional spatial symbol generation of theGSM scheme the later scheme relies on the fact that thesecond signal symbol is distinguishable from the first throughthe angle rotation This allows expanding the set of antennacombinations that can be used for transmission leading tothe reduction in the number of transmit antennas required toachieve a given spectral efficiencyThe two proposed general-izations perform close to CQSMusing QPSK and outperformit using higher order modulation schemes Also GCQSM-PC requires 10 antennas to achieve a spectral efficiency of10 bcpu per spatial symbol To achieve the same efficiencyGSM requires 46 and CQSM and QSM require 32 transmitantennas

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that there is no conflict of interestregarding the publication of this paper

Acknowledgments

This work was supported by research stipend by Korea Techin the period 2018-2019

References

[1] E Basar M Wen R Mesleh M Di Renzo Y Xiao andH Haas ldquoIndex modulation techniques for next-generationwireless networksrdquo IEEE Access vol 5 pp 16693ndash16746 2017

[2] N Ishikawa S Sugiura and L Hanzo ldquo50 years of permuta-tion spatial and index modulation from classic RF to visiblelight communications and data storagerdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 3 pp 1905ndash1938 2018

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

[4] G Kaddoum M F A Ahmed and Y Nijsure ldquoCode indexmodulation a high data rate and energy efficient communica-tion systemrdquo IEEE Communications Letters vol 19 no 2 pp175ndash178 2015

[5] J Zhang Y Wang J Zhang and L Ding ldquoPolarization shiftkeying (PolarSK) system scheme and performance analysisrdquoIEEE Transactions on Vehicular Technology vol 66 no 11 pp10139ndash10155 2017

[6] E g Basar U Aygolu E Panayirci andH V Poor ldquoOrthogonalfrequency division multiplexing with index modulationrdquo IEEETransactions on Signal Processing vol 61 no 22 pp 5536ndash55492013

[7] X Zhu Z Wang Q Wang and H Haas ldquoVirtual spatialmodulation for mimo systemsrdquo in Proceedings of the IEEE

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Wireless Communications and Mobile Computing

Global Communications Conference (GLOBECOM rsquo16) pp 1ndash6Wash DC USA 2016

[8] J Li M Wen M Zhang and X Cheng ldquoVirtual spatialmodulationrdquo IEEE Access vol 4 pp 6929ndash6938 2016

[9] P Ju M Zhang X Cheng C Wang and L Yang ldquoGeneralizedspatial modulation with transmit antenna grouping for corre-lated channelsrdquo in Proceedings of the IEEE International Con-ference on Communications (ICC rsquo16) pp 1ndash6 Kuala LumpurMalaysia 2016

[10] S AbuTayeh M Alsalahat I Kaddumi Y Alqannas S Althu-nibat and R Mesleh ldquoA half-full transmit-diversity spatialmodulation schemerdquo in Proceedings of the Broadband Com-munications Networks and Systems 9th International EAIConference Broadnets rsquo18 Faro Portugal 2018

[11] A Younis R Mesleh M Di Renzo et al ldquoGeneralized spatialmodulation for large-scale MIMOrdquo in Proceedings of the IEEECommunications Society pp 346ndash350 Eusipco Lisbon Portu-gal 2014

[12] J Wang S Jia and J Song ldquoGeneralised spatial modulationwith multiple active transmit antennas and low complexitydetection schemerdquo IEEE Transactions on Wireless Communica-tions vol 11 no 4 pp 1605ndash1615 2012

[13] T L Narasimhan P Raviteja and A Chockalingam ldquoGen-eralized Spatial Modulation in Large-Scale Multiuser MIMOSystemsrdquo IEEE Transactions on Wireless Communications vol14 no 7 pp 3764ndash3779 2015

[14] R Mesleh S S Ikki and H M Aggoune ldquoQuadrature spatialmodulationrdquo IEEE Transactions on Vehicular Technology vol64 no 6 pp 2738ndash2742 2015

[15] F R Castillo-Soria J Cortez-Gonzalez R Ramirez-GutierrezF M Maciel-Barboza and L Soriano-Equigua ldquoGeneralizedquadrature spatial modulation scheme using antenna group-ingrdquo ETRI Journal vol 39 no 5 pp 707ndash717 2017

[16] RMesleh O Hiari and A Younis ldquoGeneralized space modula-tion techniques hardware design and considerationsrdquo PhysicalCommunication vol 26 pp 87ndash95 2018

[17] M Mohaisen and S Lee ldquoComplex quadrature spatial modula-tionrdquo ETRI Journal vol 39 no 4 pp 514ndash524 2017

[18] M Mohaisen ldquoIncreasing the minimum Euclidean distance ofthe complex quadrature spatial modulationrdquo IET Communica-tions vol 12 no 7 pp 854ndash860 2018

[19] A Iqbal M Mohaisen and K S Kwak ldquoModulation set opti-mization for the improved complex quadrature SMrdquo WirelessCommunications and Mobile Computing vol 2018 Article ID6769484 12 pages 2018

[20] M de Berg O Cheong M van Kreveld and M OvermarsComputational Geometry Algorithms and Applications vol 45Springer Berlin Germany 2008

[21] M K Simon and M-S Alouini Digital Communication overFading Channels John Wiley amp Sons Inc Hoboken NJ USA2nd edition 2004

[22] M Di Renzo and H Haas ldquoBit error probability of SM-MIMO over generalized fading channelsrdquo IEEE Transactions onVehicular Technology vol 61 no 3 pp 1124ndash1144 2012

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom