## 1 Introduction
Amplitude Phase Shift Keying (APSK) was applied instead of Quadrature Amplitude modulation (QAM) to the DCSK scheme in [16] in order to enhance the Peak-to-Average Power Ratio (PAPR) and the robustness against channel estimation errors. The authors in [17] used M-ary phase shift keying (MPSK) in combination with parallelly concatenated index modulation. They integrated it with carrier interferometry (CI) codes into the OFDM-DCSK to reach a satisfactory quality of service (QoS) and low PAPR for all users. Precoding was used in [18] to encode the data of Multi-Carrier M-ary Chaotic Vector Cyclic Shift Keying (MC-M-CVCSK) to combat the noise and interferences caused by the channel. Meanwhile, one of the emerging and most efficient signal processing techniques for improving communication systems' throughput is Faster than Nyquist (FTN), which defies the limit set by Nyquist for inter-symbol spacing, thus resulting in data rate improvements. The approach has been extensively studied for possible time and frequency domains superpositions [19]. In [20], the FTN approach is proven to provide an increased data rate of 23%, when the symbol spacing is at a specific limit, named the Mazo limit, where the reduction in the spacing between symbols does not affect the BER performance. The reduction can be applied in time and frequency, providing two degrees of freedom for enhancing the performance of communication systems. FTN signaling has been studied extensively, especially for optimal detection of signal-spacings lower than the Mazo-limit [21], while traditional detection algorithms, like the Maximum Likelihood Sequence Estimation (MLSE) and Viterbi algorithm, provide decent performance for a limited decrease in sampling rates, their detection optimality is limited to a short increase of symbol rate. In addition, they induce an additional computational complexity for high-order constellations. New algorithms have been proposed to tackle the computation time issue. For instance, in [22], to achieve a sub-optimal detection while maintaining a polynomial solving time, the sequence estimation is based on the semidefinite relaxation (SDR) technique to achieve a sub-optimal detection. In [23], output-retainable convolutional codes (ORCCs) were used for channel memorization to reduce the decoding complexity. A new approach based on a deep learning-assisted sum-product detection algorithm (DL-SPA) was presented in [24] for faster detection convergence. Moreover, lately, the key research goals on FTN-based modulations focus on reaching higher orders of magnitude in the frequency domain, designing optimal filters in the time domain, Inter-Symbol-interference (ISI) and Inter-Carrier-Interference (ICI) reduction, and the elaboration of a general FTN theory. A comparative analysis of Frequency-domain equalization (FDE) and time-domain equalization (TDE) was carried out in [25] where it was shown that TDE outperforms FDE, especially for ISI elimination; however, this comes at the cost of increased complexity. In [26], FDE was improved using Expectation Propagation (EP) to achieve a performance similar to that of the Bahl Cocke Jelinek and Raviv (BCJR) receiver [27] with less complexity. However, all related FTN works introduce additional algorithmic or hardware-related computation requirements. To date, no simple yet efficient modulation scheme has been introduced in the literature. More complexity is expected to be introduced going forward with the research on a generalized FTN theory and its optimal detection. The main goal of this paper is to introduce a design that combines the architectural simplicity of DCSK and the benefits of a transform that enables a spectral gain namely, FTN. Accordingly, the spectrum would be used less, depending on the level of the filtering rate, ranging from a maximum value at Nyquist rate (\(\rho=1\)) to zero (\(\rho=0\)), for which no spacing would be introduced between symbols. This would enable spectral gains ranging from zero to 100% depending on the BER requirements.