Ergodic Performance Analysis of Double Intelligent Reflecting Surfaces-Aided NOMA–UAV Systems with Hardware Impairment

In recent years, as a promising transmission method, the new generation wireless systems can rely on IRS and UAV to enhance their spectral and energy efficiency since IRS is examined as a cost-effective deployment approach [1,2,3]. Specifically, by varying the amplitude and phase for the incident signal, an IRS leverages its low-cost reconfigurable passive elements to reflect signals to distant users effectively [4,5]. The system relying on IRS can enhance the link quality and enlarge the coverage significantly when IRS is able to adjust amplitude-reflection coefficients and phase-shift variables appropriately [6,7,8]. The authors in [7] studied secure IRS–UAV systems by integrating IRS with UAV in wireless networks, which are susceptible to eavesdropping related to air-ground line-of-sight channels. To facilitate security, IRS can be leveraged due to its capability of reconfiguring the propagation environment, and thus IRS helps to reduce the impacts of eavesdroppers. The secure transmission of an IRS-assisted UAV network is analyzed under the impact of an eavesdropper. The transmit beamforming, the trajectory of UAV and the phase shift of IRS can be optimized to further obtain maximal rates [7]. Besides, IRS has two prominent features—passive reflection with low power consumption and the operation of full-duplex (FD) mode without self-interference—and these benefits are crucial in comparison with conventional approaches—for example, relaying networks [9].

To further increase energy and spectrum efficiency, the power-domain NOMA technology has been studied as a potential technique for IoT applications [10,11]. The NOMA architecture can be described as follows. In the transmit side of a NOMA system, different amounts of power at the transmitter or the base station (BS) are assigned to multiple users when superimposing signal processing, while those users share the same orthogonal resources such as frequency, time and spreading code. At the receiver side associated with a downlink, demultiplexing the transmitted signals is conducted by employing successive interference cancellation (SIC) [12]. To decode the needed signals, the user treats all the weaker user signals as interference; then, its own signal can be decoded in the last step [13,14,15]. In contrast to the downlink, the common receiver or the BS achieves a signal in an uplink, allowing multiple users to consume the common communication. This means that all users send a superposed signal comprising the signals of these users toward the BS. For signal detection, the BS needs the assistance of SIC to decode the signals of the transmitting users [12]. In this way, a higher number of users can be served in the context of NOMA, which exhibits a significant improvement compared with the conventional orthogonal multiple access (OMA)-aided transmission approach [11].

The combination of IRS and NOMA could be a prominent technique to leverage energy-efficient and low-cost deployment [16,17,18,19,20,21,22,23,24,25]. The authors in [16] designed an IRS–NOMA system by considering both continuous phase shifting and discrete phase shifting corresponding to the ideal IRS and the non-ideal IRS circumstance. To demonstrate performance, closed-form expressions are derived to examine the average required transmit power, the outage probability and the diversity order by benefiting from the Laguerre series and the isotropic random vector. In demonstrated analytical results, the BS antenna number and the IRS element number are consistently affected by varying the transmit power. In [17], IRS-aided simultaneous wireless information and power transfer (SWIPT) NOMA networks are studied. In particular, a problem of minimizing the transmit power at the BS can be solved by jointly optimizing the BS transmit beamforming vector, power splitting (PS) ratio, SIC decoding order and IRS phase shift. This optimization is guaranteed to satisfy the energy harvested threshold of each user and the quality-of-service (QoS) constraint. To improve the performance of multiple user equipments (UEs), the BS utilizes the UAV-mounted IRS to flexibly serve ground users [18]. The direct non-line-of-sight links between the BS and UEs are required to power the IRS relaying links. The main performance metric is presented, i.e., the outage probability. The model of IRS–NOMA in [19] allows this cell-edge user to be paired with a cell-center user to form the NOMA scheme. In this way, the coverage can be improved at the cell-edge user. The phase shift plays an important role in the design of an IRS-aided NOMA system, i.e., the impacts of random phase shifting and coherent phase shifting are further investigated [20]. In [21], IRS-assisted NOMA benefits from enabling BS’s beamforming vectors. To minimize transmission power, the beamforming vectors and the IRS phase shift matrix could be optimized. One can place the IRS at preferred locations, and line-of-sight (LoS) can be adopted for the links between the transmitters and the IRS before reflecting to receivers [23,24,25]. The links can be enhanced if the system enables LoS and optimized passive beamforming vectors to implement a NOMA-assisted IRS. In [23], by considering ideal and non-ideal IRSs, the system performance was evaluated for the link from source to destination via IRS. In [24], a tremendous performance improvement could be confirmed in the approach of an IRS-assisted NOMA system. This means that IRS-assisted NOMA exhibits sufficient benefits to be incorporated into the existing IoT systems.

