3034 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 5, MAY 2020

Joint Impact of I/Q Imbalance and Imperfect CSIon SM-MIMO Systems Over Generalized

Beckmann Fading Channels: OptimalDetection and Cramer-Rao Bound

Ayse Elif Canbilen , Salama Said Ikki , Senior Member, IEEE, Ertugrul Basar , Senior Member, IEEE,

Seyfettin Sinan Gultekin, and Ibrahim Develi

Abstract— Spatial modulation (SM) has been shown to bea promising low-complexity alternative to the state-of-artmultiple-input multiple-output (MIMO) schemes due to itsnovel transmission approach. This paper investigates the per-formance of SM-MIMO systems in the presence of two practicalundesirable effects, namely in-phase (I) and quadrature-phase(Q) imbalance (IQI) and imperfect channel state information(ICSI). An optimum maximum likelihood detection (MLD)method is proposed to tackle the effects of self-interference andsignal distortion caused by IQI impairment by adapting thetraditional MLD technique in accordance with the asymmetriccharacteristics of the IQI. More particularly, upper-bounds ofthe closed-form average pairwise error probability (APEP) andthe average bit error rate (ABER) are derived for generalizedBeckmann fading channels. As erroneously interpreted channelcoefficients at the receiver (Rx) cause the error rate to increaseand the detection to fall short, Cramer-Rao bound, which is alower bound on the variance of the channel estimator, is utilizedto assess the estimation accuracy. The system performance isevaluated by analytical derivations that are corroborated withcomputer simulations. The obtained results show that ICSI andIQI should be seriously considered while designing the futureSM-based wireless communication systems.

Index Terms— Cramer-Rao bound, generalized Beckmann fad-ing, I/Q imbalance, optimal ML detection, spatial modulation.

Manuscript received May 14, 2019; revised September 12, 2019 andDecember 9, 2019; accepted January 13, 2020. Date of publication February 6,2020; date of current version May 8, 2020. This work was supported by theScientific and Technological Research Council of Turkey (TUBITAK) BIDEB-2214 International Doctoral Research Fellowship Programme. The work ofErtugrul Basar was supported by the Scientific and Technological ResearchCouncil of Turkey (TUBITAK) under Grant 218E035. The associate editorcoordinating the review of this article and approving it for publication wasR. He. (Corresponding author: Ayse Elif Canbilen.)

Ayse Elif Canbilen and Seyfettin Sinan Gultekin are with the Elec-trical and Electronics Engineering Department, Konya Technical Univer-sity, 42250 Konya, Turkey (e-mail: aecanbilen@ktun.edu.tr; ssgultekin@ktun.edu.tr).

Salama Said Ikki is with the Department of Electrical Engineer-ing, Lakehead University, Thunder Bay, ON P7B 5E1, Canada (e-mail:sikki@lakeheadu.ca).

Ertugrul Basar is with the Communications Research and Innovation Lab-oratory (CORELAB), Department of Electrical and Electronics Engineering,Koç University, 34450, Istanbul, Turkey (e-mail: ebasar@ku.edu.tr).

Ibrahim Develi is with the Department of Electrical and Electron-ics Engineering, Erciyes University, 38039 Kayseri, Turkey (e-mail:develi@erciyes.edu.tr).

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2020.2970002

I. INTRODUCTION

THE 5th generation (5G) and beyond of wireless networksare expected not only to provide enormous bandwidth

and ultrahigh data rates with considerably lower latency (lessthan 1 millisecond), but also enable a variety of new applica-tions such as Internet of Things (IoT), humanoid robots andconnected autonomous cars [1]. The report of the InternationalTelecommunications Union (ITU) on the minimum require-ments for International Mobile Telecommunications-2020(IMT-2020) specifies a downlink peak data rate of 20 Gbpsand a bandwith of at least 100 MHz [2]. Hence, novel physicallayer (PHY) techniques are required to meet the demandsof 5G and beyond. Although some effective PHY solutionssuch as millimeterwave (mmWave) communications, massivemultiple-input multiple-output (MIMO) systems and flexiblewaveform designs have already been proposed, researchersare still interested in exploring innovative ways to conveyinformation instead of traditional methods [3].

Index modulation (IM) techniques, which utilize the indicesof the building blocks (e.g., transmit antennas, subcarriers,radio frequency (RF) mirrors, transmit light emitting diodes(LEDs), and so on) of the corresponding communicationsystems to transmit additional information bits, seem to becompetitive candidates for next-generation wireless networksdue to the advantages they promise in terms of spectral andenergy efficiency as well as hardware simplicity [3], [4].Introducing spatial modulation (SM), a member of the IMfamily, hence opened a door for alternative radical digitalmodulation schemes.

SM has been proposed as a novel transmission techniqueand uses transmitter (Tx) antenna indices as extra informationsources for low-complexity and power-efficient applicationsof MIMO wireless systems [5]. In particular, it has beenshown that SM is an excellent solution for main drawbacksof MIMO systems, including inter-channel interference (ICI),inter-antenna synchronization, and multiple RF chains at theTx [6]. The basic idea of SM is to map a block of informationbits into two units: i) a symbol chosen from a complexsignal constellation diagram, and ii) a unique Tx-antennaindex chosen from the set of a Tx-antenna array. This sim-ple yet effective encoding mechanism provides a one-to-one

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CANBILEN et al.: JOINT IMPACT OF I/Q IMBALANCE AND IMPERFECT CSI ON SM-MIMO SYSTEMS 3035

mapping between blocks of information bits to be transmittedand the spatial positions of the Tx-antennas in the antennaarray [6], [7].

Hence, SM takes advantage of the location-specific charac-teristic of the wireless channel, i.e., the uniqueness of eachTx-Rx wireless link, for transmission [8]. It is known that SMcan achieve better performance than traditional modulationschemes when the Rx has perfect channel state information(CSI) [9], [10]. A comprehensive survey on the SM systemdesign and history was presented in [11] and [12]. AlthoughSM is a promising technique for next-generation communi-cation systems [13], [14], research in this field is still in itsinfancy and fundamental issues still need to be addressed toassess the possible applications of this technology in practicalscenarios.

A. Related Works

The performance of SM-based MIMO systems has beenextensively studied in the literature over Rayleigh, Rician,and Nakagami fading channels [15]–[17]. A comprehensiveframework for the analysis of SM-MIMO over generalizedfading channels was proposed in [18]. An analytical approachto the performance of SM and its simplified version, namelyspace shift keying (SSK), over Rayleigh flat fading channels inthe presence of channel estimation error was developed in [19]and [20], respectively. An exact and closed-form expression ofthe average pairwise error probability (APEP) of SM and SSKover generalized α − μ and κ − μ fading channels under theeffect of imperfect CSI (ICSI) was derived in [21] and [22],respectively.

Although all aforementioned works assume ideal trans-ceiver hardware, physical transceivers are affected by hardwareimpairments, such as phase noise and I/Q imbalance (IQI),that create distortions which degrade the performance of thecommunication systems [23]. For IoT and wearable devices,the IQI will be even more influential due to the use ofcost-constrained analog components, such as mixers and phaseshifters in their RF front-ends [24]. A dual-hop amplify-and-forward (AF) relaying system in the presence of IQI wasanalyzed and the destructive effects of IQI were addressed viamaximum likelihood (ML) detection-based methods in [25].The performance of underlay cognitive radio networks, whichconsider the effect of IQI, was studied in [26], and it wasshown that the distorting effects of IQI is maximized in thecase of ICSI.

