Application of Hypergeometric Functions of Several Variables in the Mathematical Theory of Communication: Evaluation of Error Probability in Fading Singlechannel System

Abstract

A method is proposed of evaluation of symbol and/or bit error probabilities for coherent receiving of multipositional signal constructions in communication channel with fadings, which are described with the help of classical and generalized models Multiple-Wave with Diffuse Power (MWDP) fading and of additive white Gaussian noise (AWGN). This method uses the hypergeometric functions of several variables.

Appendix A. PROBABILITY DENSITY FUNCTION FOR MWDP

Consider the integral

$$J_{n}\left(\genfrac{}{}{0.0pt}{0}{\left(a_{n}\right);}{\sigma^{2},\mu,\left(\nu_{n}\right);s}\right)=\int\limits_{0}^{\infty}x^{s-1}e^{-\sigma^{2}x^{2}/2}J_{\nu_{1}}\left(a_{1}x\right)J_{\nu_{2}}\left(a_{2}x\right)\dots J_{\nu_{n}}\left(a_{n}x\right)dx.$$

(A1)

Since $J_{\nu}\left(z\right)=\dfrac{\left(z/2\right)^{\nu}}{\Gamma\left(\nu+1\right)}_{0}F_{1}\left(\nu+1;{-}\dfrac{z^{2}}{4}\right)$ the equality (A1) can be converted to the form

$$J_{n}\left(\genfrac{}{}{0.0pt}{0}{\left(a_{n}\right);}{\sigma^{2},\mu,\left(\nu_{n}\right);s}\right)=\frac{2^{-\nu_{1}-\dots-\nu_{n}}a_{1}^{\nu_{1}}\dots a_{n}^{\nu_{n}}}{\Gamma\left(\nu_{1}+1\right)\dots\Gamma\left(\nu_{n}+1\right)}\int\limits_{0}^{\infty}x^{s+\nu_{1}+\dots+\nu_{n}-1}$$

$${}\times e^{-\sigma^{2}x^{2}/2}{}_{0}F_{1}\left(\nu_{1}+1;{-}\frac{a_{1}^{2}x^{2}}{4}\right)\dots{}_{0}F_{1}\left(\nu_{n}+1;{-}\frac{a_{n}^{2}x^{2}}{4}\right)dx,$$

After the change of variable $t=\sigma^{2}x^{2}/2$ and simple transformations we obtain

$$J_{n}\left(\genfrac{}{}{0.0pt}{0}{\left(a_{n}\right);}{\sigma^{2},\mu,\left(\nu_{n}\right);s}\right)=\frac{1}{2}\left(\frac{2}{\sigma^{2}}\right)^{s/2}\prod_{k=1}^{n}\left(\frac{a_{k}^{2}}{2\sigma^{2}}\right)^{\nu_{k}/2}\Gamma\biggl{[}\genfrac{}{}{0.0pt}{0}{\frac{s+\nu_{1}+\dots+\nu_{n}}{2}}{\nu_{1}+1,\dots,\nu_{n}+1}\biggr{]}$$

$${}\times\Psi_{2}^{(n)}\left(\frac{s+\nu_{1}+\dots+\nu_{n}}{2};\nu_{1}+1,\ldots,\nu_{n}+1;{-}\frac{a_{1}^{2}}{2\sigma^{2}},\ldots,-\frac{a_{n}^{2}}{2\sigma^{2}}\right),$$

