The hopping discrete fractional Fourier transform
Introduction
The fractional Fourier transform (FrFT), as a generalization of the Fourier transform (FT), has aroused extensive interest and been applied in the areas of optics, quantum mechanics, encryption, time-frequency representation, signal processing, etc [1], [2], [3], [4], [5], [6], [7]. As the numerical calculation method of the fractional Fourier transform, the study of the discretization of the fractional Fourier transform is particularly important. Many different forms of discrete FrFT (DFrFT) have been extensively studie [8], [9], [10]. Among them, the closed-form definition of the DFrFT proposed by Pei, is very suitable for practical applications due to its simpler and closed form of discrete fractional convolution and correlation [11]. Its definition is given bywhere , N and M are the numbers of sampling in the time domain and the fractional domain, respectively. The constrains that M ≥ N and must be satisfied.
The fractional cosine transform (FrCT), the fractional sine transform (FrST) and the fractional Hartley transform (FrHT) are very closed to the FrFT, so they have drawn much attention in recent years [12], [13], [14]. Several discrete algorithms of them are then developed and successfully applied in various areas, such as encryption and verification [15], [16], [17], [18].
As mentioned above, the discrete algorithms for the FrFT as well as the FrCT, the FrST, the FrHT have been well established and can be implemented effectively and fast for the case of calculating all the N output spectra using a block of N input samples. However, there are some real-time applications, such as the anomaly detection of network traffic and ECG signal processing, the input signals are stream data which arrive in a sequential manner. In this case, the output spectra are required recalculating when each new sample or several new samples arrive. According to the traditional discrete algorithms, for example the DFrFT algorithms [8], [9], [10], [11], the DFrFT result is calculated first at some moment, and then one DFrFT operation is performed again when the new sample arrives. The two operations are performed independently, which results in a large amount of computation and can’t meet the requirements of real-time output. For this case, the sliding window algorithm often makes a more suitable way. The sliding window algorithm can be interpreted as transforms on block data with a sliding window. It is essentially an iterative calculation method, which is based on the relationship between the transform results of signal samples at two adjacent window positions. By using the sliding window algorithm, when the window with the length of N slides along one sample at a time, the transform result for the new block which consists of the newly arrived sample and the samples before can be directly computed from the results of the previous block.
The sliding window algorithm has been successfully applied to many transforms. Among them, because of the important role of the discrete Fourier transform (DFT) in signal processing, the sliding DFT (SDFT) algorithm has been extensively researched and developed. It is first developed by Springer in 1991 [19], and then improved and popularized by Jacobsen and Lyons in 2003 [20], [21]. Then, some solutions to the spectral leakage problem and some methods to improve the stability of the SDFT system are proposed [22], [23], [24], [25]. For the discontinuous sliding of the window in time domain, a multi-point sliding DFT algorithm, called the hopping DFT (HDFT) algorithm is proposed by Park and Ko [26]. Then, the HDFT algorithm is further improved by Juang et al by using a compact recursive structure [27]. In this letter, Juang et al introduce the architecture design of the HDFT algorithm and further improved it by using some very-large-scale integration (VLSI) schemes. Based on the two implementation architectures, the computation load of the HDFT algorithm further reduced compared to the butterfly structure proposed by Park and Ko.
Although the theory of sliding DFT algorithm has been well developed, it has some limitations in the study of non-stationary signals due to the characteristics of the FT. Singular spectrum analysis (SSA) recently received a lot of attention in non-stationary signal processing because it doesn’t have to make assumption on the processed data. In order to solve the limitations of SSA algorithm in processing non-stationary multi-component signals, the sliding window algorithm has recently been applied in SSA and developed the sliding SSA algorithm [28], [29]. It operates by matching the reconstruction components in current window position with previously computed reconstruction components, and the matching is completed through minimization of a given distance metric function. The introduction of sliding window makes it more suitable for processing non-stationary sequences, especially the non-stationary signals with AM-FM components.
As the DFrFT is a powerful tool in non-stationary signals processing, so the application of sliding window algorithm in calculating DFrFT should be well studied for real-time processing of non-stationary signals. In 2008, Bhat and Vijaya proposed a pseudo sliding DFrFT algorithm, which uses a first-in-first-out (FIFO) buffer to hold the input data [30]. In 2018, Sun and Li proposed the sliding discrete linear canonical transform (DLCT) algorithm [31] and the sliding two dimensional DFrFT is proposed by Liu et al in the following year [32]. Beyond that, there are few researches on the sliding DFrFT algorithm. In this letter, we extend the single-point sliding of the SDFrFT algorithm to multi-point sliding and proposed the hopping DFrFT algorithm. Besides, the windowing methods in the sliding process are also be proposed. Then we propose the hopping discrete time fractional Fourier transform (HDTFrFT) algorithm to obtain a continuous fractional spectrum. The hopping sliding algorithm then be further extended to compute the discrete FrCT (DFrCT), the discrete FrST (DFrST) and the discrete FrHT (DFrHT). The main contributions of this paper are summarized as follows.
