# Chaotic Pulse-Shaping Filter Based on Root-Raised-Cosine Division
## 1 Introduction  
Chaos, as a deterministic system, exhibits quasi-random and unpredictable behavior from a simple but nonlinear rule. A number of characteristics of low-frequency broadband, sensitivity to initial values, and impulsive autocorrelation give chaos a good application prospect in the field of chaos-based physical-layer security [1; 2; 3; 4; 5]. In practice, chaos communication has been included in some communication standards. Many chaos-based modulation schemes have been proposed as promising solutions for the ultra-wide band radio standards in wireless personal area networks [6]. Frequency-modulated differential chaos shift keying (FM-DCSK) has been adopted as the standard IEEE 802.15.6 [7]. Furthermore, chaos has been applied in high-speed long-distance communication over a commercial fiber-optic channel [8]. However, for the high-rate narrowband wireless communication systems, such as Wi-Fi and cellular networks, there are few practical applications of chaos-based communications in the available literature. The main reasons are as follows. First, the coherent chaotic communication is difficult to be realized due to the difficulty of chaos synchronization between transmitters and receivers in multi-path time-varying channels. Second, the existing chaos-based wireless communications require complicated hardware and software support. Third, non-coherent chaotic communication, such as DCSK, requires a broadband spectrum, which is not applicable to wireless channels with limited-frequency bands. To overcome those problems, the chaotic transmission method in the universal narrowband wireless communication system framework needs to be explored. Chaotic baseband wireless communication system (CBWCS), first proposed in [9], constructs chaotic signal as baseband waveform by using a special basis function as a pulse-shaping filter. A second-order linear system with an analytic solution provides the expression of the filter [10]. Its infinitely long and exponentially increasing precursor provides determinism and a positive Lyapunov exponent. Due to the invariance of the Lyapunov exponent over the wireless channel [11], CBWCS has excellent performance when communicating in the presence of bounded noise. At the receiver, instead of complex chaotic synchronization steps, a simple matched-filter structure corresponding to the pulse-shaping filter is used to maximize the signal-to-noise ratio (SNR) [12]. Thus, the whole scheme is easily implementable in the universal wireless communication platform [9; 13; 14; 15]. Furthermore, the chaotic pulse is applicable to all scenarios where root-raised-cosine (RRC) can be used as a pulse-shaping filter. Ref. [16] used raised-cosine (RC) windowing to mitigate the effect of intercarrier interference in an orthogonal frequency division multiplexing system. Ref. [17] compared the performance of RC and RRC in filter bank multicarrier/offset quadrature amplitude modulation. In [18], RRC is used in a band-limited spatial modulation massive multi-input multi-output system. Ref. [19] presented an analysis of the influence of RC and RRC and their roll-off factors over the additive white Gaussian noise (AWGN) and the Rician fading channel on communication quality. A detailed analysis of generalized frequency division multiplexing performance under Hoyt-q, Weibull-v and log-normal shadowing fading channels with RC as a pulse-shape filter is represented in [20]. In all communication scenarios mentioned above with RC or RRC as a pulse-shaping filter, chaotic pulses can be applied. However, the aforementioned pulse-shaping filter does not satisfy the Nyquist intersymbol interference (ISI) criterion. The presence of intrinsic ISI degrades system performance. Refs. [21; 22] designed a threshold sequence in the symbol detection process to mitigate ISI from the filter itself and multi-path propagation. However, the threshold structure is complex, which requires high computational complexity in threshold calculation.