## 4 Recurrence-Analysis-Based Detector
### Detection Model  
Usually, the RQA is performed on the entire distance matrix \(D=(d_{ij})_{1\leq i,j\leq K}\) of different state vectors. As \(D\) is a symmetrical matrix, in order to reduce the computational cost, we use the upper triangular part of \(D\), without the main diagonal defined by its general coefficients:  
\[D_{ij}=d_{ij}\ ;\quad\forall\ 1\leq i<j\leq K \tag{16}\]  
We denote by \(\mathbf{u}_{d}\) the coefficients of the first upper diagonal of \(D\) and by \(\mathbf{u}_{\Delta}\) the other coefficients of the upper triangular part of \(D\), without the main diagonal. \(\mathbf{u}_{d}\) and \(\mathbf{u}_{\Delta}\) verify two great properties:  
1. For a WGN, \(\mathbf{u}_{d}\) and \(\mathbf{u}_{\Delta}\) have the same Probability Density Function (PDF), which is not the case for a noisy communication signal. Hence, to detect the presence of a communication signal, we check if \(\mathbf{u}_{d}\) is representative of \(\mathbf{u}_{\Delta}\). For this purpose, we use a statistical test of conformity to evaluate this representativeness [48].
2. The \(\mathbf{u}_{d}\) of a WGN or a communication signal has the same PDF. This remark allows us to design a detector free of noise variance estimation.  
To provide a better understanding of this notion of representativeness, we present the histogram of the \(\mathbf{u}_{d}\) and \(\mathbf{u}_{\Delta}\) of WGN and the communication signal, respectively, in Figures 6 and 7, and we define the confidence interval \(\Phi_{\Delta}\) as follows:  
\[\Phi_{\Delta}=[\overline{\mathbf{u}}_{\Delta}-\lambda;\overline{\mathbf{u}}_ {\Delta}+\lambda] \tag{17}\]  
where \(\overline{\mathbf{u}}_{\Delta}\) is the average value of \(\mathbf{u}_{\Delta}\) and \(\lambda\) is the predetermined detection threshold for RAD. Further details about \(\lambda\) can be found in Section 4.2.4.  
On the basis of the histogram of distance and by using Kolmogorov-Smirnov test [49], for a WGN only, we note that \(\mathbf{u}_{d}\) and \(\mathbf{u}_{\Delta}\) can be approximated by the same PDF (see Figure 6). Consequently, by using the statistical test of conformity based on the estimation average value, we show that \(\mathbf{u}_{d}\) is representative of \(\mathbf{u}_{\Delta}\). Indeed, \(\overline{\mathbf{u}}_{d}\in\Phi_{\Delta}\); \(\overline{\mathbf{u}}_{d}\) denotes the average value of \(\mathbf{u}_{d}\). Contrariwise, Figure 7 gives the histogram of distance for a communication signal buried in the WGN with an SNR\(=0\) dB; we can notice that \(\overline{\mathbf{u}}_{d}\
otin\Phi_{\Delta}\), so \(\mathbf{u}_{d}\) is not representative of \(\mathbf{u}_{\Delta}\).  
Using the statistical test of conformity, we propose the following test statistic \(T\):  
\[T=|\overline{\mathbf{u}}_{\Delta}-\overline{\mathbf{u}}_{d}|\begin{array}{ ccc}\mathcal{H}_{1}\\ \underset{H_{0}}{\gtrless}&\lambda\end{array} \tag{18}\]  
Figure 6: Histogram of \(\mathbf{u}_{d}\) and \(\mathbf{u}_{\Delta}\) for a White Gaussian Noise (WGN): We have \(\overline{u}_{d}=90.6\) and \(\overline{u}_{\Delta}=90\). For \(P_{fa}=0.01\), the detection threshold of the RAD becomes \(\lambda=12.5\). Based on the \(\overline{u}_{\Delta}\) value, the confidence interval becomes \(\Phi_{\Delta}=[77.5;102.5]\). We notice that \(\overline{u}_{d}\in\Phi_{\Delta}\). Consequently, using the conformity test, we can conclude that \(\mathbf{u}_{d}\) is representative of \(\mathbf{u}_{\Delta}\).  
In the course of our work, we establish the analytical expressions of the probability of detection \(P_{d}\) and the probability of false alarm \(P_{fa}\) in order to compare the theoretical and experimental results.