Cooperative spectrum sensing with relay selection

Abstract

In this paper, we evaluate the detection and false alarm probabilities for relay based spectrum sensing techniques using the energy detector. The communication process consists of two phases. In the first phase, the primary user P transmits and K relays listen. In the second phase, the relays amplify the signal to the fusion center where spectrum sensing based on the energy detector is performed. All relays transmit over orthogonal channels. We also consider cooperative spectrum sensing with best relay selection. Both opportunistic amplify and forward, partial and reactive relay selection are considered. The results are valid for Rayleigh fading channels in the absence or presence of a direct link between P and the fusion center D.

Appendix A

The Cumulative Distribution Function (CDF) of $\Gamma _{k}^{up}$ is equal to

$$\begin{aligned} F_{\Gamma _{k}^{up}}(x)= & {} 1-P(\Gamma _{k}^{up}>x) \nonumber \\= & {} 1-P(\Gamma _{PR}^{\max }>x)P(\Gamma _{R_{k}D}>x) \end{aligned}$$

(52)

We have

$$\begin{aligned} P(\Gamma _{PR}^{\max }> & {} x)=1-\prod _{j=1}^{K}P(\Gamma _{PR_{j}}\le x) \nonumber \\= & {} 1-\prod _{j=1}^{K}\left[ 1-e^{-\frac{x}{{\overline{\Gamma }}_{PR_{j}}}} \right] \nonumber \\= & {} 1-\sum _{n=0}^{2^{K}-1}(-1)^{b(n)}e^{-x\sum _{l=1}^{K}\frac{a_{n}(l)}{ {\overline{\Gamma }}_{PR_{l}}}} \end{aligned}$$

(53)

where $(a_{n}(1),\ldots ,a_{n}(K))$ is the binary representation of n and $b(n)=\sum _{l=1}^{K}a_{n}(l).$

Using (52) and (53), we deduce

$$\begin{aligned} F_{\Gamma _{k}^{up}}(x)=1-e^{-\frac{x}{{\overline{\Gamma }}_{R_{k}D}} }-\sum _{n=0}^{2^{K}-1}(-1)^{b(n)}e^{-x\sum _{l=1}^{K}\frac{a_{n}(l)}{ {\overline{\Gamma }}_{PR_{l}}}-\frac{x}{{\overline{\Gamma }}_{R_{k}D}}}. \end{aligned}$$

(54)

By a simple derivative, we deduce the PDF of $\Gamma _{k}^{up}$:

$$\begin{aligned} f_{\Gamma _{k}^{up}}(x)=\frac{e^{-\frac{x}{{\overline{\Gamma }}_{R_{k}D}}}}{ {\overline{\Gamma }}_{R_{k}D}}+\sum _{n=0}^{2^{K}-1}(-1)^{b(n)}\frac{e^{-\frac{x }{\alpha _{k,n}}}}{\alpha _{k,n}}, \end{aligned}$$

(55)

where

$$\begin{aligned} \alpha _{k,n}=\frac{1}{\frac{1}{{\overline{\Gamma }}_{R_{k}D}}+\sum _{l=1}^{K} \frac{a_{n}(l)}{{\overline{\Gamma }}_{PR_{l}}}}. \end{aligned}$$

(56)