Blind Spectrum Sensing Based on the Statistical Covariance Matrix and K-Median Clustering Algorithm

Abstract

Spectrum sensing is a fundamental function for cognitive radio systems, which can improve spectrum utilization. In this article, a blind spectrum sensing method based on the sample covariance matrix and K-median clustering algorithm is proposed to further improve the sensing performance. Specifically, to obtain a two-dimensional signal feature vector, the received signal matrix is rebuilt into two sub-matrices by the decomposition and reorganization (DAR) method. Moreover, two statistical covariance matrices are constructed by the sub-matrices, respectively. The ratios between the sum of some elements from the sample covariance matrices and the sum of diagonal elements from those matrices are used as a signal feature vector. It is demonstrated that the new signal feature vector and the feature vector based on the improved covariance absolute value method are equivalent. Furthermore, K-median clustering algorithm is trained by signal feature vectors to obtain a classifier. Indeed, this classifier can directly be used to detect whether the PU signal is absent or not. Simulation results report that the proposed algorithm has better sensing performance than some popular sensing algorithms based on random matrix theory or information geometry.

This work was supported in part by the National Natural Science Foundation of China under Grant 61971147, in part by the special funds from the central finance to support the development of local universities under Grant 400170044 and Grant 400180004, in part by the project supported by the State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences under Grant 20180106, in part by the Foundation of National and Local Joint Engineering Research Center of Intelligent Manufacturing Cyber-Physical Systems and Guangdong Provincial Key Laboratory of Cyber-Physical Systems under Grant 008, and in part by the higher education quality projects of Guangdong Province and Guangdong University of Technology.

Appendix

In this section, it will be demonstrated that the new signal feature vector and the improved CAV-based feature vector are equivalent. Assume that a $3 \times 3$ statistical covariance matrix is given by

$$\begin{aligned} \mathbf {C}=\begin{bmatrix} c_{11} &{} c_{12} &{} c_{13} \\ c_{21} &{} c_{22} &{} c_{23} \\ c_{31} &{} c_{32} &{} c_{33} \end{bmatrix}. \end{aligned}$$

(13)

The matrix $\mathbf {C}$ is rebuilt into two sub-matrices by the DAR method. Two sub-matrices can be written as

$$\begin{aligned} \mathbf {C}^1=\begin{bmatrix} c^1_{11} &{} c^1_{12} &{} c^1_{13} \\ c^1_{21} &{} c^1_{22} &{} c^1_{23} \\ c^1_{31} &{} c^1_{32} &{} c^1_{33} \end{bmatrix} \end{aligned}$$

(14)

and

$$\begin{aligned} \mathbf {C}^2=\begin{bmatrix} c^2_{11} &{} c^2_{12} &{} c^2_{13} \\ c^2_{21} &{} c^2_{22} &{} c^2_{23} \\ c^2_{31} &{} c^2_{32} &{} c^2_{33} \end{bmatrix}, \end{aligned}$$

(15)

respectively.

According to [5], the improved CAV-based feature vector can be obtain by

$$\begin{aligned} \mathbf {U}=[t^1_{\text {ICAV}},t^2_{\text {ICAV}}], \end{aligned}$$

(16)

where $t^1_{\text {ICAV}}$ and $t^2_{\text {ICAV}}$ can be formulated as

$$\begin{aligned} t^1_{\text {ICAV}}=\frac{\sum _{1\le i\le j\le 3}|c^1_{ij}|}{|c^1_{11}|+|c^1_{22}|+|c^1_{33}|} \end{aligned}$$

(17)

and

$$\begin{aligned} t^2_{\text {ICAV}}=\frac{\sum _{1\le i\le j\le 3}|c^2_{ij}|}{|c^2_{11}|+|c^2_{22}|+|c^2_{33}|}, \end{aligned}$$

