Performance analysis of receive diversity under time-varying and spatially correlated channels using partial CSI

Abstract

In this paper, a single input multiple output system is considered with L receive antennas and the underlying channels are assumed to be time varying with temporal correlation coefficient a and spatially correlated with correlation coefficient $\rho$. Further, the channel is assumed to be identically distributed using Rayleigh fading channels and characterized by first order autoregressive model. For the detection, we assume partial channel state information (CSI) available at the receiver. The partial CSI is in the form of a known preamble (one symbol) only before beginning of a frame of size N symbols. Further, the preamble is imperfectly known with a variance of error $\sigma _e^2$. For the assumed system, closed form expressions of symbol error rate (SER) are derived for M-PSK and M-QAM constellations under the compound effect of spatially correlated channels, temporally correlated channels and partial CSI at the Receiver. The derived expressions are functions of average SNR per symbol, $\rho$, a, $\sigma _e^2$, N, L and modulation order M. Further, these expressions are reduced for some special cases and compared with prevailing results in literature. The analytical expressions are also validated by comparing them with the corresponding simulation results. The derived expressions are very useful to select N, L and M to overcome the deterioration in SER due to adverse effects of $\rho$, a and $\sigma _e^2$.

Appendices

Appendix-A

Symbol error rate ($P_{s,k,PSK}$) at $k{\mathrm{th}}$ symbol position for M-PSK modulation scheme is defined, in terms of MGF as described in section-III, as follows:

$$\begin{aligned} P_{s,k,PSK}&= \frac{1}{2\pi } \int _{\mu }^{\infty } \frac{M(-g_{PSK}y)}{y\sqrt{y-1}} dy \end{aligned}$$

where, $\mu = cosec^2(\frac{\pi (M-1)}{M})$ and $g_{PSK} = sin^2(\pi /M)$.

$$\begin{aligned} P_{s,k,PSK}&= \int _{\mu }^{\infty } \frac{y^{-1}(1+\bar{\gamma }_kg_{PSK}y(1-\rho ))^{1-L}dy}{2\pi \sqrt{y-1}(1+\bar{\gamma }_kg_{PSK}y(1-\rho (1-L)))}\\&= \int _{\mu }^{\infty } \Bigg [ \frac{1}{2\pi y\sqrt{y-1}(1+\bar{\gamma }_kg_{PSK}y(1-\rho ))^{L-1}} \\&\quad -\frac{(1+\bar{\gamma }_kg_{PSK}y(1-\rho ))^{1-L}}{\sqrt{y-1}(1+\bar{\gamma }_kg_{PSK}y(1-\rho (1-L)))} \\&\quad \quad \bar{\gamma }_kg_{PSK}(1-\rho (1-L))\Bigg ] dy \\&= I_1 - I_2 \end{aligned}$$

Integration of $I_1$ and $I_2$ are derived separately to find out $P_{s,k,PSK}$ at $k{\mathrm{th}}$ symbol for M-PSK modulation. Generalized expression of symbol error rate for BPSK ($M=2$) and M-PSK ($M>2$) signals are derived separately using proposed scheme; as, it was difficult to derive the generalized expression for symbol error rate for $M\ge 2$.

Case-1: For M-PSK modulation ($M>2$),

$$\begin{aligned} I_1&= \frac{1}{2\pi } \int _{\mu }^{\infty } \Bigg [ \frac{1}{y\sqrt{y-1}(1+\bar{\gamma }_kg_{PSK}y(1-\rho ))^{L-1}} \Bigg ] dy \end{aligned}$$