On the other hand, SM becomes more attractive to be usedagainst IQI effects as it outperforms single-input multiple-output (SIMO) systems with IQI [24]. Considering this,the effect of IQI on SM-based MIMO transmission wasdiscussed in [27], and it was observed that both amplitudeand phase IQI cause performance degradation for SM-basedtransmission. An optimal ML detector (MLD) was proposedand compared with a non-optimal Rx over Rayleigh fadingchannels in [28]. The joint impact of hardware imperfectionsand co-channel interference (CCI) on SSK-MIMO systemswas investigated, and a closed-form expression of averageBER (ABER) was derived for Rayleigh channels in [29].

The performance of pilot-aided estimation and data detec-tion for MIMO systems was firstly investigated in [30], andCramer-Rao lower bound (CRLB), which is a performancecriterion that gives a lower bound to the mean square errorof estimation in the set of unbiased estimates, was comparedfor different pilot insertion schemes. Then, the principle ofCRLB was adapted for SM-based systems, and in order toassess the channel estimation accuracy, CRLB was computedfor cognitive SM-MIMO systems with channel imperfectionsover Rayleigh fading channels in [31].

B. Motivation

It is an undeniable fact that estimation errors have destruc-tive effects on the performance of wireless communicationsystems, including SM-based models [32]. Whether or not SMis more sensitive to channel imperfections than conventionalsystems has been a topic frequently discussed, since correctreception depends entirely on the uniqueness of channel coef-ficients and wrong estimation of CSI at Rx causes a BERincrement [33], [34]. The studies in literature of SM-MIMOthat considers ICSI either did not take into account the effectof any hardware impairments [21], [22], or only realized overRayleigh fading channels [20], [31], [33], [34].

The generalized Beckmann distribution is a versatile multi-path fading model that includes Rayleigh, Ricean, Beckmann,Hoyt (also known as Nakagami-q), single-sided Gaussian,κ−μ and Beaulieu-Xie distributions as special cases [35], [36].Unlike other state-of-the-art envelope fading models,the impact of imbalances in the line-of-sight (LOS) andnon-LOS (NLOS) components is considered simultaneouslyin generalized Beckmann fading [37]. Hence, this modeleffectively captures the correlation between the amplitudeand phase [37]. It has a wide range of applications in RFcommunications, underwater wireless optical communicationsand free-space optical (FSO) communications [36], [38], [39].However, the performance analyses involving Beckmanndistribution are challenging since deriving the closed-formprobability density function (PDF) and cumulative densityfunction (CDF) of Beckmann distribution with unequalvariances of the I/Q components is still an open problem [40].The performance of SM-MIMO wireless communicationsystems has not been studied over generalized Beckmannfading channels yet, either.

IQI, as another practical key parameter that affects the sys-tem performance, has not been investigated comprehensivelyin the studies on SM-based MIMO communication systems.Only Rx-side IQI was introduced in [27] and [28], while onlyTx-side IQI was considered in [24], assuming perfect CSI forSM wireless communication systems. However, as some ofthe information bits are transmitted with the active antennaindices, both Tx- and Rx-side IQI have destructive effects onthe detection procedure of SM [41]. Hence, the considerationof the IQI effect only at Tx/Rx side is not enough to interpretthis impairment for SM.

The lack of studies on SM-MIMO systems under the jointimpact of transceiver IQI and ICSI motivates our work. Thisstudy fills in the gap by analytically evaluating the error

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3036 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 5, MAY 2020

Fig. 1. The scheme of SM-MIMO wireless communication system employing a synchrodyne architecture transceiver under the effect of IQI.

performance of SM-MIMO systems while jointly taking intoaccount those critical factors. Moreover, utilizing CRLB hasprovided correct estimation of CSI, and analyzing the systemperformance over generalized Beckmann fading channels hasshed light on studies in this field.

C. Contributions

Motivated by the aforementioned factors, the contributionsof this paper can be summarized as follows:

• This study investigates the error performance ofSM-MIMO wireless communication systems over gen-eralized Beckmann fading channels.

• The APEP performance of SM-MIMO wireless commu-nication systems has been presented under the joint effectof IQI and channel impairments. The results are obtainedvia analytical derivations and computer simulations.

• The open problem of deriving a closed-form to theAPEP over generalized Beckmann fading channels forSM-MIMO communication systems has been addressedwith an upper-bound by an optimal MLD solution. Addi-tionally, an asymptotic analysis has been conducted to getinsightful findings on the impact of key parameters.

• Analytical expressions of CRLB that is specific forSM-MIMO systems have been derived for predicting andevaluating channel estimation accuracy in the presenceof IQI.

Organization: The system model of SM-MIMO underthe joint effect of transceiver IQI and ICSI is presentedin Section II, and the optimal MLD method is given inSection III. Average and asymptotic error probability arepresented in Section IV, while numerical results are providedin Section V. Finally, the study is concluded in Section VI.

Notation: Bold lower and upper case letters denote vectorsand matrices, respectively. (·)T and (·)−1 indicate matrixtranspose and matrix inverse operations. N (μ, σ2) representsreal Gaussian distribution with mean μ and variance σ2. (·)I

and (·)Q denote the I and Q components. (·)∗ is the complexconjugate and (·)H denotes the transpose conjugate, whileE{·} denoting the expectation. �{z} stands for the real partof the complex variable z, and Pr{·} denotes the probabilityof an event.

II. SYSTEM AND CHANNEL MODELS

The scheme of the utilized SM-MIMO wireless commu-nication system, which employs a synchrodyne architecturetransceiver in the RF front-end, is given in Fig. 1. The

incoming data bit stream q is divided into blocks containingm = log2(NtM) bits each (M is the modulation order ofcomplex constellations and Nt is the number of Tx antennas).These blocks are further partitioned into two sub-blocks toidentify the spatial and the conventional modulation bits atTx side. Spatial bits determine a Tx antenna to switch onfor transmission and RF switch block assigns the modulatedsignal to the activated Tx antenna. The signal passes throughan Nr × Nt sized generalized Beckmann fading channel H(Nr is the number of Rx antennas), whose columns representthe complex fading gain coefficients between the ith Txantenna and the Rx antennas as vectors, i.e., hi = hI

i + jhQi

(i = 1, . . . , Nt).The transmitted signal also experiences additive white

Gaussian noise, n = nI + jnQ, which has zero-mean inde-pendent and identical I and Q parts, i.e., σ2

nI = σ2nQ = σ2

n/2.The incoming signal to the Rx antenna is first amplified by alow-noise amplifier (LNA), and quadrature mixing is appliedby two local oscillator (LO) signals. Afterwards, the signalon each branch is passed through a low pass filter (LPF),a power amplifier (PA) and an analog to digital converter(ADC), respectively. Then, the I and Q parts of the basebandsignal are combined, and SM detector decides which symbolhas been sent and which Tx antenna has been used.

A. Generalized Beckmann Fading Model

Generalized expressions of the envelope and phase dis-tributions for two jointly correlated Gaussian random vari-ables (RVs) are of interest in various areas of communications[42]. Thus, the real and imaginary parts of the entries of hi areassumed to follow N (μI , σ

2I ), and N (μQ, σ2

Q) distributions,respectively, based on the generalized Beckmann distribution,which considers the envelope of correlated Gaussian RVswith arbitrary means and non-identical variances [36].1 Thecorrelation coefficient between hI

l,i and hQl,i, which repre-

sents the I/Q parts of the corresponding channel coefficientsbetween ith Tx antenna and lth Rx antenna, can be calculatedfrom:

ρh =E{(hI

l,i − μI)(hQl,i − μQ)}

σIσQ. (1)

Note that this extensive channel model turns into the clas-sical Beckmann envelope fading for ρh = 0, and the PDF is

1A generalized Beckmann RV was defined as the square root of the quadraticsum of N correlated Gaussian RVs in [36]. In this study, the multipath fadingchannel model is defined by assuming N = 2.

