# Impact of IQ Imbalance on RIS-Assisted SISO Communication Systems

Authors: Asma Bouhlel; Anis Sakly
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

Reconfigurable intelligent surface (RIS) for wireless networks has emerged as a promising future transmission technique to create smart radio environments that improve the system performance by turning the wireless channel into an adjustable system block. However, transceivers come with various hardware impairments, such as phase noise and in-phase/quadrature-phase imbalance (IQI). Hence, for robust configuration of RIS-based communication under practical conditions, assuming the identical performance analysis when subject to IQI, will lead to inaccurate analysis. In this paper, the implementation of this novel transmission technique is thoroughly investigated under intensive realistic circumstances. For this purpose, based on the maximum likelihood (ML) detector, a novel analytical expression of average pairwise error probability under IQI is proposed and compared to the standard ML detector. Further, the proposed analytical approaches are confirmed by numerical simulations.


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## Body

## 1. Introduction

Many attempts have been done in recent years to deliver new deployment models with high speeds, superior reliability, and negligible latency to meet the requirements of 5G standards. To achieve these goals, several transmission techniques have been used such as millimeter wave (mmWave), orthogonal frequency division multiplexing (OFDM), and massive multiple input multiple output (MIMO) [1]. Although the successful launch of the first 5G service, the introduced technologies suffer from high energy consumption and uncontrolled propagation environments effects. Therefore, for the sixth-generation (6G) mobile communication systems with very-high frequency bands and more power efficiency, researchers are already exploring new methods [2] such as reconfigurable intelligent surface (RIS) [3]. RIS, also known as large intelligent surface (LIS) [4], has attracted a significant amount of attention from researchers as a promising future transmission technique to create a smart propagation environment [5, 6]. Conventionally, only the source and the destination are controlled with coding, encoding, and many processing operations to enhance the quality of the signal. By putting a RIS between the transmitter (Tx) and the receiver (Rx), an additional propagation path appears. Thus, the created channel behavior can be software-controlled in order to achieve a smart programmable wireless environment that provides more freedom degrees and boost the performance [6].The announced technique has been compared with the massive MIMO [7], amplify-and-forward (AF) relaying [8], backscatter communication [9], mmWave communication [10], and network densification [11]. Although it is similar to other existing technologies, the RIS is based on a large number of thin passive reflectors without buffering and processing any incoming signals [4]. These reflectors are designed based on two-dimensional meta-surfaces [12, 13]. In addition, RIS is equipped with a programmable microcontroller that modifies the phase of the incident electromagnetic waves in a way that can enhance the signal quality at the Rx and improve the network coverage. Hence, RIS improves the quality of the received signal by simply reflecting and adjusting the incident signal phase shifts favorably with low cost and low energy consumption [3]. The novel proposed concept presents important theoretical and electromagnetic design challenges [14, 15].However, the use of RIS poses several new challenges for the transceiver design, and it is paramount to analyze the system performance under practical conditions [16]. Accordingly, RIS-aided communication performance can be significantly degraded by different types of realistic imperfections, including noise signal, imperfect channel state information (CSI), and transceiver hardware impairments. The effect of the hardware impairments general model on the achievable rate of RIS has been studied in [17]. In [18, 19], an asymptotic analysis of the uplink data rate in a RIS-based communication system affected by channel estimation errors and interference was investigated. The design and the implementation of RIS were also detailed in [20].To the best of the authors’ knowledge, RIS under in-phase and quadrature imbalance (IQI) has not been highlighted yet. However, for robust configuration, modeling the transceiver radio frequency (RF) front-end hardware as perfect will lead to inaccurate analysis [21]. Indeed both in-phase (I) and quadrature (Q) modulator and demodulator at the Tx and Rx may introduce phase and/or amplitude mismatch [22]. Additional harmful effects could degrade the system performance such as crosstalk and frequency interference [23–25]. Hence, taking into account IQI effects is a crucial factor for RIS effective design policies.Motivated by the aforementioned limitations of the existing literature, this paper explores the design of an optimal Rx detector which is compared to the performance of the classical ML in the presence of IQI. Accordingly, novel error probability analytical expressions are derived and proved with simulation results.