using [20], above integration is reduced to following term,

$$\begin{aligned} I_1&= \frac{(\bar{\gamma }_kg_{PSK}(1-\rho ))^{1-L}}{2\pi } \Bigg [ 2\Bigg (\frac{1}{\bar{\gamma }_kg_{PSK}(1-\rho )}+1\Bigg )^{1-L} \\&\quad \sqrt{y-1}{F}_1(\frac{1}{2};L-1,1;\frac{3}{2}; -\frac{y-1}{1+\frac{1}{\bar{\gamma }_kg_{PSK}(1-\rho )}}, 1-y) \Bigg ]^\infty _{\mu } \\ I_1&= -\frac{\sqrt{\mu -1}}{\pi (\bar{\gamma }_kg_{PSK}(1-\rho )+1)^{L-1}} {F}_1\Bigg (\frac{1}{2};L-1,1;\frac{3}{2}; \\&\quad \frac{1-\mu }{1+\frac{1}{\bar{\gamma _k}g_{PSK}(1-\rho )}},1-\mu \Bigg ) \\ I_2&= \frac{1}{2\pi } \int _{\mu }^{\infty } \Bigg [\frac{\bar{\gamma }_kg_{PSK}(1-\rho (1-L))}{\sqrt{y-1}(1+\bar{\gamma }_kg_{PSK}y(1-\rho (1-L)))} \\&\quad (1+\bar{\gamma }_kg_{PSK}y(1-\rho ))^{1-L} \Bigg ] dy \\&= \frac{\bar{\gamma }_kg_{PSK}(1-\rho (1-L))}{2\pi } \\&\quad \int _{\mu -1}^{\infty } \Bigg [\frac{(1+\bar{\gamma }_kg_{PSK}(1+t)(1-\rho ))^{1-L}}{\sqrt{t}(1+\bar{\gamma }_kg_{PSK}(1+t)(1-\rho (1-L)))}\Bigg ]dt \\&= \frac{\bar{\gamma }_kg_{PSK}(1-\rho (1-L))}{2\pi } \\&\quad \int _{\mu -1}^{\infty } \Bigg [\frac{(1+\bar{\gamma }_kg_{PSK}(1-\rho )+\bar{\gamma }_kg_{PSK}t(1-\rho ))^{1-L}}{\sqrt{t}} \\&\quad \frac{dt}{(1+\bar{\gamma }_kg_{PSK}(1-\rho (1-L)) + \bar{\gamma }_kg_{PSK}t(1-\rho (1-M)))} \Bigg ] \\&= \frac{(\bar{\gamma }_kg_{PSK}(1-\rho ))^{1-L}}{2\pi } \int _{\mu -1}^{\infty } \frac{(\beta +t)^{1-L}}{\sqrt{t}(\alpha +t)} dt \\&\quad \Bigg [where, ~\alpha =\frac{1+\bar{\gamma }_kg_{PSK}(1-\rho (1-L))}{\bar{\gamma }_kg_{PSK}(1-\rho (1-L))} ~and \\&~\beta =\frac{1+\bar{\gamma }_kg_{PSK}(1-\rho )}{\bar{\gamma }_kg_{PSK}(1-\rho )} \Bigg ] \end{aligned}$$

using [20], above integration is reduced to following term,

$$\begin{aligned} I_2&= \frac{(\bar{\gamma }_kg_{PSK}(1-\rho ))^{1-L}}{2\pi } \Bigg [ \frac{2\sqrt{t}}{\alpha \beta ^L} \Bigg ( (\beta -\alpha ){F}_1\Bigg (\frac{1}{2};L,1; \\&\quad \frac{3}{2};-\frac{t}{\beta },-\frac{t}{\alpha }\Bigg ) + \alpha _2{F}_1\Bigg (\frac{1}{2},L;\frac{3}{2};-\frac{t}{\beta }\Bigg ) \Bigg ) \Bigg ]^\infty _{\mu -1} \\ I_2&= \frac{-\sqrt{\mu -1}}{\pi (\bar{\gamma }_kg_{PSK}(1-\rho ))^{L-1}\alpha \beta ^L} \Bigg [ (\beta -\alpha ) {F}_1\Bigg (\frac{1}{2};L,1; \\&\quad \frac{3}{2};\frac{1-\mu }{\beta },\frac{1-\mu }{\alpha }\Bigg )+\alpha _2{F}_1\Bigg (\frac{1}{2},L;\frac{3}{2};-\frac{\mu -1}{\beta }\Bigg ) \Bigg ] \end{aligned}$$

