###### Abstract
the TP of the secondary network in the context of cognitive radio over Nakagami-\(m\) fading without taking shadowing into account. Recently, the authors in ref. [29] analyzed the TP and the OP of overlay networks over \(\kappa\)-\(\mu\) shadowed fading which considers fading distributions (Rayleigh [26; 27], Nakagami-\(m\)[28], Rician [27]) as special cases and accounts for shadowing and path loss. Nevertheless, only [27] considered multiple antennas at T for high energy efficiency and at S for beam-forming implementation, while [26; 28; 29] investigated the single antenna at T, therefore hardly improving energy-harvesting efficiency. Although multiple antennas at T increase considerably the quantity of harvested energy and therefore enhance the system performance, the performance analysis requires new statistics of harvested energy at S and therefore complicating analytical results. Briefly, the performance analysis for wireless communications through measures of the TP and the OP under simultaneous considerations of EHNL, HWi, multiple antennas at T, flexible-and-general \(\kappa\)-\(\mu\) fading, shadowing, and path loss has been left open in the literature. This paper will solve this open problem to swiftly evaluate and maximize the reliability of the communication before practical implementation. More specifically, our contributions are presented as follows:  
* Our work proposes WCwEH in Figure 1, wherein the power transmitter T employs an arbitrary quantity of antennas for ameliorating energy-harvesting efficiency, ultimately ameliorating communications reliability. To feature properly nonlinear circuit elements in energy harvesters, our work proposes the application of the extensively acknowledged NLEH paradigm in [18]. * To assess the reliability of the communication quickly, our work proposes the TP and the OP analyses for the recommended WCwEH under the consideration of EHNL, multi-antenna power transmitter, HWi, and divergent impairment degrees of shadowing, fading, and path loss in propagation conditions. * Our work rate maximizes the reliability of communication in diverse realistic contexts. A plurality of results illustrates that EHNL, HWi, and propagation conditions drastically deteriorate system performance. Notwithstanding, EHNL influences the system performance more severely than HWi. In addition, the desired system performance is accomplished flexibly and possibly by choosing a cluster of specifications. Remarkably, the proposed transmission scheme obtains the optimal performance with the appropriate selection of the time-splitting factor. Our work continues with Section 2, which presents the proposed WCwEH. Then, Section 3 analyzes the TP and the OP for the proposed WCwEH. Subsequently, Section 4 outlines the analyses for four extreme scenarios to facilitate quick performance comparison and highlights the impacts of EHNL, HWi, and multi-antenna deployment. Next, Section 5 provides simulated/analytical results in diverse practical settings. Finally, the paper is concluded in Section 6. Frequently used notations are tabulated in Table 1. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Interpretation** \\ \hline \(\mathcal{CN}(0,c)\) & complex Gaussian random variable with mean 0 and variance \(c\) \\ \(\Gamma(\cdot,\cdot)\) & Incomplete upper Gamma function \\ \(\mathrm{C}_{m}^{n}=\frac{n!}{m!(a-m)!}\) & Binomial coefficient \\ \(F_{N}(\cdot)\) & complementary cumulative distribution function (CCDF) of \(N\) \\ \(\Psi_{N}(\cdot)\) & Moment Generating Function (MGF) of \(N\) \\ \(F_{N}(\cdot)\) & cumulative distribution function (CDF) of \(N\) \\ \(\mathbb{P}\{\cdot\}\) & Probability operator \\ \(\mathbb{E}\{\cdot\}\) & Expectation operator \\ \(\Gamma(\cdot)\) & Complete Gamma function \\ \(\mathrm{K}_{i}(\cdot)\) & Modified Bessel function [30] \\ \(f_{N}(\cdot)\) & probability density function (PDF) of \(N\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequentlyused notations.