Optimal Cooperative Spectrum Sensing Under Faded and Bandwidth Limited Control Channels

Abstract

Most of the cooperative spectrum sensing related research assumes system models with perfect control channel (i.e. with unlimited channel bandwidth and no channel errors). However, the assumption is not realistic and can lead to incorrect conclusions regarding the performance analysis of the cooperative spectrum sensing detection capabilities. This paper proposes a novel cooperative spectrum sensing framework that mitigates the imperfect control channel features, like the limited control channel bandwidth and error proneness, and achieves the detection performances of cooperative spectrum sensing under ideal control channel. It utilizes node clustering and multi-antenna spatial multiplexing (i.e. beamforming) and provides a generic framework that can be exploited by any cooperative spectrum sensing and fusion technique. The performance analysis shows that the proposed framework alleviates the control channel bandwidth limitation and significantly decreases the control channel error rate. The performance evaluation results also show that the proposed framework achieves the upper bound detection performances, i.e. achieves the same detection performances as the conventional cooperative spectrum sensing with ideal control channel.

Appendices

Appendix 1

This Appendix derives the closed form expression for the average SNR presented in Eq. (6). The derivations is conducted from the cumulative distribution function (cdf) of the SNR elaborated in (Eq. (6), [24]):

$$\begin{aligned} \begin{aligned} {{F_{\rho }(\rho )}}&=1-\frac{2}{{\alpha }r^2}\left( \frac{P_T}{Z_t}\right) ^\frac{2}{\alpha }\left( \frac{\rho }{2}\right) ^{-\frac{2}{\alpha }}\\&\quad \times \sum ^{Z_t-1}_{u=0}\frac{1}{u!}\left\{ \Gamma \left( u+\frac{2}{\alpha }\right) -\Gamma \left( u+\frac{2}{\alpha },\frac{Z_tr^{\alpha }\rho }{2P_T}\right) \right\} , \end{aligned} \end{aligned}$$

(23)

Because the first moment of the SNR cannot be expressed in a closed form from Eq. (23), the problem is solved by introducing the Markov inequality [25]:

$$\begin{aligned} \begin{aligned} \overline{\rho }=\mathbb {E}\left[ \rho \right] \ge {c\left( 1-F_{\rho }(c)\right) }, \end{aligned} \end{aligned}$$

(24)

where $c$ denotes the average SNR upper bound in the system. The average SNR, $\overline{\rho }$, in Eq. (23) can be only derived in a closed form for small path loss exponents (i.e. $\alpha =1$ or $\alpha =2$). This work derives the average SNR for the case of $\alpha =2$:

$$\begin{aligned} \begin{aligned} \overline{\rho }\ge {c\left( 1-F_{\rho }(c)\right) }&=c\left( \frac{2P_T}{cZ_tr^2} \sum ^{Z_t-1}_{u=0}\frac{1}{u!} \left\{ \Gamma \left( u+1\right) -\Gamma \left( u+1,\frac{Z_tr^{2}c}{2P_T}\right) \right\} \right) \\&=\frac{2P_T}{Z_tr^2}\sum ^{Z_t-1}_{u=0}\frac{1}{u!}\left\{ \Gamma \left( u+1\right) -\Gamma \left( u+1,\frac{Z_tr^{2}c}{2P_T}\right) \right\} , \end{aligned} \end{aligned}$$

(25)

Appendix 2

This Appendix derives the closed form expression for the average SNR presented in Eq. (7), based on the average SNR expression in Eq. (25) from “Appendix 1”. By carefully analyzing Eq. (25) it can be proven that, if the second argument in the Incomplete Gamma function is sufficiently large (i.e. $\frac{Z_tr^{2}c}{2P_T}>>1$ ), the Incomplete Gamma function approaches to zero, and the average SNR, $\overline{\rho }$, can be expressed as:

$$\begin{aligned} \begin{aligned} \overline{\rho }&\ge \frac{2P_T}{Z_tr^2}\sum ^{Z_t-1}_{u=0}\frac{1}{u!}\left\{ \Gamma \left( u+1\right) -\underbrace{\Gamma \left( u+1,\frac{Z_tr^{2}c}{2P_T}\right) }_{0}\right\} \\&\approx \frac{2P_T}{Z_tr^2}\sum ^{Z_t-1}_{u=0}\frac{1}{u!}\Gamma \left( u+1\right) = \frac{2P_T}{Z_tr^2}\sum ^{Z_t-1}_{u=0}\frac{1}{u!}{u!}=\frac{2P_T}{r^2}. \end{aligned} \end{aligned}$$

(26)

Proof of Eq.(26):