So, symbol error rate $P_{s,k,PSK}$ for M-PSK modulation ($M>2$) can be written as

$$\begin{aligned}&P_{s,k,PSK}= \frac{\sqrt{\mu -1}}{\pi (\bar{\gamma }_kg_{PSK}(1-\rho ))^{L-1}\alpha \beta ^L} \Bigg [ (\beta -\alpha ) \\&\qquad {F}_1\Bigg (\frac{1}{2};L,1;\frac{3}{2};\frac{1-\mu }{\beta },\frac{1-\mu }{\alpha }\Bigg )\\&\qquad \qquad \quad \quad +\alpha _2{F}_1\Bigg (\frac{1}{2},L;\frac{3}{2};-\frac{\mu -1}{\beta }\Bigg ) \Bigg ] \\&\qquad \qquad \quad \quad - \frac{\sqrt{\mu -1}}{\pi (\bar{\gamma }_kg_{PSK}(1-\rho )+1)^{L-1}} \\&\quad \quad {F}_1\Bigg (\frac{1}{2};L-1,1; \frac{3}{2}; \frac{1-\mu }{1+\frac{1}{\bar{\gamma _k}g_{PSK}(1-\rho )}},1-\mu \Bigg ). \end{aligned}$$

Case-2: For BPSK modulation ($M=2$),

$$\begin{aligned} I_1&= \frac{1}{2\pi } \int _{1}^{\infty } \Bigg [ \frac{(1+\bar{\gamma }_ky(1-\rho ))^{1-L}}{y\sqrt{y-1}} \Bigg ] dy\\&= \frac{(\bar{\gamma }_k(1-\rho ))^{1-L}}{2\pi } \int _{1}^{\infty } \Bigg [\frac{(\frac{1}{\bar{\gamma }_k(1-\rho ))}+ y)^{1-L}}{y\sqrt{y-1}} \Bigg ] dy \end{aligned}$$

Above integral term is solved using [eq.(3.197), [21]],

$$\begin{aligned} I_1&= \frac{(\bar{\gamma }_k(1-\rho ))^{1-L}}{2\pi } {B}\Bigg (L-\frac{1}{2}, \frac{1}{2}\Bigg ){}_2{F}_1\Bigg (L-1, \\&\quad L-\frac{1}{2}; L; -\frac{1}{\bar{\gamma }_k(1-\rho )}\Bigg ) \\ I_2&= \int _{1}^{\infty } \Bigg [\frac{\bar{\gamma }_k(1-\rho (1-L))(1+\bar{\gamma }_ky(1-\rho ))^{1-L}}{2\pi \sqrt{y-1}(1+\bar{\gamma }_ky(1-\rho (1-L)))} \Bigg ]dy \\&= \frac{\bar{\gamma }_k(1-\rho (1-L))}{2\pi } \\&\quad \int _{0}^{\infty } \Bigg [\frac{(1+\bar{\gamma }_k(1+t)(1-\rho ))^{1-L}}{\sqrt{t}(1+\bar{\gamma }_k(1+t)(1-\rho (1-L)))}\Bigg ]dt \\&= \frac{1}{2\pi (\bar{\gamma }_k(1-\rho ))^{L-1}} \int _{0}^{\infty } t^{-\frac{1}{2}} \Bigg (\frac{1+\bar{\gamma }_k(1-\rho )}{\bar{\gamma }_k(1-\rho )} + t\Bigg )^{1-L} \\&\quad \Bigg (t+\frac{1+\bar{\gamma }_k(1-\rho (1-L))}{\bar{\gamma }_k(1-\rho (1-L)}\Bigg )^{-1} dt \end{aligned}$$

Above integral term is solved using [eq.(3.197),[21]],

$$\begin{aligned} I_2&=\frac{(1+\bar{\gamma }_k(1-\rho ))^{1-L}(\bar{\gamma }_k(1-\rho (1-L)))^{\frac{1}{2}}}{2\pi (1+\bar{\gamma }_k(1-\rho (1-L)))^{\frac{1}{2}}} {B}\Bigg (\frac{1}{2},L-\frac{1}{2}\Bigg ) \\&\quad _2{F}_1\Bigg (L-1, \frac{1}{2}; L;1-\frac{(1+\bar{\gamma }_k(1-\rho (1-L)))(1-\rho )}{(1-\rho (1-L))(1+\bar{\gamma }_k(1-\rho ))}\Bigg ) \end{aligned}$$

