Novel Semi-Blind Spectrum Sensing in Cognitive Radio Networks with Fourth-Order Statistics

Abstract

Cognitive radio (CR) system is able to exploit the bands allocated to licensed or primary users when they are not being used. Spectrum sensing is a key element of a working CR system. In this paper, enhanced algorithms are proposed for semi-blind spectrum sensing in CR networks using fourth-order statistics of the received primary user’s signal. The proposed statistic will have a value equal or close to 3 when only Gaussian noise samples exist in the received signal. This estimate is used to differentiate between the presence or absence of the primary user by comparing with a predefined threshold. Using the Neyman-Pearson criterion, an optimized threshold is established and an analytical expression for the upper bound on the average probability of miss-detection $(P_{m,avg})$ is also derived. The proposed algorithm clearly outperforms the energy detection (ED) method over the low-SNR range of $-15$ to $12$ dB. For $P_{m,avg} = 10^{-2}$ , e.g., the proposed scheme outperforms the ED method by 1.5 dB for the case of a single user. Moreover, the proposed algorithm has been extended to the cooperative spectrum sensing model. Simulation results show that the proposed scheme significantly outperforms the ED method in cooperative spectrum sensing scenarios.

Appendices

Appendix 1: Proof of (13)

$$\begin{aligned} P_{m,avg}=1-P_{m_{1},avg}-P_{m_{2},avg}, \end{aligned}$$

where,

$$\begin{aligned} \begin{aligned} P_{m_{1},avg}&=\int _{\gamma =0 }^{\infty }{\int _{\theta =0}^{2\pi }{P_{m_{1}|\gamma ,\theta }f_{\gamma ,\theta }(\gamma ,\theta )d\theta \, d\gamma }}\\&=\int _{\gamma =0 }^{\infty }{\int _{\theta =0}^{2\pi }{\frac{1}{2}\mathrm{{erfc}}\left( \frac{K}{\sigma _{1}(\gamma ,\theta )}\right) \left( \frac{1}{2\pi \bar{\gamma }}e^{-\frac{\gamma }{\bar{\gamma }}}\right) d\theta \, d\gamma }}, \end{aligned} \end{aligned}$$

(19)

and $K=\frac{(\lambda ^{}-\mu _{1})}{\sqrt{2}}$. In (19), $f_{\gamma ,\theta }(\gamma ,\theta )=f_{\gamma }(\gamma )f_{\theta }(\theta )=\frac{1}{2\pi \bar{\gamma }}e^{-\frac{\gamma }{\bar{\gamma }}}$ is the joint pdf of $\gamma$ and $\theta$, which are statistically independent. We can simplify the integral given in (19) as follows,

$$\begin{aligned} \begin{aligned} P_{m_{1},avg} =&c\int _{\gamma =0 }^{x_{0}}{\int _{\theta =0}^{2\pi }{\frac{1}{2}\mathrm{erfc}\left( \frac{K}{\sigma _{1}(\gamma ,\theta )}\right) \left( \frac{1}{2\pi \bar{\gamma }}e^{-\frac{\gamma }{\bar{\gamma }}}\right) d\theta \, d\gamma }} \\&+c\int _{\gamma =x_{0} }^{1}{\int _{\theta =0}^{2\pi }{\frac{1}{2}\mathrm{erfc}\left( \frac{K}{\sigma _{1}(\gamma ,\theta )}\right) \left( \frac{1}{2\pi \bar{\gamma }}e^{-\frac{\gamma }{\bar{\gamma }}}\right) d\theta \, d\gamma }} \\&+c\int _{\gamma =1 }^{\infty }{\int _{\theta =0}^{2\pi }{\frac{1}{2}\mathrm{erfc}\left( \frac{K}{\sigma _{1}(\gamma ,\theta )}\right) \left( \frac{1}{2\pi \bar{\gamma }}e^{-\frac{\gamma }{\bar{\gamma }}}\right) d\theta \, d\gamma }}.\\ =&I_{1}+I_{2}+I_{3}, \end{aligned} \end{aligned}$$

(20)

where $x_{0}<1, I_{1}=c\int _{\gamma =0}^{x_0}\int _{\theta =0}^{2\pi }\mathrm{erfc}\left( \frac{K}{\sigma _{1}(\gamma ,\theta )}\right) e^{-\frac{\gamma }{\bar{\gamma }}}d\theta d\gamma , I_{2}=c\int _{\gamma =x_0}^1 \int _{\theta =0}^{2\pi }\mathrm{erfc}\left( \frac{K}{\sigma _{1}(\gamma ,\theta )}\right) \, e^{-\frac{\gamma }{\bar{\gamma }}}d\theta d\gamma ,$   $I_3 = c\int _{\gamma =1}^{\infty }\int _{\theta =0}^{2\pi }\, \mathrm{erfc}\left( \frac{K}{\sigma _1(\gamma ,\theta )}\right) e^{-\frac{\gamma }{\bar{\gamma }}}d\theta d\gamma$, and $c$ is a constant that is independent of the variables of integration.