## 2. System Model and Signal Detection

### 2.1. System Model

In this section, we have adapted a general RIS-assisted single-input single output (SISO) wireless communication system as presented in Figure1. The direct signal path between the source and the destination is ignored in the rest of the paper [4], and the RIS is deployed to relay the scattered signal. Indeed, this assumption holds in the case of unfavorable propagation conditions that might be caused by an obstacle or a long distance, for example, [26–28].Figure 1
RIS under IQI system model.First, the information is conveyed from the source to the RIS. Then, the RIS software controls the amplitude and the phase of the received signal to combat the propagation environment’s harmful effects and reflect it to the destination.Although RIS is based on small passive elements and does not need any signal processing power, the font-ends of both Tx and Rx could be affected by the IQI which limits the system performance. Actually in practical conditions, due to the local oscillator (LO), filters, analog components, and up-and-down-conversion steps at Tx and Rx sides, the generated signals present a mismatch between the I and Q parts [29]. Accordingly, the Tx and Rx sides IQI parameters are introduced, respectively, using (G1, G2) and (K1, K2) which can be expressed while based on the complex LO signals [24, 30] as(1)G1=121+ξtejφt,G2=121−ξte−jφt,(2)K1=121+ξre−jφr,K2=121−ξrejφr,where ξt,φt and ξr,φr denote, respectively, Tx and Rx amplitude and phase imbalances. In the ideal case, where the IQ branch is perfectly matching these parameters that are reduced to ξt=ξr=1 and φt=φr=0∘. Consequently, we have G1=K1=1 and G2=K2=0. The up-converted signal modulated over M-ary mapper at the source affected by IQI can be written as(3)xIQ=G1x+G2x∗,where ∗ denotes the complex conjugate.Using RIS withN reflectors, the signal is firstly transmitted from the source antenna to the RIS then it is conveyed from the RIS to the destination through, respectively, the flat fading channels hk and gk for the kth reflecting meta-surface (k=1,2,⋯,N). Note that hk,gk follow zero-mean complex Gaussian distribution with unit variance.In such case, an intelligent RIS software is deployed to adjust the reflection phases based on the channel phases in order to maximize the received SNR. Hence, the adjustable signal received at the destination is given as(4)y=P∑k=1NhkejφkgkxIQ+n,where φk characterizes the adapted phase for the kth RIS reflector, P is the average of the transmitted power symbol, and n is a complex additive white Gaussian noise (AWGN) with zero mean and N0 variance. Using (3), the received signal becomes(5)y=P∑k=1NhkejφkgkG1x+G2x∗+n.Taking into account the Rx IQI and the adjusted phases, the resulting signal at the destination can be expressed as(6)yIQ=K1y+K2y∗=PK1G1∑k=1Nhkejφkgk+K2G2∗∑k=1Nhk∗e−jφkgk∗⏟Sx+PK1G2∑k=1Nhkejφkgk+K2G1∗∑k=1Nhk∗e−jφkgk∗⏟Ix∗+K1n+K2n∗=PSx+PIx∗+η=Pχ+η,where χ=Sx+Ix∗ and η=K1n+K2n∗.Note that due to Tx and Rx IQI, the baseband signalx is interfered by its complex conjugate x∗. Hence, Ix∗ represents the self-interferences.Analogous to [4], with the assistance of a software communication, the following derived expressions are analyzed based on the knowledge of the channel at the RIS in function of amplitudes and phases as hk=εke−jθk, gk=βke−jψk. In such case, the RIS adjusts the phases in order to maximize the signal-to-noise ratio (SNR) such as φk=θk+ψk.Moreover, the noise can be expressed asη=nI+jKcnI+KdnQ where nI and nQ note, respectively, the real and imaginary parts of n, Kc=K1Q+K2Q, and Kd=K1I−K2I [23].Accordingly, stating thatη=ηI+jηQ where ηI and ηQ note, respectively, the real and imaginary parts of η, it is worth noting that η is an improper Gaussian noise with unequal real and imaginary parts variances such as(7)σηI2=σn22=N02,(8)σηQ2=Kc2+Kd2σn22=N02ξr2.Further, the correlation factor between the noise components isρ=covηI,ηQ/σηIσηQ=Kcσn2/2/ξrσn2/2=−sinφr [25]. It is important to emphasize the IQI effects in changing the behavior of the noise from proper to improper. Thus, the received signal has correlated IQ components. Considering the presence of improper noise is a critical factor in analyzing the RIS performance.The received signal expression in (6) can be analyzed under several scenarios taking into account the possible values of IQ parameters:(1)
Perfect IQ Matching. In this scenario, Tx and Rx sides are assumed to be perfect and IQI parameters are defined as G1=K1=1,G2=K2=0. The received signal in (6) can be expressed as the following well known expressed used for all previous studies:(9)y=P∑k=1Nεkβkx+n.(2)
Tx Impaired by IQI. Taking into account the destructive effects of only the Tx side with perfect IQI parameters on the destination, i.e., K1=1 and K2=0, the resulting signal is given as(10)yTxIQ=P∑k=1NεkβkG1x+G2x∗+n.(3)
Rx Impaired by IQI. Considering ideal Tx IQI parameters and stating that G1=1,G2=0, (6) can be rewritten as(11)yRxIQ=P∑k=1NεkβkK1x+K2x∗+K1n+K2n∗.It can be observed from the above expressions that Tx impaired with IQI causes self-interference from the conjugate of the transmitted signal. On the other side, Rx suffering from IQI affects both the signal and the noise.As previously mentioned, with the presence of IQI, a new derived received signal expression is required which shows the presence of self-interference and new noise behavior. Hence, for an effective performance study of RIS-based communication systems, a novel design of optimal ML detector that incorporates the IQI effects is obligatory.


