Energy Efficiency in Cognitive Radio Network Using Cooperative Spectrum Sensing

Abstract

In this paper, we consider an optimization of number of secondary users (SUs) in a cooperative spectrum sensing by maximizing the energy efficiency of the cognitive radio network. We obtain the mathematical expressions for number of SUs using OR and AND fusion rules at the fusion center. We consider energy detector as an example for the analysis, based on the analysis we show that performance obtained for OR rule is better than the AND rule.

Appendices

Appendix A

Proof of Proposition 1

Let us consider a function G as

$$\begin{aligned} G(L,\rho _{OR})& = \alpha _0 + \alpha _1(1-Q_{m,OR})-\alpha _2 Q_{f,OR}\nonumber \\&-\,\rho _{OR} L(e_1+e_2)+P(H_1)e_p+\beta _{OR} e_s \end{aligned}$$

(19)

where $\rho _{OR}$ is a positive constant.

Optimal number of cooperative secondary users using OR fusion rule can be calculated by differentiating Eq. 22 with respect to L and equating to zero

$$\begin{aligned} \frac{\partial G(L,\rho _{OR})}{\partial L}\approx G(L+1,\rho _{OR})-G(L,\rho _{OR})=0 \end{aligned}$$

(20)

The above equation can be simplified as

$$\begin{aligned}&(P_m)^L (1-P_m)[\alpha _1+P(H_1)(\rho _{OR} )e_s]- P_f(1-P_f)^L\nonumber \\&\quad [\alpha _2 + P(H_0)(\rho _{OR} )e_s]-\rho (e_1+e_2)= 0 \end{aligned}$$

(21)

It is difficult to calculate the L value from the above equation. So, to derive the L value the term $\rho _{OR}$ ($e_1$ + $e_2$) is neglected.It does not make much difference and the approximated value of L is given by

$$\begin{aligned} L^*_{OR} \approx \frac{ln \left( \frac{P_f (\alpha _2 - P(H_0)\ \rho _{OR} \ e_s)}{(1-P_m)(\alpha _1 + P(H_1) \rho _{OR}\ e_s )}\right) }{ln\left( \frac{P_m}{1-P_f}\right) } \end{aligned}$$

(22)

$L^*_{OR}$ gives the optimal number of cooperative secondary users for OR fusion rule.

The optimal number of cooperative secondary users is always greater than or equal to one and $ln(P_m/1-P_f)$ is also greater than zero, then

$$\begin{aligned} ln\left[ \left( \frac{P_f}{1-P_m}\right) \left( \frac{\alpha _2- P(H_0)(\rho _{OR} )e_s}{\alpha _1 + P(H_1)(\rho _{OR} )e_s }\right) \right] >0 \end{aligned}$$

(23)

The positive constant ($\rho _{OR}$) is derived from (27) as

$$\begin{aligned} \rho _{OR} <\frac{P_f \alpha _2 -(1-P_m)\alpha _1}{P(H_1)e_s(1-P_m)+P(H_0)e_sP_f} \end{aligned}$$

(24)

According to Bi-section algorithm $\rho$ value is derived by considering $\rho _{OR} = [\rho _{1,OR}, \rho _{2,OR}]$ and assuming the initial value of $\rho _{1,OR} = 0$ and

$$\begin{aligned} \rho _{2OR} =\frac{P_f\alpha _2 -(1-P_m)\alpha _1}{P(H_1)e_s(1-P_m)+P(H_0)e_s P_f} \end{aligned}$$

Now at

$$\begin{aligned} \rho _{OR}& = {\frac{\rho _{1,OR} + \rho _{2,OR}}{2}}\nonumber \\& = {\frac{P_f\alpha _2 -(1-P_m)\alpha _1}{2 \left( P(H_1)e_s(1-P_m)+P(H_0)e_sP_f \right) }} \end{aligned}$$

(25)

The optimal number of cooperative secondary users for OR fusion rule is $L^*_{OR}$

$$\begin{aligned} L^*_{OR}=\left\lceil \frac{ln \left( \frac{P_f (\alpha _2 - P(H_0)\ \rho _{OR} \ e_s)}{(1-P_m)(\alpha _1 + P(H_1) \rho _{OR}\ e_s )}\right) }{ln\left( \frac{P_m}{1-P_f}\right) }\right\rceil \end{aligned}$$

(26)

Appendix B

Proof of Proposition 2

Let us consider a function K as

$$\begin{aligned} K(L,\rho _{AND})& = \alpha _0+\alpha _1(1-Q_{m,AND})-\alpha _2 Q_{f,AND}\nonumber \\ \quad &-\,\rho _{AND}L(e_1+e_2)+P(H_1)e_p+\beta _{AND}e_s. \end{aligned}$$

(27)

To derive the value of optimal number of cooperative secondary users L, differentiate (30) with respect to L and equate to zero

$$\begin{aligned} \frac{\partial K(L,\rho _{AND})}{\partial L}\approx K(L+1,\rho _{AND})-K(L,\rho _{AND}) = 0. \end{aligned}$$

(28)

The above equation can be simplified as

$$\begin{aligned}&(P_f)^L (1-P_f)[\alpha _2 + P(H_0)(\rho _{AND} )e_s]- P_m(1-P_m)^L \nonumber \\&\quad [\alpha _1 + P(H_1)(\rho _{AND} )e_s]-\rho _{AND}(e_1+e_2)= 0. \end{aligned}$$

(29)

By neglecting the term $\rho _{AND}$.($e_1$ + $e_2$), the approximated value of L is

$$\begin{aligned} L^*_{AND} \approx \frac{ln \frac{ P_m \left( \alpha _1 - P(H_1)\ \rho _{AND}\ e_s \right) }{\left( 1-P_f\right) \left( \alpha _0 + P(H_0)\ \rho _{AND}\ e_s \right) }}{ln\left( \frac{P_f}{1-P_m}\right) } \end{aligned}$$

(30)

$L^*_{AND}$ gives the optimal number of cooperative secondary users using AND fusion rule.

By simplifying the above equation, the positive constant ($\rho$) value is derived as

$$\begin{aligned} \rho _{AND} <\frac{P_m\alpha _1 -(1-P_f)\alpha _2}{P(H_1)e_sP_m-P(H_0)e_s(1-P_f)}. \end{aligned}$$

(31)

According to Bi-section, the $\rho _{AND}$ value is derived by considering $\rho _{AND}$ = [$\rho _{1,AND}$,$\rho _{2,AND}$] and assuming the initial value of $\rho _{1,AND}$ = 0 and

$$\begin{aligned} \rho _{2,AND}=\frac{P_m\alpha _1 -(1-P_f)\alpha _2}{P(H_1)e_sP_m-P(H_0)e_s(1-P_f)}. \end{aligned}$$

Now at

$$\begin{aligned} \rho _{AND}& = {\frac{\rho _{1,AND} + \rho _{2,AND}}{2}}\nonumber \\& = {\frac{P_m\alpha _1 -(1-P_f)\alpha _2}{P(H_1)e_sP_m-P(H_0)e_s(1-P_f)}}. \end{aligned}$$

(32)

The optimal number of cooperative secondary users using AND fusion rule $L^*_{AND}$ is

$$\begin{aligned} L^*_{AND}= \left\lceil \frac{ln \frac{ P_m \left( \alpha _1 - P(H_1)\ \rho _{AND}\ e_s \right) }{\left( 1-P_f\right) \left( \alpha _0 + P(H_0)\ \rho _{AND}\ e_s \right) }}{ln\left( \frac{P_f}{1-P_m}\right) }\right\rceil . \end{aligned}$$

(33)