So, Symbol error rate $P_{s,k,PSK}$ for BPSK modulation scheme can be written as

$$\begin{aligned}&P_{s,k,PSK}= \frac{(\bar{\gamma }_k(1-\rho ))^{1-L}}{2\pi } {B}\Bigg (L-\frac{1}{2}, \frac{1}{2}\Bigg ) \\&\quad _2{F}_1\Bigg (L-1, L-\frac{1}{2}; L; -\frac{1}{\bar{\gamma }_k(1-\rho )}\Bigg ) - \\&\quad \frac{(1+\bar{\gamma }_k(1-\rho ))^{1-L}(\bar{\gamma }_k(1-\rho (1-L)))^{\frac{1}{2}}}{2\pi (1+\bar{\gamma }_k(1-\rho (1-L)))^{\frac{1}{2}}} {B}\Bigg (\frac{1}{2},L-\frac{1}{2}\Bigg ) \\&\quad _2{F}_1\Bigg (L-1, \frac{1}{2}; L;1-\frac{(1+\bar{\gamma }_k(1-\rho (1-L)))(1-\rho )}{(1-\rho (1-L))(1+\bar{\gamma }_k(1-\rho ))}\Bigg ). \end{aligned}$$

Appendix-B

Symbol error rate ($P_{s,k,QAM}$) at $k{\mathrm{th}}$ symbol position for M-QAM modulation scheme is defined, in terms of MGF as described in section-III, as follows:

$$\begin{aligned} P_{s,k,QAM}&= \frac{2}{\pi } \Bigg (1-\frac{1}{\sqrt{M}} \Bigg ) \int _{1}^{\infty } \frac{M(-g_{QAM}y)}{y\sqrt{y-1}} dy~ \\&\quad -\frac{2}{\pi } \Bigg (1-\frac{1}{\sqrt{M}} \Bigg )^2 \int _{2}^{\infty } \frac{M(-g_{QAM}y)}{y\sqrt{y-1}} dy, \\&= I_1 - I_2 \end{aligned}$$

where, $g_{{ QAM}} = 3/2(M-1)$. Integration of $I_1$ and $I_2$ are derived separately to find out $P_{s,k,QAM}$ at $k{\mathrm{th}}$ symbol for M-QAM modulation.

$$\begin{aligned} I_1&= \int _{1}^{\infty } \frac{\frac{2}{\pi }(1-\frac{1}{\sqrt{M}})(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{y\sqrt{y-1}(1+\bar{\gamma }_kg_{QAM}y(1-\rho (1-L)))} dy \\&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg ) \int _{1}^{\infty } \Bigg [ \frac{(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{y\sqrt{y-1}} \\&\quad -\frac{\bar{\gamma }_kg_{QAM}(1-\rho (1-L))(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{\sqrt{y-1}(1+\bar{\gamma }_kg_{QAM}y(1-\rho (1-L)))} \Bigg ] dy \\&=I_{11} - I_{12} \end{aligned}$$

Integration of $I_{11}$ and $I_{12}$ are derived separately to find out solution of $I_1$.

$$\begin{aligned} I_{11}&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg ) \int _{1}^{\infty } \Bigg [ \frac{(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{y\sqrt{y-1}} \Bigg ]dy \end{aligned}$$

Above integral term is solved using [eq.(3.197),[21]],

$$\begin{aligned} I_{11}&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg ) \frac{{B}(L-\frac{1}{2},\frac{1}{2})}{(\bar{\gamma }_kg_{QAM}(1-\rho ))^{L-1}} \\&\quad _2F_1\Bigg (L-1,L-\frac{1}{2};L;\frac{-1}{\bar{\gamma }_kg_{QAM}(1-\rho )}\Bigg ) \\ I_{12}&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg ) \int _{1}^{\infty } \Bigg [ \frac{\bar{\gamma }_kg_{QAM}(1-\rho (1-L))}{\sqrt{y-1}} \\&\quad \frac{(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{(1+\bar{\gamma }_kg_{QAM}y(1-\rho (1-L)))} \Bigg ] dy \\&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )(\bar{\gamma }_kg_{QAM}(1-\rho ))^{1-L} \int _{0}^{\infty } \Bigg [ t^{-\frac{1}{2}} \\&\quad \Bigg (t+\frac{1+\bar{\gamma }_kg_{QAM}(1-\rho (1-L))}{\bar{\gamma }_kg_{QAM}(1-\rho (1-L))}\Bigg )^{-1} \\&\quad \Bigg (t+\frac{1+\bar{\gamma }_kg_{QAM}(1-\rho )}{\bar{\gamma }_kg_{QAM}(1-\rho )}\Bigg )^{1-L}\Bigg ] dt \end{aligned}$$