Here we use the formula

$$\Psi_{2}^{(n)}\left(a;c_{1},\ldots,c_{n};z_{1},\ldots,z_{n}\right)=\frac{1}{\Gamma\left(a\right)}\int\limits_{0}^{\infty}t^{a-1}e^{-t}{}_{0}F_{1}\left(c_{1};z_{1}t\right)\dots{}_{0}F_{1}\left(c_{n};z_{n}t\right)dt,$$

which follows from the relations

$$F_{A}^{(n)}\left(a,b_{1},\ldots,b_{n};c_{1},\ldots,c_{n};z_{1},\ldots,z_{n}\right)=\frac{1}{\Gamma\left(a\right)}\int\limits_{0}^{\infty}t^{a-1}e^{-t}{}{}_{1}F_{1}\left(b_{1};c_{1};z_{1}t\right)\dots{}_{1}F_{1}\left(b_{n};c_{n};z_{n}t\right)dt,$$

$${}_{0}F_{1}\left(b;z\right)=\lim_{a\to\infty}{}_{1}F_{1}\left(a;b;\frac{z}{a}\right),$$

where the Lauricella function is defined as

$$F_{A}^{(n)}\left(a,b_{1},\ldots,b_{n};c_{1},\ldots,c_{n};z_{1},\ldots,z_{n}\right)$$

$${}=\sum\limits_{k_{1},\ldots,k_{n}=0}^{\infty}\dfrac{\left(a\right)_{k_{1}+\ldots+k_{n}}\left(b_{1}\right)_{k_{1}}\ldots\left(b_{n}\right)_{k_{n}}}{\left(c_{1}\right)_{k_{1}}\ldots\left(c_{n}\right)_{k_{n}}}\dfrac{z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!},\qquad\biggl{[}\displaystyle\sum\limits_{j=1}^{n}|z_{j}|<1\biggr{]},$$

and the conluent function as

$$\Psi_{2}^{(n)}\left(a;c_{1},\ldots,c_{n};z_{1},\ldots,z_{n}\right)=\lim_{|b|\to\infty}F_{C}^{(n)}\left(a,b;c_{1},\ldots,c_{n};\frac{z_{1}}{b},\ldots,\frac{z_{n}}{b}\right)$$

$$=\lim_{\min\left(|b_{1}|,\dots,|b_{n}|\right)\to\infty}F_{A}^{(n)}\left(a,b_{1},\ldots,b_{n};c_{1},\ldots,c_{n};\frac{z_{1}}{b_{1}},\ldots,\frac{z_{n}}{b_{n}}\right)=\displaystyle\sum\limits_{k_{1},\ldots,k_{n}=0}^{\infty}\dfrac{\left(a\right)_{k_{1}+\ldots+k_{n}}}{\left(c_{1}\right)_{k_{1}}\ldots\left(c_{n}\right)_{k_{n}}}\dfrac{z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!},$$

and

$$F_{C}^{(n)}\left(a,b;c_{1},\ldots,c_{n};z_{1},\ldots,z_{n}\right)=\displaystyle\sum\limits_{k_{1},\ldots,k_{n}=0}^{\infty}\dfrac{\left(a\right)_{k_{1}+\ldots+k_{n}}\left(b\right)_{k_{1}+\ldots+k_{n}}}{\left(c_{1}\right)_{k_{1}}\ldots\left(c_{n}\right)_{k_{n}}}\dfrac{z_{1}^{k_{1}}\ldots z_{n}^{k_{n}}}{k_{1}!\ldots k_{n}!},\qquad\biggl{[}\displaystyle\sum\limits_{j=1}^{n}\sqrt{|z_{j}|}<1\biggr{]}.$$

Remark 1. From the integral representation of  the Lauricella function and identity ${}_{1}F_{1}\left(a,b;z\right)=e^{z}{}_{1}F_{1}\left(b-a,b;{-}z\right)$ with $\sum_{j=1}^{n}|z_{j}|<1$, it follows that

$$F_{A}^{(n)}\left(a,\left(b_{n}\right);\left(c_{n}\right);z_{1},\ldots,z_{n}\right)=\left(1-z_{1}-\dots-z_{n}\right)^{-a}$$

$${}\times F_{A}^{(n)}\left(a,\left(b_{n}-c_{n}\right);\left(c_{n}\right);-\frac{z_{1}}{1-z_{1}-\dots-z_{n}},\ldots,-\frac{z_{n}}{1-z_{1}-\dots-z_{n}}\right).$$