The Incomplete Gamma function, denoted as $\Gamma (a,b)$ approaches to zero, when its arguments satisfy the following assumptions: $a<<0$ and $b>>0$ . Both of the assumptions are satisfied in this system model and scenario. The first argument in the function relates to the maximal number of transmit antennas i.e. $u\in \left[ 0,Z_t-1\right] <<\infty$, which in reality is significantly smaller than infinity. In order for $b>>0$, the following expression must be satisfied:

$$\begin{aligned} \frac{Z_tr^{2}c}{2P_T}>>0 \end{aligned}$$

(27)

For easier analytical tractability assume that the number of antennas, cell radius and transmits power have unit values i.e. ($Z_t=1$ , $r=1$ and $P_T=1$), than the expression in Eq. (27) can be defined as:

$$\begin{aligned} c>>0 \end{aligned}$$

(28)

Because the argument $c$ reflects the expected upper bound of the SNR in the system, which is an arbitrarily large number i.e. $c>>0$, (e.g. an upper bound of 30 dB for the SNR is a realistic assumption, hence $c=1{,}000>>0$) the requirement in Eq. (28) is satisfied. By substituting $\frac{Z_tr^{2}c}{2P_T}$ with $R$, the Incomplete Gamma function in Eq. (26) can be expressed as:

$$\begin{aligned} \Gamma \left( u+1,R\right) =\int \limits ^{\infty }_{R}x^{u}e^{-x}dx. \end{aligned}$$

(29)

In order to prove that the expression in Eq. (29) approaches zero for arbitrarily large $R$, assume the following inequality:

$$\begin{aligned} \int \limits ^{\infty }_{R}x^{u}e^{-x}dx<<\int \limits ^{\infty }_{R}e^{\frac{x}{2}}e^{-x}dx. \end{aligned}$$

(30)

Comment For any positive number $u < \infty$ , the inequality i.e. the assumption $x^{(2u)}<<e^{x}$ i.e. $x^{u}<<e^{\frac{x}{2}}$ holds, thus the validity of Eq. (30) is correct.

Because the upper bound i.e. (the right term) in Eq. (30) converges to zero, for arbitrarily large $R$, i.e.:

$$\begin{aligned} \int \limits ^{\infty }_{R}e^{\frac{x}{2}}e^{-x}dx=2e^{-\frac{R}{2}}\rightarrow 0,\quad R>>0 \end{aligned}$$

(31)

the expression in Eq. (29) also converges to zero for arbitrarily large $R$ i.e.:

$$\begin{aligned} \Gamma \left( u+1,R\right) =\int \limits ^{\infty }_{R}x^{u}e^{-x}dx\rightarrow 0,\quad R>>0 \end{aligned}$$

(32)

thus proving the soundness of Eq. (26).

Appendix 3

This Appendix derives the average SNR on the on the CC between the CH and BS (“Appendix 3.1”) and the average SNR on the on the CC between the CSN and CH (“Appendix 3.2”).

1.1 Appendix 3.1

The proposed Node clustering method in Sect. 4.1 selects the closest CSNs for CHs. Because the CSNs are assumed to be uniformly randomly distributed over a circle around the BS [17], the radius of the circle in which all CH are located can be computed based on Eq. (13) from [16]:

$$\begin{aligned} {\mathbb {E}}\left( r_m\right) =\frac{rm^{1/2}}{(K+1)^{1/2}} \end{aligned}$$

(33)

where $m$ denotes the $m$th nearest node to the BS (i.e. the most further CH from the BS) and $\mathbb {E}\left( r_m\right)$ denotes the average radius of the circle in which all $m$ CH are located. Than the average SNR on the link between the CHs and the BS, i.e. Eq. (16), can be derived by substituting $r$, in Eq. (7), with Eq. (33).

1.2 Appendix 3.2

The average cluster radius $\overline{r}_{m,n}$, Eq. (17), when assuming $m$ clusters (i.e. CHs) and $n$ CSNs per cluster is defined as:

$$\begin{aligned} \overline{r}_{m,n}=\frac{1}{m}\sum ^{m}_{i=1}\overline{r}_{i,n}, \end{aligned}$$

(34)

where $\overline{r}_{i,n}$ denotes the average radius of the $i$th cluster (i.e. the average distance between the most distant CSN associated to cluster $i$ and its CH) and can be derived from (Eq. (18) [16]):

$$\begin{aligned} \begin{aligned} \overline{r}_{i,n}&=\left[ \frac{s^{2}_{i}}{\left( n-i\right) B\left( K-n+1,n-i\right) } \right. \\&\quad \times \left. F_1(n-i;n-K,-1;n-i+1;1,1-\left( \frac{r}{s_i}\right) ^{2})\right] -s_i, \end{aligned} \end{aligned}$$

(35)

where $B(a,b)=\Gamma (a)\Gamma (b)/\Gamma (a+b)$ refers to the beta function, $F_1(a;b_1,b_2;c;x,y)$ refers to the Appell hypergeometric function [16] of two variables and $s_i$ denotes the average distance between the $i$th CH and the BS.