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CANBILEN et al.: JOINT IMPACT OF I/Q IMBALANCE AND IMPERFECT CSI ON SM-MIMO SYSTEMS 3037

written as given in (2), shown at the bottom of this page, [42].Generalized Beckmann distribution is characterized by fourparameters2:

q2B � σ2

I

σ2Q

, r2B � μ2

I

μ2Q

, KB �μ2

I + μ2Q

σ2I + σ2

Q

, (3)

ΩB � μ2I + μ2

Q + σ2I + σ2

Q. (4)

Here, the parameters q2B and r2

B indicate the power imbalancebetween the I and Q parts of NLOS and LOS components,respectively. KB denotes the ratio between the LOS andNLOS power, and ΩB states the average received power. Theconnection between the generalized Beckmann distributionand the special cases included therein can be found in [37].

B. I/Q Imbalance Model

The I/Q mismatches mainly stem from the limited accuracyof the analog components, such as capacitors and resis-tors [43]. On the Tx side, this impairment can occur atthe I/Q up-conversion step beside filters and digital-to-analogconverters (DAC). On the Rx side, it can emerge at thedown-conversion step as well as amplification and samplingstages. For simplicity, all mismatches are referred to theI/Q up- and down-conversion steps, which is typical in theliterature in this context [43]–[45]. Considering both Tx andRx IQI effects, the corresponding complex LO signals for Txand Rx sides can be written as follows, respectively [43]:

zT (t) = cos(ωLt) + jξT sin(ωLt + οT )= G1e

jωLt + G2e−jωLt, (5)

zR(t) = cos(ωLt) − jξR sin(ωLt + οR)= K1e

−jωLt + K2ejωLt, (6)

where ωL = 2πfL, while fL is the LO frequency; {ξT , οT }and {ξR, οR} denote the total effective amplitude and phaseimbalances of the Tx and Rx sides, respectively. IQI parame-ters of Tx side (G1, G2) and Rx side (K1, K2) are given inthe form of:

G1 =12(1 + ξT ejφT ), G2 =

12(1 − ξT e−jφT ), (7)

K1 =12(1 + ξRe−jφR), K2 =

12(1 − ξRejφR). (8)

Note that for perfect I/Q matching, IQI parameters reduceto ξT = ξR = 1 and οT = οR = 0◦. Therefore, in this case,it is clear that G1 = K1 = 1 and G2 = K2 = 0.

The effects of IQI can be modelled either symmetrical orasymmetrical. In the symmetrical method, each arm (I andQ) experiences half of the phase and/or amplitude errors,while the I branch is assumed to be ideal and the errors are

2In literature, these parameters are given as q,r,K and Ω. In order to avoidconfusion with the other parameters in the paper, (·)B is added here to definegeneralized Beckmann parameters.

modelled in the Q branch in the asymmetrical method [46].It can be easily verified that these two methods are equivalent.The asymmetrical IQI structure is considered for the furtheranalyses in this study. In this case, the baseband representationof the up-converted signal in the presence of Tx-side IQI isgiven as:

xq = G1xq + G∗2x

∗q . (9)

Here, it is assumed that M -QAM modulation scheme isutilized, and xq is the modulated baseband Tx signal underperfect I/Q matching, where q ∈ {1, . . . , M}.

Considering the frequency independent IQI model in [47],the received baseband signal at lth Rx antenna is given asfollows:

yl = K1(√

Ehl,ixq + nl) + K2(√

Ehl,ixq + nl)∗, (10)

where l = {1, . . . , Nr} and E is the transmitted signal energy.(10) can be extended to the following by considering (9):

yl =√

E(K1G1hl,i + K2G2h∗l,i)xq + K1nl

+√

E(K2G∗1h

∗l,i + K1G

∗2hl,i)x∗

q + K2n∗l . (11)

As it is seen in (11), IQI causes a self-interferenceeffect,

√E(K2G

∗1h

∗l,i + K2G

∗2hl,i)x∗

q , beside distorting thesignal. In order to clarify the destructive effect of this self-interference, assuming that symbols have unit-energy, averagesignal-to-interference ratio (SIR) can be calculated by using(11) as:

SIR =E{|√E(G1K1hl,i + G2K2h

∗l,i)|2}

E{|√E(G∗1K2h∗

l,i + G∗2K1hl,i)|2}

=|G1K1|2 + |G2K2|2 + 2�{G1G

∗2K1K

∗2}(σ2

I − σ2Q)

|G2K1|2 + |G1K2|2 + 2�{G1G∗2K1K∗

2}(σ2I − σ2

Q).

(12)

In the case of perfect I/Q matching, SIR has the ideal valueof SIR = ∞ as seen in Fig. 2. On the contrary, even very smallvalues of imbalance in gain, such as ξT = ξR = 0.05, or phase,such as οT = οR = 2◦, cause degradation of the SIR value.Thus, IQI needs to be considered as an important drawbackof SM-MIMO wireless communication systems. On the otherside, it is clear from (12) that the relationship between thevariances of the I and Q components of the channel hasan effect on the SIR value. Considering Fig. 2, being theparameter q2

B , which gives the ratio between the variancesof the I/Q parts of the channel, inversely proportional to theSIR value clarifies this effect.

C. Imperfect CSI Model

In practical MIMO systems, hl,i is not known at the Rxside and a channel estimation algorithm is used to obtain the

f|hl,i|(hl,i) =hl,i

2πσIσQ

�1 − ρ2

h

� 2π

0

exp�− 1

2(1 − ρ2h)

��

hIl,i−μI

σI

�2

−�

hQl,i−μQ

σQ

�2

− 2ρh(hIl,i−μI)(h

Ql,i − μQ)

σIσQ

�dθ (2)

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3038 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 5, MAY 2020

Fig. 2. SIR values given for IQI affected SM-MIMO system with fixed valuesof phase imbalance (first column) and fixed values of amplitude imbalance(second column).

estimated channel matrix H [48]. It is assumed that the gener-alized Beckmann channel components, hl,i, and the estimatedversions of them, hl,i, are jointly ergodic and stationary.

Assuming that the actual channel and the estimation errorare orthogonal as well, a common estimation rule is consid-ered, and given as [31], [49], [50]:

hl,i = hl,i − el, (13)

where el denotes the channel estimation error, which has cor-related I and Q parts modeled as N 0, σ2

eI

�, and N 0, σ2

eQ

�respectively. Here, it is worth noting that σ2

e = E[|el|2] =σ2

eI + σ2eQ indicates the quality of the estimation and can be

chosen depending on the channel dynamics and estimationmethods [48], [51]. In this study, CRB method, which isdiscussed in Section-D, has been utilized for estimation ofσ2

eI and σ2eQ .

