A Feature-Based Compressive Spectrum Sensing Technique for Cognitive Radio Operation

Abstract

In cognitive radio systems, data throughput of the secondary user is an important performance metric used to evaluate the spectrum usage efficiency. As such, the effectiveness of the spectrum sensing process used by the secondary user, namely the spectrum sensing accuracy, its sampling time and processing time will have significant impacts on the data throughput performance. This paper presents a novel wideband spectrum sensing technique operating at low sub-Nyquist sampling rate that can achieve high sensing accuracy and high throughput without high computational cost. The proposed technique applies a novel likelihood ratio test on the learned feature information of the primary signal for efficient spectrum sensing, which is based directly on the compressive data collected by a sub-Nyquist sampler. Comprehensive analysis of the sensing-throughput performance for various commonly used spectrum sensing techniques is also presented, which are then used to compare against the proposed technique. Simulation results using real-world ATSC DTV data operating in IEEE 802.22 WRAN environment show that due to the higher detection accuracy and shorter spectrum sensing duration, the proposed technique is able to achieve better achievable secondary user’s transmission throughput compared to other well-known spectrum sensing techniques, while operating at 0.17 time of the Nyquist sampling rate.

Appendices

Appendix A

This “Appendix” shows the derivation of the $\sigma _0$ and $\sigma _1$ used in (33). First, we derive the differential expression of the (31) and (32), leading to

$$\begin{aligned} \begin{aligned}&\frac{{\partial \ln p\left( {{\varvec{\varGamma }_{y}}|{\sigma ^2},\mathcal{H}_1} \right) }}{{\partial {\sigma ^2}}} = - \frac{M}{2}\left[ \frac{\lambda _{a,1}}{ \lambda _{s,1} + \lambda _{a,1} \sigma ^2 } + \frac{ (L - 1)}{{\sigma ^2 }} \right] \\&\quad - \frac{M}{2} \left[ -\frac{\varvec{\phi }_{a,1}^T \varvec{C}_y \varvec{\phi }_{a,1} \lambda _{a,1} }{\left( \varvec{\phi }_{a,1} \varvec{C}_s^1 \varvec{\phi }_{a,1} + \lambda _{a,1}\sigma ^2 \right) ^2 } - \frac{{\frac{1}{{M }}\sum \limits _{m=1}^{M } {\varvec{y}_m^T \varvec{y}_m } - {\varvec{\phi }_{a,1}^T{\varvec{C}_y}{\varvec{\phi }_{a,1}}} } }{ \epsilon \left( {\sigma ^2 } \right) ^2 } \right] \end{aligned} \end{aligned}$$

(36)

and

$$\begin{aligned} \begin{aligned}&\frac{{\partial \ln p\left( {{\varvec{\varGamma }_{y}}|{\sigma ^2},\mathcal{H}_0} \right) }}{{\partial {\sigma ^2}}} = - \frac{M}{2}\left[ \frac{1}{{\sigma ^2 }} + \frac{ (L - 1)}{{\sigma ^2 }} \right] \\&\quad - \frac{M}{2}\left[ -\frac{{\varvec{\phi }_{a,1}^T{\varvec{C}_y}{\varvec{\phi }_{a,1}}}}{\lambda _{a,1} \left( { \sigma ^2 } \right) ^2 } - \frac{{\frac{1}{{M }}\sum \limits _{m=1}^{M } {\varvec{y}_m^T \varvec{y}_m } - {\varvec{\phi }_{a,1}^T{\varvec{C}_y}{\varvec{\phi }_{a,1}}} } }{ \epsilon \left( {\sigma ^2 } \right) ^2 } \right] \end{aligned} \end{aligned}$$

(37)

Setting $\frac{{\partial \ln p\left( {\varvec{\varGamma }_{y}|{\sigma ^2},\mathcal{H}_0} \right) }}{{\partial {\sigma ^2}}} = 0$ in (37) and we have the MLE of $\sigma ^2$ under $\mathcal{H}_0$:

$$\begin{aligned} {\hat{\sigma } }_0^2 = \frac{\varvec{\phi }_{a,1}^T \varvec{C}_y \varvec{\phi }_{a,1}}{\lambda _{a,1} L} + \frac{\sum \limits _{m=1}^{M } \varvec{y}_m^T \varvec{y}_m - \varvec{\phi }_{a,1}^T \varvec{C}_y \varvec{\phi }_{a,1} }{\epsilon L} \end{aligned}$$

