# Performance Analysis of IQI Impaired Cooperative NOMA for 5G-Enabled Internet of Things

Authors: Hui Guo; Xuejiao Guo; Chao Deng; Shangqing Zhao
Journal: Wireless Communications and Mobile Computing
 (2020)

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## Abstract

This paper investigates the joint effects of in-phase and quadrature-phase imbalance (IQI) and imperfect successive interference cancellation (ipSIC) on the cooperative Internet of Things (IoT) nonorthogonal multiple access (NOMA) networks where the Nakagami-m fading channel is taken into account. The closed-form expressions of outage probability for the far and near IoT devices are derived to evaluate the outage behaviors. For deeper insights of the performance of the considered system, the approximate outage probability and diversity order in high signal-to-noise ratio (SNR) regime are obtained. In addition, we also analyze the throughput and energy efficiency to characterize the performance of the considered system. The simulation results demonstrate that, compared with IQI, ipSIC has a greater impact on the outage performance for the near-IoT-device of the considered system. Furthermore, we also find that the outage probabilities of IoT devices can be minimized by selecting a specific power allocation scheme.


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## Body

## 1. Introduction

With the development of Internet of Things (IoT), traditional orthogonal multiple access (OMA) can no longer afford massive connections due to its low spectrum utilization [1]. The proposals of massive multiple-input multiple-output (MIMO) [2], small cell networks (SCNs) [3], millimeter wave (mmWave) [4], nonorthogonal multiple access (NOMA) [5], and some other 5G-related technologies have made it possible to implement the IoT [6]. Among these technologies, NOMA has been accepted by the Third Generation Partnership Project (3GPP) due to the fact that it can serve multiple devices simultaneously without neglecting fairness [7]. Traditional OMA is required to use orthogonal resources to support multiple devices, while relying on the power multiplexing of NOMA IoT can serve a large number of various devices in the same time/frequency domain [8, 9]. Furthermore, the physical layer security performance of NOMA systems outperforms the OMA system and improves the security rate of the system greatly [10]. In addition, to ensure the fairness among devices, different power signals are transmitted according to the channel state information (CSI) between the transmitter (TX) and receiver (RX), and the successive interference cancellation (SIC) is adopted at the RX to eliminate interdevice interference [11, 12].To further enhance the robustness and expand coverage, cooperative relay communication has been introduced into NOMA systems [13–17]. In [13], cooperative relay was taken into account in the NOMA systems, and it has been proved that the proposed scheme can reduce the outage probability and increase the diversity gain of the systems. In order to improve the spectral efficiency, the authors of [14] proposed a new cooperative NOMA transmission scheme in the cognitive radio systems and derived the exact closed-form expressions for outage probability. The outage performance of cooperative NOMA amplify-and-forward (AF) and decode-and-forward (DF) system with a single user was studied in [15], and the results showed that the relay in AF protocol outperforms the relay in DF protocol. In [16], the outage probability of cooperative NOMA system over Rayleigh channels was analyzed where the near user acted as a DF relay. Particularly, the cooperative NOMA network in the IoT was proposed in [17]; the average throughput and the diversity order were analyzed to characterize the system. In [18], the authors proposed a cooperative NOMA scheme in which the near user acts as a DF relay and the analytical expression of intercept probability was derived to evaluate the system. All of the above studies showed that the cooperative NOMA can improve the system performance. As the core technology of NOMA, SIC has well performance on signal detection, while at the expense of the complexity of RX [19]. To this end, a host of works has drawn attention to imperfect SIC (ipSIC) ([20–23] and references therein). Authors in [20] proposed a more practical network in which ipSIC was considered in the cooperative NOMA system, and the closed-form expressions of the outage probability were derived. In [21], the authors presented the closed-form expressions of outage probability and ergodic sum rate of the cooperative AF NOMA over Nakagami-m channels by considering ipSIC in the network. In addition, the effects of ipSIC on the two-way and cognitive radio NOMA system were studied in [22, 23], respectively. All of these articles are instructive for the practical NOMA systems.The aforementioned works mainly focused on the perfect radio frequency (RF) which is overidealistic. In practice, in-phase and quadrature-phase imbalance (IQI) inevitably occurs due to component mismatch or manufacturing process problems [24]. Although the influence of IQI can be mitigated by compensation algorithms and correction algorithms, it cannot be completely eliminated due to the different forms of noise [25–28]. The authors in [29] have demonstrated that IQI has negative effects on the NOMA systems and cannot be ignored. To solve this problem, authors in [30] studied the impact of IQI on the two-way cooperative AF relay systems and derived the closed-form and asymptotic expressions for the outage probability. The effects of TX IQI, RX IQI, and joint TX/RX IQI on the outage performance of the single-hop NOMA system were compared in [31]. The performance of cooperative NOMA system with IQI at the relay was analyzed by deriving closed-form expressions for the outage probability [32]. Recently, the impact of IQI on the cooperative NOMA DF network over Rayleigh channels has been considered in [33], while the influence of IQI on the source node was not taken into account. Moreover, the direct link between the source and the far user was not considered in [33].Motivated by the previous discussion, in this paper, we discuss the effect of IQI on the cooperative IoT NOMA AF relaying system over Nakagami-m fading channels, and ipSIC is taken into account as well. In the considered system, the source node can communicate with the destination nodes with the aid of an AF relay or directly. To demonstrate the performance of the considered system, the exact analytical expressions of the outage probability are derived. In order to gain more insights, asymptotic analyses and diversity orders are calculated. Finally, the throughput and energy efficiency are also performed. The main contributions of this paper are summarized as follows:
(i)
The exact outage probability expressions of the near-IoT-device and far-IoT-device of the considered system are derived by considering IQI at source, relay, and IoT devices; meanwhile, ipSIC is taken into account(ii)
In order to obtain more insights, the asymptotic outage performance is analyzed. For more intuitive evaluation of the considered system performance, the diversity order is further obtained(iii)
Finally, we deduce the throughput and energy efficiency of the considered network. The results confirm that (1) the system throughput will increase when the signal-to-noise ratio (SNR) is increasing, while approaches to a fixed value in the high SNR regime; (2) the energy efficiency will increase all the time in the ideal conditions, while in the nonideal conditions, the energy efficiency will reach the upper bound due to the existence of IQI and ipSICThe rest of this paper is organized as follows. Section2 describes the cooperative IoT NOMA systems with IQI and ipSIC; Section 3 discusses the exact and asymptotic outage performance, the diversity order, the system throughput, and the energy efficiency. The numerical results are presented in Section 4 before we conclude the paper in Section 5.