Above integral term is solved using [eq.(3.197), [21]],

$$\begin{aligned} I_{12}&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )\frac{B(\frac{1}{2},L-\frac{1}{2})}{(1+\bar{\gamma }_kg_{QAM}(1-\rho ))^{L-1}} ~_2F_1 \\&\qquad \Bigg (L- 1,\frac{1}{2};L;1-\frac{(1+\bar{\gamma }_kg_{QAM}(1-\rho (1-L)))(1-\rho )}{(1-\rho (1-L))(1+\bar{\gamma }_kg_{QAM}(1-\rho ))} \Bigg ) \\&\qquad \Bigg ( \frac{\bar{\gamma }_kg_{QAM}(1-\rho (1-L))}{1+\bar{\gamma }_kg_{QAM}(1-\rho (1-L))} \Bigg )^{\frac{1}{2}} \\ I_2&= \int _{2}^{\infty } \frac{\frac{2}{\pi }(1-\frac{1}{\sqrt{M}})^2(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{y\sqrt{y-1}(1+\bar{\gamma }_kg_{QAM}y(1-\rho (1-L)))} dy \\&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )^2 \int _{2}^{\infty } \Bigg [ \frac{(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{y\sqrt{y-1}} \\&\quad -\frac{\bar{\gamma }_kg_{QAM}(1-\rho (1-L))(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{\sqrt{y-1}(1+\bar{\gamma }_kg_{QAM}y(1-\rho (1-L)))} \Bigg ] dy \\&=I_{21} - I_{22} \end{aligned}$$

Integration of $I_{21}$ and $I_{22}$ are derived separately to find out solution of $I_2$.

$$\begin{aligned} I_{21}&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )^2 \int _{2}^{\infty } \frac{(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{y\sqrt{y-1}} dy \end{aligned}$$

using [20]], above integration is reduced to following term,

$$\begin{aligned} I_{21}&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )^2 (\bar{\gamma }_kg_{QAM}(1-\rho ))^{1-L} \Bigg [2\sqrt{y-1} \\&\quad \Bigg ( 1+\frac{1}{\bar{\gamma }_kg_{QAM}(1-\rho )} \Bigg )^{1-L} F_1 \Bigg (\frac{1}{2};L-1,1;\frac{3}{2}; \\&\quad \frac{1-y}{1+\frac{1}{\bar{\gamma }_kg_{QAM}(1-\rho )}}, 1-y \Bigg )\Bigg ]^{\infty }_{2} \\ I_{21}&= -\frac{4}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )^2 (1+\bar{\gamma }_kg_{QAM}(1-\rho ))^{1-L} \\&\quad F_1 \Bigg (\frac{1}{2};L-1,1;\frac{3}{2};\frac{-1}{1+\frac{1}{\bar{\gamma }_kg_{QAM}(1-\rho )}}, -1 \Bigg ) \\ I_{22}&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )^2 \int _{2}^{\infty } \Bigg [ \frac{\bar{\gamma }_kg_{QAM}(1-\rho (1-L))}{\sqrt{y-1}} \\&\quad \frac{(1+\bar{\gamma }_kg_{QAM}y(1-\rho ))^{1-L}}{(1+\bar{\gamma }_kg_{QAM}y(1-\rho (1-L)))} \Bigg ] dy \\&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )^2 \int _{1}^{\infty } \frac{(\bar{\gamma }_kg_{QAM}(1-\rho ))^{1-L}}{\sqrt{t}(\alpha +t)(\beta +t)^{L-1}} dt \\&\quad \Bigg [where, ~\alpha =\frac{1+\bar{\gamma }_kg_{QAM}(1-\rho (1-L))}{\bar{\gamma }_kg_{QAM}(1-\rho (1-L))} ~and \\&\quad \beta =\frac{1+\bar{\gamma }_kg_{QAM}(1-\rho )}{\bar{\gamma }_kg_{QAM}(1-\rho )} \Bigg ] \end{aligned}$$