Evaluation of $I_{1}$: The range of $\gamma$ is $0\le \gamma \le x_{0}$. Using the series expansion of $\mathrm{erfc}(.)$ [22, Eq. (8.253)],

$$\begin{aligned} \mathrm{erfc}(z)=1-\left( \frac{2}{\sqrt{\pi }}\right) \sum _{k=1}^{\infty }{\frac{(-1)^{k+1}z^{2k-1}}{(2k-1)(k-1)!}}, \end{aligned}$$

(21)

Using (21) in $I_{1}$,

$$\begin{aligned} I_{1}= \frac{c}{4\pi \bar{\gamma }}\cdot \int _{\gamma =0}^{x_{0}}{\int _{\theta =0}^{2\pi }{\left[ 1-\left( \frac{2}{\sqrt{\pi }}\right) \sum _{k=1}^{\infty }\frac{(-1)^{k+1}\left( \frac{K}{\sigma _{1}}\right) ^{2k-1}}{(2k-1)(k-1)!}\right] e^{-\frac{\gamma }{\bar{\gamma }}}d\theta d\gamma }}. \end{aligned}$$

(22)

For $0\le \gamma \le x_{0}$, $\sigma _{1}(\gamma ,\theta )$ in (40) may be approximated by

$$\begin{aligned} \sigma _{1}(\gamma ,\theta )\approx \sqrt{a_{00}+a_{10}}=\sqrt{a_{00}(1+b_{0}\gamma )} \end{aligned}$$

(23)

where $a_{00}$ and $a_{10}$ are constants which depend on the modulation scheme being employed under hypothesis $H_{1}$, and whose values are given in “Appendix 1”, for 16-QAM constellation, and $b_{0}=\frac{a_{10}}{a_{00}}$. The $k$-th term of the integrand, $t_{k}$, of $I_{1}$ in (22) is given by,

$$\begin{aligned} t_{k}=\left( -\frac{2}{\sqrt{\pi }}\right) \sum _{k=1}^{\infty }{\frac{(-1)^{k+1}\left( \frac{K}{\sigma _{1}(\gamma ,\theta )}\right) ^{2k-1}}{(2k-1)(k-1)!}}e^{-\frac{\gamma }{\bar{\gamma }}}, \end{aligned}$$

(24)

for $k\ge 1$. Using (23) and (24), this can be written as,

$$\begin{aligned} t_{k}=\left( -\frac{2}{\sqrt{\pi }}\right) \sum _{k=1}^{\infty }{\frac{(-1)^{k+1}\left( \frac{K}{\sqrt{a_{00}}\sqrt{1+b_{0}\gamma }}\right) ^{2k-1}}{(2k-1)(k-1)!}}e^{-\frac{\gamma }{\bar{\gamma }}}, \end{aligned}$$

(25)

Since $|b_0\gamma | < 1$ for the range of $\gamma$ considered above, using the binomial expansion $[$22, Eq. (1.110)$]$ for $(1+b_{0}\gamma )^{-k+\frac{1}{2}}$, (25) can be further simplified as,

$$\begin{aligned} t_{k}&= \left( -\frac{2}{\sqrt{\pi }}\right) \left( \frac{K}{\sqrt{a_{00}}}\right) ^{2k-1}\frac{(-1)^{k+1}}{(2k-1)(k-1)!}\nonumber \\&\cdot \left\{ 1+\sum _{p=1}^{\infty }\frac{(-1)^{p}(2k-1)(2k-3)\ldots (2k-(2p-3))b_{0}^{p}\gamma ^{p}}{2^{p}p!} \right\} e^{-\frac{\gamma }{\bar{\gamma }}}. \end{aligned}$$

(26)

Therefore, $I_{1}$ in (22) is a double integration with a double summation. The $(k,p)$-th term of the double summation, where $k$ and $p$ are the indices of the first and second summation, respectively, after integration yields,

$$\begin{aligned} I_{1,k,p}=\int _{\gamma =0 }^{x_{0}}{\int _{\theta =0}^{2\pi }{(\gamma )^{p}\exp \left( -\frac{\gamma }{\bar{\gamma }}\right) d\theta d\gamma }}=2\pi \gamma _{L}\left( p+1,\frac{x_{0}}{\bar{\gamma }}\right) , \end{aligned}$$