### 1.1. Notation

In this paper,E· denotes the expectation operator; fX· and FX· are the probability density function (PDF) and the cumulative distribution function (CDF) of a random variable X, respectively; Γ· is the Gamma function; Pr· represents the probability of a random variable; Kv· symbolizes the vth-order modified Bessel function of the second kind, and ≜ stands for the definition operator.


## 1.1. Notation

In this paper,E· denotes the expectation operator; fX· and FX· are the probability density function (PDF) and the cumulative distribution function (CDF) of a random variable X, respectively; Γ· is the Gamma function; Pr· represents the probability of a random variable; Kv· symbolizes the vth-order modified Bessel function of the second kind, and ≜ stands for the definition operator.


## 2. System Model

As can be seen from Figure1, we consider a single-carrier cooperative IoT NOMA network, which consists of one source S, one AF relay R, and multiple IoT devices. In the considered system, it is infeasible to study all devices due to the extremely high complexity. Therefore, under normal circumstances, all devices are divided into multiple clusters and OMA is used for the intercluster while NOMA is utilized within clusters (In this paper, we only study the devices in the same cluster.). In one cluster, the devices are classified into two groups, namely, the far-IoT -device Df and the near-IoT-device Dn (The near-IoT-device and the far-IoT-device are distinguished by the geographic locations of the devices away from the source node, and this case of the two-user downlink NOMA system has been widely considered in existing works, e.g., see Ref. [34–36].). It is assumed that all nodes are equipped with a single antenna and work in the half-duplex mode. The channel coefficients and channel gains between source and relay, source and destinations, and relay and destinations are denoted as hSR, hSDn, hSDf, hRDf, hRDn and ρSR, ρSDn, ρSDf, ρRDf, ρRDn, respectively. Without loss of generality, the channel gains between source and M destinations are sorted as ρSD1≤ρSD2≤⋯≤ρSDM.Figure 1
System model of cooperative IoT NOMA networks.


### 2.1. Signal Transmission

The communication procedure is divided into two time slots. In the first time slot,S transmits signals yS=a1PSx1+a2PSx2 to R, Df, and Dn, where PS is the transmit power of S, x1 and x2 are the desired signals of Df and Dn, satisfying with Ex12=Ex22=1, and a1 and a2 are the power allocation coefficients with a1+a2=1 and a1>a2. Thus, in the first time slot, the received signals at R, Df, and Dn can be expressed as (The subscripts S, R, Df, and Dn in μt, νt, μr, and νr represent IQIs caused by source, relay, the far-IoT-device, and the near-IoT-device, respectively.)
(1)ySR=μrRhSRμtSyS+νtSyS∗+n0+νrR×hSRμtSyS+νtSyS∗+n0∗,ySDf=μrDfhSDfμtSyS+νtSyS∗+n0+νrDf×hSDfμtSyS+νtSyS∗+n0∗,ySDn=μrDnhSDnμtSyS+νtSyS∗+n0+νrDn×hSDnμtSyS+νtSyS∗+n0∗,where n0∼CN0,N0 denotes the additive white Gaussian noise (AWGN), μr, νr and μt, νt are the IQI coefficients caused by the RX and TX which can be expressed as
(2)μt=121+ζtexpjϕt,νt=121−ζtexp−jϕt,μr=121+ζrexp−jϕr,νr=121−ζrexpjϕr,where ζt and ζr are the amplitude mismatch levels caused by TX and RX, while ϕt and ϕr are the phase mismatch levels and j=−1 denotes the unit of imaginary. Furthermore, the TX/RX image rejection ratio (IRR) is expressed as
(3)IRRt/r=μt/r2νt/r2.According to the NOMA protocol, the received signal-to-interference-plus-noise ratio (SINR) forDf and Dn to decoded x1 is expressed as
(4)γSDf=AfρSDfa1γ1a2ρSDfAfγ1+ρSDfBfγ1+Cf,(5)γSDn→f=AnρSDna1γ1a2ρSDnAnγ1+ρSDnBnγ1+Cn,where Af=μrDf2μtS2+νrDf2νtS2, An=μrDn2μtS2+νrDn2νtS2, Bf=μrDf2νtS2+νrDf2μtS2, Bn=μrDn2νtS2+νrDn2μtS2, Cf=μrDf2+νrDf2, Cn=μrDn2+νrDn2, and γ1=PS/N0 denotes the transmit SNR caused by S. Based on the ipSIC, the received SINR of Dn to decoded x2 can be expressed as
(6)γSDn=a2AnρSDnγ1a2BnρSDnγ1+a1gSDnAnγ1+Cn,where gSDn∼CN0,ερSDn and ε∈0,1 is the parameter of ipSIC which follows the Gaussian distribution [37]. Note that in this system the value of ε cannot take the value of 1, because the network is completely out of the NOMA scheme when ε=1, which is clearly contradictory to the considered system.In the second time slot, the relay amplifies and forwards the received signals to the IoT devices. Thus, the received signals atDf and Dn can be expressed as
(7)yRDf=μrDfhRDfμtRGySR+νtRGySR∗+n0+νrDfhRDfμtRGySR+νtRGySR∗+n0∗,yRDn=μrDnhRDnμtRGySR+νtRGySR∗+n0+νrDnhRDnμtRGySR+νtRGySR∗+n0∗,where G=PR/QPSρSR+N0 represents the amplifying gain factor and PR is the transmit power by R and Q=μtR2+νtR2μrR2+νrR2. The received SINRs for Df to decode x1 and for Dn to decode x1 and x2 are expressed as
(8)γRDf=a1EfρSRρRDfγ1γ2a2Ef+JfρSRρRDfγ1γ2+TfρSRγ1+LfρRDfγ2+Tf,(9)γRDn→f=a1EnρSRρRDnγ1γ2a2En+JnρSRρRDnγ1γ2+TnρSRγ1+LnρRDnγ2+Tn,(10)γRDn=a2EnρSRρRDnγ1γ2a2Jn+a1εEnρSRρRDnγ1γ2+LnρRDnγ2+TnρSRγ1+Tn,where γ1=PR/N0 represents the transmit SNR caused by R, Ef=μrR2μtS2+νrR2νtS2μrDf2μtR2+νrDf2νtR2+μrR2νtS2+μtS2νrR2μrDf2νtR2+μtR2νrDf2, En=μrR2μtS2+νrR2νtS2μrDn2μtR2+νrDn2νtR2+μrR2νtS2+μtS2νrR2μrDn2νtR2+μtR2νrDn2, Jf=μrR2νtS2+μtS2×νrR2μrDf2μtR2+νrDf2νtR2+μrDf2νtR2+μtR2νrDf2μrR2μtS2+νrR2νtS2, Jn=μrR2νtS2+μtS2νrR2μrDn2μtR2+νrDn2×νtR2+μrDn2νtR2+μtR2νrDn2μrR2μtS2+νrR2νtS2, Lf=2μrDf2μtR2+2μrDf2νtR2+2νrDf2μtR2+2νrDf2νtR2, Ln=2μrDn2μtR2+2μrDn2νtR2+2νrDn2μtR2+2νrDn2νtR2, Tf=QCf, and Tf=CnQ.


