Impact of Hardware Impairments with Imperfect Channel Estimation for Cache-Enabled UAV Relaying Networks

The effects of hardware impairments with imperfect channel estimation for cache-enabled UAV networks are studied in this paper. The effects of the setup parameters, such as the number of cached or relaying nodes, the hardware impairment factor, the channel estimation error, and the transmission SNR, on the outage probability are present by deriving the exact closed-form expressions on outage probability. Also, the asymptotic results are present when the transmission SNR is large enough. From the asymptotic analysis, we can see that there is an error floor introduced by the hardware impairments and imperfect channel estimation. Specifically, the error floor is dependent on the channel estimation error factor, the hardware impairment distortion factor, and the QoS requirements. And the diversity order of the error floor for the cached/relaying links is equal to the number of cached/relaying UAV nodes.


Introduction
In recent years, wireless throughput is growing up exponentially [1][2][3] and wireless communication network is facing more and more challenges introduced by wireless big data [4][5][6][7]. Cache technology, which can both improve the quality of service (QoS) of wireless link and relieve the pressure of wireless throughput, has demonstrated as a remarkable enabling solution for wireless networks [8][9][10]. By exploiting interference neutralization, the authors in [11] proposed an optimal content placement method to improve the system capacity. Furthermore, the maximum distance separable coding is adopted in wireless cache networks in [12], which can improve the system performance in terms of successful retrieval probability.
On the other hand, hardware impairments, which are introduced by phase noise of the RF components, show great effects on the system performance. Hardware impairments have been extensively studied and modeled as additive distortion noise [13] or nonlinear polynomial multiplicative fac-tor [14,15]. Considering amplify-and-forward (AF) relaying networks, the authors in [16] investigated the effects of hardware impairments on channel capacity by exact closed-form expression as well as the tight bounds, while Ref. [17] proved that larger hardware impairments can be used in massive MIMO due to the huge degrees of space freedom. By using iterative optimization algorithm, Ding et al. [18] proposed a solution to maximize the detection probability.
Meanwhile, due to the high mobility, there exists imperfect channel estimation since the linear estimation methods are applied on pilot signals [19]. Considering correlated Rayleigh fading with imperfect channel status information, Al-Hussaibi and Ali [20] derived expressions on the ergodic capacity with antenna selection. The authors in [21] analyzed the impact of imperfect CSI estimation for Alamouti-OSTBC Wireless Cooperative Networks in terms of symbol error probability.
However, the impacts of hardware impairments with imperfect channel estimation on relaying networks are still open questions. Considering unmanned aerial vehicle (UAV) networks, the UAV nodes work as a cache enable access point or an AF relaying node. To improve the QoS of the wireless channels, the best cached UAV nodes or AF relaying nodes are selected to help the message transmission from the macro BS to the user. The effects of the system parameters, such as the number of cached/relaying nodes, the hardware impairment factor, the channel estimation error, and the transmission SNR, on the outage probability are present by deriving the exact closed-form expressions on outage probability. Also, the asymptotic results are present when the transmission SNR is large enough. From the asymptotic analysis, we can see that there is an error floor introduced by the hardware impairments and imperfect channel estimation. Specifically, the error floor is dependent on the channel estimation error factor, the hardware impairment distortion factor, and the QoS requirements. And the diversity order of the error floor for the cached/ relaying links is equal to the number of cached/relaying UAV nodes.
The main contributions of this paper are as follows: (i) Cache-enabled UAV relaying protocol is adopted to improve the QoS of the wireless links, where the UAV nodes work as a cache enable access point or an AF relaying node (ii) We provide the deep insight on the effects of the system parameters, such as the number of cached/relaying nodes, the hardware impairment factor, the channel estimation error, and the transmission SNR, on the outage probability by deriving the exact closed-form expressions on outage probability The organization of this paper is as follows. Section 2 presents the system model of cache-enabled amplify-andforward UAV networks, while the UAV selection algorithm and its performance analysis are given in Section 3. Furthermore, the asymptotical analysis is present in Section 4 with large transmission power. In Section 5, simulation results are present to validate the theoretical analysis. Finally, conclusion is given in Section 6.

System Model
The system model of cache-enabled amplify-and-forward unmanned aerial vehicle (UAV) networks is present in Figure 1, where there is one destination user, one macro base station (BS) and N cache-enabled UAV relaying nodes. We assumed that all nodes are equipped with only one antenna. Furthermore, because of the shadow fading, there is no direct link from the macro BS to the destination user. In order to improve the QoS of the links, one of the UAV nodes is selected to assist the message transmission. Since the UAV nodes hold limited cache memory, the backhaul load from the macro BS to the UAV nodes can be relieved if the requested files from the user are hit in the UAV nodes. In this case, one of the UAV nodes is selected and directly transmits message to the user. On the other hand, if the requested file is not cached, the selected UAV node will work as an amplify-andforward relaying node to help the data transmission. It is assumed that all links are block Rayleigh fading channel and independent with each other.