Considering (13), it can be seen that the estimated channelcoefficients have correlated I and Q parts with:

ρh =E{(hI

l,i − μI)(hQl,i − μQ)} + E{eI

l eQl }�

(σ2I + σ2

eI )(σ2Q + σ2

eQ). (14)

(10) can be rewritten by using (13) as follows:

y =√

E[K1(hi − e)xq + K2(hi − e)∗x∗q ] + K1n + K2n

∗

=√

Eχp + n. (15)

Here χp = K1hixq +K2h∗i x

∗q , p ∈ {1, . . . , m}, and the noise

part of y equals to n = −√EK1exq +

√EK2e

∗x∗q +K1n+

K2n∗.

At this point, note that if a complex Gaussian RV isnot circularly-symmetric, it is referred to as improper [52].Improper characteristics of a complex Gaussian RV is basedon the existence of at least one of these two conditions:

i) real and imaginary parts are correlated, ii) real and imag-inary parts do not have the same variance (non-identical).Hence, it is seen that n in (15) has correlated I/Q components,since cov(nI

l , nQl ) �= 0, and the correlation coefficient can be

calculated form [28]:

ρn =cov(nI

l , nQl )

σnIlσnQ

l

. (16)

Although just having correlated components is enough toremark that n is an improper Gaussian noise, the secondcondition, being non-identical, is valid for it as well, sincethe variance values of I and Q parts of n are calculated asfollows:

σ2nI

l= E[|xI

q |2σ2eI + |xQ

q |2σ2eQ − 2E{eI

l eQl }xI

q xQq ] +

σ2n

2,

σ2nQ

l

= E[σ2eI |Kcx

Iq + Kdx

Qq |2 + σ2

eQ |KdxIq − Kcx

Qq |2

+2E{eIl e

Ql }(Kcx

Iq +Kdx

Qq )(Kdx

Iq−Kcx

Qq )]+

ξ2Rσ2

n

2.

(17)

where Kc = KQ1 + KQ

2 and Kd = KI1 − KI

2 . Therefore,the variances of I and Q parts of n are non-identical. Here,it is worth nothing that the traditional MLD can not providean optimum solution since it has been derived by assumingproper Gaussain noise characteristics.

D. Cramer-Rao Lower Bound (CRLB) of SM-MIMOWith IQI and ICSI

The CRLB is a performance criterion that gives a lowerbound to the variance of estimation error in a set of unbiasedestimates [30]. In this section, pilot symbols insertion isintroduced to perform channel estimation for coherent detec-tion and an exact CRLB expression, which is used not onlyto evaluate but also to predict the estimator performance,is derived.

The exact Fisher information matrix (FIM) of the channelestimation that measures the sensitivity of the channel modelto changes of hi [53], is given as [30]:

I(hi)(hi) = E

�∂ ln Py(y; hi)

∂h∗i

��∂ ln Py(y; hi)

∂h∗i

�H�

=

�I(hI

i ) I(hIi ,hQ

i )

I(hQi ,hI

i ) I(hQi )

�, (18)

where Py(y; hi) is the likelihood function of the unknownchannel hi, which is also equal to the joint PDF ofNp obsevations based on an observation vector y =[y(1) y(2) . . . y(Np)]T , and I(hI

i ,hQi ) = I(hQ

i ,hIi ). In this

expression, I(hIi ), I(hQ

i ) and I(hIi ,hQ

i ) values are calculated asfollows for the proposed SM model:

I(hIi ) =

NpE

1 − ρ2η

�(xI

q)2

σ2ηI

+(xI

qKc + xQq Kd)2

σ2ηQ

−2ρηxIq(x

IqKc + xQ

q Kd)σηI σηQ

�, (19)

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CANBILEN et al.: JOINT IMPACT OF I/Q IMBALANCE AND IMPERFECT CSI ON SM-MIMO SYSTEMS 3039

I(hQi ) =

NpE

1 − ρ2η

�(xQ

q )2

σ2ηI

+(xI

qKd − xQq Kc)2

σ2ηQ

−2ρηxQq (xI

qKd − xQq Kc)

σηI σηQ

�, (20)

I(hIi ,hQ

i ) =NpE

1 − ρ2η

�−xIq x

Qq

σ2ηI

+(xI

qKd − xQq Kc)(xI

qKc + xQq Kd)

σ2ηQ

+ρη(xQ

q (xIqKc + xQ

q Kd)+xIq(x

IqKd−xQ

q Kc))σηI σηQ

�,

(21)

where NP is the number of pilot symbols, and ρη = − sinοR.Details on the mathematical background to obtain these valuesare given in Appendix-A.

CRLB matrix can be obtained by utilizing (18), and itprovides a lower bound for the covariance matrix of theestimation error, which includes the variance values of thereal and imaginary parts, as follows:

CRLBi = [I(hi)(hi)]−1i =

�σ2

eI E{eIeQ}E{eIeQ} σ2

eQ

�. (22)

III. OPTIMAL MAXIMUM LIKELIHOOD DETECTOR DESIGN

In this section, an optimal MLD design is introducedfor the presented SM-MIMO wireless communication systemoperating under the effects of IQI and ICSI. The destructiveeffects of these impairments are modeled as improper Gaussianas mentioned in Section II-C, and this allows us to analyze theasymmetric characteristics of the IQI accurately [54]. Since itis proven that utilizing the improperness of the Gaussian noiseusing MLD method leads to a decrease in the error probability[55], the optimal MLD scheme is designed by consideringthis phenomenon. Additionally, it is assumed that the IQIparameters are known at the Rx-side during the formulation.

The joint multi-variate PDF of the real part and the imagi-nary part of the received signal, y, which is under the effectof both Tx- and Rx-side IQI as well as ICSI, can be expressedby using (15) as follows [28]:

fyI ,yQ(yI , yQ|χp)

=�

12πσnI

lσnQ

l

�1 − ρ2

n

�Nr

× exp

⎛⎝ −1

2(1 − ρ2n)

⎡⎣�yI−√

EχIp�2

σ2nI

l

+�yQ −√

EχQp �2

σ2nQ

l

−2ρn(yI −√EχI

p)T (yQ −√EχQ

p )σnI

lσnQ

l

��. (23)

Hence, the optimal MLD decision rule, assuming equiprob-able symbols, can be defined by:

p = arg maxp

{fyI ,yQ(yI , yQ|χp)}

= arg minp

⎧⎨⎩�yI −√

EχIp�2

σ2nI

l

+�yQ −√

EχQp �2

σ2nQ

l

−2ρn(yI −√EχI

p)T (yQ −√

EχQp )

σnIlσnQ

l

�, (24)

which jointly counts the error in Tx antenna indices andsymbols.

IV. ERROR PERFORMANCE ANALYSIS

A. Average Pairwise Error Probability (APEP)

Considering the optimal MLD rule given in (24), the con-ditional pairwise error probability (PEP) can be calculatedfrom (25) under the assumption that χp has been transmitted;however, χp is detected erroneously at the Rx side:

Pr (χp → χp|H�

= Pr

⎛⎝�yI −√

EχIp�2

σ2nI

l

+�yQ −√

EχQp �2

σ2nQ

l

−2ρn(yI −√EχI

p)T (yQ −√

EχQp )

σnIlσnQ

l

>�yI −√

EχIp�2

σ2nI

l

+�yQ −√

EχQp �2

σ2nQ

l

− 2ρn(yI−√EχI

p)T (yQ−√

EχQp )

σnIlσnQ

l

⎞⎠.