### 2.2. Fading Channels

Assume that all the channel gains follow Nakagami-m distribution; thus, the PDF and CDF of ρi can be expressed as [38]
(11)fρix=xαi−1Γαiβiαie−x/βi,(12)Fρix=1−∑gi=0αi−1e−x/βigi!xβigi,where αi denotes the multipath fading parameter while βi represents the control spread parameter. By utilizing the order statistics, the PDF and CDF of the m-th devices’ channel gain ρm can be expressed as
(13)fρmx=M!m−1!M−m!Fρixm−1×fρix1−FρixM−m,(14)Fρmx=M!m−1!M−m!∑z=0M−mM−mz×−1zm+zFρixm+z,where M denotes the total numbers of IoT devices.


## 2.1. Signal Transmission

The communication procedure is divided into two time slots. In the first time slot,S transmits signals yS=a1PSx1+a2PSx2 to R, Df, and Dn, where PS is the transmit power of S, x1 and x2 are the desired signals of Df and Dn, satisfying with Ex12=Ex22=1, and a1 and a2 are the power allocation coefficients with a1+a2=1 and a1>a2. Thus, in the first time slot, the received signals at R, Df, and Dn can be expressed as (The subscripts S, R, Df, and Dn in μt, νt, μr, and νr represent IQIs caused by source, relay, the far-IoT-device, and the near-IoT-device, respectively.)
(1)ySR=μrRhSRμtSyS+νtSyS∗+n0+νrR×hSRμtSyS+νtSyS∗+n0∗,ySDf=μrDfhSDfμtSyS+νtSyS∗+n0+νrDf×hSDfμtSyS+νtSyS∗+n0∗,ySDn=μrDnhSDnμtSyS+νtSyS∗+n0+νrDn×hSDnμtSyS+νtSyS∗+n0∗,where n0∼CN0,N0 denotes the additive white Gaussian noise (AWGN), μr, νr and μt, νt are the IQI coefficients caused by the RX and TX which can be expressed as
(2)μt=121+ζtexpjϕt,νt=121−ζtexp−jϕt,μr=121+ζrexp−jϕr,νr=121−ζrexpjϕr,where ζt and ζr are the amplitude mismatch levels caused by TX and RX, while ϕt and ϕr are the phase mismatch levels and j=−1 denotes the unit of imaginary. Furthermore, the TX/RX image rejection ratio (IRR) is expressed as
(3)IRRt/r=μt/r2νt/r2.According to the NOMA protocol, the received signal-to-interference-plus-noise ratio (SINR) forDf and Dn to decoded x1 is expressed as
(4)γSDf=AfρSDfa1γ1a2ρSDfAfγ1+ρSDfBfγ1+Cf,(5)γSDn→f=AnρSDna1γ1a2ρSDnAnγ1+ρSDnBnγ1+Cn,where Af=μrDf2μtS2+νrDf2νtS2, An=μrDn2μtS2+νrDn2νtS2, Bf=μrDf2νtS2+νrDf2μtS2, Bn=μrDn2νtS2+νrDn2μtS2, Cf=μrDf2+νrDf2, Cn=μrDn2+νrDn2, and γ1=PS/N0 denotes the transmit SNR caused by S. Based on the ipSIC, the received SINR of Dn to decoded x2 can be expressed as
(6)γSDn=a2AnρSDnγ1a2BnρSDnγ1+a1gSDnAnγ1+Cn,where gSDn∼CN0,ερSDn and ε∈0,1 is the parameter of ipSIC which follows the Gaussian distribution [37]. Note that in this system the value of ε cannot take the value of 1, because the network is completely out of the NOMA scheme when ε=1, which is clearly contradictory to the considered system.In the second time slot, the relay amplifies and forwards the received signals to the IoT devices. Thus, the received signals atDf and Dn can be expressed as
(7)yRDf=μrDfhRDfμtRGySR+νtRGySR∗+n0+νrDfhRDfμtRGySR+νtRGySR∗+n0∗,yRDn=μrDnhRDnμtRGySR+νtRGySR∗+n0+νrDnhRDnμtRGySR+νtRGySR∗+n0∗,where G=PR/QPSρSR+N0 represents the amplifying gain factor and PR is the transmit power by R and Q=μtR2+νtR2μrR2+νrR2. The received SINRs for Df to decode x1 and for Dn to decode x1 and x2 are expressed as
(8)γRDf=a1EfρSRρRDfγ1γ2a2Ef+JfρSRρRDfγ1γ2+TfρSRγ1+LfρRDfγ2+Tf,(9)γRDn→f=a1EnρSRρRDnγ1γ2a2En+JnρSRρRDnγ1γ2+TnρSRγ1+LnρRDnγ2+Tn,(10)γRDn=a2EnρSRρRDnγ1γ2a2Jn+a1εEnρSRρRDnγ1γ2+LnρRDnγ2+TnρSRγ1+Tn,where γ1=PR/N0 represents the transmit SNR caused by R, Ef=μrR2μtS2+νrR2νtS2μrDf2μtR2+νrDf2νtR2+μrR2νtS2+μtS2νrR2μrDf2νtR2+μtR2νrDf2, En=μrR2μtS2+νrR2νtS2μrDn2μtR2+νrDn2νtR2+μrR2νtS2+μtS2νrR2μrDn2νtR2+μtR2νrDn2, Jf=μrR2νtS2+μtS2×νrR2μrDf2μtR2+νrDf2νtR2+μrDf2νtR2+μtR2νrDf2μrR2μtS2+νrR2νtS2, Jn=μrR2νtS2+μtS2νrR2μrDn2μtR2+νrDn2×νtR2+μrDn2νtR2+μtR2νrDn2μrR2μtS2+νrR2νtS2, Lf=2μrDf2μtR2+2μrDf2νtR2+2νrDf2μtR2+2νrDf2νtR2, Ln=2μrDn2μtR2+2μrDn2νtR2+2νrDn2μtR2+2νrDn2νtR2, Tf=QCf, and Tf=CnQ.


