# Analysis of Low Probability of Detection Capability for Chaotic Standard Map-Based FH-OFDMA System
## 1 Introduction  
Frequency hopping-orthogonal frequency division multiple access (FH-OFDMA) can provide low probability of detection (LPD) and anti-jamming capabilities to authenticated users against adversaries for secure multimedia and military communication services [1]. In the OFDMA system, the entire frequency spectrum is divided into multiple narrow-band subcarriers, which are then assigned to multiple users to transmit data symbols on different subcarriers simultaneously [2]. In the FH-OFDMA system, the total number of subcarriers is partitioned into a set of clusters composed of contiguous subcarriers, and each cluster is changed to a different frequency during each period based on the FH pattern. The users are each assigned to one of the clusters to transmit data symbols over the FH-OFDMA system. Thus, the FH-OFDMA system has the advantages of both LPD due to FH pattern and high spectral efficiency of the OFDMA system [3]. The FH-OFDMA system chooses the orthogonal frequency division multiplexing (OFDM) symbols and cyclic prefix durations according to the Doppler and delay spreads of the transmitting channel, respectively; thus, subcarriers of the FH-OFDMA system retain their orthogonality and experience flat fading upon reception. The FH-OFDMA system can be combined with channel coding resulting in frequency diversity when the channel response varies over the hopping distance, and such frequency diversity has been verified in many prior studies [4; 5; 6]. In the military domain, the authors of Reference [7] consider the condition that an adversary intentionally jams parts of frequency hopping slots and select non-jammed hopping sequences to obtain low bit error performance. However, in this study, we have fundamentally different focus on theadvantage of the FH-OFDMA system that supports LPD against detection of adversaries, such as eavesdroppers or radiometers. Some previous works on FH-OFDMA systems have noted various FH pattern designs to achieve LPD capability and frequency diversity [8; 9; 10; 11; 12; 13]. In Reference [8; 9; 10; 11; 12], the transmitted symbol index is changed at each symbol period through the symbol-by-symbol (SBS) hopping strategy. In order to design an SBS-based FH pattern, orthogonal Latin squares and Welch-Costas-based FH patterns are introduced in Reference [8; 9], respectively. In Reference [10; 11], secure collision free and residue number system arithmetic assisted FH patterns are proposed to obtain high frequency diversity through SBS hopping over fast frequency selective fading. In [12], the authors propose a wide-gap FH pattern, which can be adopted to FH-OFDMA systems, to highly reduce interference between adjacent frequencies. In [13], we propose two frequency hopping strategies based on the chaotic standard map (CSM) and cat map-based FH patterns. As these two FH patterns are hopped into time and frequency domains using additional memory of the cluster size, the FH-OFDMA systems of these two proposed FH patterns have lower probabilities of intercept than the SBS-based FH pattern. Between the CSM and cat map-based FH patterns, the CSM-based FH-OFDMA system has a lower probability of intercept, since a symbol of the CSM cannot be shown in every symbol period, whereas a symbol of the cat map is mixed with the symbol index and is shown in every symbol period similar to the SBS. In Reference [13], we only analyze the complexity of regenerating the transmitted symbol sequence. It is possible that a detector already knows the symbol and frame structures, and their parameters, and the detector is synchronized to the FH-OFDMA system. Unlike our previous work, here, we analyze whether the CSM provides greater LPD capability than the cat map and SBS-based FH-OFDMA systems by detection probability using spectrum sensing technique. The spectrum sensing technique is studied to identify unused spectrum regions that may be exploited efficiently without causing harmful interference to the existing incumbent users.