(25)

Here, χp = K1ˆhi

ˆxq +K2ˆh∗iˆx∗

q and ˆhi is the complex channel

coefficient related to detected Tx antenna that transmits thedetected signal ˆxq . After some algebraic simplifications, it isobtained that:

PEP = Pr

�2ρn

√E{(χI

p − χIp)T nQ

l + (χQp − χQ

p )T nIl }

σnIlσnQ

l

−2√

E(χIp − χI

p)T nIl

σ2nI

l

− 2√

E(χQp − χQ

p )T nQl

σ2nQ

l

−E�χIp − χI

p�2

σ2nI

l

− E�χQp − χQ

p �2

σ2nQ

l

+2ρnE

χI

p − χIp

�T (χQp − χQ

p )σnI

lσnQ

l

> 0

�

= Pr(G > 0), (26)

where G, conditioned on χp, is a Gaussian RV. Now, PEP canbe defined by using the mean, μG , and variance, σ2

G , valuesof the G, and the well-known Q-function3 as:

PEP = Q

�E{G}�var(G)

�= Q

�μGσG

�. (27)

Here, μG and σ2G are obtained respectively as follows:

μG =

⎛⎝E�χI

p − χIp�2

σ2nI

l

+E�χQ

p − χQp �2

σ2nQ

l

−2ρnEχI

p − χIp

�T (χQp − χQ

p )σnI

lσnQ

l

�, (28)

σ2G = 4E(1 − ρ2

n) ×⎛⎝�χI

p − χIp�2

σ2nI

l

+�χQ

p − χQp �2

σ2nQ

l

−2ρn

χI

p − χIp

�T (χQp − χQ

p )σnI

lσnQ

l

�. (29)

3Q(x) is defined as Q(x) = 1√2π

�∞x e−t2/2dt.

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3040 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 5, MAY 2020

Substituting (28) and (29) in (27), the conditional PEP canbe given as in (30), shown at the bottom of this page. Althoughthe average PEP (APEP) can be defined by using the expectedvalue of the PEP expression of (30) as given in (31), it is notrelatively easy to find fγ(γ), which is the PDF of γ, in thefollowing expression:

APEP = E

Q

��Eγ

4(1 − ρ2n)

��

=

∞�−∞

Q

��Eγ

4(1 − ρ2n)

�fγ(γ)dγ. (31)

Therefore, the moment generating function4 (MGF),which provides an alternative solution instead of using theQ-function, is utilized to find APEP in this study. In order todefine the MGF of γ, (30) can be simplified as:

PEP = Q

⎛⎜⎝� ! E

4(1−ρ2n)

⎛⎝�ζ1�2

σ2nI

l

+�ζ2�2

σ2nQ

l

− 2ρnζT1 ζ2

σnIlσnQ

l

⎞⎠⎞⎟⎠ ,

(32)

where γ = �ζ1�2

σ2nI

l

+ �ζ2�2

σ2n

Ql

− 2ρnζT1 ζ2

σnI

lσ

nQl

, ζ1 = (χIp − χI

p) and

ζ2 = (χQp − χQ

p ), while ζ1 and ζ2 follow N (μ1, υ1) andN (μ2, υ2), respectively. Here, it should be noted that if thevalue of error variance is adjusted as proportional to 1/E in(17), the effect of error variance on the PEP in (32) would becancelled. In other words, the denominator in (32) decreasesin this case, since the transmitted energy increases, therebydecreasing the error probability.

The mean values of ζ1 and ζ2 are calculated as:

μ1 = μI(xIq − ˆxI

q) + μQ(ˆxQq − xQ

q ),μ2 = μI(Kcx

Iq + Kdx

Qq − Kc

ˆxIq − Kd

ˆxQq )

+μQ(KdxIq − Kcx

Qq − Kd

ˆxIq + Kc

ˆxQq ). (33)

Representing i as the index of the detected Tx antenna, whichis used to convey the modulated signal, the variance values υ1

and υ2 are obtained depending on two cases:Case 1: i = i

υ1 = (σ2I + σ2

eI )|xIq − ˆxI

q |2 + (σ2Q + σ2

eQ )|ˆxQq − xQ

q |2+2E{hI

l,ihQl,i}(xI

q − ˆxIq)(ˆx

Qq − xQ

q )

4Assuming that γ equals to the SNR of any γ, MGF of γ, stated byM, can be utilized as an alternative representation of Q-function as follows:�∞−∞ Q(γγ)fγ (γ)dγ = 1

π

� π2

0 M(− γ2 sin2 θ

)dθ.

υ2 = (σ2I + σ2

eI )|Kc(xIq − ˆxI

q) + Kd(xQq − ˆxQ

q )|2+(σ2

Q + σ2eQ)|Kd(xI

q − ˆxIq) + Kc(ˆxQ

q − xQq )|2

+2E{hIl,ih

Ql,i} × [Kc(xI

q − ˆxIq) + Kd(xQ

q − ˆxQq )]

×[Kd(xIq − ˆxI

q) + Kc(ˆxQq − xQ

q )] (34)

Case 2: i �= i

υ1 = (σ2I + σ2

eI )(|xIq |2+|ˆxI

q |2)+(σ2Q + σ2

eQ)(|ˆxQq |2 + |xQ

q |2)−2E{hI

l,ihQl,i}(xI

q xQq + ˆxI

qˆxQ

q )

υ2 = (σ2I + σ2

eI )(|KcxIq + Kdx

Qq |2 + |Kc

ˆxIq + Kd

ˆxQq |2)

+(σ2Q + σ2

eQ )(|KdxIq − Kcx

Qq |2 + |Kd

ˆxIq − Kc

ˆxQq |2)

+2E{hIl,ih

Ql,i} × [(Kcx

Iq + Kdx

Qq )(Kdx

Iq − Kcx

Qq )

+(KcˆxI

q +KdˆxQ

q )(KdˆxI

q−KcˆxQ

q )]. (35)

Noting that γ is a linear combination of two correlatednoncentral chi-squared RVs with one degree of freedom, γ canbe written in a quadratic form of Gaussian RVs, which isgiven in more detail in Appendix B. Thus, the MGF ofγ is given as follows under the assumption of independentchannels [56]:

Mγ(t) =

⎛⎝exp

#b21λ1t

1−2λ1t

�√

1 − 2λ1t×

exp#

b22λ2t1−2λ2t

�√

1 − 2λ2t

⎞⎠

Nr

, (36)

where b1 and b2 are the mean values and λ1 and λ2 are theeigenvalues all calculated based on the covariance matrix ofζ1 and ζ2, and given in Appendix B.

In light of this information, the APEP can be calculatedanalytically by using (36) from:

APEP = Pr (χp → χp)=1π

� π2

0

Mγ

� −E

8(1 − ρ2n) sin2 θ

�dθ.

(37)

This integral can be easily solved via numerical methods.Furthermore, considering the maximum value of sin2 θ = 1[57, p. 230], this integration can be upper-bounded as:

APEP ≤ 12Mγ

� −E

8(1 − ρ2n)

�. (38)

Consequently, a closed-form upper-bound, as given in (39),shown at the bottom of this page, is obtained by using(36) and (38).