## 2.2. Fading Channels

Assume that all the channel gains follow Nakagami-m distribution; thus, the PDF and CDF of ρi can be expressed as [38]
(11)fρix=xαi−1Γαiβiαie−x/βi,(12)Fρix=1−∑gi=0αi−1e−x/βigi!xβigi,where αi denotes the multipath fading parameter while βi represents the control spread parameter. By utilizing the order statistics, the PDF and CDF of the m-th devices’ channel gain ρm can be expressed as
(13)fρmx=M!m−1!M−m!Fρixm−1×fρix1−FρixM−m,(14)Fρmx=M!m−1!M−m!∑z=0M−mM−mz×−1zm+zFρixm+z,where M denotes the total numbers of IoT devices.


## 3. Outage Probability Analysis

In this section, the outage performances ofDf and Dn are discussed. We first analyze the considered system by deriving the exact expressions of the outage probability and then formulate the asymptotic analysis in the high SNR. In addition, the throughput of the considered system is also explored.


### 3.1. Exact Outage Probability

ForDf, the outage event will occur when Df fails to decode the expected signal x1 transmitted from S and R. Therefore, the outage probability of Df can be expressed as
(15)PoutDf=PrγSDf<γthfPrγRDf<γthf,where γthf is the target threshold at Df. The closed-form expression of outage probability of Df under IQI and ipSIC is provided in the following theorem.Theorem 1.
The exact outage probability expression ofDf in the presence of IQI and ipSIC is expressed as
(16)PoutDf=bf∑z=0M−fM−fz−1zf+z1−e−θ/βSDf∑g5=0αSDf−11g5×θβSDfg5f+z1−2ΓαRDfβRDfαRDf∑g1=0αSR−1∑t1=0αRDf−1∑t2=0g11g1!×g1g2αRDf−1t1βSRγ1Tf−g1−t1−t2+1/2φg1+αRDf−t1+t2+1/2×Tf+Lfγ2φt1+t2+1/2βRDft1−t2+1/2Lfγ2g1−t2×e−φ/βRDf−Lfγ2φ/βSRTfγ1Kt1−t2+12Tf+Lfγ2φφβSRβRDfTfγ1,where f denotes the f-th device (the far device), bf≜M!/f−1!M−f!, θ≜Cfγthf/a1Afγ1−a1Af+Bfγ1γthf with Afa1>a1Af+Bfγthf, and φ≜Tfγthf/a1Efγ2−a2Ef+Jfγ2γthf with a1Ef>a2Ef+Jfγthf.Proof.
See AppendixA.
ForDn, the outage event will occur when Dn fails to decode either the signals x1 or x2 transmitted from S or R. Therefore, the outage probability of Dn can be expressed as
(17)PoutDn=1−PrγSDn→f>γthf,γSDn>γthns×1−PrγRDn→f>γthf,γRDn>γthn,where γthn is the target threshold at Dn. The closed-form expression of outage probability of Dn under IQI and ipSIC is provided in the following theorem.Theorem 2.
The exact outage probability expression ofDn in the presence of IQI and ipSIC is expressed as
(18)PoutDn=bn∑z=0M−nM−nz−1zn+z1−∑g4=0αSDn−11g4!e−ξ/βSDn×ξβSDng4n+z1−2αRDnβRDnαRDn∑g1=0αSR−1∑q1=0αRDn−1∑q2=0g11g1!×g1q2αRDn−1q1βSRγ1Tn−g1−q1−q2+1/2τg1+αRDn−q1+q2+1/2×Tn+Lnγ2τq1+q2+1/2βRDnq1−q2+1/2Lnγ2g1−q2×e−τ/βRDn−Lnγ2τ/βSRTnγ1Kq1−q2+12Tn+Lnγ2ττβSRβRDnTfγ1,where n denotes the n-th device (the near device) and bn≜M!/n−1!M−n!, ξ≜maxξ1,ξ2, ξ1≜Cfγthf/a1Anγ1−a1An+Bnγ1γthf, ξ2≜Cnγthn/a2Anγ1−γ1γthna2Bn+a1εAnBn+a1εAn, τ=maxτ1,τ2, τ1≜Tnγ1γthf/a1Enγ2−a2En+Jnγ2γthf, and τ2≜Tnγthn/a2Enγ2−a2Jn+a1εEnγ2γthn.Proof.
See AppendixB.Remark 3.
From Theorems1 and 2, we can observe that the outage probabilities of Df and Dn are determined by IQI parameters, the performance of SIC, fading parameters, and distortion noise. Meanwhile, it is worth noting that when ςt=ςr=1, ϕt=ϕr=0∘, and ε=0 are all satisfied, the considered system reduces to ideal conditions.


