## 1 Introduction  
The linear frequency modulated (LFM) signal is a broadband signal and its bandwidth utilization is high. This characteristic makes the LFM signal receive more attention in the field of underwater acoustic communication. For example, in the orthogonal chirp division multiplexing (OCDM) communication system [1], a bank of orthogonal LFM signals acts as the carrier to modulate transmission signal. In addition, LFM signal has the lower side-lobe after pulse compression, and its Doppler frequency is not sensitive. Thus, the LFM signal is often applied in some detection or ranging systems [2, 3] by using matched filtering to compress the received signal. It is easy to detect an LFM signal in a high signal-to-noise environment. However, the detection of the LFM signal in the presence of high level noise is a tough task. Fractional Fourier transform (FrFT) is a generalization of Fourier transform (FT). In recent years, it has been attracting more attention in the signal processing field [4, 5, 6, 7]. In order to practically realize the FrFT-based engineering application, research on discrete fractional Fourier transform (DFrFT) is particularly needed. There are three main methods for calculating the DFrFT, namely, linear combination-type DfrFT [8], eigenvector decomposition-type DfrFT [9] and sampling-type FrFT [10, 11]. The simplest definition of DFrFT is linear combination-type DfrFT. The computational complexity of linear combination-type DfrFT approach is \(O(NlogN)\), which is same as the fast Fourier transform (FFT). However, this approach produces an error deviation in terms of continuous FrFT. Although the eigenvector decomposition-type DfrFT has a smaller deviation error with respect to the continuous FrFT, the computational complexity of this method reaches \(O(N^{2})\). The sampling-type FrFT provides the best approximation and has a fast computing speed. For example, the Reference [11] proposed the Pei sampling-type approach. It requires two chirp operations and one FFT operation. The computational complexity of the Pei sampling-type approach is slightly larger than FFT. However, compared with other DFrFT approaches, the sampling-type FrFT cannot perform the inverse operation, which will limit its application scope. Designers require one to select the proper method to numerically calculate DFrFT according to demands. The spectrum of infinite-length LFM signal in the optimal order fractional Fourier domain is a pulse function [12]. While the spectrum of finite-length LFM signal is a sinc function [6], it means that LFM signal can be easily detected in fractional Fourier domain. Several methods based on FrFT have been proposed to detect LFM signal with low signal-to-noise ratio. The authors in [13] compared the performance of FrFT detector and Fourier transform (FT) detector. In [14], a lower computational complexity LFM signal detector based on the integration of the 4 th-power modulus of the fractional Fourier transform is proposed. An adaptive FrFT detection algorithm is shown in [15]. It combined statistic-based and FrFT-based method to detect moving target with a low speed in heavy sea clutter. The signal to be detected at the receiving end is simply modeled as a LFM signal corrupted by noise. However, in practice, the received signal is propagated in the channel. Hence, the impacts of the channel, especially wireless channel, cannot be ignored. For example, in the underwater acoustic channel [16], due to multiple reflections from boundaries or scatters, the received signal can be viewed as the superposition of a number of amplitude-weighted and delayed replicas of the original emitted signal. In this scenario, the FrFT of LFM signal has multiple peaks [17] in the optimal order fractional Fourier domain. The presence of spurious peaks indicates that the received signal has been expanded. For FrFT detector, it is hard to decide whether the peak value is produced by noise or multi-path when the signal-to-noise ratio is low.