# Adaptive Damping Log-Domain Message-Passing Algorithm for FTN-OTFS in V2X Communications
## 1 Introduction
By reducing the subcarrier spacing in OFDM, SEFDM enables the transmission of more information symbols within the same spectral resources. Fundamentally, SEFDM can be considered a multi-carrier implementation of FTN signaling. To address the limitations of conventional OTFS and ensure reliable communication in high-mobility environments, the integration of FTN signaling with OTFS has emerged as a promising approach. This combination offers the potential to enhance the information rate and meet the demands of next-generation V2X communication systems, making it a key focus of research. In [20], an eigenvalue decomposition (EVD) precoded OTFS-based FTN architecture was proposed. In this approach, power-allocation coefficients were optimized for each frame to maximize mutual information under a total transmit power constraint. The study confirmed that the OTFS-based FTN scheme achieved higher spectral efficiency compared to the traditional OTFS scheme. In [21], a novel waveform design combining OTFS and FTN signaling was presented, employing a minimum mean square error (MMSE) equalizer to mitigate ISI introduced by FTN signaling. Furthermore, ref. [22] proposed a new FTN-OTFS system in which non-orthogonal pulses were jointly applied in the time-frequency (TF) domain. However, these studies primarily focused on system design and failed to address the development or application of effective nonlinear detection algorithms, which remained an open area for further investigation. Against this background, this paper proposes a novel FTN-OTFS modulation scheme, derives the corresponding input-output models, and designs a new detection algorithm. To better reflect practical V2X scenarios, we consider both the extended vehicular A (EVA) and high-speed train (HST) channel models, further demonstrating the application potential of the proposed solution in real-world V2X environments. Simulation results show that the proposed detection algorithm achieves superior bit error rate (BER) performance and enhanced transmission reliability in highly dynamic V2X environments, highlighting its significant potential as a physical-layer solution for future high-mobility communications. Specifically, the main contributions of our works can be summarized as follows:  
* We propose a novel FTN-OTFS modulation scheme that combines SEFDM to compress subcarrier spacing in the frequency domain and root-raised-cosine pulse-shaped FTN signaling to break the Nyquist criterion in the time domain, thereby accelerating symbol transmission. By utilizing non-orthogonal techniques, the scheme significantly boosts system throughput under conditions with severe Doppler effects and fast time-varying channels in V2X scenarios. * We derive analytically tractable input-output models for the proposed FTN-OTFS system in both the DD and TF domains, providing a solid theoretical foundation for the design of efficient receiver algorithms. * We propose a novel adaptive damping log-domain message-passing (ADL-MP) detection algorithm, which incorporates a convergence-rate adaptive damping mechanism to improve both convergence speed and numerical stability. The complexity and convergence properties of the proposed algorithm are also analyzed. The rest of this paper is organized as follows. Section 2 gives the system model, including the transmitter, channel model, and receiver. Section 3 details the proposed ADL-MP detection algorithm. Section 4 provides extensive simulation results. Section 5 comprises the conclusion.