Sub-Nyquist Sampling and Compressed Sensing in Cognitive Radio Networks

Abstract

Cognitive radio has become one of the most promising solutions for addressing the spectral under-utilization problem in wireless communication systems. As a key technology, spectrum sensing enables cognitive radios to find spectrum holes and improve spectral utilization efficiency. To exploit more spectral opportunities, wideband spectrum sensing approaches should be adopted to search multiple frequency bands at a time. However, wideband spectrum sensing systems are difficult to design, due to either high implementation complexity or high financial/energy costs. Sub-Nyquist sampling and compressed sensing play crucial roles in the efficient implementation of wideband spectrum sensing in cognitive radios. In this chapter, Sect. 6.1 presents the fundamentals of cognitive radios. A literature review of spectrum sensing algorithms is given in Sect. 6.2. Wideband spectrum sensing algorithms are then discussed in Sect. 6.3. Special attention is paid to the use of Sub-Nyquist sampling and compressed sensing techniques for realizing wideband spectrum sensing. Finally, Sect. 6.4 shows an adaptive compressed sensing approach for wideband spectrum sensing in cognitive radio networks.

Appendices

Appendix

Proof of Theorem 1

Using a variant of the Johnson-Lindenstrauss lemma as shown in Theorem 5.1 of [38], we have

$$\begin{aligned} \Pr \left[ (1 - \varepsilon )\Vert \mathbf x \Vert _2 \le \frac{ \Vert \varvec{\varPsi } \mathbf x \Vert _1}{\sqrt{2/\pi }\;r} \le (1 + \varepsilon )\Vert \mathbf x \Vert _2 \right] \ge 1 - \xi . \end{aligned}$$

(6.23)

Defining $\mathbf x \stackrel{\triangle }{=}\mathbf F ^{-1} (\mathbf X -\hat{X}_p)$ in (6.23), we obtain

$$\begin{aligned} \Pr \bigg [ (1 - \varepsilon )\Vert \mathbf F ^{-1} (\mathbf X - \hat{X}_p)\Vert _2&\le \frac{ \Vert \varvec{\varPsi } \mathbf F ^{-1} (\mathbf X - \hat{X}_p) \Vert _1}{\sqrt{2/\pi }\;r} \nonumber \\&\le (1 + \varepsilon )\Vert \mathbf F ^{-1} (\mathbf X - \hat{X}_p)\Vert _2 \bigg ] \ge 1 - \xi . \end{aligned}$$

(6.24)

The above inequality can be rewritten by using (6.12) and (6.14)

$$\begin{aligned} \small \Pr \left[ (1 - \varepsilon )\Vert \mathbf F ^{-1} (\mathbf X - \hat{X}_p)\Vert _2 \le \sqrt{\frac{\pi }{2}} \rho _p \le (1 + \varepsilon )\Vert \mathbf F ^{-1} (\mathbf X - \hat{X}_p)\Vert _2 \right] \ge 1 - \xi . \end{aligned}$$

(6.25)

Applying Parseval’s relation to (6.25), we have

$$\begin{aligned} \Pr \left[ (1 - \varepsilon )\Vert \mathbf X - \hat{X}_p\Vert _2 \le \sqrt{\frac{\pi N}{2}}\rho _p \le (1 + \varepsilon )\Vert \mathbf X - \hat{X}_p\Vert _2 \right] \ge 1 - \xi . \end{aligned}$$

(6.26)

Thus, Theorem 1 follows.