### 2.2. Signal-to-Interference Ratio (SIR)

The average SIR is calculated as follows to illustrate the harmful effects of the caused self-interference(12)SIR=ES2EI2.Note that the image rejection ratio IRR which denotes the measure of image frequency band attenuation can be defined for Tx and Rx sides, respectively, asIRRTx=G12/G22 and IRRRx=K12/K22[31]. IRR has a typical value in the range of 20-40 dB for practical analog RF front-end electronics [32]. Based on (2), the following relation can be defined K1=1−K2∗. It can be depicted that for high IRRRx values K1⟶1 and, K1K2∗=K1−K12⟶0 [33]. Supposing γk=εkβk, an approximate widely used in the literature [31–34] is obtained by assuming(13)E∑k=1Nγk2K12G12+K22G22>>E2RK1G1K2∗G2∑k=1Nγk2E∑k=1Nγk2K12G22+K22G12>>E2RK1G2K2∗G1∑k=1Nγk2.Hence, the averaged SIR can be tightly approximated as(14)SIR≃K12G12+K22G22K12G22+K22G12.It is obviously seen that even small values of IQI degrade the system performance. Further, the SIR expression is independent ofN. Hence, increasing the reflector number cannot mitigate the interference caused by the IQI. In the ideal case of perfect IQ matching SIR=∞.


### 2.3. Signal Detection

#### 2.3.1. Optimal ML Detector

In the attempt to cover the presence of an improper Gaussian noise, an optimal ML detector is designed based on the received signal expression in (6). Hence, the bivariate Gaussian random variable (RV) distribution of the real, yIQI, and imaginary, yIQQ, correlated components of the received signal vector y is described as follows:(15)fyIQI,yIQQyIQI,yIQQ∣x=12πσηIσηQ1−ρ2exp−121−ρ2yIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ,where χI and χQ represent, respectively, Rχ and Iχ components.Regarding that the transmitted symbols are assumed equally distributed, the optimal ML detector is given by maximizing the argument of the conditional joint probability density function (PDF) defined in (15) which is equivalent to the following expression:(16)x^MLop=argminxyIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ.


#### 2.3.2. Traditional ML Detector

The traditional ML detector is the classical ML detector used in the previous performance analysis of RIS in which the presence of IQI in the received signal expression is neglected. Indeed, it is simply expressed as the following well-known expression used for all previous studies of RIS performance:(17)x^MLtra=argminxyIQ−Pχ2.


