Generalized Eigenvalue Based Spectrum Sensing

Abstract

Spectrum sensing is one of the fundamental components in cognitive radio networks. In this chapter, a generalized spectrum sensing framework which is referred to as Generalized Mean Detector (GMD) has been introduced. In this context, we generalize the detectors based on the eigenvalues of the received signal covariance matrix and transform the eigenvalue based spectrum sensing detectors namely: (i) the Eigenvalue Ratio Detector (ERD) and two newly proposed detectors which are referred to as (ii) the GEometric Mean Detector (GEMD) and (iii) the ARithmetic Mean Detector (ARMD) into an unified framework of generalize spectrum sensing. The foundation of the proposed framework is based on the calculation of exact analytical moments of the random variables of the decision threshold of the respective detectors. The decision threshold has been calculated in a closed form which is based on the approximation of Cumulative Distribution Functions (CDFs) of the respective test statistics. In this context, we exchange the analytical moments of the two random variables of the respective test statistics with the moments of the Gaussian (or Gamma) distribution function. The performance of the eigenvalue based detectors is compared with the several traditional detectors including the energy detector (ED) to validate the importance of the eigenvalue based detectors and the performance of the GEMD and the ARMD particularly in realistic wireless cognitive radio network. Analytical and simulation results show that the newly proposed detectors yields considerable performance advantage in realistic spectrum sensing scenarios. Moreover, the presented results based on proposed approximation approaches are in perfect agreement with the empirical results.

The authors would like to thank Center for Communication Systems Research (CCSR), University of Surrey, UK. Our sincere thanks to Dr. M. A. Imran, Dr. K. Arshad, W. Tang and X. Liu for useful discussions, valuable cooperation and providing support in producing some of the simulation results. We also would like to extend our sincere thanks to A. Rao, KAUST.

Appendices

Appendix A

1.1 Proof of Theorem 3.1

Proof

From (6.5), let us consider the numerator first

$$ \begin{aligned} {\mathfrak{M}}_r(r)&=\left(\frac{1}{K}\sum_{k=1}^K\lambda_k^r\right)^{1/r}\\ \lim_{r\rightarrow \infty} \log {\mathfrak{M}}_r(r)&=\lim_{r\rightarrow \infty} \frac{1}{r}\log \left(\frac{1}{K}\sum_{k=1}^K\lambda_k^r\right)\\ &=\lim_{r\rightarrow \infty} \frac{1}{r}\log \left(\lambda_1^r \frac{\sum_{k=1}^K\lambda_k^r}{K \lambda_1^r}\right)\\ &=\lim_{r\rightarrow \infty} \left(\frac{\log\left(\lambda_1^r\right)}{r}\right)+\lim_{r\rightarrow \infty}\left( \frac{1}{r} \log\left(\frac{\sum_{k=1}^K\lambda_k^r}{K \lambda_1^r}\right)\right) \end{aligned} $$

applying limits,

$$ \log {\mathfrak{M}}_r(\infty)=\log\left(\lambda_1\right)+0 $$

here, the Squeeze theorem has been employed [34]. Finally, by assuming that the exponential function is continuous,

$$ \begin{aligned} \lim_{r\rightarrow \infty} {\mathfrak{M}}_r(r)&= \lim_{r\rightarrow \infty} {\text{exp}}\left(\frac{1}{r} \log\left(\sum_{k=1}^K \lambda_k^r\right) \right)\\ {\mathfrak{M}}_r(\infty)&=\lambda_1 \end{aligned} $$

Similarly, it can be shown that

$$ \begin{aligned} \lim_{q\rightarrow -\infty} {\mathfrak{M}}_q(q)=& \lim_{q\rightarrow -\infty} {\text{exp}}\left(\frac{1}{q} \log\left(\sum_{k=1}^K \lambda_k^q\right) \right)\\ {\mathfrak{M}}_q(-\infty)=&\lambda_K \end{aligned} $$

Hence, the largest and the smallest eigenvalues are the limits of the Generalized mean of the eigenvalues of received signal covariance matrix at respectively, ${+}\infty$ and ${-}\infty$ such that [25].

$$ \lambda_1 \leq{{\mathfrak{M}}(\cdot)}\leq \lambda_K. $$

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Appendix B

2.1 Proof of Theorem 3.2

Proof

From (6.5), let us consider the denominator as

$$ \begin{aligned} {\mathfrak{M}}_q(q)&=\frac{1}{K}\left(\sum_{k=1}^{K}\lambda_k^q\right)^{1/q}\\ \lim_{q\rightarrow 0}\,\log {\mathfrak{M}}_q(q)&=\lim_{q\rightarrow 0} \frac{1}{q} \log \left(\frac{1}{K}\sum_{k=1}^K\lambda_k^q\right)\\ \end{aligned} $$

Using L’Hopital’s rule its limits becomes

$$ =\lim_{q\rightarrow 0} \frac{D_q \log \left(\frac{1}{K}\sum_{k=1}^K\lambda_k^q\right)}{D_q q}\\ $$

where the operator $D_q=\frac{\LARGE{d}}{\LARGE{dq}}$

$$ =\lim_{q\rightarrow 0} \frac{\sum_{k=1}^K \lambda_k^q \log \lambda_k}{\sum_{k=1}^K \lambda_k^q}\\ $$

applying limits,

$$ \begin{aligned} =&\frac{1}{K}\sum_{k=1}^K \log \lambda_k =\frac{1}{K} \log \prod_{k=1}^K \lambda_k\\ =&\log \left(\prod_{k=1}^K \lambda_k\right)^{1/K}\\ \end{aligned} $$

it follows immediately that

$$ \lim_{q\rightarrow 0}{\mathfrak{M}}_q(q)= {\mathfrak{M}}_q(0)=\left(\prod_{k=1}^K \lambda_k\right)^{1/K} $$