(27)

where $\gamma _{L}(\cdot ,\cdot )$ is the lower incomplete Gamma function $[$22, Eq. (8.350.1)$]$. Therefore, $I_{1}$ in (22) can be written as,

$$\begin{aligned} \begin{aligned} I_1=&\sum _{k=1}^{\infty }{\sum _{p=1}^{\infty }{I_{1,k,p}}}\\ =&\sum _{k=1}^{\infty }{\frac{2c}{\sqrt{\pi }}\frac{(-1)^{k+1}\left( \frac{K}{\sqrt{a_{00}}}\right) ^{2k-1}}{(2k-1)(k-1)!}}\,\left\{ \frac{1}{2\bar{\gamma }^{p}}\gamma _{L}\left( p+1,\frac{x_{0}}{\bar{\gamma }}\right) \delta (p)\right. \\&+\sum _{p=1}^{\infty }\left. \frac{(-1)^{p}(2k-1)(2k-3)\ldots (2k-(2p-3))b_{0}^{p}\bar{\gamma }^p}{2^p p!} \gamma _{L}\left( p+1,\frac{x_{0}}{\bar{\gamma }}\right) \right\} , \end{aligned} \end{aligned}$$

(28)

where $\delta (.)$ is the Kronecker delta function.

Evaluation of $I_{2}:$ The range of $\gamma$ is $x_{0}\le \gamma \le 1$. In this range of $\gamma$, since $a_{10}\gamma$ is the dominant term in the expression for $\sigma (\gamma , \theta )$ in (40), it can be approximated as,

$$\begin{aligned} \sigma _{1}(\gamma ,\theta )=\sqrt{a_{10}\gamma }, \end{aligned}$$

(29)

where $a_{10}$ is a constant which depends on the modulation format and is given in “Appendix 2”. Using the series expansion of $\mathrm{erfc}(.)$ in $I_{2}$ in (22) and proceeding in a similar way as for $I_{1}$, the $k$-th term integrand for $I_{2}$in (22) can be written as,

$$\begin{aligned} t_{k}=\frac{\left( -\frac{2}{\sqrt{\pi }}\right) (-1)^{k+1}\left( \frac{K}{\sqrt{a_{10}}}\right) ^{2k-1}}{(2k-1)(k-1)!}\gamma ^{-k+\left( \frac{1}{2}\right) }\exp \left( -\frac{\gamma }{\bar{\gamma }}\right) . \end{aligned}$$

(30)

Using the series expansion for $\exp (-\frac{\gamma }{\bar{\gamma }})\, [$22, Eq. (1.211.1)$]$, (30) reduces to,

$$\begin{aligned} t_{k}=\frac{\left( -\frac{2}{\sqrt{\pi }}\right) (-1)^{k+1}\left( \frac{K}{\sqrt{a_{10}}}\right) ^{2k-1}}{(2k-1)(k-1)!}\gamma ^{-k+\left( \frac{1}{2}\right) }\sum _{p=0}^{\infty }{\frac{\left( -\frac{\gamma }{\bar{\gamma }}\right) ^{p}}{p!}}. \end{aligned}$$

(31)

Therefore, the $(k,p)$-th term in the integrand in $I_{2}$ will be

$$\begin{aligned} t_{k,p}=\frac{\left( \frac{2}{\sqrt{\pi }}\right) (-1)^{k+p+1}\left( \frac{K}{\sqrt{a_{10}}}\right) ^{2k-1}}{(2k-1)(k-1)!p!\bar{\gamma }^{p}}\gamma ^{p-k+\left( \frac{1}{2}\right) }. \end{aligned}$$

(32)

Therefore, using (32), the expression for $I_{2}$ becomes,

$$\begin{aligned} I_{2}=\sum _{k=1}^{\infty }{\sum _{p=0}^{\infty }{\frac{\left( \frac{c}{\sqrt{\pi }}\right) (-1)^{k+p+1}\left( \frac{K}{\sqrt{a_{10}}}\right) ^{2k-1}\left( 1-x_{0}^{p-k+\frac{3}{2}}\right) }{(2k-1)(k-1)!p!\left( p-k+\frac{1}{2}\right) \bar{\gamma }^{p+1}}}}. \end{aligned}$$

(33)

