# Analysis of Low Probability of Detection Capability for Chaotic Standard Map-Based FH-OFDMA System
## 2 FH Patterns for FH-OFDMA System  
The total bandwidth is divided into a number of subcarriers \(N_{\text{FFT}}\) according to the length of the fast Fourier transform (FFT) signal of the FH-OFDMA system, which is composed of contiguous OFDMA symbols consisting of a sum of subcarriers modulated by multiple phase shift keying (PSK) or quadrature amplitude modulation (QAM). Without loss of generality, we assume the sampling rate between the number of effective symbols \(T_{d}\), and the number of subcarriers \(N_{\text{FFT}}\), after the inverse FFT (IFFT) as 1. The size of each subcarrier is constructed using several clusters with the number of clusters \(N_{L}\), which are composed of contiguous subcarriers whose number is denoted as \(N_{s}\). Thus, we achieve \(N_{L}=N_{\text{FFT}}/N_{s}\). The FH-OFDMA signal converts from serial to parallel by an IFFT block. Let \(D_{n}(0),D_{n}(1),\ldots,D_{n}(N_{\text{FFT}}-1)\) be complex multiple PSK or QAM modulated data that are transmitted on \(N_{\text{FFT}}\) subcarriers at the \(n\)-th cluster index; then, the output of the IFFT as the symbol of the transmitted FH-OFDMA frame is  
\[s_{n}(t)=\frac{1}{\sqrt{N_{\text{FFT}}}}\sum_{f=0}^{N_{\text{FFT}}-1}D_{n}(f)e ^{\frac{t^{2\pi tf}}{N_{\text{FFT}}}}, \tag{1}\]  
where \(t\) is a discrete time index, \(t\!=\!0,\ldots,T_{d}\!-\!1\), \(n\) is cluster index from 0 to \(N_{L}\!-\!1\), and \(f\) is a discrete subcarrier index. Let \(T_{c}\) be the number of symbols for the cyclic prefix; we define cyclic prefix symbols of the \(n\)-th cluster as \(s_{n}(T_{d}\!-\!T_{c}),\ldots,s_{n}(T_{d}\!-\!1)\), and the cyclic prefix symbols are added to the FH-OFDMA symbols. In the time domain, the FH-OFDMA time block with duration \(T_{s}=T_{d}+T_{c}\) is constructed as \([s_{n}(T_{d}\!-\!T_{c}),\ldots,s_{n}(T_{d}\!-\!1),s_{n}(0),s_{n}(1),\ldots,s_{ n}(T_{d}-1)]\). In the FH-OFDMA system, the modulated data with \(N_{\text{FFT}}\) size before an IFFT block are constructed by a specific FH pattern given by  
\[\sum_{f=0}^{N_{\text{FFT}}-1}\!\!\!D_{n}(f)\!=\!\left\{\sum_{k}\sum_{i=0}^{N_{ L}-1}\sum_{t=i\times N_{n}}^{b\times N_{i}+(N_{s}-1)}\!\!\!d_{n}^{k}(t)\,|\,k\!\in\! \{H_{a,x}^{k}(j)\!=\!i,\!H_{a,y}^{k}(j)\!=\!n\}\right\}, \tag{2}\]  
where \(d_{n}^{k}(t)\) denotes the transmitted data of \(k\)-th user for \(n\)-th cluster at \(t\)-th symbol time index. \(H_{a,x}^{k}(j)\) and \(H_{a,y}^{k}(j)\) are the \(x\)-axis and \(y\)-axis of the FH pattern for the \(j\)-th generated data of \(k\)-th user, respectively. Each user generates data of size \(N_{L}\), and the data is mixed and modulated by the FH pattern, such as CSM, cat map, and SBS-based FH pattern. \(x\)-axis and \(y\)-axis indicate the range of the symbol and cluster indexes, respectively. Since we consider that a user can be assigned only one cluster index in the FH-OFDMA system, the maximum number of users is limited to \(N_{L}\). With a seed number of \(\alpha\), we assign the value of \(\{H_{a,x}^{k}(j),H_{a,y}^{k}(j)\}\) to \(\{L_{a,x}^{k}(j),L_{a,y}^{k}(j)\}\), \(\{T_{a,x}^{k}(j),T_{a,y}^{k}(j)\}\), and \(\{C_{a,x}^{k}(j),C_{a,y}^{k}(j)\}\) for the SBS, cat map, and CSM-based hopping strategies, respectively. In Figure 1, we set \(N_{s}=4\) and show the symbol of a transmitted FH-OFDMA frame as an example. The same-colored blocks are transmitted contiguously by the same user. As shown in Figure 1, the number of symbols for cyclic prefix, \(T_{c}\), is copied from the number of effective symbols, \(T_{d}\), and is added in front of the effective symbols to absorb the multipath delay of the FH-OFDMA system. Before we introduce three FH patterns to verify the LPD capabilities of the FH-OFDMA system, we address that the CSM and cat map-based FH patterns have practical limitations that those FH patterns must have symbol buffers with size \(N_{L}\) unlike the SBS-based FH pattern. Because the transmitting data sequences of the two FH patterns are mixed to increase the complexity of detection probability against eavesdroppers.