Enhancing Throughput Over Varied Fading Channels for Cluster-Based Cooperative Spectrum Sensing in Cognitive Wireless Networks

Abstract

This paper reports throughput maximization over various fading channels for cluster-based cooperative spectrum sensing in cognitive radio. For that purpose, the secondary users are grouped into clusters and effect of different fusion rules on throughput over Nakagami, Weibull, Rayleigh and Rician fading channels is investigated. Inside each cluster, i.e., for intra fusion, OR ($m=1$) and AND ($m=K$) fusion rules are considered. Further, a formula is proposed for optimal number of clusters which maximize throughput over fading channels. This expression is derived for two different inter-cluster fusion rules (fusion rule between cluster head and fusion centre) given as OR ($p=1$) and AND ($p=N$). Simulation results are carried out for OR-AND ($m=1$, $p=N$) and AND-OR ($m=K$, $p=1$) fusion rules for given cluster size (number of users inside a cluster), and number of clusters are optimized over Weibull and Nakagami fading channels. The proposed expressions for the optimal number of clusters are validated by the results. It is found that for given fixed parameters, the number of clusters at which throughput is maximum are $N=6$ and $N=5$ over Nakagami and Weibull fading channels respectively for AND-OR fusion rule. In case of OR-AND fusion rule, maximum throughput is achieved for $N=5$ and $N=6$ over Nakagami and Weibull fading channels respectively.

Appendices

Appendix

Proof of Proposition

2.1 a.1 Proof of Optimal Number of Clusters for OR Inter-Fusion Rule (${{\varvec{N}}}_{{\varvec{o}}{\varvec{p}},{\varvec{O}}{\varvec{R}}}$)

In order to determine optimal number of clusters, for OR inter-fusion rule, Eqs. (12) and (13) can be written as [30]

$${\widetilde{Q}}_{f,OR}=1-{\left(1-{Q}_{f}\right)}^{N},$$

(23)

$${\widetilde{Q}}_{d.OR}=1-{\left(1-{Q}_{d}\right)}^{N},$$

(24)

As ${\widetilde{Q}}_{m}=1-{\widetilde{Q}}_{d},$ we can write Equation (24) as

$${\widetilde{Q}}_{m,OR}={\left({Q}_{m}\right)}^{N},$$

(25)

${Q}_{m}$ is given by Eq. (11).

Let us define a function $\fancyscript{g}$ given as

$$\fancyscript{g}\left(N\right)={\varphi }_{0}+{\varphi }_{1}{\widetilde{Q}}_{d,OR}(N)-{\varphi }_{2}{\widetilde{Q}}_{f,OR}(N),$$

(26)

Optimal number of clusters (${N}_{op}$) that maximize throughput are found by differentiating Eq. (26) with respect to $N$ and then equating it to zero, i.e.,

$$\frac{\partial \fancyscript{g}(N)}{\partial N}\approx \fancyscript{g}\left(N+1\right)-\fancyscript{g}\left(N\right)=0,$$

(27)

Or,

$${\varphi }_{1}\left({\widetilde{Q}}_{d,OR}\left(N+1\right)-{\widetilde{Q}}_{d,OR}\left(N\right)\right)={\varphi }_{2}\left({\widetilde{Q}}_{f,OR}\left(N+1\right)-{\widetilde{Q}}_{f,OR}\left(N\right)\right),$$

(28)

By substituting ${\widetilde{Q}}_{m,OR}$ in Eq. (28) and solving we get,

$${\varphi }_{1}{{Q}_{m}}^{N}(1-{Q}_{m})={\varphi }_{2}{(1-{Q}_{f})}^{N}{Q}_{f},$$

(29)

Equation (29) is evaluated as

$${N}_{op,OR}=\lceil\frac{\mathrm{ln}\left(\frac{{\varphi }_{1}}{{\varphi }_{2}}\right)+ln\left(\frac{1-{Q}_{m}}{{Q}_{f}}\right)}{ln\left(\frac{1-{Q}_{f}}{{Q}_{m}}\right)}\rceil,$$

(30)

${Q}_{f}$ and ${Q}_{m}$ are computed by Eqs. (9) and (11). ${\varphi }_{1}$ and ${\varphi }_{2}$ are given by Eqs. (18) and (19).

By putting $ln\left(\frac{1-{Q}_{m}}{{Q}_{f}}\right)={\rm A}$ and $ln\left(\frac{1-{Q}_{f}}{{Q}_{m}}\right)={\rm B}$ in Equation (30) we get,

$${N}_{op,OR}=\lceil\frac{\mathrm{ln}\left(\frac{{\varphi }_{1}}{{\varphi }_{2}}\right)+{\rm A}}{\rm B}\rceil,$$

(31)

${N}_{op,OR}$ gives the optimal number of clusters for OR inter-fusion rule.

2.2 a.2 Proof of Optimal Number of Clusters for AND Inter-Fusion Rule (${{\varvec{N}}}_{{\varvec{o}}{\varvec{p}}, {\varvec{A}}{\varvec{N}}{\varvec{D}}}$)

For AND inter-cluster fusion rule, Eqs. (12) and (13) can be written as [30]

$${\widetilde{Q}}_{f,AND}=1-{\left(1-{Q}_{f}\right)}^{N},$$

(32)

$${\widetilde{Q}}_{d,AND}={\left({Q}_{d}\right)}^{N},$$

(33)

${\widetilde{Q}}_{m,AND}=1-{\widetilde{Q}}_{d,AND}$, (34)

Or,

$${\widetilde{Q}}_{m,AND}=1-{\left(1-{Q}_{m}\right)}^{N},$$

(35)

To find optimal $N$ for this case, let us write function $h$ as

$$h(N)={\varphi }_{0}+{\varphi }_{1}{\widetilde{Q}}_{d,AND}(N)-{\varphi }_{2}{\widetilde{Q}}_{f,AND}(N),$$

(36)

Equation (36) is differentiated with respect to $N$ and equated to zero,

$$\frac{\partial h(N)}{\partial N}\approx h\left(N+1\right)-h\left(p,N\right)=0,$$

(37)

Equation (37) can be expressed as

$${\varphi }_{1}\left({\widetilde{Q}}_{d,AND}\left(N+1\right)-{\widetilde{Q}}_{d,AND}\left(N\right)\right)={\varphi }_{2}\left({\widetilde{Q}}_{f,AND}\left(N+1\right)-{\widetilde{Q}}_{f,AND}\left(N\right)\right)$$

(38)

${\widetilde{Q}}_{m,AND}$ (Eq. 35) is substituted in Eq. (38) and evaluated as

$${\varphi }_{1}(1-{{Q}_{m})}^{N}{Q}_{m}={\varphi }_{2}{{Q}_{f}}^{N}(1-{Q}_{f}),$$

(39)

By solving Eq. (39) we get

$${N}_{op,AND}=\lceil\frac{\mathrm{ln}\left(\frac{1-{Q}_{f}}{{Q}_{m}}\right)-ln\left(\frac{{\varphi }_{1}}{{\varphi }_{2}}\right)}{ln\left(\frac{1-{Q}_{m}}{{Q}_{f}}\right)}\rceil,$$

(40)

Or,

$${N}_{op,AND}=\lceil\frac{B-ln\left(\frac{{\varphi }_{1}}{{\varphi }_{2}}\right)}{A}\rceil,$$

(41)

here, ${N}_{op,AND}$ are the optimal number of clusters for AND inter-fusion rule.