## 2.1. System Model

In this section, we have adapted a general RIS-assisted single-input single output (SISO) wireless communication system as presented in Figure1. The direct signal path between the source and the destination is ignored in the rest of the paper [4], and the RIS is deployed to relay the scattered signal. Indeed, this assumption holds in the case of unfavorable propagation conditions that might be caused by an obstacle or a long distance, for example, [26–28].Figure 1
RIS under IQI system model.First, the information is conveyed from the source to the RIS. Then, the RIS software controls the amplitude and the phase of the received signal to combat the propagation environment’s harmful effects and reflect it to the destination.Although RIS is based on small passive elements and does not need any signal processing power, the font-ends of both Tx and Rx could be affected by the IQI which limits the system performance. Actually in practical conditions, due to the local oscillator (LO), filters, analog components, and up-and-down-conversion steps at Tx and Rx sides, the generated signals present a mismatch between the I and Q parts [29]. Accordingly, the Tx and Rx sides IQI parameters are introduced, respectively, using (G1, G2) and (K1, K2) which can be expressed while based on the complex LO signals [24, 30] as(1)G1=121+ξtejφt,G2=121−ξte−jφt,(2)K1=121+ξre−jφr,K2=121−ξrejφr,where ξt,φt and ξr,φr denote, respectively, Tx and Rx amplitude and phase imbalances. In the ideal case, where the IQ branch is perfectly matching these parameters that are reduced to ξt=ξr=1 and φt=φr=0∘. Consequently, we have G1=K1=1 and G2=K2=0. The up-converted signal modulated over M-ary mapper at the source affected by IQI can be written as(3)xIQ=G1x+G2x∗,where ∗ denotes the complex conjugate.Using RIS withN reflectors, the signal is firstly transmitted from the source antenna to the RIS then it is conveyed from the RIS to the destination through, respectively, the flat fading channels hk and gk for the kth reflecting meta-surface (k=1,2,⋯,N). Note that hk,gk follow zero-mean complex Gaussian distribution with unit variance.In such case, an intelligent RIS software is deployed to adjust the reflection phases based on the channel phases in order to maximize the received SNR. Hence, the adjustable signal received at the destination is given as(4)y=P∑k=1NhkejφkgkxIQ+n,where φk characterizes the adapted phase for the kth RIS reflector, P is the average of the transmitted power symbol, and n is a complex additive white Gaussian noise (AWGN) with zero mean and N0 variance. Using (3), the received signal becomes(5)y=P∑k=1NhkejφkgkG1x+G2x∗+n.Taking into account the Rx IQI and the adjusted phases, the resulting signal at the destination can be expressed as(6)yIQ=K1y+K2y∗=PK1G1∑k=1Nhkejφkgk+K2G2∗∑k=1Nhk∗e−jφkgk∗⏟Sx+PK1G2∑k=1Nhkejφkgk+K2G1∗∑k=1Nhk∗e−jφkgk∗⏟Ix∗+K1n+K2n∗=PSx+PIx∗+η=Pχ+η,where χ=Sx+Ix∗ and η=K1n+K2n∗.Note that due to Tx and Rx IQI, the baseband signalx is interfered by its complex conjugate x∗. Hence, Ix∗ represents the self-interferences.Analogous to [4], with the assistance of a software communication, the following derived expressions are analyzed based on the knowledge of the channel at the RIS in function of amplitudes and phases as hk=εke−jθk, gk=βke−jψk. In such case, the RIS adjusts the phases in order to maximize the signal-to-noise ratio (SNR) such as φk=θk+ψk.Moreover, the noise can be expressed asη=nI+jKcnI+KdnQ where nI and nQ note, respectively, the real and imaginary parts of n, Kc=K1Q+K2Q, and Kd=K1I−K2I [23].Accordingly, stating thatη=ηI+jηQ where ηI and ηQ note, respectively, the real and imaginary parts of η, it is worth noting that η is an improper Gaussian noise with unequal real and imaginary parts variances such as(7)σηI2=σn22=N02,(8)σηQ2=Kc2+Kd2σn22=N02ξr2.Further, the correlation factor between the noise components isρ=covηI,ηQ/σηIσηQ=Kcσn2/2/ξrσn2/2=−sinφr [25]. It is important to emphasize the IQI effects in changing the behavior of the noise from proper to improper. Thus, the received signal has correlated IQ components. Considering the presence of improper noise is a critical factor in analyzing the RIS performance.The received signal expression in (6) can be analyzed under several scenarios taking into account the possible values of IQ parameters:(1)
Perfect IQ Matching. In this scenario, Tx and Rx sides are assumed to be perfect and IQI parameters are defined as G1=K1=1,G2=K2=0. The received signal in (6) can be expressed as the following well known expressed used for all previous studies:(9)y=P∑k=1Nεkβkx+n.(2)
Tx Impaired by IQI. Taking into account the destructive effects of only the Tx side with perfect IQI parameters on the destination, i.e., K1=1 and K2=0, the resulting signal is given as(10)yTxIQ=P∑k=1NεkβkG1x+G2x∗+n.(3)
Rx Impaired by IQI. Considering ideal Tx IQI parameters and stating that G1=1,G2=0, (6) can be rewritten as(11)yRxIQ=P∑k=1NεkβkK1x+K2x∗+K1n+K2n∗.It can be observed from the above expressions that Tx impaired with IQI causes self-interference from the conjugate of the transmitted signal. On the other side, Rx suffering from IQI affects both the signal and the noise.As previously mentioned, with the presence of IQI, a new derived received signal expression is required which shows the presence of self-interference and new noise behavior. Hence, for an effective performance study of RIS-based communication systems, a novel design of optimal ML detector that incorporates the IQI effects is obligatory.


## 2.2. Signal-to-Interference Ratio (SIR)

The average SIR is calculated as follows to illustrate the harmful effects of the caused self-interference(12)SIR=ES2EI2.Note that the image rejection ratio IRR which denotes the measure of image frequency band attenuation can be defined for Tx and Rx sides, respectively, asIRRTx=G12/G22 and IRRRx=K12/K22[31]. IRR has a typical value in the range of 20-40 dB for practical analog RF front-end electronics [32]. Based on (2), the following relation can be defined K1=1−K2∗. It can be depicted that for high IRRRx values K1⟶1 and, K1K2∗=K1−K12⟶0 [33]. Supposing γk=εkβk, an approximate widely used in the literature [31–34] is obtained by assuming(13)E∑k=1Nγk2K12G12+K22G22>>E2RK1G1K2∗G2∑k=1Nγk2E∑k=1Nγk2K12G22+K22G12>>E2RK1G2K2∗G1∑k=1Nγk2.Hence, the averaged SIR can be tightly approximated as(14)SIR≃K12G12+K22G22K12G22+K22G12.It is obviously seen that even small values of IQI degrade the system performance. Further, the SIR expression is independent ofN. Hence, increasing the reflector number cannot mitigate the interference caused by the IQI. In the ideal case of perfect IQ matching SIR=∞.


