Collaborative Spectrum Sensing of Cognitive Radio Networks with Simple and Effective Fusion Scheme

Abstract

Cognitive radio (CR) provides a mechanism for effective spectrum usage. Spectrum sensing is an essential component of CR which comprises intelligent signal processing algorithms to identify the spectrum holes. In this paper, we propose a novel soft combining scheme, called mean cumulative sum (MCS), at the cognitive base station or Fusion center (FC), and we have investigated our proposed method through closed form expressions of probability of detection and probability of false alarm, respectively. We have also provided a method of obtaining an optimum threshold to minimize the sensing error which occurs in FC. Our simulation results show that proposed MCS combining scheme outperforms single user spectrum sensing and performs equally well as that of equal gain soft combining scheme. In addition, our results show that sensing error is reduced when optimum threshold is chosen at the FC.

Appendices

Appendix 1

Proof for optimum threshold

$$\begin{aligned} \begin{array}{rcll} \dfrac{\mathrm{d}}{\mathrm{d}\Lambda }\left\{ P_e \right\} &{}=&{} 0\\ \dfrac{\mathrm{d}}{\mathrm{d}\Lambda }\left\{ 1+\mathbf {Q}\left( \dfrac{ \Lambda - \mu _{H_0}}{\sqrt{\Sigma _{H_0}}}\right) - \mathbf {Q}\left( \dfrac{ \Lambda - \mu _{H_1}}{\sqrt{\Sigma _{H_1}}}\right) \right\} &{}=&{}0\\ \dfrac{\mathrm{d}}{\mathrm{d}\Lambda }\left\{ 1+\frac{1}{2} - \frac{1}{2}erf\left( \dfrac{ \Lambda - \mu _{H_0}}{\sqrt{2\Sigma _{H_0}}}\right) - \frac{1}{2}+\frac{1}{2}erf\left( \dfrac{ \Lambda - \mu _{H_1}}{\sqrt{2\Sigma _{H_1}}}\right) \right\} &{}=&{}0\\ \end{array} \end{aligned}$$

where $erf(\cdot )$ is an error function. Using the relation $\frac{d}{\mathrm{d}x}erf(x) = \frac{2}{\sqrt{\pi }}\exp ^{-x^2}$, and after substituting from (8), (9), (10) and (11), the above equation becomes

$$\begin{aligned} \begin{array}{rcll} \left( \sqrt{1+2\gamma }\right) \exp {^{\dfrac{{-\left( \Lambda - \dfrac{M(N+1)}{2}\right) ^2}}{\dfrac{2M\left( 2N^2+3N+1\right) }{6N}}}} &{}=&{} \exp ^{\dfrac{{-\left( \Lambda - \dfrac{M(N+1)(1+\gamma )}{2}\right) ^2}}{\dfrac{2M\left( 1+2\gamma \right) \left( 2N^2+3N+1\right) }{6N}}} \\ \\ \end{array} \end{aligned}$$

taking natural logarithm on both sides, and solving the quadratic equation for $\Lambda$, we get optimum threshold as,

$$\begin{aligned} \begin{array}{lcll} \Lambda&= \dfrac{-M\left( N+1\right) \gamma \pm \sqrt{M^2\left( N+1\right) ^2\gamma ^2 + 8\gamma X}}{-4\gamma } \end{array} \end{aligned}$$

where we let $X \!=\! 2M\left( 1\!+\!2\gamma \right) \left( \dfrac{2N^2\!+\!3N\!+\!1}{6N}\right) \ln \left( \sqrt{1\!+\!2\gamma }\right) \!+\! \left( \dfrac{M\left( 1\!+\!N\right) \gamma }{2}\right) ^2$

Appendix 2

Proof for convexity of $P_e$

We continue from our previous derivation to find $\dfrac{\mathrm{d}^2P_e}{\mathrm{d}\Lambda ^2}$ as,

$$\begin{aligned} \begin{array}{rcll} \dfrac{\mathrm{d}^2P_e}{\mathrm{d}\Lambda ^2}&{}=&{}\dfrac{\left( \Lambda - \mu _{H_{0}}\right) }{\Sigma _{H_0}\sqrt{2 \pi \Sigma _{H_0}}}\exp ^{\dfrac{-\left( \Lambda -\mu _{H_0}\right) ^2}{2\Sigma _{H_0}}} - \dfrac{\left( \Lambda - \mu _{H_{1}}\right) }{\Sigma _{H_1}\sqrt{2 \pi \Sigma _{H_1}}}\exp ^{\dfrac{-\left( \Lambda -\mu _{H_1}\right) ^2}{2\Sigma _{H_1}}} \\ \end{array} \end{aligned}$$

From (8), (9), (10) and (11), we know that $\mu _{H_1}>\mu _{H_0}, \Sigma _{H_1}>\Sigma _{H_0}$, and in addition $\Sigma _{H_1}>\mu _{H_1}$. Therefore, the second derivative is $\dfrac{\mathrm{d}_e}{\mathrm{d}\Lambda ^2} \ge 0$. Hence, the function $P_e$ is convex.