# 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
These can be easily obtained by standard procedures, which after being evaluated at \(\theta=0\) are reduced to:  
\[\Lambda^{(1)}(0)=\sum_{n=0}^{N-1}\left\{\frac{2\sigma_{d}^{2}|x[n]|^{2}}{ \sigma^{4}}-\frac{2|t[n-\tau]|^{2}}{\sigma^{2}}\right\}, \tag{21}\]  
and  
\[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]0  
Therefore, \(p^{(2)}(\mathbf{x};\theta,\phi)\) evaluated at \(\theta=0\) is given by  
\[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]1 \[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]2  
Next, by substituting (23) in (14) and then including the resulting constant terms along with \(\lambda_{st}\) in a new threshold \(\lambda_{st}^{\prime}\), the second derivative-based version of the ST-Det (or simply ST-Det) that depends on the unknown values \(\phi\) and \(\tau\) is:  
\[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]3After substituting the estimate of \(\tau\) modulo-\(P\), which is obtained solving (7), and then, the estimate of \(\phi\) given by (19), the enhanced ST-Det test statistic is finally given by  
\[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]4  
where \(\Phi_{\text{ed}}\) denotes the energy detection metric of the ST-Det and \(|\Phi_{\text{t}}|^{2}\) denotes the ST related metric of the ST-Det. It is important to mention that the optimality of the detector in (25) might have been lost due to the estimates \(\hat{\tau}\) and \(\hat{\phi}\). In order to evaluate the difference between the test statistic based on the exact values of \(\tau\) and \(\phi\) (i.e., \(\Phi_{\text{ST}}(\mathbf{x})\)) and the test statistic based on the estimated values \(\hat{\tau}\) and \(\hat{\phi}\) (i.e., \(\Phi_{\text{ST}}(\mathbf{x})\)), the power of each test defined respectively as \(\Pr\{\Phi_{\text{ST}_{2}}(\mathbf{x},\phi,\tau)>\lambda_{st}^{\prime}|\mathcal{ H}_{1}\}\) and \(\Pr\{\Phi_{\text{ST}}(\mathbf{x})>\lambda_{st}^{\prime}|\mathcal{H}_{1}\}\), can be considered. For this, 5000 Monte Carlo simulations are performed to quantify the mean squared error between the power of the test statistic in (24) and the power of the test statistic in (25) as shown in \[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]5 It can be seen in this figure that the mean squared error is very close to zero when the number of samples \(N\) used for spectrum sensing is greater than 4000. For example, the mean squared error is \(2.53\times 10^{-6}\) when \(N=6000\), which is the number of samples required to attain a low synchronization error as shown in \[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]6 Therefore, since simulations show that the difference between the power of both test statistics is very small, the proposed detector in (25) continues to be referred to as locally optimum. Thus, in the second sample processing period, an enhanced version of ST-training based detection is used, since both the training and data sequences are exploited. Interestingly, in the very low SNR region, the ST related term of the ST-Det plays a major role in the detection of ST signals, since \(\sigma_{d}^{2}\) is very small with respect to \(\sigma^{2}\). Moreover, the ST-Det can also be used to detect non-ST signals, given that it reduces to  
\[\Phi(\mathbf{x})=\frac{p^{(i)}(\mathbf{x};\theta,\phi)\Big{|}_{\theta=0}}{p( \mathbf{x};0,\phi)}>\lambda_{st}, \tag{14}\]7 Mean squared error of the power function of the ST-based detector (ST-Det) test statistic against the number of samples \(N\). the energy detection metric when \(t[n-\tau]=0\ \forall n\).