# Superimposed Training Combined Approach for a Reduced Phase of Spectrum Sensing in Cognitive Radio
## 3 SCAR-Phase of Spectrum Sensing
### Second Sample Processing Period: Enhanced ST-Based Detection  
The second processing period is enabled when the SU does not detect the PU presence. This might be caused for several reasons, including its operation in the low SNR, the hidden terminal problem or missed detections of the energy detector. For the second processing period, it is proposed to take advantage of the PU's training information once the training sequence synchronization process has been completed. For this task, an enhanced ST-based detector (ST-Det) is designed. From the received signal model in (2), note that each sample of the set is distributed according to \(x[n]\sim\mathcal{CN}(\theta e^{j\phi}t[n-\tau],\ \theta^{2}\sigma_{d}^{2}+\sigma^{2})\). Since the received samples are considered to be i.i.d., the joint multivariate probability density function (or simply PDF) of \(\mathbf{x}\) is given by  
\[p(\mathbf{x};\theta,\phi)=\frac{1}{\pi^{N}(\theta^{2}\sigma_{d}^{2}+\sigma^{2} )^{N}}\exp\left(-\sum_{n=0}^{N-1}\frac{|x[n]-\theta e^{j\phi}t[n-\tau]|^{2}}{ \theta^{2}\sigma_{d}^{2}+\sigma^{2}}\right), \tag{13}\]  
which is said to be parameterized by the amplitude \(\theta\) and the phase \(\phi\) of the unknown channel gain. The goal of a detector is to decide either \(\mathcal{H}_{0}\) or \(\mathcal{H}_{1}\) based on an observed set of data \(\mathbf{x}\). This is a mapping of each data set value into a decision [21]. In this sense, the locally optimum detection (LOD) criterion maximizes the slope of the power function (i.e., the probability of detection) amongst the detectors with the same probability of false alarm. Given that LOD produces relatively simple structures that work well in the low SNR region [22], it is used here in the design of the ST-Det that takes into account the PU's ST information to enhance the overall detection of the SCAR-Phase of spectrum sensing. Hence, according to LOD, the SU decides \(\mathcal{H}_{1}\) is true when [23]  
\[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]  
where \(p^{(i)}(\mathbf{x};\theta,\phi)\) is the \(i\)-th derivative of \(p(\mathbf{x};\theta,\phi)\) with respect to \(\theta\), \(p(\mathbf{x};0,\phi)\) is the PDF of \(\mathbf{x}\) evaluated at \(\theta=0\) and \(\lambda_{st}\) is a threshold against which the test statistic \(\Phi(\mathbf{x})\) is compared. It is worth noting that \(p(\mathbf{x};\theta,\phi)\) evaluated at \(\theta=0\) is equal to  
\[p(\mathbf{x};0)=\frac{1}{(\pi^{N}\sigma^{2N})}\exp\left(-\sum_{n=0}^{N-1}\frac {|x[n]|^{2}}{\sigma^{2}}\right), \tag{15}\]  
which does not depend on the value of \(\phi\). Next, the first derivative of \(p(\mathbf{x};\theta,\phi)\) with respect of \(\theta\) is obtained, which after some algebraic manipulations and rearranging some terms is found to be  
\[p^{(1)}(\mathbf{x};\theta,\phi)=p(\mathbf{x};\theta,\phi)\left\{\Lambda(\theta,\phi)+\Omega(\theta)\right\}, \tag{16}\]  
where \(\Lambda(\theta,\phi)\) is equal to  
\[\Lambda(\theta,\phi)= \sum_{n=0}^{N-1}\left\{\frac{2\theta^{3}\sigma_{d}^{2}|t[n-\tau] |^{2}-4\theta^{2}\sigma_{d}^{2}\Re\{e^{i\phi}t[n-\tau]x^{*}[n]\}+2\theta\sigma _{d}^{2}|x[n]|^{2}}{(\theta^{2}\sigma_{d}^{2}+\sigma^{2})^{2}}\right. \tag{17}\] \[\left.-\frac{2\theta|t[n-\tau]|^{2}-2\Re\{e^{i\phi}t[n-\tau]x^{*}[ n]\}}{\theta^{2}\sigma_{d}^{2}+\sigma^{2}}\right\},\]  
and \(\Omega(\theta)\) is given by  
\[\Omega(\theta)=-\frac{2N\theta\sigma_{d}^{2}}{\theta^{2}\sigma_{d}^{2}+\sigma ^{2}}. \tag{18}\]  
Note that \(\Re\{\cdot\}\) and \({}^{*}\) denote the real part and conjugate of a complex variable, respectively. Then, after evaluating (16) at \(\theta=0\) and substituting the result in (14), the first derivative-based test statistic is \(\Phi_{\mathrm{ST}_{1}}(\mathbf{x},\phi,\tau)=(2/\sigma^{2})\sum_{n=0}^{N-1} \Re\{e^{i\phi}t[n-\tau]x^{*}[n]\}\). In order to compute \(\Phi_{\mathrm{ST}_{1}}(\mathbf{x},\phi)\), the parameters \(\tau\) and \(\phi\) must be estimated.