- 1.
We first propose the sliding DFrFT algorithm with sliding step p, termed as the hopping DFrFT algorithm. It’s essentially an iterative algorithm, which makes full use of the overlapping information between two windows. Besides, to reduce spectral leakage caused by truncation with the rectangular window, we propose two windowing methods, which are realized in the fractional domain.
- 2.
Then, we propose the hopping DTFrFT algorithm to get a continuous fractional spectrum. It is a sliding algorithm which is realized by windowing the DTFrFT with an infinite-length, causal exponential sliding window.
- 3.
Finally, the sliding window algorithm is also applied in computing the DFrCT, the DFrST and the DFrHT. Based on the character of trigonometric function, we proposed two different forms of the sliding DFrCT (SDFrCT) algorithm, the sliding DFrST (SDFrST)algorithm and the sliding DFrHT (SDFrHT) algorithm respectively.
The outline of this paper is as follows. Section 2 introduces the sliding process and the definition of the SDFT algorithm. Section 3 presents a multi-point sliding algorithm for computing the DFrFT, namely the hopping DFrFT algorithm. In Section 4, the hopping DTFrFT algorithm is presented, which is a sliding algorithm to compute the DTFrFT. Then, based on the relations among the DFrCT, the DFrST, the DFrHT and the DFrFT, the sliding window algoritm is also applied in computing them in Section 5. Finally, in Section 6, the computational complexity is analyzed and the computational accuracy is illustrated by simulations.
Section snippets
The sliding process and the SDFT algorithm
The SDFT, a recursive algorithm in essence, is a method to compute the DFT on stream data. For a N length window, the SDFT computes N-point DFT on the first N samples initially, then as the window advances one sample, the new DFT for the current block of data can be obtained directly from the result of the previous DFT without having to compute the new DFT from scratch.
For a more intuitive explanation of the sliding process, a diagram as shown in Fig. 1 is presened to describe this process. In
The hopping DFrFT algorithm
As the SDFT algorithm introduced above, we can also use sliding skill to compute the DFrFT. That is when the window slides along time samples, the DFrFT of the new block can be computed using the DFrFT of earlier block. And we can see, the SDFT algorithm introduced above is for the case where the window slides point by point, namely single-point sliding. But sometimes, the sliding is discontinuous, i.e., the sliding step isn’t equal to 1. For this case, we can directly derive a multi-point
The hopping DTFrFT algorithm
To compute the fractional Fourier transform for discrete signals, besides using the DFrFT, we can also use the DTFrFT. What’s more, by using the DTFrFT algorithm, we can obtain a continuous fractional spectrum. In this section, we apply the sliding window algorithm in computing the DTFrFT output and propose the hopping DTFrFT algorithm.
The DTFrFT is defined by [36]where T is the sampling interval in the time domain, and Xα(u) satisfies
Discrete fractional cosine/sine transform (DFrCT/DFrST)
Just as the FT can be generalized into the FrFT, the cosine transform and the sine transform also can be generalized into the fractional domain because they are very closed to the FT. In this section, we will give the definitions of the fractional cosine/sine transform and their discrete forms.
First, based on the method of eigenfunction decomposition did in the FrFT, Pei used a similar method to derive the expressions of the FrCT and the FrST and obtained the following relations between the
Complexity comparison
In Section 3, we have compared the computation complexity between the proposed HDFrFT algorithm and the closed-form DFrFT algorithm to illustrate the advantages and disadvantages of the HDFrFT algorithm. In this section, we will analyze the computation time and computing resources of various DFrFT algorithms with regard to the hopping issue.
Table 3 compares the computing resources of various DFrFT algorithms in terms of multipliers, adders, coefficient-ROM size and registers when the numbers of
Conclusions
In this paper, a new calculation method of DFrFT, named HDFrFT, was presented for the fast implementation of the DFrFT on a sliding window. It can directly compute the DFrFT output at time using the DFrFT at time n when the sliding step of window is p. The proposed HDFrFT algorithm has shown to be more effective than other classical DFrFT algorithms by fully utilizing the similarity among blocks of samples. One advantage of the HDFrFT algorithm is that it has the lowest computational
CRediT authorship contribution statement
Yu Liu: Conceptualization, Methodology, Software, Writing - original draft. Feng Zhang: Validation, Writing - review & editing, Funding acquisition. Hongxia Miao: Formal analysis, Data curation. Ran Tao: Supervision, Project administration.
Declaration of Competing Interest
The authors have no conflict of interest, financial or otherwise. In detail, firstly, no support, financial or otherwise, has been received from any organization that may have an interest in their manuscript, and secondly, there are no other relationships or activities that could appear to have influenced the manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants 61731023, 61571042, 61421001 and U1833203.
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