# Analysis of Low Probability of Detection Capability for Chaotic Standard Map-Based FH-OFDMA System
## 3 Analysis of Detection Probability  
In this section, our goal is to analyze detection probability of a CSM-based FH-OFDMA system compared to detection probabilities of the other FH-OFDMA systems from the point of view of an adversary detector. To increase detection probability, the detector should observe over several FH-OFDMA symbols, i.e., \(y(0),y(1),\ldots,y(M+T_{d}-1)\). The number of observing FH-OFDMA symbols, \(M\), must be composed of multiplications of the redundant number of FH-OFDMA frames, \(N_{R}\), and the number of symbols in a frame, i.e., \(M=N_{R}\times N_{L}\). Our analysis assumes that the adversary detector already knows the effective symbol duration \(T_{d}\) and its cyclic prefix \(T_{c}\). The knowledge of symbol duration is specified in standards or can be estimated by knowledge of the implementable IFFT and FFT sizes. Even if the exact value is not known, the symbol duration can be estimated by the receipt of sufficient symbols. We analyze the detection probability using the spectrum sensing technique based on hypothesis testing. First, let \(h_{0}\) be the null hypothesis, i.e., a transmitting user is absent in a FH-OFDMA system, and \(h_{1}\) be the alternate hypothesis, i.e., a transmitting user exists in the FH-OFDMA system. Thus, the hypothesis testing problem may be written as  
\[ \begin{cases}h_{0}:y(t)=w(t)\\ h_{1}:y(t)=s(t)+w(t)\end{cases} , \tag{7}\]  
where \(y(t)\) is the received complex FH-OFDMA signal, and \(w(t)\) is the complex circular additive white Gaussian noise. Invoking the central limit theorem under the assumption of sufficiently large IFFT size, we have  
\[ \begin{split}& s(t)\sim\mathcal{N}_{c}(0,\sigma_{s}^{2})\\ \text{and}& w(t)\sim\mathcal{N}_{c}(0,\sigma_{w}^{2}) \end{split} , \tag{8}\]  
where \(\mathcal{N}_{c}(. )\) denotes the Gaussian distribution for a complex random variable. Therefore, for different hypotheses, we have  
\[ \begin{cases}h_{0}:y(t)\sim\mathcal{N}_{c}(0,\sigma_{w}^{2})\\ h_{1}:y(t)\sim\mathcal{N}_{c}(0,\sigma_{w}^{2}+\sigma_{w}^{2}).\end{cases} . \tag{9}\]  
We consider \(y(t)=y_{r}(t)+iy_{t}(t)\) as a circularly symmetric Gaussian random variable composed of real and imaginary parts of \(y\), and these real and imaginary parts are obtained as:  
\[ \begin{split}& y_{r}(t)\sim\mathcal{N}_{r}(0,\sigma_{w}^{2}/2)\\ \text{and}& y_{i}(t)\sim\mathcal{N}_{r}(0,\sigma_{y}^{2}/2) \end{split} , \tag{10}\]  
where \(\mathcal{N}_{r}(\cdot)\) denotes the Gaussian distribution for a real random variable. For the FH-OFDMA signal using cyclic prefix, the values of the auto-correlation coefficient \(\rho(\tau)=E[y(t)y^{*}(t+\tau)]/E[y(t)y^{*}(t)]\) for delay \(\tau=\pm T_{d}\) under the two hypotheses are  
\[ \begin{cases}h_{0}:\rho(\pm T_{d})=0\\ h_{1}:\rho(\pm T_{d})=\rho_{D}\end{cases} , \tag{11}\]  
where  
\[ \begin{split}\rho_{D}&=\frac{T_{c}}{T_{d}+T_{c}}\frac{ \sigma_{s}^{2}}{\sigma_{s}^{2}+\sigma_{w}^{2}}\\ &=\frac{T_{c}}{T_{d}+T_{c}}\frac{\Gamma}{1+\Gamma}\end{split} , \tag{12}\]  
where \(\Gamma\) denotes the signal-to-noise ratio (SNR) as \(\sigma_{s}^{2}/\sigma_{w}^{2}\) and \(0<\rho_{D}<1\). For a real valued coefficient, \(\rho(\tau)=\rho(-\tau)\). Hence, without loss of generality, we consider only \(\tau=T_{d}\). We can form two real random vectors from the observations such that \(\mathbf{z}_{1}=[y_{r}(0)y_{i}(0)\)\(y_{r}(1)y_{i}(1)\)\(\ldots\)\(y_{r}(M-1)y_{i}(M-1)]\) and \(\mathbf{z}_{2}=[y_{r}(T_{d})y_{i}(T_{d})y_{r}(T_{d}+1)y_{i}(T_{d}+1)\ldots y_{r} (M+1)y_{i}(M-1)]\)\(T_{d}-1\)) \(y_{i}(M+T_{d}-1)\)], where \(y_{r}(t)\) and \(y_{i}(t)\) are the real and imaginary parts of \(y(t)\). Due to the circular symmetry assumption, the zero mean random variables \(y_{r}(t)\) and \(y_{i}(t)\) are independent and identically distributed (i.i.d.). We define \(\hat{\rho}_{ML}\) as the maximum likelihood estimate from the vectors \(\mathbf{z}_{1}\) and \(\mathbf{z}_{2}\).