using [20], above integration is reduced to following term,

$$\begin{aligned} I_{22}&= \frac{2}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )^2 (\bar{\gamma }_kg_{QAM}(1-\rho ))^{1-L} \Bigg [ \frac{2\sqrt{t}}{\alpha \beta ^L}\Bigg ((\beta -\alpha ) \\&F_1\Bigg (\frac{1}{2};L,1;\frac{3}{2};-\frac{t}{\beta },-\frac{t}{\alpha }\Bigg ) +\alpha _2F_1\Bigg (\frac{1}{2},L;\frac{3}{2};\frac{-t}{\beta }\Bigg ) \Bigg ]^{\infty }_{1} \\ I_{22}&= -\frac{4}{\pi }\Bigg (1-\frac{1}{\sqrt{M}}\Bigg )^2 \frac{(\bar{\gamma }_kg_{QAM}(1-\rho ))^{1-L}}{\alpha \beta ^L} \Bigg [ (\beta -\alpha ) \\&F_1\Bigg (\frac{1}{2};L,1;\frac{3}{2};-\frac{1}{\beta },-\frac{1}{\alpha }\Bigg ) + \alpha _2F_1\Bigg (\frac{1}{2},L;\frac{3}{2};\frac{-1}{\beta }\Bigg ) \Bigg ] \end{aligned}$$

using $I_{11}$, $I_{12}$, $I_{21}$, $I_{22}$ terms, $P_{s,k,QAM}$ can be written as follows:

$$\begin{aligned}&P_{s,k,QAM}= \frac{2(1-1/\sqrt{M})}{\pi (\bar{\gamma }_kg_{QAM}(1-\rho ))^{L-1}} {B}\Bigg (L-\frac{1}{2}, \frac{1}{2}\Bigg ) ~_2{F}_1 \\&\quad \quad \Bigg (L- 1, L-\frac{1}{2}; L; -\frac{1}{\bar{\gamma }_k(1-\rho )}\Bigg ) \\&\qquad \qquad \quad \quad -\frac{2(1-1/\sqrt{M})}{\pi (1+\bar{\gamma }_kg_{QAM}(1-\rho ))^{L-1}}\\&\quad \quad \Bigg (\frac{\bar{\gamma }_kg_{QAM}(1-\rho (1-L))}{1+\bar{\gamma }_kg_{QAM}(1-\rho (1-L))}\Bigg )^{\frac{1}{2}} {B}\Bigg (\frac{1}{2},L-\frac{1}{2}\Bigg )\\&\quad \quad _2{F}_1\Bigg (L-1, \frac{1}{2}; L; 1-\frac{(1+\bar{\gamma }_kg_{QAM}(1-\rho (1-L)))(1-\rho )}{(1-\rho (1-L))(1+\bar{\gamma }_kg_{QAM}(1-\rho ))}\Bigg ) \\&\qquad \qquad \quad \quad + \frac{4(1-1/\sqrt{M})^2}{\pi (1+\bar{\gamma }_kg_{QAM}(1-\rho ))^{L-1}} F_1\Bigg ( \frac{1}{2}; L-1, 1; \frac{3}{2}; \\&\quad \quad \quad \quad \quad \quad \quad \frac{-1}{1+\frac{1}{\bar{\gamma }_kg_{QAM}(1-\rho )}},-1\Bigg ) \\&\qquad \qquad \quad \quad + \frac{4(1-1/\sqrt{M})^2}{\pi (\bar{\gamma }_kg_{QAM}(1-\rho ))^{L-1}\alpha \beta ^L}\\&\quad \quad \Bigg [(\beta -\alpha ) F_1\Bigg (\frac{1}{2};L,1;\frac{3}{2};\frac{-1}{\beta },\frac{-1}{\alpha }\Bigg )\\&\qquad \qquad \quad \quad + \alpha _2F_1\Bigg (\frac{1}{2},L;\frac{3}{2};\frac{-1}{\beta }\Bigg )\Bigg ]. \end{aligned}$$