###### Abstract  
A new fractional-order multi-carrier M-ary differential chaos shift keying system with multilevel code-shifted modulation (MC-MDCSK-MCS) is presented in this paper. The proposed system adopts multiple subcarriers, on which multiple MCS-MDCSK-modulated signals are transmitted simultaneously. Moreover, M-ary modulation has been combined with the proposed system to achieve a higher bit rate. On the receiver side, the recovered reference signal is first averaged and then used for MCS-MDCSK demodulation, which helps improve performance. We analyze the bit error rate (BER) of the proposed system and verify our theoretical derivations with the simulation results over additive white Gaussian noise (AWGN) and Rayleigh fading channels. Finally, related comparisons are completed, which show that the MC-MDCSK-MCS system is excellent and promising.  
chaos-based communications; multilevel code-shifted differential chaos shift keying; M-ary modulation; multi-carrier; fractional-order chaos system; bit error rate +
Footnote †: journal: 24 September 2021  
keying system with multilevel code-shifted modulation (MCS-MDCSK) was studied [14] for increased bandwidth efficiency.  
Alternatively, for increasing the data rate and the energy efficiency, the multi-carrier technique offers another important solution. A multi-carrier differential chaos shift keying (MC-DCSK) system was proposed in [15], which is more energy-efficient and has a higher data rate. Multi-carrier chaos shift keying (MC-CSK) was presented in [16]. When data symbols of an M-ary are mapped into normalized orthogonal chaotic signals, the data-bearing signals are generated in the MC-CSK system [16]. In [17], a DCSK with an M-ary modulation system was presented, in which multiple carriers carried out differential modulation and demodulation in the frequency domain. For alleviating channel noise as well as improving BER performance, a subcarrier-allocated MC-DCSK system was presented in [18]. To achieve higher performance, a multi-carrier M-ary DCSK system with code index modulation was proposed in [19], where the index modulation based on Walsh codes was used.  
In recent years, fractional calculus has been a useful mathematical tool. The utilization of fractional calculus in scientific and technological disciplines has attracted significant interest, in areas such as signal processing, image processing, biological systems, and automatic control [20; 21; 22]. Moreover, applications of fractional-order chaotic systems in electrical engineering and secure communication have become more and more attractive [23; 24; 25].  
This paper, inspired by MC-DCSK and MCS-MDCSK modulations, proposes a new MC-MDCSK-MCS system, where different orthogonal Walsh code sequences carry the in-phase and quadrature components of the M-ary constellation symbols. In this system, the information bits are split into several parts, and each part is modulated by the MCS-MDCSK and transmitted over different subcarriers. In the receiver, an average filter is required before correlation detection. The reference signal of each part is averaged by the filter, which helps reduce the interference of noise. The bit error performance is noticeably improved by the average filter. Compared with the conventional MCS-MDCSK, the proposed system can improve the energy efficiency, the data rate, and the confidentiality.  
There are six sections in this paper. The system model of MC-MDCSK-MCS is given in Section 2. The bit error performance of the system is analyzed in Section 3. The simulation results are shown in Section 4. The discussions are presented in Section 5. Finally, some conclusions are shown in Section 6.