(38)

Similarly, setting $\frac{{\partial \ln p\left( {{\varvec{\varGamma }_{y}}|{\sigma ^2},\mathcal{H}_1} \right) }}{{\partial {\sigma ^2}}} = 0$ in (36), and with the $\frac{{\partial \ln p\left( {{\varvec{\varGamma }_{y}}|{\lambda _{a,1}},{\sigma ^2},\mathcal{H}_1} \right) }}{{\partial \lambda _{a,1}}} = - \frac{{{M}}}{2}\left( \frac{\sigma ^2}{{{\lambda _{s,1}} + { \lambda _{a,1}\sigma ^2}}} - \frac{\sigma ^2 \varvec{\phi }_{a,1}^T \varvec{C}_y \varvec{\phi }_{a,1}}{{\left( \varvec{\phi }_{a,1} \varvec{C}_s^1 \varvec{\phi }_{a,1} + \lambda _{a,1} \sigma ^2 \right) }^2} \right) =0$, we have MLE of $\sigma ^2$ under $\mathcal{H}_1$:

$$\begin{aligned} {\hat{\sigma } }_1^2 = \frac{{\frac{1}{{M }}\sum \limits _{m=1}^{M } \varvec{y}_m^T \varvec{y}_m - \varvec{\phi }_{a,1}^T \varvec{C}_y \varvec{\phi }_{a,1} } }{ \epsilon (L-1) } \end{aligned}$$

(39)

Substituting the estimated $\hat{\sigma }_0^2$ and $\hat{\sigma }_1^2$ back to the (32) and (31), we obtained the updated logarithm likelihood function as:

$$\begin{aligned} \begin{aligned} \ln p\left( {{\varvec{\varGamma }_{y}}|\mathcal{H}_1} \right) =&- \frac{{L{M}}}{2}\ln 2\pi - \frac{{{M}}}{2}\left[ {\ln \left( \lambda _{s,1} + \lambda _{a,1} \sigma _1^2 \right) + \left( {L - 1} \right) \ln \left( {{ \epsilon \sigma _1^2}} \right) } \right] \\&- \frac{{{M}}}{2}\left[ \frac{\varvec{\phi }_{a,1}^T \varvec{C}_s^1 \varvec{\phi }_{a,1} + \lambda _{a,1} \sigma _1^2 }{\lambda _{s,1} + \lambda _{a,1}\sigma _1^2} + (L-1) \right] \end{aligned} \end{aligned}$$

(40)

and

$$\begin{aligned} \begin{aligned} \ln p\left( {{\varvec{\varGamma }_{y}}|\mathcal{H}_0} \right) = - \frac{{L{M}}}{2}\ln 2\pi - \frac{{{M}}}{2}\left[ {\ln \left( {\lambda _{a,1} \sigma _0^2} \right) + \left( {L - 1} \right) \ln \left( \epsilon \sigma _0^2 \right) } +L \right] \\ \end{aligned} \end{aligned}$$

(41)

With (40) and (41), the test function for the proposed spectrum sensing (33) becomes:

$$\begin{aligned} \begin{aligned} T(\varvec{y}) = \ln \frac{\lambda _{a,1} \sigma _0^2}{\lambda _{s,1} + \lambda _{a,1} \sigma _1^2} + (L-1) \ln \frac{\sigma _0^2}{\sigma _1^2} +\frac{\lambda _{s,1} - \varvec{\phi }_{a,1}^T \varvec{C}_s^1 \varvec{\phi }_{a,1} }{\lambda _{s,1} + \lambda _{a,1} \sigma _1^2} \underset{\mathcal{H}_0}{\overset{\mathcal{H}_1}{\gtrless }} \eta \end{aligned} \end{aligned}$$

(42)

Appendix B

This “Appendix” shows a notation table which includes all vectors and matrices used in the derivation of the proposed SS in Sect. 5.3 (Table 3).

Table 3 Notation table of vectors and matrices used for the derivation of proposed SS