We aim to design a IoT network by enabling IRS, NOMA and UAV techniques, where a single antenna source $\left(S\right)$ powers the signal transmission by IRS to serve many users at destinations. The IRSs are mounted in UAVs for better transmission from the base station to ground users. These ground users are divided into many groups, and a considered group contains two users $(D_i;i=1,2)$. The system can maximize the benefits of IRS if multiple I IRSs are deployed, as shown in Figure 1. It is assumed that there is no direct path between the source S and ground users (destinations). To reduce the cost of the design, we refer to the first scenario where two IRSs, i.e., $I_1$ and $I_2$, possess reflecting elements $N_1$ and $N_2$. Those IRSs are placed at specific locations (i.e., buildings) to reflect passive beamforming signals towards the destinations (IoT devices). We need to answer how many IRSs are required to achieve the best performance at destinations. Therefore, we move our attention to the second scenario (the benchmark) by assigning three IRSs, i.e., $I_1$, $I_2$ and $I_3$, which are installed with reflecting elements $N_1$, $N_2$ and $N_3$, respectively (it is reasonable to design double-IRS due to cost efficiency in deployment. Although multi-IRS was developed in [31], the numerical result demonstrated a small gap between the double-IRS and three-IRSs case. Therefore, in this study, we emphasize the performance for two scenarios of IRSs).

We denote the (scalar) channels from the S to the IRSs, and from the IRSs to $D_i$, respectively, as $h_{S{I}_u,n}^{}$ and $h_{I_u{D}_i,n}$. The hardware impairment at the S is denoted as ${\tau }_S\sim CN\left(0,{\mathrm{{\rm Y}}}_S^{2}P\right)$, where ${\mathrm{{\rm Y}}}_S$ represents the proportionality coefficients, which describe the severity of the distortion noises at S. The hardware impairments at the $D_i$ is ${\tau }_{D_{ui}}\sim CN\left(0,{\mathrm{{\rm Y}}}_{D_i}^{2}P{\left|{\displaystyle \sum _{u=1}^I}{\displaystyle \sum _{n=1}^{N_u}}{h}_{S{I}_u,n}^{}{h}_{I_u{D}_i,n}{\eta }_{un}{e}^{j{\theta }_{un}^{\left(D_i\right)}}\right|}^2\right)$ [32], where ${\mathrm{{\rm Y}}}_{D_i}$ represents the proportionality coefficients. Here, ${\mathrm{{\rm Y}}}_{D_i}$ is associated with the distortion noises at $D_i$. Then, the overall received signal at the user $D_i$ with multiple I IRSs is given as [30,33] where P is the total transmit power at the source, $x_i$ is the transmitted signal by source, ${\chi }_i$ denotes the power allocation factor for message $x_i$ with ${\chi }_1+{\chi }_2=1$, ${\chi }_1>{\chi }_2$ [34], ${\theta }_{un}^{\left(D_i\right)}$ is the adjustable phase applied by the n-th reflecting element of the $S-I-D_i$, ${\eta }_{un}$ is the reflection coefficients of I with ${\eta }_{un}\in \left(0,1\right]$, and $n_{D_i}$ stands for the circularly symmetric additive White Gaussian noise (AWGN) with zero mean and variance ${\sigma }_n^2$, i.e., $n_{D_i}\sim \mathcal{CN}\left(0,{\sigma }_n^2\right)$.

In this section, two types of fading channels are considered for $S\to I_1$: $I_1\to D_i$, $S\to I_2$, $I_2\to D_i$, $S\to I_3$, and $I_3\to D_i$, i.e., Rayleigh and Rician fading channels (similar to [31], these channel distributions are preferred to examine ergodic performance, while the Nakagami-m demonstrated similar performance, and hence we do not want to evaluate the case of Nakagami-m in the scope of this study). In the following sections, upper bounds using Rayleigh and Rician fading channels are derived for our proposed scheme.