Proof of Theorem 2

The best spectral approximation $\mathbf X ^{\star }$ means that $\Vert \mathbf X ^{\star }-\mathbf X \Vert _2$ is sufficiently small. Without loss of generality, we approximate $\mathbf X ^{\star }$ by $\mathbf X$. Thus, if $\hat{X}_p$ is the best spectral approximation, the validation parameter can be rewritten by using (6.18)

$$\begin{aligned} \rho _p = \frac{\Vert \mathbf V - \varvec{\varPsi } \mathbf F ^{-1} \hat{X}_p \Vert _1}{r} = \frac{\Vert \mathbf n \Vert _1}{r} = \frac{\sum _{i = 1}^{r}|n^i |}{r}. \end{aligned}$$

(6.27)

As the measurement noise $n^i \sim \mathcal CN (0, \delta ^2)$, its absolute value $|n^i|$ follows the Rayleigh distribution with mean $\sqrt{\frac{\pi }{2}}\delta$ and variance $\frac{4-\pi }{2}\delta ^2$. Using the cumulative distribution function of the Rayleigh distribution, we have $\Pr (|n^i| \le x)=1-\exp (-\frac{x^2}{2\delta ^2})$. Further, as the measurement noise level has an upper-bound $\bar{n}$ in practice, there exists a sufficiently large parameter $\nu$ that makes $|n^i| \le \bar{n} \le (\nu +1)\sqrt{\frac{\pi }{2}}\delta$ almost surely. If we define a new variable $D_i = |n^i|-\sqrt{\frac{\pi }{2}}\delta$, we obtain $\mathbb E [D_i]=0$, $\mathbb E [ D_i^2 ]=\frac{4-\pi }{2}\delta ^2$, and $|D_i|\le \sqrt{\frac{\pi }{2}}\delta \nu$. Based on the Bernstein’s inequality [39], the following inequality holds

$$\begin{aligned} \Pr \left[ \left| \sum _{i = 1}^{r} D_i \right| >\varepsilon \right]&= \Pr \left[ \left| \sum _{i = 1}^{r} |n^i| - r \sqrt{\frac{\pi }{2}}\delta \right| >\varepsilon \right] \nonumber \\&\le \ 2\exp \left( -\frac{\varepsilon ^2/2}{ \sum _{i = 1}^r \mathbb E [ D_i^2 ] + \overline{D} \varepsilon /3}\right) \nonumber \\&\le \ 2\exp \left( -\frac{3\varepsilon ^2}{3(4 - \pi )r\delta ^2 + \sqrt{2\pi } \varepsilon \delta \nu }\right) \end{aligned}$$

(6.28)

where $\overline{D}=\sqrt{\frac{\pi }{2}}\delta \nu$ denotes the upper-bound on $|D_i|$.

Simply replacing $\varepsilon$ by $r\epsilon$ in (6.28) while using (6.27), we can rewrite (6.28) as

$$\begin{aligned} \Pr \left[ \left| \rho _p - \sqrt{\frac{\pi }{2}}\delta \right| > \epsilon \right] \le 2\exp \left( -\frac{3r\epsilon ^2}{3(4-\pi )\delta ^2+\sqrt{2\pi } \epsilon \delta \nu } \right) \!. \end{aligned}$$

(6.29)

Using (6.29), we end up with

$$\begin{aligned} \Pr \left[ \left| \rho _p - \sqrt{\frac{\pi }{2}}\delta \right| \le \epsilon \right] > 1- 2\exp \left( -\frac{3r\epsilon ^2}{3(4-\pi )\delta ^2+\sqrt{2\pi } \epsilon \delta \nu } \right) \!. \end{aligned}$$

(6.30)

To derive the required $r$, we set the lower probability bound in (6.30) as

$$\begin{aligned} 1- 2\exp \left( -\frac{3r\epsilon ^2}{3(4-\pi )\delta ^2+\sqrt{2\pi } \epsilon \delta \nu } \right) =1-\varrho . \end{aligned}$$

(6.31)

Solving the above equation, we obtain

$$\begin{aligned} r=\ln \left( \frac{2}{\varrho }\right) \frac{3(4-\pi )\delta ^2+\sqrt{2\pi } \epsilon \delta \nu }{3\epsilon ^2}. \end{aligned}$$

(6.32)

This completes the proof of Theorem 2.