# Superimposed Training Combined Approach for a Reduced Phase of Spectrum Sensing in Cognitive Radio
## 3 SCAR-Phase of Spectrum Sensing
### First Sample Processing Period  
#### 3.1.1 Training Sequence Synchronization  
The training sequence synchronization process represents a challenge for the SU, who must synchronize with the PU's ST sequence and then decide the presence or absence of the PU in a very short time. The synchronization problem of ST-based transmissions has been previously studied in some investigations [17; 18], which have proposed robust algorithms that can be used in the proposed approach to complete this task. In this paper, a synchronization algorithm is implemented based on a particular case of the method proposed in [18], which exploits the characteristics of the cyclic mean of the received signal and has lower computational complexity than other methods. Hence, for the SU's received signal model in (2), the time offset \(\tau\) is estimated by choosing the integer \(\hat{\tau}=\tau\) modulo-\(P\) (with \(0\leq\hat{\tau}\leq P-1\)) that satisfies the objective function given by  
\[\mathcal{J}(\hat{\tau}):=||(\mathbf{C}_{\hat{\tau}}^{-1}\mathbf{y})_{[P-1]_{ \tau}}||=0, \tag{3}\]  
where the matrix \(\mathbf{C}_{\hat{\tau}}\) is given by \(\mathbf{C}_{\hat{\tau}}=circ(t[-\hat{\tau}],t[-\hat{\tau}-1],\ldots,t[-\hat{ \tau}-P+1])\), with \(circ(\cdot)\) producing a circulant matrix. Moreover, the subscript \([P-1]_{\tau}\) indicates the last \(P-1\) rows of a matrix and \(\mathbf{y}=[y[0],y[1],\ldots,y[P-1]]^{\mathrm{T}}\) (with the superscript \({}^{\mathrm{T}}\) denoting transpose) represents the period-P cyclic mean of Equation (2) given by  
\[y[j]:=\mathrm{E}[x[iP+j]]=\theta e^{j\hat{\rho}t}[j-\tau], \tag{4}\]  
where \(\mathrm{E}[\cdot]\) denotes the expected value and \(j=0,1,\ldots,P-1\). This last sequence is expressed in matrix form as follows:  
\[\mathbf{y}=\mathbf{C}_{\tau}^{[1]c}\theta e^{j\hat{\rho}}, \tag{5}\]  
Figure 2: Block diagram of the proposed ST combined approach for a reduced phase (SCAR-Phase) of spectrum sensing. where the matrix \(\mathbf{C}_{\tau}\) is given by \(\mathbf{C}_{\tau}=circ(t[-\tau],t[-\tau-1],\ldots,t[-\tau-P+1])\). Moreover, the superscript \([1]c\) indicates the first column of the matrix \(\mathbf{C}_{\tau}\). Since \(\tau\) is unknown, \(\mathbf{y}\) must be estimated from the received signal at the SU. The proposed estimate \(\hat{y}[j]\) (with \(j=0,1,\ldots,P-1\)) is:  
\[\hat{y}[j]=\frac{1}{(N/P)}\sum_{i=0}^{(N/P)-1}x[iP+j], \tag{6}\]  
which is used in (3) instead of (5). Given that (3) is only satisfied under ideal conditions (i.e., when \(\theta\) and \(\phi\) are perfectly known in a noiseless channel), \(\hat{\tau}\) is therefore obtained by performing a linear search over \(0\leq\hat{\tau}\leq P-1\) and then looking for the argument that minimizes \(\mathcal{J}(\hat{\tau}):=||(\mathbf{C}_{\hat{\tau}}^{-1}\hat{\mathbf{y}})_{[P -1]_{\tau}}||\). That is,  
\[\underset{0\leq\tau\leq P-1}{\text{arg min}}\ \mathcal{J}(\hat{\tau}):=||( \mathbf{C}_{\hat{\tau}}^{-1}\hat{\mathbf{y}})_{[P-1]_{\tau}}||. \tag{7}\]  
Note that, since \(\mathbf{y}\) is estimated using the received signal \(\mathbf{x}\), the correct estimation of \(\tau\) depends on the number of received samples \(N\). Hence, the greater the number of received samples, the better the estimate of \(\tau\). The performance of the ST-based synchronization method is presented in Figure 3, where the probability of synchronization error is show against different values of SNR. The probability of synchronization error is found using 5000 Monte Carlo trials and a different number of samples. The received signal is modeled as in (2) with the value \(0\leq\tau\leq P-1\) (\(\tau\in\mathbb{N}\)) varying in each Monte Carlo trial. Moreover, the training sequence is obtained as follows [15]:  
\[t[n]=\sigma_{t}e^{j\frac{\pi}{P}[n(n+u)]}\quad u=1,P\text{ odd; }u=2,P\text{ even;}\quad n=0,1,\ldots,P-1, \tag{8}\]  
which has an unity peak-to-average power ratio, with period \(P=10\) and \(\sigma_{t}^{2}\) chosen such that \(\sigma_{t}^{2}+\sigma_{d}^{2}=1\) and the TIR \(\alpha=0.2\).