PEP = Q

⎛⎜⎝� ! E

4 (1 − ρ2n)

⎛⎝�χI

p − χIp�2

σ2nI

l

+�χQ

p − χQp �2

σ2nQ

l

− 2ρn

χI

p − χIp

�T (χQp − χQ

p )σnI

lσnQ

l

⎞⎠⎞⎟⎠ = Q

��Eγ

4(1 − ρ2n)

�(30)

APEP ≤ 12

⎛⎜⎜⎝

8(1 − ρ2n) exp

�− b21λ1E

8(1−ρ2n)+2λ1E

− b22λ2E

8(1−ρ2n)+2λ2E

��

[8(1 − ρ2n) + 2λ1E][8(1 − ρ2

n) + 2λ2E]

⎞⎟⎟⎠ (39)

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CANBILEN et al.: JOINT IMPACT OF I/Q IMBALANCE AND IMPERFECT CSI ON SM-MIMO SYSTEMS 3041

B. Asymptotic Analysis

Assuming E 1, an asymptotic APEP is provided byutilizing (36) and (37) as follows:

APEPasy =1π

�exp(− b21+b22

2 )[E/4(1 − ρ2

n)]√

λ1λ2

�Nr � π2

0

(sin θ)2Nrdθ.

(40)

This expression proves that APEP depends on the numberof Rx antennas, the mean values based on the generalizedBeckmann fading channel, the energy of the transmitted signal,the correlation between the I/Q parts of the improper Gaussiannoise and the eigenvalues that calculated based on the covari-ance matrix of ζ1 and ζ2.

Considering that B(Nr + 12 , 1

2 ) = 2$ π

20 (sin θ)2Nrdθ

[58, Eq. 8.308(2)] and B(Nr + 12 , 1

2 ) = Γ(Nr+ 12 )Γ( 1

2 )

Γ(Nr+1)

[58, Eq. 8.384(1)], one has$ π

20 (sin θ)2Nrdθ =

√π Γ(Nr+ 1

2 )

2Γ(Nr+1) ,where B(·, ·) and Γ(·) are the Beta function and the Gammafunction, respectively. Considering also that Γ(Nr +1) = Nr![59, Eq. 1.1.7] and Γ(Nr + 1

2 ) =√

π(2Nr−1)!!2Nr [60, p.509], (40)

can be rewritten as:

APEPasy = Ξ�

E

4(1 − ρn)

�−Nr

, (41)

where Ξ = (2Nr−1)!!2Nr+1Nr!

�exp(− b21+b22

2 )√λ1λ2

�Nr

can be calculated

as a constant value, and E4(1−ρn) is equal to signal-to-noise

ratio (SNR) that can be seen in (30).Now, the diversity gain can be calculated from:

g = − limSNR→∞

log(APEPasy)log(SNR)

. (42)

Substituting (41) in (42) proves that the diversity gain is equalto the number of Rx antennas, Nr, while the error variancedecreases with the energy, as such in the case of perfect I/Qmatch.

C. Average Bit Error Rate

After the evaluation of the APEP, the ABER of the proposedscheme can be given under the assumption of equiprobablesymbols by using well-known tight union bound method asfollows [28]:

ABER ≤ 12m

2m%n=1

2m%k=1n�=k

Pr (χp → χp) en,k

m, (43)

where en,k is the number of bit errors relevant to the corre-sponding pairwise error event.

V. NUMERICAL RESULTS

In this section, the joint effect of IQI and ICSI on the per-formance of the SM-MIMO wireless communication systemis evaluated over generalized Beckmann fading channels byusing the proposed optimal MLD design through computersimulations and analytical derivations. It is assumed that atleast 5 × 105 symbols have been transmitted for each SNR

Fig. 3. APEP results of the proposed optimal MLD design for SM systemunder the effect of CSI impairment with perfect I/Q match (q2

B = 0.01, r2B =

0.09, KB = 0.5, ρh = 0.9).

value, which is defined as E/σ2n, between 0−30 dB. 4-QAM

modulation scheme is utilized on SM wireless communicationsystem with one Rx antenna because of the simplicity, unlessotherwise stated; however, all results can be generalized forany number of Rx antennas and higher order modulationschemes.

First of all, the APEP results are given for the SM wirelesscommunication system under the effect of only ICSI impair-ment assuming ideal hardware in Fig. 3. It is observed thateven very small values of the channel estimation error havesignificant destructive impacts on the system performance. Forinstance, approximately a 7 dB power loss as well as an errorfloor are observed while APEP = 4×10−3 for the estimationerror value of just σ2

e = 0.01. On the other side, it can benoted that variable channel estimation (σ2

e = 1/SNR) insteadof constant values solves the error floor problem. However,this is not a complete solution since deterioration effect isnot avoided, e.g., still ∼ 6 dB power loss is seen whileAPEP = 2 × 10−4 compared to perfect CSI case.

The effect of IQI on the APEP performance can be analyzedby Fig. 4, which is given under the assumption of perfectCSI. In this figure, the results are presented for the varyingvalues of amplitude imbalance between 0.5 to 1, and phaseimbalance among 0◦ to 15◦. Although it is clear that bothamplitude and phase imbalance cause deterioration comparedto the ideal case, i.e. ξT = ξR = 1 and οT = οR = 0◦,it is observed that an imbalance in phase has minor effects onthe system performance, since changing οT = οR = 10◦ toοT = οR = 15◦ for fixed value of ξT = ξR = 0.5 providesalmost the same results.

Fig. 5 is given to analyze the joint effect of IQI and ICSI onthe APEP performance. It is observed that boosting the effectof estimation error by changing the variance of the error fromσ2

e = 0.01 to σ2e = 0.03 for the fixed values of ξT = ξR = 0.9

and οT = οR = 5◦, results in a remarkable signal distor-tion. Contrary to this, increasing the IQI effect by changingξT = ξR = 0.9 and οT = οR = 5◦ to ξT = ξR = 0.5

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3042 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 5, MAY 2020

Fig. 4. APEP results of the proposed optimal MLD design for SM systemunder the effect of IQI with perfect CSI (q2

B = 0.01, r2B = 0.09, KB =

0.5, ρh = 0.9).

Fig. 5. APEP results of the proposed optimal MLD design for SMsystem under the joint effect of IQI and imperfect CSI (q2

B = 0.01, r2B =

0.09, KB = 0.6, ρh = 0.9).

and οT = οR = 10◦ for fixed σ2e also impairs the system

performance; however, this degradation impact is less than theone caused by the estimation error.

Considering both Figs. 3 and 5, it is easy to say thatestimation at the Rx-side is very important for SM-MIMOsystems over generalized Beckmann channels. Thus, in orderto decrease the deterioration arising from estimation errors,CRLB, which calculates a lower bound for σ2

e , is utilized forthe proposed system. The positive effect of using CRLB isproven with Fig. 6, since the error floor observed for fixedvalues of σ2

e in Figs. 3 and 5, is eliminated. Moreover, con-sidering this figure, it can also be demonstrated that increasingthe number of pilot symbols used in CRLB improves the APEPresults as well. For instance, using 10 pilot symbols insteadof 1 provides ∼ 3 dB gain in power when APEP = 10−4.

The effect of the characteristic properties of the generalizedBeckmann fading parameters on the system performance isclarified with Fig. 7. As seen from this figure, increasingthe correlation between the real and imaginary parts of the

Fig. 6. APEP results of SM with CRLB under the joint effect of IQI andimperfect CSI (q2

B = 0.01, r2B = 0.09, KB = 0.5, ρh = 0.9, ξT = ξR =

0.9 and φT = φR = 5◦).