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Appendix C

3.1 Gaussian Copula: Dependency Between Extreme Eigenvalues

In this section, the dependency analysis has been discussed between the random variables $(\lambda_1,\lambda_K)$ by plotting their Copula. A Copula is a multivariate distribution function with known marginal cumulative distribution functions (CDFs) [22]. More specifically, a bivariate joint distribution function $F_{\lambda_1,\,\lambda_K}(x,y)=Pr\lbrace \lambda_1 \leq x,\lambda_K \leq y \rbrace$ of two random variables $\lambda_1$ and $\lambda_K,$ may be represented by a Copula $C$ as a function of their marginal CDFs $F_{\lambda_1}(x)=Pr\lbrace {\lambda_1} \leq x \rbrace$ and $F_{\lambda_K}(y)=Pr\lbrace {\lambda_K} \leq y \rbrace$ and therefore may be expressed as [22, 35]

$$ F_{\lambda_1,\,\lambda_K}(x,y) = C(F_{\lambda_1}(x),F_{\lambda_K}(y)) \triangleq C(u,v) $$

(C.35)

where $u = F_{\lambda_1}(x)$ and $v = F_{\lambda_K}(y); C(u,v)$ is the associated Copula distribution function. Thus,

$$ C(u,v) = F_{\lambda_1,\,\lambda_K}(F_{\lambda_1}^{-1}(u),F_{\lambda_K}^{-1}(v)) $$

(C.36)

By exploiting the chain rule, the corresponding joint PDF $f_{\lambda_1,\,\lambda_K}(x,y)$ may be decomposed as

$$ f_{\lambda_1,\,\lambda_K}(x,y) = \frac{\partial^2 F_{\lambda_1,\,\lambda_K}(x,y)}{\partial x \partial y} = \frac{\partial^2 C(F_{\lambda_1}(x),F_{\lambda_K}(y))}{\partial x \partial y} $$

$$ = \frac{\partial^2 C(u,v)}{\partial u \partial v}\frac{\partial F_{\lambda_1}(x)}{\partial x}\frac{\partial F_{\lambda_K}(y)}{\partial y}\triangleq c(u,v)f_{\lambda_1}(x)f_{\lambda_K}(y) $$

(C.37)

It is obvious that the joint PDF is the product of the marginal PDFs $f_{\lambda_1}(x)$ and $f_{\lambda_K}(y)$ and Copula density function $c(u, v).$ The definition of Copula identifies a strong relationship which provides link between the marginal PDFs/CDFs and the respective joint PDF/CDF. Also, $c(u, v) = 1,$ for independent random variables [22].

Let $\Upphi_{\lambda_1}(x)$ and $\Upphi_{\lambda_K}(y)$ are the marginal distribution functions of the approximated Gaussian random variables $\lambda_1$ and $\lambda_K$ respectively. Using the statistical values of $\mu_{\lambda_1}, \mu_{\lambda_K}, \sigma_{\lambda_1}^2, \sigma_{\lambda_K}^2,$ the Copula distribution of (6.31) is given by [22]

$$ C(u,v)=\int_{-\infty}^{\Upphi^{-1}_{\lambda_{1}}(u)}\int_{-\infty}^{\Upphi^{-1}_{\lambda_{K}}(v)} f_{\lambda_{1},\,\lambda_{K}}\left(x,y\right) dxdy $$

(C.38)

where $\Upphi^{-1}_{(\cdot)}(\cdot)$ is the inverse of the standard univariate Gaussian distribution function.

It is to note that the closed-form of (C.38) is exceptionally cumbersome to compute. However, the integral may be solved numerically using Matlab built-in functions. The plots of Copula between $\lambda_1$ and $\lambda_K$ based on the empirical and Gaussian marginal distribution functions are showing in Fig. 6.13a and Fig. 6.13 b, respectively, for $(K,N)=(2,10).$ It can be seen that the structure of Copula based on the empirical distribution functions appears same as the Copula structure based on the Gaussian distribution functions for $\rho=0.3.$ Similar kind of observation can be made from Fig. 6.13c, d when $(K,N)=(4,50)$ and correlation reduces to $\rho=0.1.$ It can also be noticed that the Copula structure is distinctive for small value of $K$ and $N,$ i.e. the extreme eigenvalues are more dependent for reasonably small values of $K$ and $N.$ However, with the increase in number of $K$ and $N,$ the dependency between $\lambda_1$ and $\lambda_K$ decreases. The same is illustrated in Fig. 6.13. Therefore, the dependency between $\lambda_1$ and $\lambda_K$ can not be ignored if accurate spectrum sensing is required.

Fig. 6.13

Plots of Copula between random variables $\lambda_1$ and $\lambda_K$ based on empirical and Gaussian marginal distribution functions for selected values of $K$ and $N.$ a $K=2; N=10;$ b $K=2; N=10; \rho=0.36;$ c $K=4; N=50;$ d $K=4; N=50;\rho=0.17$