## 2.3. Signal Detection

### 2.3.1. Optimal ML Detector

In the attempt to cover the presence of an improper Gaussian noise, an optimal ML detector is designed based on the received signal expression in (6). Hence, the bivariate Gaussian random variable (RV) distribution of the real, yIQI, and imaginary, yIQQ, correlated components of the received signal vector y is described as follows:(15)fyIQI,yIQQyIQI,yIQQ∣x=12πσηIσηQ1−ρ2exp−121−ρ2yIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ,where χI and χQ represent, respectively, Rχ and Iχ components.Regarding that the transmitted symbols are assumed equally distributed, the optimal ML detector is given by maximizing the argument of the conditional joint probability density function (PDF) defined in (15) which is equivalent to the following expression:(16)x^MLop=argminxyIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ.


### 2.3.2. Traditional ML Detector

The traditional ML detector is the classical ML detector used in the previous performance analysis of RIS in which the presence of IQI in the received signal expression is neglected. Indeed, it is simply expressed as the following well-known expression used for all previous studies of RIS performance:(17)x^MLtra=argminxyIQ−Pχ2.


## 2.3.1. Optimal ML Detector

In the attempt to cover the presence of an improper Gaussian noise, an optimal ML detector is designed based on the received signal expression in (6). Hence, the bivariate Gaussian random variable (RV) distribution of the real, yIQI, and imaginary, yIQQ, correlated components of the received signal vector y is described as follows:(15)fyIQI,yIQQyIQI,yIQQ∣x=12πσηIσηQ1−ρ2exp−121−ρ2yIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ,where χI and χQ represent, respectively, Rχ and Iχ components.Regarding that the transmitted symbols are assumed equally distributed, the optimal ML detector is given by maximizing the argument of the conditional joint probability density function (PDF) defined in (15) which is equivalent to the following expression:(16)x^MLop=argminxyIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ.


## 2.3.2. Traditional ML Detector

The traditional ML detector is the classical ML detector used in the previous performance analysis of RIS in which the presence of IQI in the received signal expression is neglected. Indeed, it is simply expressed as the following well-known expression used for all previous studies of RIS performance:(17)x^MLtra=argminxyIQ−Pχ2.


## 3. Performance Analysis

### 3.1. Conditional Error Probability

#### 3.1.1. Optimal ML Detector

Assumingχ is transmitted, the pairwise error probability (PEP) of deciding in favor of χ~ is given from the optimal ML detector expression in (16) as(18)PEPopt=PryIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ>yIQI−Pχ~I2σηI2+yIQQ−Pχ~Q2σηI2−2ρyIQI−Pχ~IyIQQ−Pχ~QσηIσηQ=Prβ>0,where χ~=Sx~+Ix~∗ and β is obtained after simple mathematical operations as(19)β=2ρPχI−χ~IηQ+χQ−χ~QηIσηIσηQ−2PχI−χ~IηIσηI2−2PχQ−χ~QηQσηQ2−EχI−χ~I2σηI2−EχQ−χ~Q2σηQ2+2ρEχI−χ~IχQ−χ~QσηIσηQ.Without loss of generalityβ, conditioned on χ, is a Gaussian RV with the following mean and variance values(20)μβ=EχI−χ~I2σηI2+EχQ−χ~Q2σηQ2−2ρEχI−χ~IχQ−χ~QσηIσηQ,(21)σβ2=4E1−ρ2EχI−χ~I2σηI2+EχQ−χ~Q2σηQ2−2ρχI−χ~IχQ−χ~QσηIσηQ.Using (18) and (20), the conditional PEP (CPEP) can be written as in (22) on the top of the page, where Qx denotes the Q-function defined as Qx=1/2π∫x∞exp−u2/2du.(22)CPEPopt=QP41−ρ2χI−χ~I2σηI2+χQ−χ~Q2σηQ2−2ρχI−χ~IχQ−χ~QσηIσηQ=QPY21−ρ2N0,


#### 3.1.2. Traditional ML Detector

Note that the improper behavior of the noise is not considered for the traditional ML detector. Thus, assumingx is transmitted, the PEP of deciding in favor of x~ is given from the classical ML detector expression in (17) as(23)PEPtra=PryIQ−Pχ2>yIQ−Pχ~2=PrD>Eχ−χ~2,where D=−2RPχ−χ~η∗ which has zero mean and σD2=4PχI−χ~I2σηI2+4PχQ−χ~Q2σηQ2+8ρσηIσηQPχI−χ~IχQ−χ~Q as variance. Hence, the CPEP using Q function is presented in (24).(24)CPEPtra=QPχI−χ~I44χI−χ~I2σηI2+4χQ−χ~Q2σηQ2+8ρσηIσηQχI−χ~IχQ−χ~Q.


