Optimization of Time-Domain Combining Cooperative Spectrum Sensing in Cognitive Radio Networks

Abstract

In cognitive radio networks, the secondary users take chances to access the spectrum without causing interference to the primary users so that the spectrum access is dynamic and somewhat opportunistic. Therefore, spectrum sensing is of significant importance. In this paper, we propose a novel time-domain combining cooperative spectrum sensing framework, in which the time consumed by reporting for one secondary user is also utilized for other secondary users’ sensing. We focus on the optimal sensing settings of the proposed sensing scheme to maximize the secondary users’ throughput and minimize the average sensing error probability under the constraint that the primary users are sufficiently protected. Some simple algorithms are also derived to calculate the optimal solutions. Simulation results show that fundamental improvement of the achievable throughput and sensing performance can be obtained by optimal sensing settings. In addition, our proposed scheme outperforms the general frame structure on either achievable throughput or the performance of average sensing error probability.

Appendices

Appendix A

1.1 Proof of Theorem 1

According to (9) and (10), taking the first derivative of $p_f$ with respect to $p_d$, we have

$$\begin{aligned} \frac{dp_f }{dp_d }\!&= \!{\frac{dp_f }{d\lambda }}/{\frac{dp_d }{d\lambda }} \nonumber \\ \!&= \!\sqrt{2\gamma \sum _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} \!+\!1}\exp \left\{ \frac{T_s f_s }{4\left( {2\gamma \sum \nolimits _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} \!+\!1} \right) }\left[ {\frac{\lambda }{\sigma ^{2}}\!-\!\sum _{i=1}^N {\omega _i \left( {\left| {h_i } \right| ^{2}\gamma \!+\!1} \right) } } \right] ^{2}\right. \nonumber \\&\quad \left. -\frac{T_s f_s }{4}\left( {\frac{\lambda }{\sigma ^{2}}-\sum _{i=1}^N {\omega _i }} \right) ^{2} \right\} \nonumber \\&> 0 \end{aligned}$$

(44)

Thus, $p_f$ is an increasing function of $p_d$.

According to (32), $\frac{dQ_f }{dQ_d }>0$. Then, according to (31), $\frac{d\mathfrak R }{dQ_d }<0$. Therefore, Theorem 1 has been proven that $\mathfrak R$ is a decreasing function of $Q_d$.

Appendix B

1.1 Proof of Theorem 2

According to (42), taking the second derivative of $p_f$ with respect to $p_d$, we have

$$\begin{aligned} \frac{d^{2}p_f }{dp_d^2 }&= \frac{d}{dp_d }\left( {\frac{dp_f }{dp_d }} \right) =\frac{d}{d\lambda }\left( {\frac{dp_f }{dp_d }} \right) 1\Big /\frac{dp_d }{d\lambda } \nonumber \\&= \frac{dp_f }{dp_d }\sqrt{\pi T_s f_s \left( {2\gamma \sum _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} +1} \right) }\left[ {\left( {\frac{\lambda }{\sigma ^{2}}-\sum _{i=1}^N {\omega _i } } \right) \!-\!\frac{\left( {\frac{\lambda }{\sigma ^{2}}\!-\!\sum \nolimits _{i=1}^N {\omega _i \left( {\left| {h_i } \right| ^{2}\gamma +1} \right) } } \right) }{2\gamma \sum \nolimits _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} +1}} \right] \nonumber \\&\quad \times \exp \left\{ {\frac{\left[ {\frac{\lambda }{\sigma ^{2}}-\sum _{i=1}^N {\omega _i \left( {\left| {h_i } \right| ^{2}\gamma +1} \right) } } \right] ^{2}T_s f_s }{4\left( {2\gamma \sum _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} +1} \right) }} \right\} \end{aligned}$$

(45)

According to (43), we can rewrite (45) as

$$\begin{aligned} \frac{d^{2}p_f }{dp_d^2 }&= \frac{dp_f }{dp_d }\sqrt{\pi }\gamma \left[ 2\sqrt{2}\mathbb{Q }^{-1}\left( {p_d } \right) \sum _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}}\right. \nonumber \\&\left. +\sqrt{T_s f_s \left( {2\gamma \sum _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} +1} \right) }\sum _{i=1}^N {\omega _i \left| {h_i } \right| ^{2}} \right] \nonumber \\&\quad \times \exp \left\{ {\frac{1}{2}\left[ {\mathbb{Q }^{-1}\left( {p_d } \right) } \right] ^{2}} \right\} \end{aligned}$$

(46)

For $0\!<\!p_d \!<\!0.5, \mathbb{Q }^{-1}({p_d})\!>\!0$. Thus, $2\sqrt{2}\mathbb{Q }^{-1}\left( {p_d } \right) \!+\!\sqrt{T_s f_s ({2\gamma +1})}>0$; For $0.5<p_d <1, \mathbb{Q }^{-1}({p_d})<0$. Since $u=T_s f_s$ is a large value, there exists a point $p_d^*$, for $0.5<p_d <p_d^*, 2\sqrt{2}\mathbb{Q }^{-1}({p_d })+ \sqrt{T_s f_s ({2\gamma +1})}>0$, where $p_d^*$ approximates to 1 but less than 1. Therefore, we have $2\sqrt{2}\mathbb{Q }^{-1}\left( {p_d } \right) +\sqrt{T_s f_s ({2\gamma +1})}>0$ for $p_d \in ({0,p_d^*})$.

