Performance analysis of digital transmission in interference-limited networks

Abstract

This paper aims at investigating the impact of interference on the performance of a wireless communication system operating over the Fisher–Snedecor $\mathcal {F}$ composite fading channel. In particular, we consider an interference-limited network scenario, where both links: the main link and interfere links, are modeled by $\mathcal {F}$ fading channels. Then, closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) are derived for such scenario. Based on the obtained PDF and CDF, closed-form expressions for the outage probability, average bit error rates (BERs) for several modulation schemes, and average channel capacity are derived. Furthermore, we derive the expressions for the asymptotic performance for these above-mentioned metrics in order to provide more insights into the impact of different channel parameters on the system performance. To provide further insights into the impact of interference, the expressions for the performance metrics under the no-interference setup have been also derived and compared to its interference-limited counterparts. The excellent agreement between the analytical and Monte-Carlo simulations validates our analysis.

Appendix

In this appendix, we provide some preliminary definitions and identities that help in the analysis of the system under consideration.

The Meijer’s G-function is defined by means of a one-fold contour integral in the complex plane [37, Eq. (8.2.1.1)], i.e.,

$$\begin{aligned}&\mathrm {G}_{p,q}^{m,n}\left( x\left| \begin{matrix} a_{1},\cdots ,a_{p}\\ b_{1},\cdots ,b_{q}\end{matrix}\right. \right) =\frac{1}{2\pi i}\int _{L}{\varPsi }_1(s)x^{-s}ds,&\end{aligned}$$

(A1)

where

$$\begin{aligned}&{\varPsi }_1\left( s\right) ={\prod _{j=1}^{m}{\varGamma }(b_{j}+s)\prod _{k=1}^{n}{\varGamma }(1-a_{k}-s)\over \prod _{k=n+1}^{p}{\varGamma }(a_{k}+s) \prod _{j=m+1}^{q}{\varGamma }(1-b_{j}-s)}.&\end{aligned}$$

The contour L is a suitable closed contour in the complex s-plane which can be chosen among three types of integration paths. The poles ${\varGamma }(b_{j}+s)$ must not coincide with the poles of ${\varGamma }(1-a_{k}-s)$ (with $j=1,\ldots ,m$ and $k=1,\ldots ,n$).

The Fox H-function [37, Eq. (8.2.1.1)] is defined by means of a one-fold contour integral in the complex plane, similar to the contour integral of the Meijer’s G-function, as

$$\begin{aligned} \begin{aligned} H \begin{array}{l}{m,}{n}\\ {p,}{q}\\ \end{array}\left( {x}\Big |\begin{array}{l}{[a_p,A_p]}\\ {[b_q,B_q]}\end{array}\right)&\equiv H \begin{array}{l}{m,}{n}\\ {p,}{q}\\ \end{array}\left( {x}\Big |\begin{array}{l}{(a_1,A_1), \ldots ,(a_p,A_p)}\\ {(b_1,B_1), \ldots ,(b_q,B_q)}\\ \end{array}\right) \\&= \frac{1}{2\pi i}\int _{L}{\varPsi }_2(s)x^{-s}ds, \end{aligned} \end{aligned}$$

(A2)

where

$$\begin{aligned}&{\varPsi }_2\left( s\right) ={\prod _{j=1}^{m}{\varGamma }(b_{j}+B_j s)\prod _{k=1}^{n}{\varGamma }(1-a_{k}-A_k s)\over \prod _{k=n+1}^{p}{\varGamma }(a_{k}+A_k s) \prod _{j=m+1}^{q}{\varGamma }(1-b_{j}-B_j s)}.&\end{aligned}$$

The Meijer’s G-function is a special case of the Fox H-function. In particular, the Fox-H function reduces to the standard Meijer’s G-function when $A_1 = \ldots = A_p = B_1 = \ldots = B_q = 1$ [37, Eq. (8.3.2.21)]

$$\begin{aligned} H \begin{array}{l}{m,}{n}\\ {p,}{q}\\ \end{array}\left( {x}\Big |\begin{array}{l}{[a_p,1]}\\ {[b_q,1]}\end{array}\right) = G \begin{array}{l}{m,}{n}\\ {p,}{q}\\ \end{array}\left( {x}\Big |\begin{array}{l}{(a_p)}\\ {(b_q)}\end{array}\right) \end{aligned}$$

(A3)

The Gaussian hypergeometric function is defined as [37, Eq. (7.2.1.1)]

$$\begin{aligned} \begin{aligned} { }_{2} F_{1}(a, b ; c ; z)=\sum _{k=0}^{\infty } \frac{(a)_{k}(b)_{k}}{(c)_{k} k !} z^{k}, \end{aligned} \end{aligned}$$

(A4)

The relation between the Gaussian hypergeometric function is related to the Meijer’s G-function through [37, Eq. (8.4.49.13)]

$$\begin{aligned} { }_{2} F_{1}(a, b ; c ;-x)=\frac{{\varGamma }(c)}{{\varGamma }(a){\varGamma }(b)} G\begin{array}{l}{1,}{2}\\ {2,}{2}\\ \end{array}\left( {x}\Big |\begin{array}{l}{1-a,1-b}\\ {0,1-c} \end{array}\right) \end{aligned}$$

(A5)

Two important identities that are used to reduce the order of the Meijer’s G-function are given by (A6) [37, Eq. (8.2.2.8)] and (A7) [37, Eq. (8.2.2.9)], respectively

$$\begin{aligned} G \begin{array}{l}{m,}{n}\\ {p,}{q}\\ \end{array}\left( {z}\Big |\begin{array}{l}{(a_p)}\\ {(b_{q-1}),a_1}\end{array}\right) = G \begin{array}{l}{m,}{n-1}\\ {p-1}{q-1}\\ \end{array}\left( {z}\Big |\begin{array}{l}{a_{2}, \ldots , a_{p}}\\ {(b_{q-1})}\end{array}\right) \end{aligned}$$

(A6)

$$\begin{aligned} G \begin{array}{l}{m,}{n}\\ {p,}{q}\\ \end{array}\left( {z}\Big |\begin{array}{l}{(a_p)}\\ {(b_{q-1}),a_1}\end{array}\right) = G \begin{array}{l}{m-1,}{n}\\ {p-1}{q-1}\\ \end{array}\left( {z}\Big |\begin{array}{l}{(a_{p-1})}\\ {b_2, \ldots ,b_q}\end{array}\right) \end{aligned}$$

(A7)

The $\ln (1+x)$ and $\text {erfc}(\sqrt{x})$ functions can be written in terms of the Meijer’s G-function using [37, Eq. (8.4.6.5)] and [37, Eq. (8.4.14.2)], respectively as follows

$$\begin{aligned} \ln (1+x) = G \begin{array}{l}{1,}{2}\\ {2,}{2}\\ \end{array}\left( {x}\Big |\begin{array}{l}{1,1}\\ {1,0} \end{array}\right) \end{aligned}$$

(A8)

$$\begin{aligned} \text {erfc}(\sqrt{x}) = \frac{1}{\sqrt{\pi }} G \begin{array}{l}{2,}{0}\\ {1,}{2}\\ \end{array}\left( {x}\Big |\begin{array}{l} {1}\\ {0,1/2}\end{array}\right) \end{aligned}$$

(A9)