# 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
#### 3.2.1 ST-Det Probability of a False Alarm  
The probability of a false alarm of the ST-Det, denoted by \(P_{\text{fa}}^{st}\) is defined as the probability of deciding \(\mathcal{H}_{1}\) when \(\mathcal{H}_{0}\) is true, i.e.,  
\[P_{\text{fa}}^{st}\triangleq\Pr\{\Phi_{\text{ST}}(\mathbf{x})>\lambda_{st}^{ \prime}|\mathcal{H}_{0}\}=\int_{\lambda^{\prime}}^{\infty}p_{\Phi_{\text{ST}}} (x|\mathcal{H}_{0})dx, \tag{26}\]  
where \(p_{\Phi_{\text{ST}}}(x|\mathcal{H}_{0})\) is the PDF of \(\Phi_{\text{ST}}(\mathbf{x})\) when \(\mathcal{H}_{0}\) is true. In order to obtain an analytical expression for the \(P_{\text{fa}}^{st}\), a statistical analysis must be carried out on \(\Phi_{\text{ST}}(\mathbf{x})\) when \(\mathcal{H}_{0}\) is true. Note that, for a large \(N\), the central limit theorem (CLT) can be invoked. Hence, \(\Phi_{\text{ed}}\) is Gaussian distributed under \(\mathcal{H}_{0}\). Therefore, after analyzing its statistical properties it can be determined that  
\[\Phi_{\text{ed}}\sim\mathcal{N}\left(N\sigma^{2},N\sigma^{4}\right). \tag{27}\]  
Similarly, \(\Phi_{\text{t}}\) is a complex Gaussian random variable (r.v.) Then, after computing its mean and variance it can be shown that  
\[\Phi_{\text{t}}\sim\mathcal{CN}\left(0,N\sigma^{2}\sigma_{t}^{2}\right). \tag{28}\]  
Thereby, \(|\Phi_{\text{t}}|^{2}\) can be expressed in terms of a chi-squared distribution with two degrees of freedom (i.e., exponential distribution). Therefore, under \(\mathcal{H}_{0}\), the ST-Det test statistic can also be written as:  
\[\Phi_{\text{ST}}(\mathbf{x})=\psi\mathcal{G}+N\sigma^{2}+v\mathcal{E}, \tag{29}\]  
where \(\mathcal{G}\sim\mathcal{N}(0,1)\), \(\psi=\sigma^{2}\sqrt{N}\), \(\mathcal{E}\) is an exponential r.v with parameter \(\rho=1/2\) (i.e. \(\mathcal{E}\sim\text{Exp}(1/2)\)) and \(v=N\sigma^{2}\alpha\). In strict sense, \(\mathcal{G}\) and \(\mathcal{E}\) are not independent, which complicates obtaining the exact distribution of \(\Phi_{\text{ST}}(\mathbf{x})\). However, since it can be shown that correlation amongst these random variables is low, independence can be assumed without affecting the results significantly. Considering this, the expression for \(P_{\text{FA}}\) is found as:  
\[P_{\text{fa}}^{st}=\int_{0}^{\infty}\int_{\lambda_{st}^{\prime}}^{\infty} \frac{1}{\psi\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{\mathcal{G}-N\sigma^{2}-v \mathcal{E}}{\varphi}\right)^{2}}\frac{1}{2}e^{-\mathcal{E}/2}d\mathcal{G}d \mathcal{E}. \tag{30}\]  
Solving the integral, the expression for the probability of false alarm is:  
\[P_{\text{fa}}^{st}=f(z):=\exp\left(\frac{\psi^{2}-4 vz}{8v^{2}}\right)\ \mathcal{Q}\left(\frac{\psi^{2}-2 vz}{2v\psi}\right)+\mathcal{Q}\left(\frac{z} {\psi}\right), \tag{31}\]  
where \(z=\lambda_{st}^{\prime}-N\sigma^{2}\). Considering that \(f(z)\) is monotonically decreasing in \(z\), the threshold can be obtained as  
\[\lambda_{st}^{\prime}=f^{-1}\left(P_{\text{fa}}^{st}\right)+N\sigma^{2}, \tag{32}\]  
where \(f^{-1}(\cdot)\) is the inverse function of \(f(\cdot)\).