### 3.2. Average Error Probability

#### 3.2.1. Optimal ML Detector

To derive the average PEP (APEP), (22) should be averaged over the PDF of Y which can be depicted as:(25)APEPopt=∫0∞QPY41−ρ2fYYdY.However, note thatY=Ψ12+Ψ22−2ρΨ1Ψ2 where Ψ1=χI−χ~I and Ψ2=χQ−χ~Q/ξr. Indeed, Ψ1 and Ψ2 depend on real and imaginary parts of S and I given in (6). Thus, based on central limit theorem (CLT) and for large number of reflectors N, Y is considered as a combination of correlated noncentral chi-squared random variables. The determination of the APEP using the Y PDF is very hard. Hence, the APEP is numerically computed by averaging PEP over large number of channel realization.


#### 3.2.2. Traditional ML Detector

Traditional ML detector APEP based on (24) is difficult and even impossible to determinate. However, it will be numerically calculated with the same method used for the optimal ML detector.


## 3.1. Conditional Error Probability

### 3.1.1. Optimal ML Detector

Assumingχ is transmitted, the pairwise error probability (PEP) of deciding in favor of χ~ is given from the optimal ML detector expression in (16) as(18)PEPopt=PryIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ>yIQI−Pχ~I2σηI2+yIQQ−Pχ~Q2σηI2−2ρyIQI−Pχ~IyIQQ−Pχ~QσηIσηQ=Prβ>0,where χ~=Sx~+Ix~∗ and β is obtained after simple mathematical operations as(19)β=2ρPχI−χ~IηQ+χQ−χ~QηIσηIσηQ−2PχI−χ~IηIσηI2−2PχQ−χ~QηQσηQ2−EχI−χ~I2σηI2−EχQ−χ~Q2σηQ2+2ρEχI−χ~IχQ−χ~QσηIσηQ.Without loss of generalityβ, conditioned on χ, is a Gaussian RV with the following mean and variance values(20)μβ=EχI−χ~I2σηI2+EχQ−χ~Q2σηQ2−2ρEχI−χ~IχQ−χ~QσηIσηQ,(21)σβ2=4E1−ρ2EχI−χ~I2σηI2+EχQ−χ~Q2σηQ2−2ρχI−χ~IχQ−χ~QσηIσηQ.Using (18) and (20), the conditional PEP (CPEP) can be written as in (22) on the top of the page, where Qx denotes the Q-function defined as Qx=1/2π∫x∞exp−u2/2du.(22)CPEPopt=QP41−ρ2χI−χ~I2σηI2+χQ−χ~Q2σηQ2−2ρχI−χ~IχQ−χ~QσηIσηQ=QPY21−ρ2N0,


### 3.1.2. Traditional ML Detector

Note that the improper behavior of the noise is not considered for the traditional ML detector. Thus, assumingx is transmitted, the PEP of deciding in favor of x~ is given from the classical ML detector expression in (17) as(23)PEPtra=PryIQ−Pχ2>yIQ−Pχ~2=PrD>Eχ−χ~2,where D=−2RPχ−χ~η∗ which has zero mean and σD2=4PχI−χ~I2σηI2+4PχQ−χ~Q2σηQ2+8ρσηIσηQPχI−χ~IχQ−χ~Q as variance. Hence, the CPEP using Q function is presented in (24).(24)CPEPtra=QPχI−χ~I44χI−χ~I2σηI2+4χQ−χ~Q2σηQ2+8ρσηIσηQχI−χ~IχQ−χ~Q.


## 3.1.1. Optimal ML Detector

Assumingχ is transmitted, the pairwise error probability (PEP) of deciding in favor of χ~ is given from the optimal ML detector expression in (16) as(18)PEPopt=PryIQI−PχI2σηI2+yIQQ−PχQ2σηQ2−2ρyIQI−PχIyIQQ−PχQσηIσηQ>yIQI−Pχ~I2σηI2+yIQQ−Pχ~Q2σηI2−2ρyIQI−Pχ~IyIQQ−Pχ~QσηIσηQ=Prβ>0,where χ~=Sx~+Ix~∗ and β is obtained after simple mathematical operations as(19)β=2ρPχI−χ~IηQ+χQ−χ~QηIσηIσηQ−2PχI−χ~IηIσηI2−2PχQ−χ~QηQσηQ2−EχI−χ~I2σηI2−EχQ−χ~Q2σηQ2+2ρEχI−χ~IχQ−χ~QσηIσηQ.Without loss of generalityβ, conditioned on χ, is a Gaussian RV with the following mean and variance values(20)μβ=EχI−χ~I2σηI2+EχQ−χ~Q2σηQ2−2ρEχI−χ~IχQ−χ~QσηIσηQ,(21)σβ2=4E1−ρ2EχI−χ~I2σηI2+EχQ−χ~Q2σηQ2−2ρχI−χ~IχQ−χ~QσηIσηQ.Using (18) and (20), the conditional PEP (CPEP) can be written as in (22) on the top of the page, where Qx denotes the Q-function defined as Qx=1/2π∫x∞exp−u2/2du.(22)CPEPopt=QP41−ρ2χI−χ~I2σηI2+χQ−χ~Q2σηQ2−2ρχI−χ~IχQ−χ~QσηIσηQ=QPY21−ρ2N0,


