LCR and AFD of the Products of Nakagami-m and Nakagami-m Squared Random Variables: Application to Wireless Communications Through Relays

Abstract

The paper considers level crossing rate (LCR) and average fade duration (AFD) of the product of independent and identically distributed (i.i.d) Nakagami-m (NM) and double NM squared (also known as gamma-gamma) random variables (RVs). The derived statistics are then directly applied to the radio-frequency (RF) - free space optical (FSO), dual-hop (DH), amplify-and-forward (AF) relaying system over non turbulent-induced-fading channels (nTIFCs) and turbulent-induced-fading channels (TIFCs). The obtained results for LCR and AFD of DH-AF, RF-FSO system over TIFCs and nTIFCs are numerically evaluated and graphically presented for various system model parameters.

Appendix A

The variance of $\dot{Z}_{SD}$ denoted as $\sigma _{\dot{Z}_{SD}}^{2}$ and given by (8) is obtained under the assumption that $\dot{Z}_{SD}$ is a zero-mean Gaussian RV [45]. Based on (1), the first derivative of ${Z}_{SD}$ can be written as:

$$\begin{aligned} \dot{Z}_{SD}= & y_{NM,2}^{2}y_{NM,3}^{2}\dot{y}_{NM,1}+2y_{NM,1}y_{NM,2}y_{NM,3}^{2}\dot{y}_{NM,2} \nonumber \\&{+}\,2y_{NM,1}y_{NM,2}^{2}y_{NM,3}\dot{y}_{NM,3} \end{aligned}$$

(A1)

where $\dot{Y}_{NM,1}$, $\dot{Y}_{NM,2}$ and $\dot{Y}_{NM,3}$ are the first derivatives of ${Y}_{NM,1}$,${Y}_{NM,2}$, and ${Y}_{NM,3}$, respectively. Since the linear transformation of the Gaussian RVs is a zero mean Gaussian RV, the variance of $\sigma _{\dot{Z}_{SD}}^{2}$ can be expressed through the variances of $\dot{Y}_{NM,1}$, $\dot{Y}_{NM,2}$ and $\dot{Y}_{NM,3}$ denoted as $\sigma _{\dot{Y}_{NM,1}}^{2}, \sigma _{\dot{Y}_{NM,2}}^{2}$ and $\sigma _{\dot{Y}_{NM,3}}^{2}$, respectively:

$$\begin{aligned} \sigma _{\dot{Z}_{SD}}^{2}= & y_{NM,2}^{4}y_{NM,4}^{4}\sigma _{\dot{Y}_{NM,1}}^{2}+4y_{NM,1}^{2}y_{NM,2}^{2}y_{NM,3}^{4}\sigma _{\dot{Y}_{NM,2}}^{2} \nonumber \\&{+}\,4y_{NM,1}^{2}y_{NM,2}^{4}y_{NM,3}^{2}\sigma _{\dot{Y}_{NM,3}}^{2} \end{aligned}$$

(A2)

After using substitution $y_{NM,1}=\frac{z_{SD}}{y_{NM,2}^{2}y_{NM,3}^{2}}$ and some algebra $\sigma _{\dot{Z}_{SD}}^{2}$ is obtained as given by (8).