Evaluation of $I_{3}$: Here, $\gamma >1$. In this range of $\gamma$ we can neglect the lower order terms retaining the highest order term of $\gamma$ in the expression for $\sigma _{1}(\gamma ,\theta )$ in (40). Hence

$$\begin{aligned} \sigma _{1}(\gamma ,\theta )&\approx \sqrt{[a_{40}+a_{41}\sin ^{2}2\theta -a_{41}\sin ^{4}4\theta ]\gamma ^{4}}\nonumber \\&\approx \sqrt{[a_{40}+a_{41}\sin ^{2}4\theta (1-\sin ^{2}2\theta )]\gamma ^{4}}\nonumber \\&\approx \sqrt{a_{40}[1+b_{1}\sin ^{2}4\theta ]\gamma ^{4}}, \end{aligned}$$

(34)

where $b_{1}= \frac{a_{41}}{a_{40}}$, and $a_{40}$, $a_{41}$ are constants, given in “Appendix 2”. Therefore, the $k$-th term in the integrand of $I_{3}$ can be written as

$$\begin{aligned} t_{k}=\frac{\frac{2}{\sqrt{\pi }}(-1)^{k+1}\left( \frac{K}{\sqrt{a_{40}}}\right) ^{2k-1}\left[ 1+b_{1}\sin ^{2}4\theta \right] ^{-k+\frac{1}{2}} \gamma ^{-4k+2}e^{-\frac{\gamma }{\bar{\gamma }}}}{(2k-1)(k-1)!} \end{aligned}$$

(35)

Since $|b_{1}|<1$ the term $[1+b_{1}\sin ^{2}4\theta ]^{-k+\frac{1}{2}}$ can be expanded as a binomial series,

$$\begin{aligned}&\left[ 1+b_{1}\sin ^{2}4\theta \right] ^{-k+\frac{1}{2}}\nonumber \\&\quad = 1 +\sum _{p=1}^{\infty } \left\{ \frac{(-1)^{p}(2k-1)(2k-3)\ldots \left( 2k-(2p-3)\right) }{2^{p}p!}\right\} b_{0}^{p}\sin ^{2p}4\theta . \end{aligned}$$

(36)

The $(k,p)$-th integral term of $I_{3}$ will be

$$\begin{aligned} I_{3,p,k}&= \int _{\gamma =1 }^{\infty }{\int _{\theta =0}^{2\pi }{\sin ^{2p}4\theta \gamma ^{-4k+2} e^{-\frac{\gamma }{\bar{\gamma }}} d\theta d\gamma }}\nonumber \\&= \frac{2\sqrt{\pi } \Gamma \left( p+\frac{1}{2}\right) }{\Gamma (p+1)}E_{4k-2}\left( \frac{1}{\bar{\gamma }}\right) , \end{aligned}$$

(37)

where $\Gamma (.)$ is the Gamma function $[$22, Eq. (8.310.1)$]$ and $E_{(.)}(.)$ is the Exponential integral $[$22, Eq. (12), pp. xxxv$]$. Therefore,

$$\begin{aligned} I_{3}&= \sum _{k=1}^{\infty }{\frac{\left( \frac{c}{\bar{\gamma }\pi ^{\frac{3}{2}}}\right) (-1)^{k+1}\left( \frac{K}{\sqrt{a_{10}}}\right) ^{2k-1}\Gamma \left( p+\frac{1}{2}\right) }{(2k-1)(k-1)!\Gamma (p+1)}}\left\{ E_{4k-2}\left( \frac{1}{\bar{\gamma }}\right) \delta (p)+\frac{1}{2\bar{\gamma }\sqrt{\pi }} \right. \nonumber \\&\quad \times \sum _{p=1}^{\infty }\left( \frac{(-1)^{p}(2k-1)(2k-3)\ldots (2k-(2p-3))}{2^{p}p!} \right. \nonumber \\&\quad \left. \left. \times \frac{b^{p}\Gamma (p+\frac{1}{2})}{\Gamma (p+1)} E_{4k-2}\left( \frac{1}{\bar{\gamma }}\right) \right) \right\} . \end{aligned}$$

(38)

An expression for $P_{m_{2},avg}$ can be derived by following the same analysis with $K=\frac{\lambda ^{}+\mu _{1}}{2}$. Therefore, the expression for $P_{m,avg}$ can be written as,

$$\begin{aligned} P_{m,avg}(\lambda )=1-\sum _{k=1}^{\infty } A_{k}\left[ \left( \frac{\lambda ^{}-\mu _{1}}{2}\right) ^{2k-1}+\left( \frac{\lambda ^{}+\mu _{1}}{2}\right) ^{2k-1}\right] , \end{aligned}$$

(39)

where $A_{k}$ is a term independent of $\lambda$.