### 3.2. Asymptotic Outage Probability

In this subsection, the asymptotic outage probabilities ofDf and Dn are analyzed to obtain more insights on the outage behavior of the considered system. The asymptotic CDF of the channel gain ρi in the case of ordering and nonordering can be expressed as [39]
(19)fρi∞x=xαiαi!βiαi,(20)Fρi∞x=M!M−m!m!1αimxβimαi.The asymptotic expressions of outage probabilities ofDf and Dn are described in the following corollaries.Corollary 4.
The asymptotic closed form of outage probability ofDf in the high SNR regime is expressed as
(21)PoutDf,∞=bf∞1αSDf!fθβSDfαSDff×φ′αSRαSR!βSRαSR+φ″αRDfαRDf!βRDfαRDf,where bf∞=M!/f!M−f!, φ′=Lfγthf/a1Efγ1−a2Ef+Jfγ1γthfEfγ1−a2Ef+Jfγ1γthf, and φ″=Tfγthf/a1γ2Ef−a2Ef+Jfγ2γthf.Proof.
See AppendixC.Corollary 5.
The asymptotic closed form of outage probability ofDn in the high SNR regime is expressed as
(22)PoutDn,∞=bn∞1αRDn!nξβRDnnαSDn×τ′αSRαSR!βSRαSR+τ″αRDnαRDn!βRDnαRDn,where bn∞=M!/n!M−n!, τ′=maxτ1′,τ2′, τ″=maxτ1″,τ2″, τ1′≜Lnγthf/a1Enγ1−γ1γthfa2En+Jn, τ1″≜Tnγthf/a1Enγ2−γ2γthfa2En+Jn, τ2′≜Lnγthn/a2Enγ1−a2Jnγthnγ1, and τ2″≜Tnγthn/a2Enγ2−a2Jnγthnγ2.Proof.
See AppendixD.Remark 6.
The results of Corollaries4 and 5 show the effects of channel fading parameters, IQI parameters, the performance of SIC, and distortion noise on asymptotic outage performances of Df and Dn. In addition, we can also observe that the asymptotic outage probability is related to the order of user arrangement directly.


### 3.3. Diversity Order

The diversity order can reflect the trend of the outage probability intuitively. To this end, the diversity order of the considered system is explored in this subsection. The diversity order is defined as [40]
(23)d=−limγ→∞logPoutDm,∞logγ,where PoutDm,∞ denotes asymptotic outage probability of m‐th device and γ∈γ1,γ2 represents the transmitted SNR.Corollary 7.
Based on (23), the diversity orders of Df and Dn in ideal conditions (ςt=ςr=1, ϕt=ϕr=0∘, and ε=0) and nonideal conditions (ςt,ςr≠1, ϕt,ϕr≠0∘, and ε≠0) can be written as
(24)dfid=dfnid=minfαSDfαSR,fαSDfαRDf,dnid=dnnid=minnαSDnαSR,nαSDnαRDn.Remark 8.
The results show that the diversity order ofDf is the minimum of fαSDfαSR and fαSDfαRDf, which indicates that the diversity order of Df is relative to αSDf, αSR, αRDf, and f, while the value is affected by the multipath fading parameters αSR and αRDf of S⟶R and R⟶Df. Similarly, the diversity order of Dn is the minimum of nαSDnαSR and nαSDnαRDn, which is related to αSDn, αSR, αRDn, and n, and the final value is jointly determined by αSR and αRDn. The results also show that the outage probabilities of Df and Dn of the considered system will always decrease with the increase of SNR. In addition, we can also observe that although the outage probabilities of IoT devices in ideal and nonideal conditions are different, the trends are the same due to the diversity orders.


### 3.4. Throughput Analysis

The system throughput is another measure of system performance, which is the number of signals transmitted per unit of time successfully. Thus, the system throughput is formulated as [41]
(25)T=∑m=1M1−PoutDmRthm,where PoutDm is the outage probability of Dm which can be obtained from (16) and (18), Rthm=1/2log1+γthm denotes the target rate of Dm, and 1/2 represents that the communication transmit process is divided into two time slots.


