Impact of Imperfect Channel Information on the Performance of Underlay Cognitive DF Multi-hop Systems

Abstract

This paper presents an analysis framework for performance evaluation of underlay cognitive decode-and-forward (DF) multi-hop systems over Rayleigh fading channel under imperfect channel information. Specifically, we derive the exact closed-form bit error rate (BER) and interference probability (i.e., the probability that the interference power constraint is invalid) expressions. The derived expressions are well supported by simulations and serve as useful tools for fast system performance evaluation under different aspects. To reduce the interference probability, we consider the back-off power control mechanism. Various results demonstrate the effect of channel information imperfection on the system performance and the trade-off between the interference probability and BER. Also, the performance of underlay cognitive DF multi-hop systems depends both network topology and the number of hops.

Appendix

This appendix derives $P_t$ in (7). Let $x = \left| {{{\hat{h}}_{tP}}} \right|$ and $y = \left| {{h_{tP}}} \right|$. Then the joint pdf of $x$ and $y$ is expressed as [26]

$$\begin{aligned} {f_{x,y}}\left( {x,y} \right) = \frac{{4xy{e^{ - \frac{{{\eta _y}{x^2} + {\eta _x}{y^2}}}{{{\eta _x}{\eta _y}\left( {1 - {\rho _{xy}}} \right) }}}}}}{{{\eta _x}{\eta _y}\left( {1 - {\rho _{xy}}} \right) }}{I_0}\left( {\frac{{2\sqrt{{\rho _{xy}}} xy}}{{\sqrt{{\eta _x}{\eta _y}} \left( {1 - {\rho _{xy}}} \right) }}} \right) , \end{aligned}$$

(18)

where ${\eta _x} = E\left\{ {{x^2}} \right\} = {\eta _{tP}}-{\sigma _{tP}},\,{\eta _y} = E\left\{ {{y^2}} \right\} ={\eta _{tP}}$, and ${\rho _{xy}}$ is the power correlation coefficient.

Using (2) and the definition of ${\rho _{xy}} = \mathrm{cov}\left( {{x^2},{y^2}} \right) \Big /\sqrt{\mathrm{var} \left\{ x^{2}\right\} \mathrm{var} \left\{ y^{2} \right\} }$, we finally obtain

$$\begin{aligned} {\rho _{xy}} = 1 - \frac{{{\sigma _{tP}}}}{{{\eta _{tP}}}}. \end{aligned}$$

(19)

The joint pdf of $z=x^2$ and $w=y^2$ can be achieved from that of $x$ and $y$ after the variable transformation. After some simplifications, we get the joint pdf of $z$ and $w$ as

$$\begin{aligned} {f_{{z},{w}}}\left( {z,w} \right) = \frac{{{e^{ - \frac{{{\eta _{tP}}z + \left( {{\eta _{tP}} - {\sigma _{tP}}} \right) w}}{{\left( {{\eta _{tP}} - {\sigma _{tP}}} \right) {\sigma _{tP}}}}}}}}{{\left( {{\eta _{tP}} - {\sigma _{tP}}} \right) {\sigma _{tP}}}}{I_0}\left( {\frac{{2\sqrt{zw} }}{{{\sigma _{tP}}}}} \right) , \end{aligned}$$

(20)

where $z,w>0$ and $I_0()$ is the zeroth-order modified Bessel function of the first kind [27, eq. (8.431.1)].

We express $P_t$ as

$$\begin{aligned} {P_t} = \Pr \left\{ {\rho w > z} \right\} = \int \limits _0^\infty {\int \limits _0^{\rho y} {{f_{z,w}}\left( {x,y} \right) dx} } dy = \int \limits _0^\infty {\int \limits _0^{\rho y} {\frac{{{e^{ - \frac{{{\eta _{tP}}x + \left( {{\eta _{tP}} - {\sigma _{tP}}} \right) y}}{{\left( {{\eta _{tP}} - {\sigma _{tP}}} \right) {\sigma _{tP}}}}}}}}{{\left( {{\eta _{tP}} - {\sigma _{tP}}} \right) {\sigma _{tP}}}}{I_0}\left( {\frac{{2\sqrt{xy} }}{{{\sigma _{tP}}}}} \right) dx} } dy.\nonumber \\ \end{aligned}$$

(21)

After changing the variable of $t = \sqrt{x}$ and applying [28, eq. (10)], we simplify (21) as

$$\begin{aligned} {P_t} = 1 - \frac{1}{{{\eta _{tP}}}}\int \limits _0^\infty {{e^{ - \frac{y}{{{\eta _{tr}}}}}}Q\left( {\sqrt{\frac{{2\left( {{\eta _{tP}} - {\sigma _{tP}}} \right) }}{{{\eta _{tP}}{\sigma _{tP}}}}} \sqrt{y},\quad \sqrt{\frac{{2\rho {\eta _{tP}}}}{{\left( {{\eta _{tP}} - {\sigma _{tP}}} \right) {\sigma _{tP}}}}} \sqrt{y} } \right) } dy, \end{aligned}$$

(22)

where $Q(a,b)$ is the first-order Marcum Q-function [28, eq. (1)].

Finally, we reduce (22) to (7) after changing the variable of $t = \sqrt{y}$ and applying [28, eq. (55)].