(18)

respectively. The Manhattan distance is used to analyze the similarity between two points in this article. Note that $\mathbf {U}=[1,1]$ when the PU signal is absent or $\mathbf {U}=[t^1_{\text {ICAV}},t^2_{\text {ICAV}}]$ when the PU signal is present, where $t^1_{\text {ICAV}}>1$ and $t^2_{\text {ICAV}}>1$. Therefore, the Manhattan distance between points which belong to $\mathcal {H}_0$ and $\mathcal {H}_1$, respectively, can be written as

$$\begin{aligned} d(\mathbf {U}_{\mathcal {H}_1},\mathbf {U}_{\mathcal {H}_0})&=|t^1_{\text {ICAV}}-1|+|t^2_{\text {ICAV}}-1| \nonumber \\&=\bigg |\frac{\sum _{1\le i\le j\le 3}|c^1_{ij}|}{|c^1_{11}|+|c^1_{22}|+|c^1_{33}|}-1\bigg |+\bigg |\frac{\sum _{1\le i\le j\le 3}|c^2_{ij}|}{|c^2_{11}|+|c^2_{22}|+|c^2_{33}|}-1\bigg | \nonumber \\&=\frac{|c^1_{12}|+|c^1_{13}|+|c^1_{23}|}{|c^1_{11}|+|c^1_{22}|+|c^1_{33}|} +\frac{|c^2_{12}|+|c^2_{13}|+|c^2_{23}|}{|c^2_{11}|+|c^2_{22}|+|c^2_{33}|} . \end{aligned}$$

(19)

Moreover, the new signal feature vector can be obtained by

$$\begin{aligned} \mathbf {T}=[t^1,t^2], \end{aligned}$$

(20)

where $t^1$ and $t^2$ are written as

$$\begin{aligned} t^1=\frac{|c^1_{12}|+|c^1_{13}|+|c^1_{23}|}{|c^1_{11}|+|c^1_{22}|+|c^1_{33}|} \end{aligned}$$

(21)

and

$$\begin{aligned} t^2=\frac{|c^2_{12}|+|c^2_{13}|+|c^2_{23}|}{|c^2_{11}|+|c^2_{22}|+|c^2_{33}|}, \end{aligned}$$

(22)

respectively. Note that $\mathbf {T}=[0,0]$ when the authorized channel is idle or $\mathbf {T}=[t^1,t^2]$ when the authorized channel is busy, where $t^1>0$ and $t^2>0$. Hence, the Manhattan distance between points which belong to $\mathcal {H}_0$ and $\mathcal {H}_1$, respectively, can be calculated by

$$\begin{aligned} d(\mathbf {T}_{\mathcal {H}_1},\mathbf {T}_{\mathcal {H}_0})&=|t^1-0|+|t^2-0| \nonumber \\&=\bigg |\frac{|c^1_{12}|+|c^1_{13}|+|c^1_{23}|}{|c^1_{11}|+|c^1_{22}|+|c^1_{33}|}-0\bigg | +\bigg |\frac{|c^2_{12}|+|c^2_{13}|+|c^2_{23}|}{|c^2_{11}|+|c^2_{22}|+|c^2_{33}|}-0\bigg | \nonumber \\&=\frac{|c^1_{12}|+|c^1_{13}|+|c^1_{23}|}{|c^1_{11}|+|c^1_{22}|+|c^1_{33}|} +\frac{|c^2_{12}|+|c^2_{13}|+|c^2_{23}|}{|c^2_{11}|+|c^2_{22}|+|c^2_{33}|} . \end{aligned}$$

(23)

Based on (19) and (23), it is easy job for us to find that $d(\mathbf {U}_{\mathcal {H}_1},\mathbf {U}_{\mathcal {H}_0})= d(\mathbf {T}_{\mathcal {H}_1},\mathbf {T}_{\mathcal {H}_0})$. Consequently, it has been demonstrated that the new signal feature vector and the improved CAV-based feature vector are equivalent.