# Analysis of Low Probability of Detection Capability for Chaotic Standard Map-Based FH-OFDMA System
## 4 Simulation Results  
The simulation results are provided to analyze whether the CSM-based FH-OFDMA system provides greater LPD capability than the cat map and SBS-based FH-OFDMA systems. We set common parameters of simulation results as the number of subcarriers, \(N_{\text{FFT}}=220\), excluding guard subcarriers of 36, number of symbols for cyclic prefix, \(T_{c}=T_{d}/4\), number of clusters as the number of maximum transmitting users, \(N_{L}=11\), and \(N_{s}=20\). The transmitted data is modulated as QPSK or 16-QAM, and the data is coded by convolutional coding at \(1/2\) coding rate. The FH-OFDMA signal is sent over a wireless six-ray channel impulse model of the Universal Mobile Telecommunications System--Terrestrial Radio Access System (UTRA) vehicular model A [20]. For the SBS-based FH pattern, we select a Latin squares-based FH pattern of \(\alpha=4\). For the cat map-based FH pattern, we consider \(\alpha=4\), \(b_{11}=b_{12}=b_{21}=1\), and \(b_{22}=2\). For the CSM-based FH pattern, we consider \(\alpha=2\), \(r_{x}=1\), \(r_{y}=3\), and \(K_{C}=3000\). Before we analyze the detection probabilities of FH-OFDMA systems, we verify that the FH-OFDMA systems based on SBS, cat map, and CSM patterns can achieve frequency diversities by bit error rate (BER) performance. Figure 3 shows the BER results of an OFDMA system without hopping and FH-OFDMA systems based on SBS, cat map, and CSM patterns, in which we adopt two modulation rates as QPSK and 16-QAM over the same channel response. As shown in Figure 3, all the three FH-OFDMA systems can obtain better BER results than the OFDMA system operated without hopping. This result verifies that the SBS, cat map, and CSM-based FH-OFDMA systems can achieve frequency diversities over the channel model. In addition, since the BER results by adopting QPSK are better than those by adopting 16-QAM, we observe that BER performance increases as the modulation rate decreases. Through the simulation results of Figure 3, we show that frequency hopping is a significant and effective method to achieve frequency diversity and improve BER performance in the OFDMA system. Figure 4 shows the detection probabilities of SBS, cat map, and CSM-based FH-OFDMA systems for logarithmic scales as the SNR increases. We set common parameters for the three FH-OFDMA systems as false alarm probability, \(P_{fa}=0.05\), redundant number of FH-OFDMA frames, \(N_{R}=100\), number of clusters, \(N_{L}=11\), and number of transmitting users, \(K=11\). As we mentioned previously, since the detection probability of the FH-OFDMA system is the product of the symbol detection probability and the intercept probability, the gap among three FH patterns is sustained as constant as SNR increases. As a result, since the CSM-based FH pattern has the lowest intercept probability, the CSM always has a lower detection probability than other FH-OFDMA systems. In addition, we can observe that all the detection probabilities are increased when SNR is varied from \(-35\) to \(-15\) dB. However, the detection probabilities of the three FH-OFDMA systems converge to a fixed value when the SNR exceeds \(-14\) dB. Figure 5 shows the relationship between the detection probability and redundant number of FH-OFDMA frames for detecting for the SBS, cat map, and CSM-based FH-OFDMA systems. We set common parameters as false alarm probability, \(P_{fa}\) = 0.05, number of clusters, \(N_{L}\) = 11, number of transmitting users, \(K\) = 11, and SNR = 10 dB. In Figure 5, as the redundant number of FH-OFDMA frames for detecting increases, the detection probability also increases. This result shows that, if a seed number changes after \(N_{L}\) periods and the hopping cluster index, thus, changes every FH-OFDMA frame, the LPD capability is enhanced extremely. Figure 5 also shows that the gap among three FH patterns is sustained as constant regardless of the value of \(N_{R}\), since the detection probability of the FH-OFDMA system is the product of the symbol detection probability and the intercept probability.