## 4 Recurrence-Analysis-Based Detector
### Analytical Expression of the Probability of False Alarm
#### 4.2.1 The PDF of \(\overline{\textbf{u}}_{\Delta}\) under Hypothesis \(\mathcal{H}_{0}\)  
Under \(\mathcal{H}_{0}\), the average value \(\overline{\textbf{u}}_{\Delta}\) of \(\textbf{u}_{\Delta}\) is defined as follows:  
\[\overline{\textbf{u}}_{\Delta/\mathcal{H}_{0}}=\frac{1}{K_{\Delta}}\sum_{k=1 }^{K_{\Delta}}u_{\Delta/\mathcal{H}_{0}}^{(k)} \tag{19}\]  
where \(K_{\Delta}=(K-1)(K/2-1)\) and \(u_{\Delta/\mathcal{H}_{0}}^{(k)}\in\{d_{ij};1\leq i\leq K-2;i+2\leq j\leq K\}\).  
Based on the central limit theorem for independent random variables and for a large \(K\), \(\overline{\textbf{u}}_{\Delta/\mathcal{H}_{0}}\) is asymptotically normally distributed with mean \(\overline{\textbf{u}}_{\Delta/\mathcal{H}_{0}}\) and variance \(\sigma_{\Delta/\mathcal{H}_{0}}^{2}\):  
\[\overline{\textbf{u}}_{\Delta/\mathcal{H}_{0}}\mapsto\mathcal{N}\Big{(}\mu_{ \Delta/\mathcal{H}_{0}},\sigma_{\Delta/\mathcal{H}_{0}}^{2}\Big{)} \tag{20}\]  
where  
\[\mu_{\Delta/\mathcal{H}_{0}}=\mathbb{E}\big{[}\overline{\textbf{u}}_{\Delta/ \mathcal{H}_{0}}\big{]} \tag{21}\]  
\[\sigma_{\Delta}^{2}=var\big{[}\overline{\textbf{u}}_{\Delta/\mathcal{H}_{0}} \big{]} \tag{22}\]  
To calculate \(\mu_{\Delta/\mathcal{H}_{0}}\) and \(\sigma_{\Delta/\mathcal{H}_{0}}^{2}\), we can use the PDF \(f_{\textbf{u}_{\Delta/\mathcal{H}_{0}}}\) (u) of \(\textbf{u}_{\Delta/\mathcal{H}_{0}}\). An outcome \(u_{\Delta/\mathcal{H}_{0}}\) of \(\textbf{u}_{\Delta/\mathcal{H}_{0}}\) is given by  
\[u_{\Delta/\mathcal{H}_{0}}=d_{ij}=\sqrt{\sum_{k=0}^{m-1}\Big{(}b_{i+k\tau}-b_{ j+k\tau}\Big{)}^{2}} \tag{23}\]  
As \(b(n)\mapsto\mathcal{N}\big{(}0,\sigma_{b}^{2}\big{)}\), then \(b_{i+k\tau}-b_{j+k\tau}\mapsto\mathcal{N}\big{(}0,2\sigma_{b}^{2}\big{)}\) and \(v_{k}=\Big{(}\frac{b_{i+k\tau}-b_{j+k\tau}}{\sigma_{b}\sqrt{2}}\Big{)}\ \mapsto\ \mathcal{N}(0,1)\).  
By setting \(Z=\sqrt{\sum_{k=0}^{m-1}\sigma_{k}^{2}}\), we can conclude that \(Z\) follows a Chi distribution \(\chi(m)\) with \(m\) degrees of freedom [50, 51]:  
\[Z=\frac{\textbf{u}_{\Delta/\mathcal{H}_{0}}}{\sigma_{b}\sqrt{2}}\mapsto\succ (m) \tag{24}\]  
The expectation value \(\mathbb{E}[Z]\) and variance \(var[Z]\) of \(Z\) are given as follows [50, 51]:  
\[\mathbb{E}[Z]=\mu_{Z}=\sqrt{2}.\frac{\Gamma\Big{(}\frac{m+1}{2}\Big{)}}{\Gamma \Big{(}\frac{m}{2}\Big{)}} \tag{25}\]  
\[var[Z]=m-\mu_{Z}^{2} \tag{26}\]  
where \(\Gamma(m)=\int_{0}^{+\infty}x^{m-1}e^{-x}dx\) is the Gamma function.  
From Equation (24), we show that the PDF \(f_{\textbf{u}_{\Delta/\mathcal{H}_{0}}}\) of \(\textbf{u}_{\Delta/\mathcal{H}_{0}}\) is defined as follows: [52]:  
\[f_{\textbf{u}_{\Delta/\mathcal{H}_{0}}}(u)=\frac{u^{m-1}}{2^{m-1}\sigma_{b}^{m} \,\Gamma(\frac{m}{2})}\exp\left[-\frac{u^{2}}{4\sigma_{b}^{2}}\right] \tag{27}\]The mean \(\mu_{\Delta/\mathcal{H}_{0}}\) and variance \(\sigma^{2}_{\Delta/\mathcal{H}_{0}}\) of \(\widehat{X}_{\Delta/\mathcal{H}_{0}}\) can be determined by using Equation (24) as follows:  
\[\mu_{\Delta/\mathcal{H}_{0}}=\sigma_{b}\sqrt{2}\mathbb{E}[Z] \tag{28}\]  
\[\sigma^{2}_{\Delta/\mathcal{H}_{0}}=\frac{2\sigma^{2}_{b}}{K_{\Delta}}\,var[Z] \tag{29}\]  
Finally, we have \(\overline{\mathbf{u}}_{\Delta/\mathcal{H}_{0}}\mapsto\mathcal{N}\left(\mu_ {\Delta/\mathcal{H}_{0}},\sigma^{2}_{\Delta/\mathcal{H}_{0}}\right)\) with  
\[\mu_{\Delta/\mathcal{H}_{0}}=2\sigma_{b}\frac{\Gamma\left(\frac{m+1}{2}\right) }{\Gamma\left(\frac{m}{2}\right)} \tag{30}\]  
\[\sigma^{2}_{\Delta/\mathcal{H}_{0}}=\frac{2\sigma^{2}_{b}}{K_{\Delta}}\left[m -2\left(\frac{\Gamma\left(\frac{m+1}{2}\right)}{\Gamma\left(\frac{m}{2}\right) }\right)^{2}\right] \tag{31}\]