Selection Algorithm and
Performance Analysis 3.1. Cached Links. If the requested file is cached by the UAV nodes, the power normalized transmission signal from the UAV node can be modeled as [13] whereŝ i is the information bearing signal with unit power, η i ∈ CN ð0, 1Þ is the distortion noise introduced by hardware impairments [22][23][24][25], and κ η is the hardware impairment distortion factor. Since the classical MMSE channel estimation algorithm is adopted, there exists some estimation error between the real channel fading coefficient and the estimated version [26][27][28][29]. It is assumed that the estimation error is independent with the imperfect CSI [30][31][32][33][34][35]. Thus, the relationship between the imperfect channel status information and the ideal version is given as where h u,i ∈ CN ð0, 1Þ is the CSI from the ith UAV node to the user, whileĥ u,i ∈ CN ð0, 1Þ denotes the imperfect estimation of h i , e i ∈ CN ð0, 1Þ is the difference betweenĥ u,i and h u,i , and κ e is the channel estimation error factor. Specifically, κ e = 0 means the perfect channel estimation.

Wireless Communications and Mobile Computing
Given that the ith UAV relaying node is selected, the requested file is cached by the selected node. Then, the received signal of the user can be given as where P R is the transmission power of the UAV node, and n i ∈ CN ð0, σ 2 Þ is the received white Gauss noise. Then, the signal to interference plus noise ratio (SINR) can be given as where Obviously, γ c,i is a monotone increasing function of ρ u,i . In this case, the optimal UAV node can be given as follows: Due to the independence between different ρ u,i 's, according to the ordered statistics, we have where N R is the number of relaying nodes. Then, the outage probability of the cached links can be given as where γ 0 = 2 C 0 − 1 and C 0 are the required capacity. Substituting (7) into (8), we obtain

Relaying Links.
On the other hand, if the requested file is not cached in the selected UAV node, the original message will be transmitted from the macro BS to the user with the help of the UAV node by using amplify-and-forward relaying protocol.
In this case, the received signal at the ith UAV node can be given as where P S is the transmission power of the macro BS, h r,i ∈ CN ð0, 1Þ denotes the channel fading coefficient from the macro BS to the ith UAV node, and n r,i ∈ CN ð0, σ 2 Þ is the received Gauss noise at the UAV node. Then, the UAV node will amplify the received signal with power P R and retransmit to the user. Thus, the received signal at the user through AF protocol can be given as where ζ i is the power amplify factor as follows and ρ r,i = jh r,i j 2 is the channel fading power of the first hop. By substituting (2) and (12) into (11), we can obtain the SINR at the user through AF protocol where In order to obtain the best SINR, an UAV selection algorithm can be employed. Note that if the transmission power 3 Wireless Communications and Mobile Computing is large enough, i.e., λ S → 0, λ R → 0, the high order items can be ignored. Then, γ r,i in (13) can be simplified as To simply the complexity of theoretical analysis, we use the following approximation. Recall the following fact that [36] 1 To obtain the best SINR through AF protocol, the selection criterion can be given as where ξ = c r /c u . Theorem 1. The cumulative distribution functions of ρ u,n * 2 and ρ r,n * 2 are given as for N R ≥ 2.
for N R = 1, where Taking the derivative of equations in Theorem 1, we can obtain the following corollary. are given as for N R ≥ 2, and for N R = 1.
Proof. See Appendix VI-A.
Considering the SINR expression in (13), the outage occurs when γ r,n * 2 falls below the quality-of-service requirement, i.e., where γ 0 = 2 C 0 − 1. Substituting (13) into (24), we have Applying conditional probability formula, we have Wireless Applying (18) on (25), we can obtain To derive the exact expression of P 3 , firstly we give the following definition Applying (18) on (29), we can obtain Thus, the item P 3 in equation (25) can be given as By using necessary mathematical derivation and applying equation (3.324-1) in [37], i.e., the closed-form expression of P 3 can be obtained. By substituting (27), (28), and (31) into (25), we can obtain the exact closed-form expression of P O,R .