Fig. 7. APEP results of SM wireless system with CRLB under the jointeffect of IQI and imperfect CSI (KB = 1, ξT = ξR = 0.9, φT = φR =5◦, NP = 2).

channel coefficients by changing ρh from 0.1 to 0.7, provides1 dB gain in power. The ascending imbalance between themean (rB = 0.1 to rB = 0.5) and variance (q2

B = 0.25 toqB = 0.81) values of the real and imaginary parts of thechannel coefficients also gives approximately 2 dB and 3 dBbetter results. The effect of the parameter KB, which stands forthe ratio between the summation of the means and variancesof I/Q parts of the channel coefficients, can be interpreted bycomparing Figs. 4 and 5 for ξT = ξR = 0.9, οT = οR = 5◦

case. Considering ∼ 2 dB power gain provided with KB =0.6 in Fig. 5 as compared to KB = 0.5 case in Fig. 4,it can be understood that increasing KB improves the APEPperformance as well.

Finally, Fig. 8 is given not only to show that the proposeddesign is applicable for higher order modulation schemes,i.e., higher data rates, but also to prove that the diversitygain is equal to the number of Rx antennas. Using 16−QAMinstead of 4−QAM increases the data rate from 3 bps/Hz to5 bps/Hz; however, it decreases the SNR around 3 dB for both

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CANBILEN et al.: JOINT IMPACT OF I/Q IMBALANCE AND IMPERFECT CSI ON SM-MIMO SYSTEMS 3043

Fig. 8. APEP results of 2×1 and 2×2 SM wireless systems with CRLB inthe presence of both IQI and imperfect CSI for different modulation schemes(q2

B = 0.01, r2B = 0.01, KB = 0.5, ρh = 0.1, ξT = ξR = 0.9 and

φT = φR = 5◦, NP = 10).

2 × 1 and 2 × 2 configurations. Considering APEP = 10−2

case, 2 × 2 SM structure doubles the power of 2 × 1, whichproves the comment on diversity gain. The accuracy of theasymptotic analysis could also be proved by considering Figs.7 and 8 together, since the analytical and simulation resultsare in perfect agreement with the asymptotic results. Duringthe simulations, it is assumed that σ2

I + σ2Q = 1. Considering

this case beside (3) and (4), the effect of the parameters KB

and ΩB would be parallel.

VI. CONCLUSION

The destructive effects of IQI, which are highlighted interms of SIR, as well as channel estimation error on theperformance of SM-MIMO wireless communication systemhave been addressed via computer simulations and analyticalderivations in this study. An upper-bound of the closed-formof APEP has been derived for the proposed optimal MLDmethod, and the performance results have been obtainedover generalized Beckmann fading channels, which has beenconsidered as the first time for SM-MIMO systems. Theobtained results point out that both IQI and ICSI are majordrawbacks for SM-MIMO systems, and the proposed optimalMLD design successfully mitigates the destructive effects ofthem. A further performance improvement has been achievedby utilizing CRLB for the channel estimation process. Asincreasing data rates cost more performance degradations,the effects of IQI has to be considered seriously while design-ing next-generation wireless systems. Future work can focusto the extention of the presented analysis to more challengingpropagation environments and frequency-domain effects ofIQI, while also extending the work towards cooperative MIMOsystems offers an interesting topic for future studies.

APPENDIX ACRLB FOR SM-MIMO SYSTEM

The received signal at the Rx side, yl, can be written in theform of:

yl = yIl + jyQ

l . (44)

Expanding (10), yQl and yI

l can be obtained as follows,respectively:

yQl =

√E[hI

l,i(xIqKc + xQ

q Kd) + hQl,i(x

IqKd − xQ

q Kc)]

+KcnIl + Kdn

Ql

yIl =

√E(hI

l,ixIq − hQ

l,ixQq ) + nI

l . (45)

Here, the noise parts of yIl and yQ

l are given as ηI = nIl

and ηQ = KcnIl + Kdn

Ql of which the variance values are

calculated as σ2ηI = σ2

n/2 and σ2ηQ = (ξ2

Rσ2n)/2, respectively.

In light of this information, the joint PDF of yIl and yQ

l

is obtained as in (46), shown at the top of the next page.Considering this PDF, the likelihood function is also writtenas given in (47), shown at the top of the next page. In orderto find the CRLB matrix, the log-likelihood function, whichis necessary to get the values of I(hI

i ), I(hQi ) and I(hI

i ,hQi ) of

FIM in (18), is achieved as (48), shown at the top of the nextpage, while the second derivative of ln(Py(y; hi)) is given in(49), shown at the top of the next page. Now, the value ofI(hI

i ) can be calculated by using the derivation in (49) from:

I(hIi ) = −E

∂2(ln(Py(y; hi)))

∂(hIl,i)2

�. (50)

Substituting (49) in (50) ends up with (19). Similarly, I(hQi )

and I(hIi ,hQ

i ), which is equal to I(hQi ,hI

i ), can be calculated byutilizing the derivations in (53) and (54), shown at the top ofthe next page, in the following expressions, respectively:

I(hQi ) = −E

∂2(ln(Py(y; hi)))

∂(hQl,i)2

�, (51)

I(hIi ,hQ

i ) = −E

∂2(ln(Py(y; hi)))

∂hIl,i∂hQ

l,i

�. (52)

Using these values in (22) gives the CRLB matrix.

APPENDIX BMGF OF QUADRATIC FORM CORRELATED NONCENTRAL

CHI-SQUARED DISTRIBUTION

Linear combination of two correlated noncentral chi-squaredRVs with one degree of freedom, γ, can be written in aquadratic form of Gaussian RVs as follows [56]:

γ = δT Aδ =2%

j=1

λj(Uj + bj)2, (55)

where Uj represents independently distributed standard normalvariables and δT =

&ζ1 ζ2

'. λj are the eigenvalues of

Ο12 AΟ

12 , where Ο is the covariance matrix of ζ1 and ζ2, and

A is given as:

A =

⎡⎢⎢⎢⎣

υ1

σ2nI

l

−ρ√

υ1υ2

σnIlσnQ

l−ρ√

υ1υ2

σnIlσnQ

l

υ2

σ2nQ

l

⎤⎥⎥⎥⎦ . (56)

Additionally, in (48), (b1, b2) ≡ b = LTΣ− 12 μ is the

reorganized mean vector where (μ1, μ2) ≡ μ is the mean

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3044 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 5, MAY 2020

fyl(yI

l , yQl ) =

1

2πσηI σηQ

�1 − ρ2

η

× exp

�−1

2(1 − ρ2η)

�|yI

l −√E(hI

l,ixIq − hQ

l,ixQq )|2

σ2ηI

+|yQ

l −√E[hI

l,i(xIqKc + xQ

q Kd) + hQl,i(x

IqKd − xQ

q Kc)]|2σ2

ηQ

−2ρη[yIl −√

E(hIl,ix

Iq − hQ

l,ixQq )]{yQ

l −√E[hI

l,i(xIqKc + xQ

q Kd) + hQl,i(x

IqKd − xQ

q Kc)]}σηI σηQ

��. (46)

Py(y; hi) =

⎛⎝ 1

2πσηI σηQ

�1 − ρ2

η

⎞⎠

Np

× exp

⎡⎣ −1

2(1 − ρ2η)

Np−1%r=0

�|yI

l −√E(hI

l,ixIq − hQ

l,ixQq )|2

σ2ηI

+|yQ

l −√E[hI

l,i(xIqKc + xQ

q Kd) + hQl,i(x

IqKd − xQ

q Kc)]|2σ2

ηQ

−2ρη[yIl −√

E(hIl,ix

Iq − hQ

l,ixQq )]{yQ

l −√E[hI

l,i(xIqKc + xQ

q Kd) + hQl,i(x

IqKd − xQ

q Kc)]}σηI σηQ

��. (47)

ln(Py(y; hi)) = −Np ln(2πσηI σηQ

�1 − ρ2

η) − Np

2(1 − ρ2η)

