Spectrum Sensing in Multi-antenna Cognitive Radio Systems via Distributed Subspace Tracking Techniques

Abstract

Among the many different techniques that have been suggested for spectrum sensing, the eigenvalue-based spectrum sensing (EBSS) techniques exhibit some important advantages. Specifically, they can operate in a totally blind manner while they offer remarkably improved performance for specific types of signals, especially when compared to energy-based methods. Until recently, most of the cooperative EBSS techniques that could be found in the literature were batch and centralized ones, thus suffering from limitations that render them impractical in several cases. Practical cooperative adaptive versions of typical EBSS techniques, which could be applied in a completely distributed manner, have been proposed very recently. The aim of this chapter is (a) to briefly review existing cooperative EBSS techniques of the batch and centralized type and (b) to present in more detail adaptive and distributed versions of typical EBSS techniques. Focusing on the latter case, at first, we present adaptive EBSS techniques for the maximum eigenvalue detector (MED), the maximum-minimum eigenvalue detector (MMED), and the generalized likelihood ratio test (GLRT) scheme, respectively, for a single-user (noncooperative) case. Then, a distributed subspace tracking method is presented which enables the cooperating nodes to track the joint subspace of their received signals. Based on this method, cooperative distributed versions of the adaptive EBSS techniques have been developed that overcome the limitations of the previous batch centralized approaches. Numerical results show that the distributed techniques exhibit good performance, even though they require reduced computational complexity compared to their batch and centralized counterparts.

Appendix: Derivation of the Distribution of the MMED Test Statistic Under the \(\mathcal{H}_{0}\) Hypothesis

Let us assume that two RVs T ₁ and T ₂ follow the same distribution with that of the MED test statistic. The CDF of the later distribution is given by (16). The corresponding probability density function (PDF) is computed by taking the derivative of (16). That is,

$$\displaystyle{ f_{T_{i}}(\tau _{i}) = \frac{\xi ^{\rho }} {\varGamma (\rho )}\tau _{i}^{\rho -1}e^{-\rho \tau _{i} }. }$$

(32)

We are interested in the distribution of the random variable Y ₁ = T ₁∕T ₂. Let us also define the auxiliary random variable Y ₂ = T ₂. Thus we have,

$$\displaystyle\begin{array}{rcl} f_{1}(t_{1},t_{2})& =& \frac{t_{1}} {t_{2}} \\ f_{2}(t_{1},t_{2})& =& t_{2},{}\end{array}$$

(33)

where $t_{i} \in \mathbb{R}^{+}$. The inverse functions of the ones of (33) are given by

$$\displaystyle\begin{array}{rcl} f_{1}^{-1}(t_{ 1},t_{2})& =& t_{1}t_{2} \\ f_{2}^{-1}(t_{ 1},t_{2})& =& t_{2},{}\end{array}$$

(34)

The joint PDF of variables Y ₁ and Y ₂ is given by

$$\displaystyle{ f_{Y _{1},Y _{2}}(y_{1},y_{2}) = f_{T_{1},T_{2}}\big(f_{1}^{-1}(y_{ 1},y_{2}),f_{2}^{-1}(y_{ 1},y_{2})\big)\vert J(y_{1},y_{2})\vert, }$$

(35)

where $J(y_{1},y_{2}) = \frac{\partial (t_{1},t_{2})} {\partial (y_{1},y_{2})}$ is the Jacobian matrix of the transformation and | J(y ₁, y ₂) | = y ₂ is its determinant.

Observe now that, since the eigenvalues are estimated via (11), there are statistically independent. That is, the joint PDF of the variables under consideration T ₁ and T ₂ can be computed as the product of the corresponding marginal ones (16). Therefore, from (35) the joint PDF of Y ₁ and Y ₂ is given by

$$\displaystyle\begin{array}{rcl} f_{Y _{1},Y _{2}}(y_{1},y_{2})& =& f_{T_{1}}(y_{1}y_{2})f_{T_{2}}(y_{2})y_{2} \\ & =& \frac{\xi ^{2\rho }} {\varGamma ^{2}(\rho )}y_{1}^{\rho -1}y_{ 2}^{2\rho -1}e^{-\xi y_{2}(y_{1}+1)}{}\end{array}$$

(36)

In order to compute the marginal PDF of RV Y ₁, we integrate the joint one of (36) with respect to y ₂. That is

$$\displaystyle\begin{array}{rcl} f_{Y _{1}}(y_{1})& =& \frac{y_{1}^{\rho -1}} {\xi ^{2\rho }\varGamma ^{2}(\rho )} \int _{0}^{+\infty }y_{ 2}^{2\rho -1}e^{-\xi y_{2}(y_{1}+1)}dy_{ 2} \\ & =& \frac{y_{1}^{\rho -1}\xi ^{2\rho }\varGamma (2\rho )} {(1 + y_{1})^{2\rho }\xi ^{2\rho }\varGamma ^{2}(\rho )} = \frac{y_{1}^{\rho -1}} {B(\rho,\rho )(1 + y_{1})^{2\rho }},{}\end{array}$$

(37)

where the following property of the beta function [20] was used

$$\displaystyle{ B(\rho _{1},\rho _{2}) =\int _{ 0}^{1}x^{\rho _{1} }(1 - x)^{\rho _{2}-1}dx = \frac{\varGamma (\rho _{1} +\rho _{2})} {\varGamma (\rho _{1})\varGamma (\rho _{2})}. }$$

(38)

By integrating (37) we derive the corresponding CDF of the beta prime distribution given by (18) of Lemma 1, and the proof is completed . □