Asymptotical Analysis
To get a deep insight on the impact of hardware impairments with imperfect channel estimation, we will conduct the asymptotical analysis of outage probability. Note that if the transmission power is large enough, i.e., 1/λ S → 0, 1/λ R → 0, and κ e → 0, κ η → 0.
Thus, we have Applying the approximation that 1 − e −x ≃ x, if x → 0 on (9), we have According to the asymptotical expression on outage probability for cached links, we can conclude the following remarks:  Considering the SINR expression in (15), we have where Remark 6. There also exists an error floor for the relaying links when the transmission power is large enough.
Remark 7. The diversity order of error floor for the relaying links is equal to the number of relaying UAV nodes.

Simulation Results
In this section, simulation results are provided to verify the accuracy of the theoretical analysis. For the sake of simplicity, we set κ e = κ η = σ 2 . The impacts of the system parameters, such as the number of UAV nodes N, the transmission SNR of the cached links and the relaying links λ S , λ R , and the capacity threshold C 0 , are investigated. Figure 2 shows the impacts of transmission SNR λ S on outage performance for cached links. In this simulation, we have the following system parameters: N C = 2, κ e = κ η = 0:01, C 0 = 1 bps/Hz, and the transmission SNR λ S changes from 12 dB to 39 dB. The simulation results and the theoretical analysis as well as the asymptotic analysis are compared in this figure. As shown in this figure, in all SNR regions, the simulation results match well with the theoretical analysis. Furthermore, an error floor appears when the transmission SNR grows large, which coincides with the asymptotic performance analysis. The reason is that when SNR is large enough, the bottleneck of the system performance is the hardware impairments and the channel estimation error. The detailed relationship between the error floor and κ η and κ e is given as in equations (34) and (36). When the requested file is not hit by the UAV nodes, amplify-andforward protocol is adopted and the simulation results are present in Figure 3, where similar conclusion can be obtained for relaying links with N R = 2. Figures 4 and 5 show the effects of the capacity threshold C 0 on outage probability for cached links and relaying links, respectively. The system parameters are set as follows: N C = N R = 2, κ e = κ η = 0:01, and the capacity threshold C 0 changed from 1 bps/Hz to 3 bps/Hz. From the two figures, we can observe that C 0 shows significant impact on outage probability for all of the SNR regions. Specifically, the diversity order of outage probability is zero with respect to the transmission SNR λ S or λ R . Figures 6 and 7 show the effects of the hardware impairment factor κ η on outage probability for cached links and relaying links, respectively. The system parameters are set as follows: N C = N R = 2, κ e = κ η = 0:01, C 0 = 1 bps/Hz, and   Wireless Communications and Mobile Computing the hardware impairment factor κ η changed from 0.01 to 0.03. From these figures, we can observe that C 0 shows considerable impact on outage probability especially in high SNR regions. Specifically, the larger κ η will introduce higher error floor. The reason is that, in the considered scenario, the bottleneck of the system performance is the hardware   impairments. When the hardware impairment factor κ η grows larger, the noise power will be higher, which will deteriorate the system performance.
The effect of the number of cached UAV nodes is present in Figure 8

Conclusions
The impacts of hardware impairments with imperfect channel estimation for cache-enabled UAV relaying networks are investigated in this paper. To improve the QoS of the wireless channels, the best cached UAV nodes or AF relaying nodes are selected to help the message transmission from the macro BS to the user. The impacts of the setup parameters, such as the number of cached/relaying nodes, the hardware impairment factor, the channel estimation error, and the transmission SNR, on the outage probability are present by deriving the exact closed-form expressions on outage probability. Also, the asymptotic results are present when the transmission SNR is large enough. From the asymptotic analysis, we can see that there is an error floor introduced by the hardware impairments and imperfect channel estimation. Specifically, the error floor is dependent on the channel estimation error factor, the hardware impairment distortion factor, and the QoS requirements. In future works, we will introduce deep learning-based [38][39][40] or Q-learning-based algorithms [41,42] to further improve the system performance. Moreover, we will apply the considered wireless techniques into some practical IoT systems [43,44] to achieve the green energy applications.

Appendix Proof of Theorem 1
For the case N R = 1, the theorem obviously holds.
We will focus on the case that N R ≥ 2. Firstly, we give the following definition Then, the PDF of z i can be derived as ðA:2Þ Due to the independence between ρ u,i and ρ r,i , we obtain ðA:8Þ As such, the PDF of θ m can be obtained as Similarly, we can obtain the CDF of ρ r,n * 2 as follows: ðA:13Þ Thus, Theorem 1 is proved.

Data Availability
The data of this work can be available through the request on the corresponding author by e-mail.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.