�|yI

l −√E(hI

l,ixIq − hQ

l,ixQq )|2

σ2ηI

+|yQ

l −√E[hI

l,i(xIqKc + xQ

q Kd) + hQl,i(x

IqKd − xQ

q Kc)]|2σ2

ηQ

−2ρη[yIl −√

E(hIl,ix

Iq − hQ

l,ixQq )]{yQ

l −√E[hI

l,i(xIqKc + xQ

q Kd) + hQl,i(x

IqKd − xQ

q Kc)]}σηI σηQ

�. (48)

∂2(ln(Py(y; hi)))∂(hI

l,i)2=

−NpE

1 − ρ2η

�

(xIq)

2

σ2ηI

+(xI

qKc + xQq Kd)2

σ2ηQ

− 2ρηxIq(x

IqKc + xQ

q Kd)σηI σηQ

�(49)

∂2(ln(Py(y; hi)))

∂(hQl,i)2

=−NpE

1 − ρ2η

�

(xQq )2

σ2ηI

+(xI

qKd − xQq Kc)2

σ2ηQ

+2ρηxQ

q (xIqKd − xQ

q Kc)σηI σηQ

�. (53)

∂2(ln(Py(y; hi)))

∂hIl,i∂hQ

l,i

=−NpE

1 − ρ2η

×�−xI

q xQq

σ2ηI

+(xI

qKd − xQq Kc)(xI

qKc + xQq Kd)

σ2ηQ

+ρη(xQ

q (xIqKc + xQ

q Kd) + xIq(x

IqKd − xQ

q Kc))σηI σηQ

�(54)

vector of ζ1 and ζ2, and L is the eigenvector that is calculatedby using λ1 and λ2.

The moment generation function (MGF) of γ can be givenby the following expression [56]:

Mγ(t) =s,

j=1

(1 − 2tλj)−12 exp

⎧⎨⎩

s%j=1

b2j tλj

1 − 2tλj

⎫⎬⎭ , (57)

where s is the number of correlated noncentral chi-squaredvariables. This equation reaches to (36) for SM-MIMO wire-less communication systems under the assumption of indepen-dent channels.

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Ayse Elif Canbilen received the B.S. and M.S.degrees from Selçuk University, Turkey, in 2012 and2015, respectively, and the Ph.D. degree from KonyaTechnical University, Turkey, in 2019, all in electri-cal and electronics engineering. From 2017 to 2018,she was with the Department of Electrical Engi-neering, Lakehead University, Thunder Bay, ON,Canada, as a Visitor Researcher. She is currently aResearch Assistant with the Department of Electri-cal and Electronics Engineering, Konya TechnicalUniversity. Her primary research interests include

MIMO systems, spatial modulation techniques, cooperative communications,cognitive radio systems, and reconfigurable intelligent surfaces. She has beenserving as a Reviewer for the IEEE TRANSACTIONS ON COMMUNICATIONS,the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY, the IEEE COMMUNICA-TIONS LETTERS, and IEEE ACCESS.

Salama Said Ikki (Senior Member, IEEE) receivedthe B.S. degree from Al-Isra University, Amman,Jordan, in 1996, the M.Sc. degree from the ArabAcademy for Science and Technology and MaritimeTransport, Alexandria, Egypt, in 2002, and the Ph.D.degree from the Memorial University of Newfound-land, St. John’s, in 2009, all in electrical engineering.He was a Research Assistant with INRS, Universityof Quebec, Montreal, from 2010 to 2012, and aPost-Doctoral Fellow with the University of Water-loo, Waterloo, ON, Canada, from 2009 to 2010.

He is currently an Associate Professor in wireless communications with theDepartment of Electrical Engineering, Lakehead University. He has authoredor coauthored more than 100 articles in peer-reviewed IEEE internationaljournals and conferences with more than 3 500 citations and has a currentH-index of 30. He has been carrying out research in communications andsignal processing for over 10 years. He was a recipient of the Best PaperAward published in the EURASIP Journal on Advances in Signal Processing,the IEEE COMMUNICATIONS LETTERS and the IEEE WIRELESS COM-MUNICATIONS LETTERS Exemplary Reviewer Certificate in 2012, and theTop Reviewer Certificate from the IEEE TRANSACTIONS ON VEHICULAR

TECHNOLOGY in 2015. He serves on the Editorial Board of the IEEECOMMUNICATIONS LETTERS and the IET Communications Proceeding. He iswidely recognized as an expert in the field of wireless communications.

Ertugrul Basar (Senior Member, IEEE) receivedthe B.S. degree (Hons.) from Istanbul Univer-sity, Turkey, in 2007, and the M.S. and Ph.D.degrees from Istanbul Technical University, Turkey,in 2009 and 2013, respectively. He is currentlyan Associate Professor with the Department ofElectrical and Electronics Engineering, Koç Uni-versity, Istanbul, Turkey, and also the Directorof the Communications Research and InnovationLaboratory (CoreLab). His primary research inter-ests include MIMO systems, index modulation,

waveform design, visible light communications, and signal processing forcommunications.

Recent recognition of his research include the Science Academy (Turkey)Young Scientists (BAGEP) Award in 2018, the Mustafa Parlar FoundationResearch Encouragement Award in 2018, the Turkish Academy of SciencesOutstanding Young Scientist (TUBA-GEBIP) Award in 2017, and the first-everIEEE Turkey Research Encouragement Award in 2017. He currently servesas an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS andPhysical Communication (Elsevier), and as an Associate Editor of the IEEECOMMUNICATIONS LETTERS. He served as an Associate Editor for IEEEACCESS from 2016 to 2018.

Seyfettin Sinan Gultekin received the B.S. andM.S. degrees from Erciyes University, Turkey,in 1988, and 1992, respectively, and the Ph.D. degreefrom Selçuk University in 2002. He was a ResearchAssistant, an Instructor, and then was an AssistantProfessor with the Department of Electrical andElectronics Engineering, Selçuk University, Konya,Turkey, from 1989 to 1994, from 1994 to 2009, andfrom 2009 to 2018. He is currently an AssociateProfessor with the Department of Electrical andElectronics Engineering, Konya Technical Univer-

sity, Konya, Turkey. He teaches lectures on microwave, antenna, commu-nications and engineering mathematics. His current research areas includemicrostrip antennas for wearable and biomedical applications, and optimiza-tion of microstrip antenna parameters. He served as a Local OrganizingCommittee Member for the International Union of Radio Science (URSI)Turkey Conference in 2018.

Ibrahim Develi received the B.S., M.S., and Ph.D.degrees from Erciyes University, Turkey, in 1995,1997, and 2003, respectively. From 1995 to 2003,he was a Research Assistant with the Department ofElectronics Engineering, Erciyes University, wherehe is currently a Professor. He teaches courses inwireless communications, and his current researchinterests are in cooperative communications,MIMO systems, spread spectrum communications,multiuser communications, wireless networks, andthe applications of neural networks to multiuser

communication systems. He has been a member of the TPC for a largenumber of major international conferences, such as IEEE ICC, IEEEGLOBECOM, IEEE BROADNETS, IEEE RWS, and IEEE WCNC.He currently serves as an Associate Editor for the EURASIP Journal onWireless Communications and Networking. He is also on the Editorial Boardof the International Journal of Mobile Communications.

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