# Analysis of Low Probability of Detection Capability for Chaotic Standard Map-Based FH-OFDMA System
## 3 Analysis of Detection Probability
When using the log likelihood ratio test (LLRT), we decide the alternative hypothesis for the present hypothesis test above the threshold, \(\eta_{l}\) of the adversary detector. For sufficiently large \(M\), we approximate the distribution of the test statistic described in Reference [18] as  
\[ \begin{cases}h_{0}:\hat{\rho}_{ML}\sim\mathcal{N}_{r}\left(0,\frac{1}{2M} \right)\\ h_{1}:\hat{\rho}_{ML}\sim\mathcal{N}_{r}\left(\rho_{D},\frac{\left(1-\rho_{D} ^{2}\right)^{2}}{2M}\right).\end{cases}  \tag{13}\]  
This is a classical detection problem with test statistic \(\hat{\rho}_{ML}\). To detect and intercept the desired signal, detectors can adopt one of the well-known detection strategies. We consider that the detectors adopt the detection strategy of Neyman-Pearson detection to satisfy a constant false alarm rate constraint [19]. For a Gaussian random variable \(r\sim\mathcal{N}_{r}(\mu_{r},\sigma_{r}^{2})\), we have  
\[P(r>\eta_{r})=\frac{1}{2}\sigma\!fc\left(\frac{\eta_{r}-\mu_{r}}{\sqrt{2}\sigma _{r}}\right), \tag{14}\]  
where \(\sigma\!fc(\cdot)\) is the complementary error function. Using (14) for \(2M\) real symbols, the required false alarm probability \(P_{fa}\) is given by  
\[P_{fa}=P(\hat{\rho}_{ML}>\eta_{l}\mid h_{0})=\frac{1}{2}\sigma\!fc\left(\sqrt{ M}\cdot\eta_{l}\right). \tag{15}\]  
Thus, the threshold at the detector can be calculated as  
\[\eta_{l}=\frac{1}{\sqrt{M}}\cdot\sigma\!fc^{-1}(2P_{fa}). \tag{16}\]  
The detection probability of the FH-OFDMA system is composed of the product of the auto-correlation-based symbol detection probability, that we have analyzed, and the intercept probability of FH patterns from the point of view of an adversary detector, as the two events are independent. Thus, the detection probability, \(P_{d}^{FT}\), of the FH-OFDMA system using different FH patterns, such as SBS, cat map, and CSM-based strategies, for \(2M\) real symbols can be obtained using (14), and we get:  
\[P_{d}^{FT}\!\!=P_{I}^{FH}.P(\hat{\rho}_{ML}\!\!>\eta_{l}\mid h_{1})=P_{I}^{ FH}.\frac{1}{2}\sigma\!fc\left(\sqrt{M}\cdot\frac{\eta_{l}-\rho_{D}}{1-\rho_{D} ^{2}}\right), \tag{17}\]  
where \(P_{I}^{FH}\) denotes the intercept probability for the FH-OFDMA system using different FH patterns. Based on Reference [13], \(P_{I}^{FH}\) has a different value according to the SBS, cat map, CSM-based FH patterns, and FH can, thus, be rewritten by SBS, CAT, and CSM, respectively. Without loss of generality, we assume that the hopping sequence does not change during the \(M\) OFDMA symbols, i.e., \(2M\) real symbols. For the SBS FH pattern, the detector is able to intercept the transmitted OFDMA symbol by selecting one cluster from \(N_{L}\) clusters, except the previously selected cluster at each period. For \(K\) transmitting users (\(K\!<\!N_{L}\)), the detector can select one cluster from \(K\) clusters at each period. Hence, the \(P_{I}^{\text{SBS}}\) can be given as  
\[P_{I}^{\text{SBS}}= \begin{cases}\frac{2N_{\text{R}}}{N_{\text{L}}\!\!\times\! \!(N_{L}\!-\!1)^{N_{L}-1}},&K\!=\!N_{L},\\ \frac{2N_{\text{R}}}{K^{N_{\text{L}}}},&K\!<\!N_{L}.\end{cases}  \tag{18}\]  
For the cat map-based FH pattern, transmitted symbols of each user appear at each symbol period once with mixed order in the time domain. The detector can intercept these symbols by selecting one cluster from \(K\) clusters at each period and finding the correct order from possible permutations. Hence, the \(P_{I}^{CAT}\) can be given by  
\[P_{I}^{CAT}= \begin{cases}\dfrac{2N_{R}}{N_{l}\times(N_{L}-1)^{N_{L}-1}\times N_{ L}!},&K\!=\!N_{L},\\ \dfrac{2N_{R}}{K^{N_{L}}\times N_{L}!},&K\!<\!N_{L}.\end{cases}  \tag{19}\]  
For the CSM-based FH pattern, the detector should select \(N_{L}\) clusters from \(N_{L}\times K(K\leq N_{L})\) symbols and find the order of these symbols after detecting, receiving, and storing the symbols during the period \(N_{L}\).