## 1 Introduction  
In order to meet the unprecedented requirement for high-data-rate transmission and to achieve higher spectral efficiency, non-orthogonal transmission techniques in time- and/or frequency domain have attracted worldwide attention in the academia and the industry, especially during the standardization process of the 5G new radio (NR) [1]. Faster-than-Nyquist (FTN) was originally proposed by [PERSON] in 1975 [2]. As a kind of non-orthogonal transmission technology, FTN has broken the traditional Nyquist criterion and the data rate can be increased by 25% at the expanse of introducing ineluctable inter-symbol interference (ISI) artificially. Therefore, FTN transmission technology has attracted wide attention in recent years as a potential time-domain solution to effectively improve spectral efficiency. Furthermore, FTN signaling has also been studied in a frequency domain [3; 4; 5; 6] and then extended to both time and frequency domains [7; 8; 9; 10]. The principle of frequency domain FTN signaling is that the modulated symbols are multiplexed onto the non-orthogonal sub-carriers which are placed at closer spacing compared to an OFDM system, i.e., bandwidth compression, thus less bandwidth resource can be occupied. At the receiver, complex detectors such as the maximum likelihood (ML) detector should be adopted to cancel the artificially introducedinter-carrier interference (ICI), however, with the complexity growing exponentially with the system size, e.g., the number of subcarriers or the constellation size. To tackle the high transceiver complexity problem, a number of efficient transceiver schemes have been developed for the FTN signaling transmission [11; 12; 13; 14]. For example, Ref. [13] proposes a circulated block transmission scheme, i.e., CB-FTN, in which circular convolution instead of the linear convolution is utilized to conduct the pulse shaping. Thus, low-complexity frequency domain equalization can be employed at the receiver to compensate for the channel impairment and the inherent ISI. Ref. [14] proposes the discrete Fourier transform (DFT) based block transmission for faster-than-Nyquist signaling (DBT-FTN) scheme where the implementation complexity of DBT-FTN system can be reduced significantly by DFT based signal generation and frequency-domain equalization (FDE) in the transmitter and receiver, respectively. Ref. [8] develops a new multicarrier FTN signaling transceiver with an FTN mapper and a linear minimum mean square error (LMMSE) equalizer. There are also works on other low complexity detectors like nonlinear detection algorithms based on maximum a posteriori (MAP) [15], Turbo-based frequency domain equalization [16; 17] and improved linear equalization algorithms [18]. To the best of our knowledge, these approaches, however, are mainly designed for the FTN system with one-dimensional compression and perform efficiently only for the additive white Gaussian noise (AWGN) channel scenario. In terms of fading channels, the inherent ISI and/or ICI of the FTN signaling combined with the ISI induced by the fading channel will even challenge the accuracy of the channel estimation. Under such circumstances, the efficient single-tap frequency domain channel estimation schemes for the OFDM system cannot be utilized directly due to the existence of the mixed interference [19; 20; 21]. Thus, the exploration of the efficient channel estimation schemes for FTN signaling is of great importance. The state-of-the art studies on the channel estimation in the FTN system can be divided into two categories, i.e., pilot-aided channel estimation schemes and the ones without the pilot. To save the pilot overhead, the channel estimation schemes without an assisted pilot are proposed by using differential modulation and correlation coefficient matrix to carry on non-coherent detection [22]. However, it is different from the current mainstream modulation scheme and cannot be compatible with the existing multi-carrier FTN transmission schemes.