# An LDPC-RS Concatenation and Decoding Scheme to Lower the Error Floor for FTN Signaling
## 1 Introduction  
Faster-than-Nyquist (FTN) signaling dates back to the 1970s and has attracted increasing interest in the past two decades due to the lack of research into frequency resources and bandwidth in digital communication, such as wireless communication, satellite communication, and optical communication [1]. One common method to improve link capacity and spectrum efficiency (SE) is to adopt high-order modulation in an orthogonal (Nyquist) modulation scheme, for example, by using quadrature amplitude modulation (QAM), such as 16-QAM, 64-QAM, or even 4096-QAM. However, high-order modulation formats often require higher power and sensitivity on radio frequency (RF) devices. Moreover, a higher modulation order is more likely to suffer from noise since constellation symbols converging in 256-QAM compared to 16-QAM and requires improved linearity of RF devices, thus being a challenge for symbol recovery and decisions. Another attractive method is to use a lower modulation order but non-orthogonal modulation scheme, i.e., FTN signaling. By accelerating the symbol speed of transmission with a faster-than-Nyquist speed of no inter-symbol interference (ISI), controlled ISI and colored noise are introduced after match filtering, and they can be eliminated in many ways, such as with a whitening filter, orthogonal basis model (OBM), precoding, turbo equalizer, or Bahl-Cock-Jelinek-Raviv (BCJR) detector [2]. In band-limited scenarios like high-speed optical, wireless, and satellite communication, FTN signaling provides higher SE with the same modulation order at the price of a higher symbol rate (higher sampling rate) and additional digital signal processing to mitigate ISI [3]. In recent years, many ISI mitigation methods have been studied, including linear equalization, decision feedback equalization (DFE), frequency domain equalization at thereceiver side, and precoding and pre-equalization at the transmitter side. Although these methods are relatively simple, their performance will degrade with severe ISI when applying the high-acceleration factor of FTN signaling. The BCJR algorithm performs trellis searching to find survivors and make decisions for each bit. However, when dealing with a high modulation order or long ISI taps, BCJR detection requires considerable memory space to store a large number of states and routes in trellises, which increases exponentially with ISI length. To solve this problem, reduced-state M-BCJR detection was studied in [2], and the authors showed that only 3-state BCJR is required for FTN signaling with acceleration factor \(\tau=0.5\). In a coded system with iterative decoding, we can apply FTN signaling as an internal ISI mechanism and use BCJR detection to generate soft decision bits, i.e., log likelihood ratio (LLR), and exchange LLRs among the BCJR detector and soft channel decoder, which is called turbo equalization. In this work, we applied the M-BCJR algorithm together with LDPC-RS concatenation code to form a coded FTN signaling scheme and mitigate ISI through turbo equalization. LDPC code was first proposed by [PERSON] in 1962 [4]. Originally, the LDPC received little attention due to limited theoretical knowledge and computer processing capacity. Over the years, with the increasing improvement in relevant basic theories and the vigorous development of computer science and technology, researchers have devoted increasing effort to the study of LDPC code. LDPC code has become one of the most popular coding technologies and has been widely used in various communication systems, such as wireless, satellite, and fiber communication. With sufficient code length, LDPC code can approach the Shannon limit [5] and be capable of error correction at a low signal-to-noise ratio (SNR). Among soft decision-decoding methods, belief propagation (BP) is a promising decoding scheme for the LDPC and for other codes that need soft decisions and iterative decoding.