# Adaptive Damping Log-Domain Message-Passing Algorithm for FTN-OTFS in V2X Communications
## 1 Introduction  
With the rapid advancement of intelligent transportation systems [1] and vehicle-to-everything (V2X) communication [2, 3, 4], wireless communication systems in high-mobility environments face increasingly stringent demands for high reliability, low latency, and high spectral efficiency [5]. In typical V2X scenarios, vehicles often travel at high speeds and navigate complex urban landscapes, resulting in rapidly time-varying wireless channels with significant Doppler shifts [6]. These characteristics present substantial challenges for designing physical-layer transmission schemes. Orthogonal frequency division multiplexing (OFDM) is well known for its excellent performance in time-invariant channels and is widely adopted in modern wireless systems due to its high spectral efficiency [7]. However, the performance of OFDM significantly degrades in contexts with rapidly time-varying channels, such as those encountered in high-mobility V2X communications [8]. In high-mobility scenarios, rapid Doppler shifts disrupt subcarrier orthogonality, leading to severe intercarrier interference (ICI) and inter-symbol interference (ISI) [9]. These impairments degrade OFDM system throughput and reliability, posing significant challenges in meeting the requirements of V2X applicationsfor low latency and high reliability. To overcome these limitations, [10] introduced orthogonal time frequency space (OTFS), a two-dimensional modulation technique that maps information symbols onto the delay-Doppler (DD) domain. In [11], the discrete-time model of OTFS was derived, demonstrating that OTFS can be implemented by incorporating appropriate preprocessing and postprocessing operations into the conventional OFDM framework. By exploiting the reduced channel variability in the DD domain, OTFS can harness both time and frequency diversity more effectively than can OFDM, resulting in enhanced transmission performance. Furthermore, in the DD domain, time-varying channels are transformed into quasi-static and sparse representations, making OTFS highly suitable for high-mobility applications. However, mapping symbols to the DD domain inevitably introduces inter-Doppler interference (IDI), which poses new challenges for receiver design. To achieve full time-frequency diversity and robustly suppress interference from various sources, efficient detection algorithms are essential. The symbol-wise maximum a posteriori (MAP) algorithm achieves optimal performance in interference channels [12], but its computational complexity grows exponentially with the OTFS frame size, making it impractical for real-world systems. To address this issue, a message-passing detection algorithm based on the factor graph framework was proposed in [13]; in that algorithm, inter-symbol interference is modeled using Gaussian approximations. It has been shown that the message-passing algorithm can achieve near-optimal performance with significantly reduced complexity. Furthermore, by employing appropriate phase shifts, the message-passing algorithm can effectively eliminate ICI and ISI, and through focus on the most significant interference terms, the impact of IDI can be mitigated [14; 15]. Consequently, the message-passing algorithm can effectively compensate for a wide range of channel Doppler spreads [16]. Despite these advances in detection algorithms, the data rate of conventional OTFS modulation remains substantially limited by its reliance on ideal bi-orthogonal pulses. Faster-than-Nyquist (FTN) signaling is a prominent non-orthogonal transmission technique that deliberately violates the Nyquist criterion by compressing the symbol interval during pulse shaping [17]. This approach enables the transmission of more symbols within a fixed bandwidth and improves the information rate but introduces ISI as a trade-off [18]. In [19], FTN signaling was extended to the frequency domain by applying it to OFDM systems in a method referred to as spectrally efficient frequency division multiplexing (SEFDM).