# Design of a New Non-Coherent Cross-QAM-Based \(M\)-ary DCSK Communication System
## 1 Introduction  
Differential chaos shift keying (DCSK) modulation is a non-coherent chaotic modulation [1] which is resistant to multipath fading and has low complexity. Due to these advantages, DCSK systems have been extensively investigated in different communication scenarios, e.g., underwater acoustic communications [2], simultaneous wireless information and power transfer systems [3, 4], and power line communication [5, 6]. In addition, similar to [7, 8, 9], some DCSK-based multiple-input multiple-output (MIMO) systems have been proposed to improve the transmission reliability under multipath fading channels [10, 11, 12].  
The coherent chaos shift keying (CSK) system requires a complex synchronization circuit and channel estimator, which are difficult to implement [1]. However, in a DCSK system, the synchronization circuit and channel estimator are not required. Distinguished from the CSK system, the DCSK system represents each information bit by two chaotic signals, i.e., the chaotic-reference signal and the information-bearing signal [1, 3]. Hence, compared with the CSK system, the DCSK system needs to transmit the reference signal so as to waste half a bit of energy, thus leading to low data rate and low energy efficiency. To boost the data rate and the performance, a quadrature chaos shift keying (QCSK) system has been studied in [13]. In such a system, it can obtain approximate BER performance with the DCSK system and double data rate. Furthermore, a phase shift keying (PSK)-based \(M\)-ary DCSK (PSK-MDCSK) system and its multiresolution version have been designed in [14], while its adaptive scheme has been designed in [15]. For the PSK-MDCSK system, it inherits non-coherent feature of the DCSK system and can provide excellent error performance and high data rate. To further improve performance, a square quadrature amplitude modulation (QAM)-based \(M\)DCSK (SQAM-MDCSK) system and its hierarchical version have been conceived in [16] and [17], respectively. Although the SQAM-MDCSK system can obtain better BER performance compared with the PSK-MDCSK system, the formerhas a higher PAPR and requires channel state information (CSI) to demodulate the received signal. To reduce the PAPR, a star QAM-based _M_DCSK (star QAM-_M_DCSK) system has been designed in [18], which can offer close or better BER performance than the SQAM-_M_DCSK system. Although the star QAM-_M_DCSK system can achieve the rate and channel adaptation, the minimal Euclidean distance of the star QAM constellation is relatively small, thus resulting in worse BER performance. Moreover, the SQAM-_M_DCSK system is inadequate for changing channel conditions and rate requirements, thus introducing odd powers of 2 constellations. Cross QAM (XQAM) with an odd power of 2 constellations is a good choice which provides better performance over rectangular QAM [19]. From the above discussion, the existing framework needs to send the pilot signals at the transmitter and requires a channel estimator at the receiver, such as star QAM-_M_DCSK and SQAM-_M_DCSK systems.  
With the above motivation, to bypass the channel estimation and achieve better performance, a non-coherent XQAM-based _M_DCSK (XQAM-_M_DCSK) system is designed in this paper. The proposed framework utilizes an autocorrelator to obtain the channel compensation value. This framework not only requires not any channel estimator and can well adapt to the varying channel, but also can be applied to the existing star QAM-_M_DCSK and SQAM-_M_DCSK systems. Moreover, the theoretical expression of BER for the proposed XQAM-_M_DCSK system over multipath Rayleigh fading channels is derived. The results indicate that, compared with the star QAM-_M_DCSK system, the proposed XQAM-_M_DCSK system has better BER performance and lower PAPR.