#### 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.

#### 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 ipSIC

The rest of this paper is organized as follows. Section 2 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, denotes the expectation operator; and are the probability density function (PDF) and the cumulative distribution function (CDF) of a random variable , respectively; is the Gamma function; represents the probability of a random variable; symbolizes the th-order modified Bessel function of the second kind, and stands for the definition operator.

#### 2. System Model

As can be seen from Figure 1, we consider a single-carrier cooperative IoT NOMA network, which consists of one source , one AF relay , 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 and the near-IoT-device (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 , , , , and , , , , , respectively. Without loss of generality, the channel gains between source and destinations are sorted as .

##### 2.1. Signal Transmission

The communication procedure is divided into two time slots. In the first time slot, transmits signals to , , and , where is the transmit power of , and are the desired signals of and , satisfying with , and and are the power allocation coefficients with and . Thus, in the first time slot, the received signals at , , and can be expressed as (The subscripts , , , and in , , , and represent IQIs caused by source, relay, the far-IoT-device, and the near-IoT-device, respectively.) where denotes the additive white Gaussian noise (AWGN), , and , are the IQI coefficients caused by the RX and TX which can be expressed as where and are the amplitude mismatch levels caused by TX and RX, while and are the phase mismatch levels and denotes the unit of imaginary. Furthermore, the TX/RX image rejection ratio (IRR) is expressed as

According to the NOMA protocol, the received signal-to-interference-plus-noise ratio (SINR) for and to decoded is expressed as where , , , , , , and denotes the transmit SNR caused by . Based on the ipSIC, the received SINR of to decoded can be expressed as where and is the parameter of ipSIC which follows the Gaussian distribution [37]. Note that in this system the value of cannot take the value of 1, because the network is completely out of the NOMA scheme when , 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 at and can be expressed as where represents the amplifying gain factor and is the transmit power by and . The received SINRs for to decode and for to decode and are expressed as where represents the transmit SNR caused by , , , , , , , , and .

##### 2.2. Fading Channels

Assume that all the channel gains follow Nakagami-*m* distribution; thus, the PDF and CDF of can be expressed as [38]
where denotes the multipath fading parameter while represents the control spread parameter. By utilizing the order statistics, the PDF and CDF of the -th devices’ channel gain can be expressed as
where denotes the total numbers of IoT devices.

#### 3. Outage Probability Analysis

In this section, the outage performances of and 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

For , the outage event will occur when fails to decode the expected signal transmitted from and . Therefore, the outage probability of can be expressed as where is the target threshold at . The closed-form expression of outage probability of under IQI and ipSIC is provided in the following theorem.

Theorem 1. *The exact outage probability expression of in the presence of IQI and ipSIC is expressed as
where denotes the -th device (the far device), , with , and with .*

*Proof. *See Appendix A.

For , the outage event will occur when fails to decode either the signals or transmitted from or . Therefore, the outage probability of can be expressed as
where is the target threshold at . The closed-form expression of outage probability of under IQI and ipSIC is provided in the following theorem.

Theorem 2. *The exact outage probability expression of in the presence of IQI and ipSIC is expressed as
where denotes the -th device (the near device) and , , , , , , and .*

*Proof. *See Appendix B.

*Remark 3. *From Theorems 1 and 2, we can observe that the outage probabilities of and are determined by IQI parameters, the performance of SIC, fading parameters, and distortion noise. Meanwhile, it is worth noting that when , , and are all satisfied, the considered system reduces to ideal conditions.

##### 3.2. Asymptotic Outage Probability

In this subsection, the asymptotic outage probabilities of and are analyzed to obtain more insights on the outage behavior of the considered system. The asymptotic CDF of the channel gain in the case of ordering and nonordering can be expressed as [39]

The asymptotic expressions of outage probabilities of and are described in the following corollaries.

Corollary 4. *The asymptotic closed form of outage probability of in the high SNR regime is expressed as
where , , and .*

*Proof. *See Appendix C.

Corollary 5. *The asymptotic closed form of outage probability of in the high SNR regime is expressed as
where , , , , , , and .*

*Proof. *See Appendix D.

*Remark 6. *The results of Corollaries 4 and 5 show the effects of channel fading parameters, IQI parameters, the performance of SIC, and distortion noise on asymptotic outage performances of and . 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] where denotes asymptotic outage probability of device and represents the transmitted SNR.

