Performance analysis of UAV-relaying with time-switching protocol and imperfect channel conditions for cooperative NOMA networks

Abstract

This paper presents a study that evaluates the performance of cooperative non-orthogonal multiple access (C-NOMA) using decode-and-forward unmanned aerial vehicle (UAV) relays in a downlink scenario, where the channel gains follow the $\kappa -\mu$ generalized fading model. The research investigates the use of C-NOMA in both indoor and outdoor user cases, with particular attention paid to practical issues such as imperfect successive interference cancellation and channel estimation errors. The study also considers the use of energy harvesting in the UAV relay node using the time-switching-based relaying protocol. Closed-form expressions are derived for the outage probability, ergodic capacity, throughput, and energy efficiency for indoor/outdoor NOMA users, with a focus on satisfying quality of service (QoS) requirements for individual users. The research examines the impacts of key parameters, such as fading channel parameters, power allocation, and splitting power factor, on the performance of the system. The results indicate that the precise selection of power allocation coefficients, QoS requirements, and channel fading parameters are critical factors in achieving high system performance. Extensive simulation results are presented to confirm the analytical expressions.

Appendices

Appendix A: Proof of Proposition 1

Proof

The instances of system outage for the indoor user $U^I$ arise when certain conditions are met. Firstly, it occurs when the UAV-relay node incorrectly decodes the signal $x_1$ during the first time slot. Secondly, it occurs when $U^I$ fails to decode its own signal in the second time slot. Hence, the probability of outage for $U^I$ can be mathematically represented as follows:

$$\begin{aligned} {\text {P}_{u_1}^{out}}&={{P_{r}}\left( {\gamma _{r}^{{x^{I}}}}<{\gamma _{th_1}}, {\gamma _{u_1}^{x_1}}<{\gamma _{th_1}}\right) }\nonumber \\&=1-\underbrace{{P_{r}}\left( {\gamma _{r}^{{x^{I}}}}>{\gamma _{th_1}}\right) }_{E_{11}}\underbrace{{P_{r}} \left( {\gamma _{u_1}^{{x^{I}}}}>{\gamma _{th_1}}\right) }_{E_{12}} \end{aligned}$$

(A1)

The threshold value of $\gamma _{th_1}$ is represented by ${2^{2R_{1}}}-1$, where $R_{1}$ is the target data rate of user $U^I$. By substituting (11) into (A1) for $E_{11}$, the resulting expression can be obtained.

$$\begin{aligned} {E_{11}}&= {P_{r}}\left( \gamma _{r}^{{x^{I}}}>{\gamma _{th_1}}\right) \nonumber \\&={P_{r}}\left( |\widehat{h}_{0}|^{2}>\frac{\gamma _{th_1}({\sigma _{e_{0}}^2}\rho +1)}{\rho (\alpha ^{I}-\alpha ^{O}\gamma _{th_1})}\overset{\Delta }{=}\varepsilon _{1,1}\right) \nonumber \\&=\intop _{0}^{\varepsilon _{1,1}}{f_{|\widehat{h}_{0}|^{2}}}(\gamma )dx = {F_{|\widehat{h}_{0}|^{2}}}\left( \varepsilon _{1,1}\right) \end{aligned}$$

(A2)

Now, we substitute (6) in (A2) to compute the first term of (A1), thus, we have the following.

$$\begin{aligned} {E_{11}}=\frac{1}{\textrm{e}^{\left( \mu \kappa \right) }}\sum _{n=0}^\infty \frac{(\kappa \mu )^{n}~}{n!~\Gamma (n+\mu )}\gamma _{inc}\left( n+\mu ~,~\frac{\mu (1 +\kappa )\varepsilon _{1,1}}{\tilde{\gamma }_{1,1}}\right) \end{aligned}$$

(A3)

where ${\tilde{\gamma }_{1,1}}$ indicates the average power link between BS and UAV-relay. Similarly, $E_{12}$ in (A1) can be evaluated by recalling (16) and (6) as follow.

$$\begin{aligned} {E_{12}}&= {P_{r}}\left( {\gamma _{u_1}^{x_1}}>{\gamma _{th_1}}\right) \nonumber \\&={P_{r}}\left( |\widehat{h}_{1}|^{2}>\frac{\gamma _{th_1}({\sigma _{e_{1}}^2}\rho | \widehat{h}_{0}|^{2}+1)}{|\widehat{h}_{0}|^{2}(\psi {1}-\psi {2}\gamma _{th_1})} \overset{\Delta }{=}\varepsilon _{1,2}\right) \nonumber \\&= \intop _{0}^{\varepsilon _{1,2}}{f_{|\widehat{h}_{1}|^{2}}}(\gamma )dx = {F_{| \widehat{h}_{1}|^{2}}}\left( \varepsilon _{1,2}\right) \end{aligned}$$

(A4)

Therefore, (A4) is regenerated as

$$\begin{aligned} {E_{12}}= \frac{1}{\textrm{e}^{\left( \mu \kappa \right) }}\sum _{n=0}^\infty \frac{(\kappa \mu )^{n}~}{n!~\Gamma (n+\mu )}\gamma _{inc}\left( n+\mu ~,~\frac{\mu (1+\kappa ) \varepsilon _{1,2}}{\tilde{\gamma }_{1,2}}\right) \end{aligned}$$

(A5)

where ${\tilde{\gamma }_{1,2}}$ is the average power link between UAV-relay and $U^I$. Now, we submit (A3) into (A5) in (A1) in order to obtain the proposition 1. Hence, the proof is completed. $\square$

Appendix B: Proof of Proposition 2

Proof

The emergence of outage events related to the outdoor user, denoted by $U^O$, can occur due to various factors. The first factor is the incorrect decoding of either signal $x_1$ or $x_2$ by the UAV-relay in the first time slot. The second factor is the inability of $U^O$ to detect either signal $x_1$ or $x_2$ following the successive interference cancellation (SIC) procedure in the second time slot. Therefore, the outage probability of $U^O$ mainly hinges on the successful decoding of signals $x_1$ and $x_2$ over both links, i.e., from the base station (BS) to the UAV and from the UAV to $U^O$, after executing the SIC process. This probability can be expressed as follows:

$$\begin{aligned} {\text {P}_{U^O}^{out}}&=1-\underbrace{{P_{r}}\left( {\gamma _{r}^{{x^{I}}}}>{\gamma _{th_1}}\right) }_{E_{21}} \underbrace{{P_{r}}\left( {\gamma _{r}^{{x^{O}}}}>{\gamma _{th_2}}\right) }_{E_{22}}\nonumber \\&\quad \underbrace{{P_{r}}\left( {\gamma _{u_2}^{x_1\rightarrow {x_2}}}>{\gamma _{th_1}}\right) }_{E_{23}}\underbrace{{P_{r}} \left( {\gamma _{u_2}^{{x^{O}}}}>{\gamma _{th_2}}\right) }_{E_{24}} \end{aligned}$$

(B6)

where $\gamma _{th_2}={2^{2R_{2}}}-1$, and $R_{2}$ is the target data rate of $U^O$. Hence, ${E_{11}}={E_{21}}$. To eliminate any duplication, the residual occurrences can be explained in the same way as the aforementioned, which were obtained from Eqs. (A1)–(A5), concerning their SINR magnitudes and thresholds. Consequently, the demonstration is concluded. $\square$