## 3 Performance Evaluation
### Error Probability  
3.1.1. Analytical Approach  
The conditional PDF of the phase of the composite signal at the receiver input can be presented in the form of  
\[p_{\Delta|R}(\delta|r)=\frac{1}{2\pi}+\sum_{n=0}^{+\infty}a_{n}(r)\cos(n \delta),\quad|\delta|\leq\pi, \tag{14}\]  
where coefficients \(a_{n}(r)\) are defined by [3; 4]  
\[a_{n}(r)=\frac{1}{n!\pi}\Gamma\Big{(}1+\frac{n}{2}\Big{)}\bigg{(}\frac{r}{ \sqrt{2}\sigma}\bigg{)}^{\!n}\exp\!\bigg{(}-\frac{r^{2}}{2\sigma^{2}}\bigg{)} _{1}\Gamma_{\!1}\!\bigg{(}\frac{n}{2}+1;n+1;\frac{r^{2}}{2\sigma^{2}}\bigg{)}. \tag{15}\]  
Unlike in our previous paper [4] in which the coefficients \(a_{n}\) were given as a function of the instantaneous SNR, here, we offer those coefficients as a function of the signal envelope. The average PDF of the composite signal phase can be written as  
\[p_{\Delta}(\delta)=\frac{1}{2\pi}+\sum_{n=0}^{+\infty}b_{n}\cos(n\delta),\quad |\delta|\leq\pi, \tag{16}\]  
where coefficients \(b_{n}\) can be determined based on  
\[b_{n}=\int\limits_{0}^{+\infty}a_{n}(r)p_{R}(r)\,\mathrm{d}r. \tag{17}\]  
The result is due to the fact that integrals and countable sums are interchangeable for non-negative functions, such as \(a_{n}(r)p_{R}(r)\), according to [PERSON]'s theorem. Obviously, \(a_{n}(r)\) is nonnegative, and the same is true by definition for any PDF \(p_{R}(r)\) on \([0,+\infty)\). The CDF of the signal envelope can be presented in the form of [19]  
\[F_{R}(r)=\frac{r^{2}\e^{-\frac{r^{2}}{2\sigma_{F}^{2}}}}{2\sigma_{F}^{2}}\sum _{m=0}^{+\infty}\frac{(-1)^{m}}{m!}h_{1}^{2m}F_{1}\!\Big{(}-m,-m;1;h_{2}^{2} \Big{)}_{1}F_{1}\!\!\left(1-m;2;\frac{r^{2}}{2\sigma_{F}^{2}}\right), \tag{18}\]  
where \(h_{1}^{2}=\frac{V_{1}^{2}}{2\sigma_{F}^{2}}\), \(h_{2}^{2}=\frac{V_{2}^{2}}{V_{1}^{2}}\). Using [20], we write the expansion using Laguerre orthogonal polynomials:  
\[F_{R}(r) = \frac{r^{2}}{2\sigma_{F}^{2}}\e^{-\frac{r^{2}}{2\sigma_{F}^{2}}} \Bigg{[}_{1}F_{1}\!\!\left(1;2;\frac{r^{2}}{2\sigma_{F}^{2}}\right) \tag{19}\] \[+ \sum_{m=1}^{+\infty}\frac{(-1)^{m}}{m}\frac{h_{1}^{2m}}{m!}_{2}F_ {1}\!\!\left(-m,-m;1;h_{2}^{2}\right)\!\!\L_{m-1}^{1}\!\left(\frac{r^{2}}{2 \sigma_{F}^{2}}\right)\Bigg{]}.\]Using integration by part, we obtain  
\[b_{n}=\lim_{r\rightarrow+\infty}a_{n}(r)F_{R}(r)-\lim_{r\to 0^{+}}a_{n}(r)F_{R}(r)- \int\limits_{0}^{+\infty}\frac{\mathrm{d}}{\mathrm{d}r}\Big{[}a_{n}(r)\Big{]}F _{R}(r)\,\mathrm{d}r. \tag{20}\]  
Limits \(\lim_{r\rightarrow+\infty}F_{R}(r)=1\) and \(\lim_{r\to 0^{+}}F_{R}(r)=0\) exist, and thus  
\[b_{n}=\frac{1}{\pi}-\int\limits_{0}^{+\infty}\frac{\mathrm{d}}{\mathrm{d}r} \Big{[}a_{n}(r)\Big{]}F_{R}(r)\,\mathrm{d}r. \tag{21}\]  
The derivative from the previous equation can be expressed as  
\[\frac{\mathrm{d}a_{n}}{\mathrm{d}r}=\frac{n\exp\left(-\frac{r^{2}}{4\sigma^{2} }\right)\left(\frac{r}{\sigma}\right)^{n-1}}{\sqrt{\pi}\sigma^{3}2^{3(2+n)/2}} \left[8\sigma^{2}\frac{\mathrm{i}\Gamma_{\!\left(\right)}\!\left(\frac{n+1}{2} ;\frac{r^{4}}{64\sigma^{4}}\right)}{\Gamma\!\left(\frac{n+1}{2}\right)}-r^{2} \frac{\mathrm{o}\Gamma_{\!\left(\right)}\!\left(\frac{n+3}{2};\frac{r^{4}}{64 \sigma^{4}}\right)}{\Gamma\!\left(\frac{n+3}{2}\right)}\right]. \tag{22}\]  
Using \(x=r/(2\sigma)\) and ([21], 07.17.27.0003.01 ), the derivative expression is shortened to  
\[a_{n}(r)=\frac{1}{n!\pi}\Gamma\Big{(}1+\frac{n}{2}\Big{)}\bigg{(}\frac{r}{ \sqrt{2}\sigma}\bigg{)}^{\!n}\exp\!\bigg{(}-\frac{r^{2}}{2\sigma^{2}}\bigg{)} _{1}\Gamma_{\!1}\!\bigg{(}\frac{n}{2}+1;n+1;\frac{r^{2}}{2\sigma^{2}}\bigg{)}. \tag{15}\]0  
It can be shown that the derivative is nonnegative, \(\mathrm{d}a_{n}/\,\mathrm{d}r\geq 0,r\geq 0\). Next, we proceed to prove that \(F_{R}(r)\) converges uniformly. We start with Weierstrass \(M\)-test on series (19) by finding the supremum over \(r\), or \(\sup_{x}x^{2}\,\mathrm{e}^{-x^{2}}\,\mathrm{|L^{\mathrm{L}}_{m-1}(x^{2})|}\). Using ([22], 10.18(14)), ([23], (18.14.8)), we notice that \(x\mathrm{e}^{-x}\big{|}L^{\mathrm{L}}_{m-1}(x)\big{|}\leq m\mathrm{xe}^{-x/2} \leq 2m/\mathrm{e}\).