### 3.5. Energy Efficiency

To further evaluate the performance of the considered system, the energy efficiency is analyzed in this subsection. Energy efficiency refers to the useful signals that IoT devices received for each unit of energy consumed byS. Hence, the energy efficiency of the considered system can be expressed as [42]
(26)ηee=TPEE,where PEE=PS+PR+PC denotes total energy consumption and PC is the fixed energy consumption which is caused by transmitter and the IoT devices and T is the system throughput which can be obtained from (25).


## 3.1. Exact Outage Probability

ForDf, the outage event will occur when Df fails to decode the expected signal x1 transmitted from S and R. Therefore, the outage probability of Df can be expressed as
(15)PoutDf=PrγSDf<γthfPrγRDf<γthf,where γthf is the target threshold at Df. The closed-form expression of outage probability of Df under IQI and ipSIC is provided in the following theorem.Theorem 1.
The exact outage probability expression ofDf in the presence of IQI and ipSIC is expressed as
(16)PoutDf=bf∑z=0M−fM−fz−1zf+z1−e−θ/βSDf∑g5=0αSDf−11g5×θβSDfg5f+z1−2ΓαRDfβRDfαRDf∑g1=0αSR−1∑t1=0αRDf−1∑t2=0g11g1!×g1g2αRDf−1t1βSRγ1Tf−g1−t1−t2+1/2φg1+αRDf−t1+t2+1/2×Tf+Lfγ2φt1+t2+1/2βRDft1−t2+1/2Lfγ2g1−t2×e−φ/βRDf−Lfγ2φ/βSRTfγ1Kt1−t2+12Tf+Lfγ2φφβSRβRDfTfγ1,where f denotes the f-th device (the far device), bf≜M!/f−1!M−f!, θ≜Cfγthf/a1Afγ1−a1Af+Bfγ1γthf with Afa1>a1Af+Bfγthf, and φ≜Tfγthf/a1Efγ2−a2Ef+Jfγ2γthf with a1Ef>a2Ef+Jfγthf.Proof.
See AppendixA.
ForDn, the outage event will occur when Dn fails to decode either the signals x1 or x2 transmitted from S or R. Therefore, the outage probability of Dn can be expressed as
(17)PoutDn=1−PrγSDn→f>γthf,γSDn>γthns×1−PrγRDn→f>γthf,γRDn>γthn,where γthn is the target threshold at Dn. The closed-form expression of outage probability of Dn under IQI and ipSIC is provided in the following theorem.Theorem 2.
The exact outage probability expression ofDn in the presence of IQI and ipSIC is expressed as
(18)PoutDn=bn∑z=0M−nM−nz−1zn+z1−∑g4=0αSDn−11g4!e−ξ/βSDn×ξβSDng4n+z1−2αRDnβRDnαRDn∑g1=0αSR−1∑q1=0αRDn−1∑q2=0g11g1!×g1q2αRDn−1q1βSRγ1Tn−g1−q1−q2+1/2τg1+αRDn−q1+q2+1/2×Tn+Lnγ2τq1+q2+1/2βRDnq1−q2+1/2Lnγ2g1−q2×e−τ/βRDn−Lnγ2τ/βSRTnγ1Kq1−q2+12Tn+Lnγ2ττβSRβRDnTfγ1,where n denotes the n-th device (the near device) and bn≜M!/n−1!M−n!, ξ≜maxξ1,ξ2, ξ1≜Cfγthf/a1Anγ1−a1An+Bnγ1γthf, ξ2≜Cnγthn/a2Anγ1−γ1γthna2Bn+a1εAnBn+a1εAn, τ=maxτ1,τ2, τ1≜Tnγ1γthf/a1Enγ2−a2En+Jnγ2γthf, and τ2≜Tnγthn/a2Enγ2−a2Jn+a1εEnγ2γthn.Proof.
See AppendixB.Remark 3.
From Theorems1 and 2, we can observe that the outage probabilities of Df and Dn are determined by IQI parameters, the performance of SIC, fading parameters, and distortion noise. Meanwhile, it is worth noting that when ςt=ςr=1, ϕt=ϕr=0∘, and ε=0 are all satisfied, the considered system reduces to ideal conditions.


## 3.2. Asymptotic Outage Probability

In this subsection, the asymptotic outage probabilities ofDf and Dn are analyzed to obtain more insights on the outage behavior of the considered system. The asymptotic CDF of the channel gain ρi in the case of ordering and nonordering can be expressed as [39]
(19)fρi∞x=xαiαi!βiαi,(20)Fρi∞x=M!M−m!m!1αimxβimαi.The asymptotic expressions of outage probabilities ofDf and Dn are described in the following corollaries.Corollary 4.
The asymptotic closed form of outage probability ofDf in the high SNR regime is expressed as
(21)PoutDf,∞=bf∞1αSDf!fθβSDfαSDff×φ′αSRαSR!βSRαSR+φ″αRDfαRDf!βRDfαRDf,where bf∞=M!/f!M−f!, φ′=Lfγthf/a1Efγ1−a2Ef+Jfγ1γthfEfγ1−a2Ef+Jfγ1γthf, and φ″=Tfγthf/a1γ2Ef−a2Ef+Jfγ2γthf.Proof.
See AppendixC.Corollary 5.
The asymptotic closed form of outage probability ofDn in the high SNR regime is expressed as
(22)PoutDn,∞=bn∞1αRDn!nξβRDnnαSDn×τ′αSRαSR!βSRαSR+τ″αRDnαRDn!βRDnαRDn,where bn∞=M!/n!M−n!, τ′=maxτ1′,τ2′, τ″=maxτ1″,τ2″, τ1′≜Lnγthf/a1Enγ1−γ1γthfa2En+Jn, τ1″≜Tnγthf/a1Enγ2−γ2γthfa2En+Jn, τ2′≜Lnγthn/a2Enγ1−a2Jnγthnγ1, and τ2″≜Tnγthn/a2Enγ2−a2Jnγthnγ2.Proof.
See AppendixD.Remark 6.
The results of Corollaries4 and 5 show the effects of channel fading parameters, IQI parameters, the performance of SIC, and distortion noise on asymptotic outage performances of Df and Dn. In addition, we can also observe that the asymptotic outage probability is related to the order of user arrangement directly.


