Cooperative spectrum sensing with energy harvesting

Abstract

In cognitive radio networks, secondary nodes should sense the channel using spectrum sensing algorithms to detect primary activity. When primary user (PU) is idle, secondary users opportunistically transmit over this frequency hole. In this paper, we provide the detection probability (DP) of the energy detector (ED) for wireless energy harvesting systems. PU and relays harvest energy from radio frequency signal received from another node H. The harvested energy is used by PU to transmit its signal and by the relay to amplify PU signal to a fusion center (FC). The FC uses the relayed signal to detect PU activity using the ED. The major contribution of the paper is to show that the DP can be lower bounded using the cumulative distribution function of signal to noise ratio. This new approach is applied to compute the DP of the ED of cooperative spectrum sensing. The derived expressions of DP are valid for relays with arbitrary positions. The theoretical derivations are confirmed using simulation results obtained with MATLAB.

Appendix

If $X_{1}$ and $X_{2}$ are exponential r.v. with mean 1/$\lambda _{1}$ and 1/$\lambda _{2}.$

The CDF of $X=X_{1}X_{2}$ is expressed as

$$\begin{aligned} P_{X}(x)= & {} P(X_{1}X_{2}\le x)\nonumber \\= & {} \int _{0}^{+\infty }P\left( X_{1}\le \frac{x}{y} \right) \lambda _{2}e^{-\lambda _{2}y}dy. \end{aligned}$$

(56)

We deduce

$$\begin{aligned} P_{X}(x)= & {} \int _{0}^{+\infty }\left[ 1-e^{-\lambda _{1}\frac{x}{y}}\right] \lambda _{2}e^{-\lambda _{2}y}dy \nonumber \\= & {} 1-\int _{0}^{+\infty }e^{-\lambda _{1}\frac{x}{y}}\lambda _{2}e^{-\lambda _{2}y}dy \end{aligned}$$

(57)

We have

$$\begin{aligned}&\int _{0}^{+\infty }e^{-\frac{c}{y}}e^{-\frac{y}{d}}dy\nonumber \\&\quad =2\sqrt{\frac{c}{d}} K_{1}\left( 2\sqrt{\frac{c}{d}}\right) . \end{aligned}$$

(58)

We use (57) and (58) with $c=\lambda _{1}x$ and $d=\frac{1}{ \lambda _{2}}$, we obtain

$$\begin{aligned} P_{X}(x)=1-2\sqrt{\lambda _{1}\lambda _{2}x}K_{1}(2\sqrt{\lambda _{1}\lambda _{2}x}). \end{aligned}$$

(59)

We deduce the PDF of SNR

$$\begin{aligned} p_{X}(x)= & {} -\frac{\sqrt{\lambda _{1}\lambda _{2}}}{\sqrt{x}}K_{1}(2\sqrt{ \lambda _{1}\lambda _{2}x})\nonumber \\&-2\sqrt{\lambda _{1}\lambda _{2}x}K_{1}^{^{\prime }}(2\sqrt{\lambda _{1}\lambda _{2}x})\frac{\sqrt{\lambda _{1}\lambda _{2}}}{ \sqrt{x}}. \end{aligned}$$

(60)

Using

$$\begin{aligned} K_{1}^{^{\prime }}(z)=-K_{0}(z)-\frac{K_{1}(z)}{z}, \end{aligned}$$

(61)

we obtain

$$\begin{aligned} p_{X}(x)=2\lambda _{1}\lambda _{2}K_{0}(2\sqrt{\lambda _{1}\lambda _{2}x}), \end{aligned}$$

(62)