Uplink Performance Analysis of Wireless Energy Harvesting-Enabled NOMA-based Networks

Abstract

This article presents a performance analysis of wireless energy harvesting (WEH)-enabled sensor networks that extract energy from ambient radio frequency (RF) signals prior to uplink transmission. A time-switching (TS)-based protocol is utilized to alternate sensor nodes between energy harvesting (EH) and data transmission modes. Implementing the non-orthogonal multiple access (NOMA) technique aims to boost the sensor network’s performance regarding uplink sum rate and outage probability. To optimize resource allocation, we propose an unequal operating time frame (OTF) scheme that determines data transmission and energy harvesting intervals based on channel gain quality. Simulation results affirm the superiority of NOMA over orthogonal multiple access (OMA), with NOMA enabling higher sum rates by accommodating more signals within the same frequency band, though at the expense of slightly degraded outage performance.

Appendix A Proof of Theorem 1

the optimization problem formulated in Eq. 8a is solved applying the Lagrangian and KKT methods, where the Lagrangian function can be expressed as

$$\begin{aligned} {\begin{matrix} L(\varvec{\alpha }, \varvec{\lambda }, \varvec{\mu }) = &{} \sum _{i=1}^{N}\frac{(1-\alpha _i)T}{2}\log _2\left( 1+\frac{P_i|h_i|^2}{\sigma ^2}\right) + \\ {} &{}\sum _{i=1}^{N}\lambda _i\left( \frac{(1-\alpha _i)T}{2}\log _2\left( 1+\frac{P^B_i|g_i|^2}{\sigma ^2}\right) - R_{th}\right) +\\ {} &{} \sum _{i=1}^{N}\mu _i(\alpha _i)(1-\alpha _i) \end{matrix}} \end{aligned}$$

(A1)

where $\varvec{\alpha } = [\alpha _1, \alpha _2, \ldots , \alpha _N]$ is the vector of variables, $\varvec{\lambda } = [\lambda _1, \lambda _2, \ldots , \lambda _N]$ and $\varvec{\mu } = [\mu _1, \mu _2, \ldots , \mu _N]$ are are symbolized as Lagrange multipliers for Eqs. 8b and 8c, respectively. Now, find the partial derivatives with respect to $\alpha _i$, $\lambda _i$, and $\mu _i$ and assign them to zero

$$\begin{aligned} \frac{\partial L}{\partial \alpha _i} = \frac{T}{2}\log _2\left( 1+\frac{P_i|h_i|^2}{\sigma ^2}\right) - \lambda _i \frac{T}{2}\log _2\left( 1+\frac{P^B_i|g_i|^2}{\sigma ^2}\right) - 2\mu _i\alpha _i=0 \end{aligned}$$

(A2)

$$\begin{aligned} \frac{\partial L}{\partial \lambda _i} = \frac{(1-\alpha _i)T}{2}\log _2\left( 1+\frac{P^B_i|g_i|^2}{\sigma ^2}\right) - R_{th} = 0 \end{aligned}$$

(A3)

$$\begin{aligned} \frac{\partial L}{\partial \mu _i} = \alpha _i(1-\alpha _i)=0 \end{aligned}$$

(A4)

To determine the values of $\alpha _i$ from $\frac{\partial L}{\partial \alpha _i}$, it’s essential to ascertain the Lagrange multipliers $\lambda _i$ and $\mu _i$. This process typically involves iterative techniques like gradient descent or Newton’s method. Alternatively, we can derive $\alpha _i$ from the Lagrangian’s derivative concerning $\lambda _i$. Ultimately, the solution for $\alpha _i$ can be obtained through this method.

$$\begin{aligned} \alpha _i = 1 - \frac{2R_{th}}{T\log _2\left( 1+\frac{P^B_i|g_i|^2}{\sigma ^2}\right) } \end{aligned}$$

(A5)