## 1 Introduction  
Chaotic signals are random-like, non-periodic, and spread spectrum signals, which make them a good carrier of communication information [1; 2; 3]. Chaotic digital modulation techniques have simple circuits, which are not easily intercepted and are resistant to the adverse effects of the channel, so it has potential applications in secure communications [4; 5]. Differential chaotic shift keying systems have become a hot research topic due to the use of non-coherent modulation techniques [6]. However, in a DCSK system, the bandwidth efficiency, energy efficiency, and data rate are inadequate because the reference signal is transmitted by fifty percent of the bit duration [7]. An efficient differential chaotic shift keying (HE\(-\)DCSK) scheme is proposed, which can carry 2 bits of data in a data modulated sample sequence, improving double efficiency [8]. With a time reversal operation generating a reference signal orthogonal to the data carrier signal, an improved DCSK (I\(-\)DCSK) system demonstrates [9]; the two sequences of data and reference are combined into a single time slot, which doubles the data rate and improves spectral efficiency. To improve the bit error rate performance of the OFDM\(-\)DCSK system, ref. [10] presents a long short-term memory based OFDM\(-\)DCSK (LSTM\(-\)OFDM\(-\)DCSK) system. In order to combat the eavesdropping of malicious attacks due to the broadcast characteristics of wireless communication system, a MIMO communication system assisted by FH\(-\)OFDM\(-\)DCSK is designed [11] to advance the security of the MIMO communication system. A multi-user OFDM-DCSK system reduces the integration complexity of the system and improves the spectral efficiency [12]. To enhance data rate and energy efficiency of the DCSK system, a series of improved schemes are implemented [13; 14; 15]. Several systems improve the security of DCSK systems [16; 17]. Multi\(-\)carrier Differential Chaos Shift Keying (MC\(-\)DCSK) is proposed in [18], which can be considered as a parallel extension of the DCSK system. To improve the bit rate of the MC\(-\)DCSK system, a series of optimization schemes are proposed in [19]. In [20], the peak-to-average power ratio (PAPR) of this system is reduced. In [21], an analog network coding multi-user MC\(-\)DCSK (ANC\(-\)MU\(-\)MC\(-\)DCSK) system is demonstrated. Compared to the traditional DCSK system, the MC\(-\)DCSK system achieves better spectral efficiency and more effective energy consumption while reducing inter-signal interference. In [22], a MC\(-\)DCSK system with sparse code spreading (SCS\(-\)MC\(-\)DCSK) is proposed to improve the efficiency and reliability of multi-user transmission. In [23], a multi-carrier chaotic modulation scheme is proposed, which has good robustness in time-varying underwater acoustic (UWA) channels. The main contributions of this paper are as follows:  
Firstly, we propose a new multi-carrier orthogonal double-speed differential chaotic shift keying (MC\(-\)ODBR\(-\)DCSK) communication system. In the system, the transmission information is divided into two parts: \(A\) and \(B\). Each part corresponds to \(L\) group of information, and each group of information corresponds to \(M\) subcarriers. The symbol \(A\) of one carrier is compounded with the chaotic reference signal, and the symbol \(B\) and the chaotic signal through the orthogonal chaotic signal generator (OCG) are compounded and then superimposed with the symbol \(A\) to form a carrier, carrying the symbol \(A\) and symbol \(B\) information, respectively. To realize the information transmission of the two symbols, each subcarrier transmits 1 bit more information than the MC-DCSK communication system. Secondly, DBR of the MC\(-\)ODBR\(-\)DCSK system is derived, and energy efficiency of system are analyzed. Compared with DCSK and MC\(-\)DCSK system, MC\(-\)ODBR\(-\)DCSK saves transmitted bit energy and increases data rate.