Obviously, in the range $p_d \in ({0,p_d^*})$,

$$\begin{aligned} 2\sqrt{2}\mathbb{Q }^{-1}\left( {p_d } \right) \sum _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} +\sqrt{T_s f_s \left( {2\gamma \sum _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} +1} \right) }\sum _{i=1}^N {\omega _i \left| {h_i } \right| ^{2}} >0 \end{aligned}$$

(47)

Therefore, $\frac{d^{2}p_f }{dp_d^2 }>0$, and $p_f$ is a concave function of $p_d$ in the range $p_d \in \left( {0,p_d^*} \right)$.

From Fig. 10, we can see that $\frac{dp_f }{dp_d }-\frac{p_f }{p_d }>0, \frac{1-p_f }{1-p_d }-\frac{dp_f }{dp_d }>0$. Obviously, it can be derived geometrically.

Fig. 10

$p_f$ versus $p_d$ in the range $p_d \in ({0,p_d^*})$

According to (40), $\frac{d\Omega (p_d )}{dp_d }>0$. Therefore, Theorem 2 has been proven that $\Omega (p_d )$ is an increasing function of $p_d$.

Appendix C

1.1 Proof of Theorem 3

It is easy to find that

$$\begin{aligned} \mathop {\lim }\limits _{p_d \rightarrow p_d^*} \frac{1-p_f }{1-p_d }\approx \mathop {\lim }\limits _{p_d \rightarrow p_d^*} \frac{dp_f }{dp_d } \end{aligned}$$

(48)

According to (39), we have

$$\begin{aligned} \mathop {\lim }\limits _{p_d \rightarrow p_d^*} \Omega (p_d )&\approx \left( {\mathop {\lim }\limits _{p_d \rightarrow p_d^*} \frac{dp_f }{dp_d }} \right) ^{N-n}\mathop {\lim }\limits _{p_d \rightarrow p_d^*} \frac{dp_f }{dp_d }\frac{p(H_0 )}{p(H_1 )}-1\nonumber \\&= \left( {\mathop {\lim }\limits _{p_d \rightarrow p_d^*} \frac{dp_f }{dp_d }} \right) ^{N-n+1}\frac{p(H_0 )}{p(H_1 )}-1 \end{aligned}$$

(49)

Since

$$\begin{aligned} \mathbb{Q }^{-1}(p_d^*)=-\frac{\sqrt{T_s f_s \left( {2\gamma \sum \nolimits _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} +1} \right) }\sum \nolimits _{i=1}^N {\omega _i \left| {h_i } \right| ^{2}} }{2\sqrt{2}\sum \nolimits _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} } \end{aligned}$$

(50)

According to (42),

$$\begin{aligned} \mathop {\lim }\limits _{p_d \rightarrow p_d^*} \frac{dp_f }{dp_d }=\sqrt{2\gamma \sum _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} +1}\exp \left\{ {\frac{T_s f_s \gamma }{8}\frac{\left( {\sum \nolimits _{i=1}^N {\omega _i \left| {h_i } \right| ^{2}} } \right) ^{2}}{\sum \nolimits _{i=1}^N {\omega _i^2 \left| {h_i } \right| ^{2}} }} \right\} >1 \end{aligned}$$

(51)

Therefore,

$$\begin{aligned} \mathop {\lim }\limits _{p_d \rightarrow p_d^*} \Omega (p_d )=\left( {\mathop {\lim }\limits _{p_d \rightarrow p_d^*} \frac{dp_f }{dp_d }} \right) ^{N-n+1}\frac{p(H_0 )}{p(H_1 )}-1>0 \end{aligned}$$

(52)

If $\Omega (\mu _1 )>0$ is satisfied, according to Theorem 2, in the range $p_d \in \left( {\mu _1 ,p_d^*} \right) , \Omega (p_d )>0$. According to (38), $Q_e$ is a increasing function in the range $p_d \in \left( {\mu _1 ,p_d^*} \right)$. Therefore, $\mu _1$ is the optimal solution for problem (35).

If $\Omega (\mu _1 )<0$ is satisfied, according to Theorem 2, in the range $p_d \in \left( {\mu _1 ,\mu _2 } \right) , \Omega (p_d )<0$; in the range $p_d \in \left( {\mu _2 ,p_d^*} \right) , \Omega (p_d )>0$. According to (38), $Q_e$ is a decreasing function in the range $p_d \in \left( {\mu _1 ,\mu _2 } \right)$, and is an increasing function in the range $p_d \in \left( {\mu _2 ,p_d^*} \right)$. Therefore, $\mu _2$ is the optimal solution for problem (35).

Thus, Theorem 3 is proved.