Spectrum Sensing for Full-Duplex Cognitive Radio Systems

Abstract

Full-Duplex (FD) transceiver has been proposed to be used in Cognitive Radio (CR) in order to enhance the Secondary User (SU) Data-Rate. In FD CR systems, in order to diagnose the Primary User activity, SU can perform the Spectrum Sensing while operating. Making an accurate decision about the PU state is related to the minimization of the Residual Self Interference (RSI). RSI represents the error of the Self Interference Cancellation (SIC) and the receiver impairments mitigation such as the Non-Linear Distortion (NLD) of the receiver Low-Noise Amplifier (LNA). In this manuscript, we deal with the RSI problem by deriving, at the first stage, the relation between the ROC curves under FD and Half-Duplex (HD) (when SU stops the transmission while sensing the channel). Such relation shows the RSI suppression to be achieved in FD in order to establish an efficient Spectrum Sensing relatively to HD. In the second stage, we deal with the receiver impairments by proposing a new technique to mitigate the NLD of LNA. Our results show the efficiency of this method that can help the Spectrum Sensing to achieve a closed performance under FD to that under HD.

A Appendix

1.1 A.1 i.i.d. property in Frequency Domain

Let r(n) be an i.i.d. time-domain signal. The DFT, R(m), of the r(n) is defined as follows:

$$\begin{aligned} R(m)=\sum _{n=1}^{L} r(n)e^{-j2\pi m\frac{n}{L}} \end{aligned}$$

(18)

Where L is the number of samples of r(n). According to the Central Limit Theorem (CLT), R(m) follows asymptotically a Gaussian distribution for a large L. Based on [18], two normal variables are independent iff they are uncorrelated.

Let $C(m_1,m_2)$ the correlation of $R(m_1)$ and $R(m_2)$ $\forall$ $m_1 \ne m_2$.

$$\begin{aligned} C(m_1,m_2)&=E[R(m_1)R^*(m_2)\nonumber \\&=E\Bigg [\sum _{n_1,n_2=1}^{L} r(n_1)r^*(n_2)e^{-j2\pi \frac{n_1m_1-m_2n_2}{L}}\Bigg ]\nonumber \\&=\sum _{n_1=n_2=1}^L E\Bigg [|r(n_1)|^2\Bigg ]e^{-j2\pi \frac{(m_1-m_2)n_1}{L}}\nonumber \\&\ \ +\sum _{n_1 \ne n_2=1}^L \underbrace{E\Bigg [r(n_1)r^*(n_2)\Bigg ]}_{=0,\ since\ r(n)\ is\ i.i.d.}e^{-j2\pi \frac{(n_1m_1-n_2m_2)}{L}}\nonumber \\&=E\left[ |r(n_1)|^2\right] \underbrace{\sum _{n_1=1}^L e^{-j2\pi (m_1-m_2)\frac{n_1}{L}}}_{=0}=0 \end{aligned}$$

(19)

$C(m_1,m_2)=0$ $\forall$ $m_1 \ne m_2$, therefore $R(m_1)$ and $R(m_2)$ are uncorrelated and they become independent because of their Gaussianity distribution.

1.2 A.2 Probility of Detection and Probability of False Alaram

As by our assumption $\xi (m)$, W(m) and X(m), are asymptotically Gaussian i.i.d., then $\hat{Y}(m)$ is also Gaussian and i.i.d.. Therefore the TS, T, of Eq. (3) follows a normal distribution according to CLT for a large N. Under $H_0$ (i.e. X(m) does not exist), the mean, $\mu _0$, and the variance, $V_0$ of T can be obtained as follows:

$$\begin{aligned} \mu _0=E[T]=E\Bigg [\frac{1}{N}\sum _{m=1}^{N}|\xi (m)+W(m)|^2\Bigg ]=\sigma _w^2+\sigma _d^2 \end{aligned}$$

(20)

$$\begin{aligned} V_0&=E[T^2]-E^2[T]=\frac{1}{N^2}E\Bigg [\left( \sum _{m=1}^N|\hat{Y}(m)|^2\right) ^2\Bigg ]-(\sigma _w^2+\sigma _d^2)^2\nonumber \\&=\frac{1}{N^2}E\Bigg [\sum _{m_1=m_2=1}^N|\hat{Y}(m_1)|^4\Bigg ]\nonumber \\&\ \ +\frac{1}{N^2}E\Bigg [\sum _{m_1 \ne m_2=1}^N|\hat{Y}(m_1)|^2\hat{Y}(m_2)|^2\Bigg ] -(\sigma _w^2+\sigma _d^2)^2\nonumber \\&=\frac{1}{N^2}\sum _{m_1=m_2=1}^NE\Bigg [|\hat{Y}(m_1)|^4\Bigg ]-\frac{1}{N}(\sigma _w^2+\sigma _d^2)^2 \end{aligned}$$

(21)

Since $\hat{Y}(m)$ is Gaussian, then its kurtosis $kurt(\hat{Y}(m))$ is zero.

$$\begin{aligned} kurt(\hat{Y}(m))=E[|\hat{Y}(m)|^4]-E[\hat{Y}^2(m)]-2E^2[|\hat{Y}(m)|^2]=0 \end{aligned}$$

(22)

Assuming that the real and the imaginary parts of $\hat{Y}(m)$ are independent and of the same variance then $E[\hat{Y}^2(m)]$ becomes zero. Therefore: $E[|\hat{Y}(m)|^4]=2E^2[|\hat{Y}(m)|]^2=2(\sigma _w^2+\sigma _d^2)^2$. Back to Eq. (21), the variance, $V_0$ becomes:

$$\begin{aligned} V_0=\frac{1}{N}(\sigma _w^2+\sigma _d^2)^2 \end{aligned}$$

(23)

By following the same procedure, $\mu _1$ and $V_1$ can be obtained as follows under $H_1$ (X(m) exists):

$$\begin{aligned} \mu _1=\sigma _w^2+\sigma _d^2+\sigma _x^2 \end{aligned}$$

(24)

$$\begin{aligned} V_1=\frac{1}{N}(\sigma _w^2+\sigma _d^2+\sigma _x^2)^2 \end{aligned}$$

(25)