# Performance Tradeoff Analysis of Hybrid Signaling SWIPT Systems with Nonlinear Power Amplifiers
## 3 Calculation of PA Efficiency  
The average efficiency of PA is defined as the ratio of the average power of the generated RF signal for actual transmission, which is denoted as \(P_{\text{RF}}\), and the direct current  
Figure 4: Normalized AM-AM characteristics. (DC) input power supplied by the DC source to the transistor PA circuit, which is denoted as \(P_{\text{DC}}\). Both \(P_{\text{RF}}\) and \(P_{\text{DC}}\) are functions of the output envelope level \(r_{o}\). It follows that the average PA efficiency is given by [41; 56]:  
\[\eta_{\text{av}}=\frac{P_{o,\text{av}}}{P_{\text{i,av}}}=\frac{E[P_{\text{RF}}( r_{o})]}{E[P_{\text{DC}}(r_{o})]}. \tag{14}\]  
The RF PAs commonly used in communication systems include Class A, Class B, and Class AB types. In recent years, some new PA circuit structures have emerged, such as envelope tracking PAs and Doherty PAs. These new types of PAs can effectively improve the efficiency at the cost of higher nonlinear distortion. In communication systems, the nonlinear distortion in PAs can be corrected by linearization techniques [57; 58; 59; 60]. In this paper, we restrict our discussions on classic Class A and Class B PAs, noting that the efficiency of Class AB PAs is between Class A and Class B. The empirical efficiency models for Class A and Class B PAs are presented in [61]. The maximum average efficiency achieved by these two types of PAs can be written as [53]  
\[\eta_{\text{av},A}=\frac{1}{2}\frac{E[r_{o}^{2}]}{r_{o,\text{max}}^{2}},\]  
\[\eta_{\text{av},B}=\frac{\pi}{4}\frac{E[r_{o}^{2}]}{r_{o,\text{max}}E[r_{o}]}. \tag{15}\]  
This shows that the efficiency function is the first and second non-central moments of the output signal envelope [53]. Moreover, the standardized \(n\)th non-central moment is defined as [52]  
\[M_{n}\triangleq\frac{E[r_{o}^{n}]}{r_{o,\text{max}}^{n}}=\frac{1}{\left(Ar_{ \text{max}}\right)^{n}}\int_{0}^{\infty}g^{n}(r)f_{r}(r)dr \tag{16}\]  
where \(f_{r}(r)\) is the probability density function (PDF) of signal envelope \(r\). It follows that we can rewrite (15) as  
\[\eta_{\text{av},A}=\frac{1}{2}M_{2}(r_{\text{max}}),\eta_{\text{av},B}=\frac{ \pi}{4}\frac{M_{2}(r_{\text{max}})}{M_{1}(r_{\text{max}})}. \tag{17}\]  
In this paper, we assume that the information signal and energy signal are a multi-carrier OFDM signal and multitone sine signal, respectively. When the subcarrier numbers are large enough, both signals can be approximated as complex Gaussian signals in the baseband. Thus, the composite signal can also be treated as a complex Gaussian signal. The PDF of its envelope is given by \(\left(2r/P_{\text{in}}\right)/\exp\left(-r^{2}/P_{\text{in}}\right)\). Given such a complex Gaussian signal approximation, the first and second non-central moments with the Saleh model are given by [53]  
\[M_{1}(\xi)=4\sqrt{\pi\xi}-8\pi\xi\exp(4\xi)\text{erfc}\left(2\sqrt{\xi}\right), \tag{18}\]  
\[M_{2}(\xi)=16\xi[(1+4\xi)\exp(4\xi)E_{1}(4\xi)-1] \tag{19}\]  
where \(\text{erfc}(x)=1-\text{erf}(x)\) is the complementary error function and \(E_{n}(x)=\int_{1}^{\infty}\frac{\exp(-xt)}{t^{n}}dt\) is the exponential integral. For the polynomial model, substituting Equation (10) into Equation (16), the moments can be calculated as  
\[M_{1}(\xi)=\left(\frac{\sqrt{\pi}}{2\sqrt{\xi}}+\frac{3c_{1}\sqrt{\pi}}{4\xi \sqrt{\xi}}+\frac{15c_{2}\sqrt{\pi\xi}}{8\xi^{2}}\right)\text{erf}\left(\sqrt{ \xi}\right)-\frac{6c_{1}+10c_{2}\xi+15c_{2}}{4\xi^{3}\exp(\xi)}, \tag{20}\]\[ \begin{split}& M_{2}(\xi)=\frac{(1-\exp(-\xi))}{\xi}-\frac{4c_{1}(1-( \xi+1)\exp(-\xi))}{\xi^{2}}\\ &-\frac{\big{(}12c_{2}+6c_{1}^{2}+2c_{2}\big{)}\big{(}1-\big{(}\xi ^{2}+2\xi+2\big{)}\exp(-\xi)\big{)}}{2\xi^{3}}\\ &-\frac{16c_{1}c_{2}}{\xi^{4}}\bigg{(}3-\frac{\big{(}\xi^{3}+3 \xi^{2}+6\xi+6\big{)}\exp(-\xi)}{2}\bigg{)}\\ &-\frac{5c_{2}^{2}\big{(}1-\big{(}\xi^{4}+4\xi^{3}+12\xi^{2}+24 \xi+24\big{)}\exp(-\xi)\big{)}}{\xi^{5}}\end{split}  \tag{21}\]  
respectively. We note that Equation (21) is associated with the specific polynomial model introduced in Equation (10).