## 3.1.2. Traditional ML Detector

Note that the improper behavior of the noise is not considered for the traditional ML detector. Thus, assumingx is transmitted, the PEP of deciding in favor of x~ is given from the classical ML detector expression in (17) as(23)PEPtra=PryIQ−Pχ2>yIQ−Pχ~2=PrD>Eχ−χ~2,where D=−2RPχ−χ~η∗ which has zero mean and σD2=4PχI−χ~I2σηI2+4PχQ−χ~Q2σηQ2+8ρσηIσηQPχI−χ~IχQ−χ~Q as variance. Hence, the CPEP using Q function is presented in (24).(24)CPEPtra=QPχI−χ~I44χI−χ~I2σηI2+4χQ−χ~Q2σηQ2+8ρσηIσηQχI−χ~IχQ−χ~Q.


## 3.2. Average Error Probability

### 3.2.1. Optimal ML Detector

To derive the average PEP (APEP), (22) should be averaged over the PDF of Y which can be depicted as:(25)APEPopt=∫0∞QPY41−ρ2fYYdY.However, note thatY=Ψ12+Ψ22−2ρΨ1Ψ2 where Ψ1=χI−χ~I and Ψ2=χQ−χ~Q/ξr. Indeed, Ψ1 and Ψ2 depend on real and imaginary parts of S and I given in (6). Thus, based on central limit theorem (CLT) and for large number of reflectors N, Y is considered as a combination of correlated noncentral chi-squared random variables. The determination of the APEP using the Y PDF is very hard. Hence, the APEP is numerically computed by averaging PEP over large number of channel realization.


### 3.2.2. Traditional ML Detector

Traditional ML detector APEP based on (24) is difficult and even impossible to determinate. However, it will be numerically calculated with the same method used for the optimal ML detector.


## 3.2.1. Optimal ML Detector

To derive the average PEP (APEP), (22) should be averaged over the PDF of Y which can be depicted as:(25)APEPopt=∫0∞QPY41−ρ2fYYdY.However, note thatY=Ψ12+Ψ22−2ρΨ1Ψ2 where Ψ1=χI−χ~I and Ψ2=χQ−χ~Q/ξr. Indeed, Ψ1 and Ψ2 depend on real and imaginary parts of S and I given in (6). Thus, based on central limit theorem (CLT) and for large number of reflectors N, Y is considered as a combination of correlated noncentral chi-squared random variables. The determination of the APEP using the Y PDF is very hard. Hence, the APEP is numerically computed by averaging PEP over large number of channel realization.


## 3.2.2. Traditional ML Detector

Traditional ML detector APEP based on (24) is difficult and even impossible to determinate. However, it will be numerically calculated with the same method used for the optimal ML detector.