Appendix 2: Kurtosis Estimate Statistics

The expressions for the mean and variance of the cumulants of a complex random variable, $r$, has been derived in [23]. Using those expressions, the following is obtained:

$$\begin{aligned} \begin{aligned} \mu _1&= E\left[ \vert r\vert ^{4}\right] -2\left( E[\vert r\vert ^{2}]\right) ^{2},\\ \sigma _{1}^{2}(\gamma ,\theta )&= a_{00}+a_{10}\gamma +[a_{20}+a_{21}\sin ^2 2\theta ]\gamma ^2+[a_{30}+a_{31}\sin ^2 2\theta ]\gamma ^{3}\\&+[a_{40}+a_{41}\sin ^{2}2\theta +a_{42}\sin ^{4}2\theta ]\gamma ^4,\\ \sigma _{0}^{2}(\gamma ,\theta )&= c_{00} \end{aligned} \end{aligned}$$

(40)

where $a_{ij}$s are constants which depend on the type of modulation scheme being employed and number of observation samples used to estimate the kurtosis. $\gamma =|h|^{2} \frac{\sigma _{s}^{2}}{\sigma _{n}^{2}}$ is the instantaneous SNR and $\theta =\tan ^{-1}\frac{Re[h]}{Im[h]}$ is the instantaneous phase angle of the channel. The values of $a_{ij}$s for 16-QAM constellation are evaluated and given in Table 1. It is important to notice that unlike second order statistics which require only the instantaneous SNR, the higher order statistics depend on the phase information of the channel as well.

Table 1 Modulation constants for 16-QAM constellation

Appendix 3: Unbiased Property of the Kurtosis Estimate

The kurtosis estimate in (5) is an unbiased estimate. This can be proved by taking the expectation on both sides of (5),

$$\begin{aligned} E\left[ \hat{k}\right]&= E\left[ \frac{1}{N_{obs}}\sum _{i=1}^{N_{obs}}{\left| r(i)\right| ^4-2\left( \frac{1}{N_{obs}}\sum _{i=1}^{N_{obs}}{\left| r(i)\right| ^2}\right) ^2}\right] \nonumber \\&= E\left[ \frac{1}{N_{obs}}\sum _{i=1}^{N_{obs}}\left| r(i)\right| ^4-2\left( \frac{1}{N_{obs}}\sum _{i=1}^{N_{obs}}{\left| r(i)\right| ^2}\right) \left( \frac{1}{N_{obs}}\sum _{j=1}^{N_{obs}}{\left| r(i)\right| ^2}\right) \right] \nonumber \\&= E \left[ \frac{1}{N_{obs}}\sum _{i=1}^{N_{obs}}\left| r(i)\right| ^4 - 2\left( \frac{1}{N_{obs}^{2}}\sum _{i=1}^{N_{obs}}{\sum _{j=1}^{N_{obs}}{\left| r(i)\right| ^2 \left| r(j)\right| ^2}}\right) \right] , \end{aligned}$$

(41)

where, $E[.]$ is the expectation operator. Taking the expectation operator inside in the summation, in (refc1), and combining the like terms yields,

$$\begin{aligned} E\left[ \hat{k}\right] \!=\!\frac{1}{N_{obs}}\sum _{i=1}^{N_{obs}} E\left[ |r(i)|^4\right] \!-\!\frac{2}{N_{obs}^2} \left[ N_{obs}E\left[ |r(i)|^{4}\right] \!+\!(N_{obs}\!-\!1) \left\{ E\left[ \left| r(i)\right| ^2\right] \right\} ^2 \right] \end{aligned}$$

(42)

Since the expectation operator is independent of the index $i$, we can rewrite (refc2) as,

$$\begin{aligned} E\left[ \hat{k}\right] =\left( 1-\frac{2}{N_{obs}}\right) E\left\{ |r|^4\right\} -2\left( 1-\frac{1}{N_{obs}}\right) \left\{ E\left[ |r|^2\right] \right\} ^2. \end{aligned}$$

(43)

For large $N_{obs}$, (43) can be approximated as $E[\hat{k}] \approx k$, and hence, $\hat{k}$ is an asymptotically unbiased estimate. It has been verified by simulations that for low to medium SNR values, choosing around $N_{obs}=50$ samples is sufficient to produce an almost unbiased estimate, $(E[\hat{k}]=k).$