## 3.3. Diversity Order

The diversity order can reflect the trend of the outage probability intuitively. To this end, the diversity order of the considered system is explored in this subsection. The diversity order is defined as [40]
(23)d=−limγ→∞logPoutDm,∞logγ,where PoutDm,∞ denotes asymptotic outage probability of m‐th device and γ∈γ1,γ2 represents the transmitted SNR.Corollary 7.
Based on (23), the diversity orders of Df and Dn in ideal conditions (ςt=ςr=1, ϕt=ϕr=0∘, and ε=0) and nonideal conditions (ςt,ςr≠1, ϕt,ϕr≠0∘, and ε≠0) can be written as
(24)dfid=dfnid=minfαSDfαSR,fαSDfαRDf,dnid=dnnid=minnαSDnαSR,nαSDnαRDn.Remark 8.
The results show that the diversity order ofDf is the minimum of fαSDfαSR and fαSDfαRDf, which indicates that the diversity order of Df is relative to αSDf, αSR, αRDf, and f, while the value is affected by the multipath fading parameters αSR and αRDf of S⟶R and R⟶Df. Similarly, the diversity order of Dn is the minimum of nαSDnαSR and nαSDnαRDn, which is related to αSDn, αSR, αRDn, and n, and the final value is jointly determined by αSR and αRDn. The results also show that the outage probabilities of Df and Dn of the considered system will always decrease with the increase of SNR. In addition, we can also observe that although the outage probabilities of IoT devices in ideal and nonideal conditions are different, the trends are the same due to the diversity orders.


## 3.4. Throughput Analysis

The system throughput is another measure of system performance, which is the number of signals transmitted per unit of time successfully. Thus, the system throughput is formulated as [41]
(25)T=∑m=1M1−PoutDmRthm,where PoutDm is the outage probability of Dm which can be obtained from (16) and (18), Rthm=1/2log1+γthm denotes the target rate of Dm, and 1/2 represents that the communication transmit process is divided into two time slots.


## 3.5. Energy Efficiency

To further evaluate the performance of the considered system, the energy efficiency is analyzed in this subsection. Energy efficiency refers to the useful signals that IoT devices received for each unit of energy consumed byS. Hence, the energy efficiency of the considered system can be expressed as [42]
(26)ηee=TPEE,where PEE=PS+PR+PC denotes total energy consumption and PC is the fixed energy consumption which is caused by transmitter and the IoT devices and T is the system throughput which can be obtained from (25).