## 4. Numerical Results and Discussions

In this part, the APEP performance of the optimal and traditional ML detector for RIS under IQI based on the proposed analytic scenarios will be presented and proved with simulation results.First, the determined expression of SIR is evaluated in Figures2 and 3 for different values, respectively, of gains εt,εr and phases ϕt,ϕr of IQI and IRR=20dB. It can be depicted that a considerable SIR value degradation occurs even for small imbalance values which highlight the importance of taking into account the IQI in RIS performance analysis.Figure 2
SIR values variation of RIS under IQI in function of phases with fixed gainsεt=εr=0.8,0.9,1.Figure 3
SIR values variation of RIS under IQI in function of gains with fixed phasesϕt=ϕr=0∘,10∘,20∘.Figures4 and 5 are presented to show the effects of, respectively, amplitudes and phases of Tx and Rx IQI on APEP performance. Indeed, in Figure 4, the APEP is analyzed for fixed amplitudes values εt=εr=3dB and for variable values of phase imbalance ϕt=ϕr=10∘,15∘,25∘. Accordingly, the effects of increasing the amplitude imbalance from 1dB to 5dB with fixed phases values ϕt=ϕr=25∘ is given in Figure 5. It can be depicted that increasing the phases and amplitudes of IQI cause a considerable performance loss. Further, the analytical analysis matches perfectly the simulation results for both ML detectors, which validates the proposed analysis. Additionally, it is observed that the proposed ML detector outperforms the classical ML detector for all IQI values and a considerable gain is achieved by using the optimal detector.Figure 4
RIS APEP of optimal and traditional ML detector under IQI in function of phases with fixed gainsεt=εr=3dB using N=32 and 4-QAM.Figure 5
RIS APEP of optimal and traditional ML detector under IQI in function of gains with fixed phasesϕt=ϕr=10∘ using N=32 and 4-QAM.Figure6 shows APEP simulation results of standard ML and optimal ML under only Tx imbalance for different values of εt,ϕt with N=32. In such case, εr=ϕr=0∘. It is worth mentioning that the optimal design improves the system performance and decreases the IQI effects. The destructive effects of Tx imbalance degrade the performance of perfect RIS. Actually, comparing the ideal RIS with the optimal ML detector performance, the system degrades about 6 dB when εt=5dB and ϕt=25∘, while the degradation is about 1 dB when εt=2dB and ϕt=15∘ at APEP=10−2 .Figure 6
RIS APEP of optimal and traditional ML detector in the presence of Tx IQI withN=32 using 4-QAM.In Figure7, the impact of Rx IQI on RIS with perfect Tx IQI parameters (εt=0,ϕt=0∘) is carried for different εr,ϕr and fixed N=32. For comparison reasons, the case of ideal IQ is also presented. It can be seen the harmful effects of increasing the IQI values on the system performance. Indeed, a 2 dB performance degradation is shown when εr=5dB and ϕr=25∘ for optimal ML compared to the ideal case at APEP=10−2. It can be noticed that there is a performance gain of the designed ML detector compared with the traditional one.Figure 7
RIS APEP of optimal and traditional ML detector in the presence of Rx IQI withN=32 using 4-QAM.In order to put in evidence the impact of RIS reflector number, Figure8 presents the numerical and simulation results of optimal and traditional ML detectors under joint Tx and Rx IQI when the number of reflectors changes among 16, 32, and 64. This figure shows the accuracy of the presented analysis for various reflectors numbers. A remarkable performance gain is achieved by using the optimal ML detector. However, even if we increase the number of reflectors, IQI will harm the system performance. For instance, Figure 8 illustrates that if the system has 16 reflectors, the optimal performance degrades by 12 dB when εt=εr=5dB and ϕt=ϕr=20∘ compared with the perfect IQ matching system at 10−3 APEP. This degradation has less impact when increasing the number of reflectors. Hence, it can be concluded that increasing the SNR or the number of reflectors enhances the system performance.Figure 8
RIS APEP of optimal and traditional ML detector in the presence of joint Tx and Rx IQI withεt=εr=5dB, ϕt=ϕr=20∘, and N=16,32,64 using 4-QAM.Figure9 shows the effects of changing IQI parameters at joint Tx and Rx on optimal and standard ML detector performances. The results are carried for fixed IQI parameters at the Tx and Rx εt=εr=5dB, ϕt=ϕr=10∘ and for different values such as εt=5dB,εr=3dB and ϕt=20∘,ϕr=10∘ with N=32. It can be depicted that the optimal ML detector under equal Tx and Rx IQI parameters outperforms the optimal ML under different values due to the additional mismatch.Figure 9
RIS APEP of optimal and traditional ML detector in the presence of joint Tx and Rx IQI withεt=εr=5dB, ϕt=ϕr=10∘ and εt=5dB,εr=3dB, ϕt=20∘,ϕr=10∘, and N=32 using 4-QAM.Figure10 validates the derived expressions of optimal and traditional ML detectors for a variable number of modulation order with fixed RIS-IQI parameters εt=εr=3dB, ϕt=ϕr=10∘, and N=32. It can be shown that RIS using conventional ML detector with increased modulation order suffers from an error floor in the high SNR region due to the fact of ignoring the noise behavior. Consequently, in the presence of IQI, RIS with a traditional ML detector could not support high order modulation which is not acceptable in practice. A careful study for high data rate RIS-aided communication systems in the presence of IQI is required. However, the designed optimal ML detector prevents the error floor and decreases the IQI impairments.Figure 10
RIS APEP of optimal and traditional ML detector in the presence of joint Tx and Rx IQI withεt=εr=3dB, ϕt=ϕr=10∘, and N=32 using different modulation order.


## 5. Conclusion

This paper investigates the harmful effects of IQI on RIS-aided communication. Hence, novel analytical expressions of PEP and the APEP for optimal and traditional ML detectors are derived. Consequently, the behavior of the designed detectors is analyzed in function of Tx and Rx IQ parameters to further characterize the effects of IQI. Finally, numerical results are illustrated to prove the efficiency of the proposed analysis. The carried results highlight RIS performance degradation’s with the presence of IQI even in high SNR regions and an increased number of reflectors. It was proved for the different carried cases that optimal ML detector performs far better than conventional detection method. However, it is worth noting that an optimal detector presents a significant computational complexity. Hence, studying IQI compensation algorithms with low complexity for RIS systems could be potential future works. Moreover, we intend to analyze the performance of RIS under IQI with the presence and absence of direct links between the source and the destination.



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*Source: 6673807-2021-08-15.xml*