Corollary 7. *Based on (23), the diversity orders of and in ideal conditions (, , and ) and nonideal conditions (, , and ) can be written as
*

*Remark 8. *The results show that the diversity order of is the minimum of and , which indicates that the diversity order of is relative to , , , and , while the value is affected by the multipath fading parameters and of and . Similarly, the diversity order of is the minimum of and , which is related to , , , and , and the final value is jointly determined by and . The results also show that the outage probabilities of and 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] where is the outage probability of which can be obtained from (16) and (18), denotes the target rate of , and 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 by . Hence, the energy efficiency of the considered system can be expressed as [42] where denotes total energy consumption and is the fixed energy consumption which is caused by transmitter and the IoT devices and is the system throughput which can be obtained from (25).

#### 4. Numerical Results

In this section, the correctness of analytical results in Section 3 is demonstrated by some computer simulations. Unless otherwise noted, the simulation parameters are provided in Table 1.

Figure 2 shows the outage probabilities of IoT devices versus transmit SNR in ideal conditions (, , and ) and nonideal conditions (, , and ) with and . 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 are larger than those of . The reason for this phenomenon can be explained by the fact that ipSIC has no impact on . 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 3 illustrates the outage probabilities of the IoT devices versus IRR in perfect I/Q and IQI conditions with , , , and and for power allocation. As can be seen from Figure 3, with the increase of IRR, the outage performances of devices and improve continuously, and when , 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 4 plots the variations of outage probabilities of the IoT devices with the increase of SNR in the presence of different nonideal factors with and . In the simulation, we assume three cases: (1) only IQI (, , and ), (2) only ipSIC (, , and ), and (3) both IQI and ipSIC (, , and ). 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 , 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 is not affected by SIC.

As a further development, Figure 5 presents the outage performances of IoT devices versus with . For nonideal conditions, we set and . It can be observed from Figure 5 that, for the far-IoT-device , the outage probabilities in either ideal or nonideal conditions will not change with the variation of , which is due to the fact that the outage performance of is independent of the performance of SIC, while for the outage probability almost increases linearly with the increase of . This indicates that SIC is vital in the outage performance of . From Figure 5, we can also see that when , 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 in Section 2, the maximum value of here is taken as 0.1. The reason is that and are all needed to be satisfied from (18). Thus, when , the outage probability of is almost 1.

Figure 6 analyzes the impact of amplitude and phase mismatch levels on the outage probabilities versus SNR for different IoT devices with , , and and for power allocation. For simplicity, in this simulation, we set and . As can be seen from Figure 6, when , the outage probabilities decrease with the increase of ; however, when , 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 , and when the phase mismatch levels reach to about , the considered system turns meaningless due to the high outage probability.

Figure 7 investigates the outage probabilities of IoT devices versus power allocation with in ideal conditions and nonideal conditions for , , and . The outage probability of first decreases and then increases with the increase of , and when is about 0.68, the outage probability of reaches the minimum. This can be deciphered that when , the considered network is always in an off-line state due to the insufficient power level for , and when , the considered system begins to perform NOMA; after that, the outage performance of begins to deteriorate as the power allocation to decreases. Unlike , the outage probability of keeps decreasing as the power allocated to it increases.

Figure 8 shows the exact and asymptotic system throughput versus SNR. In this simulation, we set , , and for nonideal factors and and 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.

In Figure 9, 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 , 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.

#### 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.

#### Appendix

#### A. Proof of Theorem 1

By substituting (4) and (8) into (15), the outage probability of is rewritten as

Utilizing the PDF and CDF in (11)–(14), and can be further expressed as

Combining (A.2) and (A.3) with (A.1), the outage probability of can be obtained.

#### B. Proof of Theorem 2

By substituting (5), (6), (9), and (10) into (17), the outage probability of is rewritten as

Utilizing the PDF and CDF in (11)–(14), and can be further expressed as

Combining (B.2) and (B.3) with (B.1), the outage probability of can be obtained.

#### C. Proof of Corollary 1

Based on (A.1), the asymptotic outage probability of can be expressed as

Utilizing the asymptotic CDF in (19), can be further expressed as

Using the inequality [43] and the asymptotic CDF in (20), can be further expressed as

Substituting (C.2) and (C.3) into (C.1), the asymptotic outage probability of can be obtained.

#### D. Proof of Corollary 2

Based on (B.1), the asymptotic outage probability of can be expressed as

Similar to (C.2) and (C.3), and can be further expressed as

Substituting (D.2) and (D.3) into (D.1), the asymptotic outage probability of can be obtained.

#### Data Availability

All data generated or analyzed during this study are owned by all the authors and will be used to our further research. The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 41904078) and the Science and Technology Research Project of Henan Province of China (No. 202102310560).