## 4. Numerical Results

In this section, the correctness of analytical results in Section3 is demonstrated by some computer simulations. Unless otherwise noted, the simulation parameters are provided in Table 1.Table 1
Simulation parameters.
ParametersValueParametersValueαi2βi1n2f1γthf1.5γthn3M2N01PC0.1 WFigure2 shows the outage probabilities of IoT devices versus transmit SNR in ideal conditions (ςt=ςr=1, ϕt=ϕr=0∘, and ε=0) and nonideal conditions (ςt=ςr=1.2, ϕt=ϕr=10∘, and ε=0.01) with a1=0.8 and a2=0.2. The perfect coincidence of the theoretical analysis value and the Monte Carlo simulation value curves in Figure 2 verifies our derivations in (16), (18), (21), and (22). From Figure 2, we can observe that the gaps caused by ideal and nonideal conditions of Dn are larger than those of Df. The reason for this phenomenon can be explained by the fact that ipSIC has no impact on Df. As can also be seen from Figure 2, the curves of different IoT devices are almost parallel in the high SNR region. Meanwhile, we also notice that the outage probability of the far-IoT-device outperforms that of the near-IoT-device, because of the large power allocation factor. Additionally, the results also show that the outage performance of the considered system can be greatly prompted by improving the transmitting SNR.Figure 2
Outage probability of IoT devices versus transmit SNR in different conditions.Figure3 illustrates the outage probabilities of the IoT devices versus IRR in perfect I/Q and IQI conditions with IRRt=IRRr, SNR=15dB, ε=0.01, and a1=0.8 and a2=0.2 for power allocation. As can be seen from Figure 3, with the increase of IRR, the outage performances of devices Df and Dn improve continuously, and when IRR>32dB, the outage probabilities of the devices almost coincide with those of the ideal IQ matching. In addition, in terms of the gaps between the perfect IQ matching and IQI curves in the figure, IQI has different effects on the outage performances of different IoT devices in the considered NOMA system. Furthermore, we can also observe that the outage performance of the far-IoT-device is better than that of the near device, which is due to the more power allocation obtained by the far device.Figure 3
Outage probability of IoT devices versus IRR in different conditions.Figure4 plots the variations of outage probabilities of the IoT devices with the increase of SNR in the presence of different nonideal factors with a1=0.8 and a2=0.2. In the simulation, we assume three cases: (1) only IQI (ςt=ςr≠1.05, ϕt=ϕr≠9∘, and ε=0), (2) only ipSIC (ςt=ςr=1, ϕt=ϕr=0∘, and ε=0.05), and (3) both IQI and ipSIC (ςt=ςr=1.05, ε=0.05, and ϕt=ϕr=9∘). The reason for setting the nonideal factors in this way is to ensure that the level of IQ mismatch is at the same extent as the level of ipSIC. For the near-IoT-device Dn, the case of only IQI outperforms the case of only ipSIC in the outage performance which indicates that ipSIC has a severe negative impact on the considered system. Interestingly, the curves of only IQI and both ipSIC and IQI are coincident, since in the NOMA system, the outage performance of Df is not affected by SIC.Figure 4
Outage probability of IoT devices versus SNR under different nonideal factors.As a further development, Figure5 presents the outage performances of IoT devices versus ε with SNR=15dB. For nonideal conditions, we set ςt=ςr=1.2 and ϕt=ϕr=10∘. It can be observed from Figure 5 that, for the far-IoT-device Df, the outage probabilities in either ideal or nonideal conditions will not change with the variation of ε, which is due to the fact that the outage performance of Df is independent of the performance of SIC, while for Dn the outage probability almost increases linearly with the increase of ε. This indicates that SIC is vital in the outage performance of Dn. From Figure 5, we can also see that when ε=0, there are gaps between the outage probabilities of ideal and nonideal conditions, which is due to the existence of IQI in the considered system. It is worth noting that although we have mentioned ε∈0,1 in Section 2, the maximum value of ε here is taken as 0.1. The reason is that a2An>a2Bn+a1εAnγthn and a2En>a2Jn+a1εEnγthn are all needed to be satisfied from (18). Thus, when ε=0.1, the outage probability of Dn is almost 1.Figure 5
Outage probability of IoT devices versusε in different conditions.Figure6 analyzes the impact of amplitude and phase mismatch levels on the outage probabilities versus SNR for different IoT devices with SNR=15dB, ε=0.01, and a1=0.7 and a2=0.3 for power allocation. For simplicity, in this simulation, we set gtS=gtR=grR=grD and ϕtS=ϕtR=ϕrR=ϕrD. As can be seen from Figure 6, when 0.6<g<1, the outage probabilities decrease with the increase of g; however, when 1<g<1.3, the opposite phenomenon appears. This can be explained that when the amplitude mismatch levels are closer to 1, the system is closer to the ideal conditions. Another observation is that the outage performances of IoT devices deteriorate with the increase of ϕ, and when the phase mismatch levels reach to about 20∘, the considered system turns meaningless due to the high outage probability.Figure 6
Outage probability of IoT devices versusg and ϕ.Figure7 investigates the outage probabilities of IoT devices versus power allocation a1 with SNR=15dB in ideal conditions and nonideal conditions for ςt=ςr=1.2, ϕt=ϕr=10∘, and ε=0.01. The outage probability of Dn first decreases and then increases with the increase of a1, and when a1 is about 0.68, the outage probability of Dn reaches the minimum. This can be deciphered that when 0.5<a1<0.6, the considered network is always in an off-line state due to the insufficient power level for Df, and when 0.6<a1<0.68, the considered system begins to perform NOMA; after that, the outage performance of Dn begins to deteriorate as the power allocation to Dn decreases. Unlike Dn, the outage probability of Df keeps decreasing as the power allocated to it increases.Figure 7
Outage probability of IoT devices versusa1 in different conditions.Figure8 shows the exact and asymptotic system throughput versus SNR. In this simulation, we set ςt=ςr=1.3, ϕt=ϕr=15∘, and ε=0.05 for nonideal factors and a1=0.7 and a2=0.3 for power allocation. When the SNR is lower than 5 dB, the throughput of the considered system is almost 0 due to the high outage probability. And the system throughput tends to be a fixed value in the high SNR regime. This is because when the SNR is large enough, the outage probabilities of IoT devices are almost close to 0. It is worth mentioning that there exists an intersection between the green curve (only IQI in the considered system) and the black curve (only ipSIC in the considered system), which indicates that in the case of low SNR, IQI has a greater impact on the system throughput, while ipSIC will have a more severe impact on the considered system in the moderate and high SNR region. From Figure 8, we can also find that these nonideal factors play a negative role in the considered system.Figure 8
System throughput versus SNR in different conditions.In Figure9, the energy efficiency of the considered system versus SNR is plotted. From the figure, we can see that in both ideal and nonideal conditions, the energy efficiency of the considered system first increases and then decreases, which is due to the fact that when the SNR is low, most of the transmission power is used for signal transmission. From the figure, we can also observe that the energy efficiency of the system which is only affected by IQI is higher than that only affected by ipSIC. This indicates that compared with IQI, the energy efficiency of the considered system is more dependent on the performance of SIC. In addition, we can also observe that when SNR>20dB, the energy efficiencies of the considered system in the four cases are almost identical. Finally, we can also conclude that the overall performance of the system cannot be improved by simply increasing the SNR in practical communication networks.Figure 9
Energy efficiency versus SNR under different nonideal factors.


## 5. Conclusion

In this paper, the performance of cooperative AF IoT NOMA system in the presence of IQI and ipSIC is studied. The exact and asymptotic outage probability expressions are derived to evaluate the considered system. Furthermore, the diversity order in the high SNR region, the system throughput, and the energy efficiency are also presented. The results show that IQI and ipSIC play negative roles in the considered system performance and the outage performance of the considered system will be greatly improved with the increase of SNR. Compared with IQI, identical degree of ipSIC plays a greater impact on the outage performance. Particularly, tuning the power allocation scheme properly can improve the outage performances of the IoT devices. The results also show that, according to the inherent characteristics of the NOMA scheme, the outage probability of the far-IoT-device is independent of the SIC performance. In addition, the simulation results of the system throughput and the energy efficiency show that the system performance cannot be improved by simply improving the SNR.



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*Source: 3812826-2020-07-13.xml*