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摘要
分数傅里叶变换是傅里叶变换的广义形式，提供了介于时域和频域之间的多分数域信号表征，为非平稳信号处理和线性时变系统分析开辟了新途径，应用十分广泛。本文首先总结近年来分数傅里叶变换的理论研究成果，包括分数傅里叶变换的数值计算、衍生的离散分数变换、分数域采样、分数域滤波与参数估计、多分数域分析。然后介绍分数傅里叶变换在工程和实践中的应用，包括雷达、通信、图像加密、光学干涉测量、医学、生物、机械仪器等。最后对分数傅里叶变换理论及其应用的未来研究方向进行展望。
Abstract
The fractional Fourier transform (FRFT) is a generalization of the Fourier transform. The FRFT can characterize signals in multiple fractional domains and provide new perspectives for non-stationary signal processing and linear time variant system analysis, thus it is widely used in reality applications. We first review recent developments of the FRFT in theory, including discretization algorithms of the FRFT, various discrete fractional transforms, sampling theorems in fractional domains, filtering and parameter estimation in fractional domains, joint analysis in multiple fractional domains. Then we summarize various applications of the FRFT, including radar and communication signal processing in fractional domains, image encryption, optical interference measurement, medicine, biology, and instrument signal processing based on the FRFT. Finally we discuss the future research directions of the FRFT, including fast algorithm of the FRFT, sparse sampling in fractional domains, machine learning utilizing the FRFT, graph signal processing in fractional domains, and discrete FRFT based on quantum computation.
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Overview
Overview:The fractional Fourier transform (FRFT) is a generalization of the Fourier transform. It has been received much attention since Namias provided its definition in the perspective of eigendecomposition and its application in quantum in 1980. FRFT can be interpreted as decomposition of a signal into chirp signals or rotation of the time-frequency plane with angle α. After years of research, the theoretical system of the FRFT has been relatively completed. Efficient and accurate discretization algorithms and sampling theorem associated with the FRFT make the digital signal processing based on discrete FRFT possible. Filtering and parameter estimation in fractional domains greatly promote applications of the FRFT in practice. Analysis of a signal in multiple fractional domains jointly distinguish signal processing utilizing FRFT from traditional signal processing, this is because with the rotation angle α changing from 0 to π/2, the FRFT of a signal can provide characteristics of the signal in many fractional domains, including time domain and frequency domain. Meanwhile, with the development of theoretical research, the FRFT also shows great values in practice. In addition to traditional areas such as quantum and optical, FRFT has also been applied in the area of signal processing, especially in radar signal processing, communication signal processing, image processing, medical signal processing, biology signal processing, and mechanical signal processing, et al. In this paper, we first provide definitions of the FRFT and its basic properties. We then review recent developments of the FRFT in theory, including discretization algorithms of the FRFT, various discrete fractional transforms derived from the discrete FRFT, sampling theory associated with the FRFT, filtering and parameter estimation in fractional domains, and joint analysis in multiple fractional domains. We next summarize progress in several application areas utilizing FRFT, including radar, communication, image encryption, optical measurement, health care, biology, and instrument. We also provide several future research directions of the FRFT, for example, fast algorithm and sparse sampling associated with the FRFT can be studied further to reduce complexity, existing applications of the FRFT can be promoted to improve the system performance further, FRFT can also be applied to machine learning because FRFT can provide characteristic of images in multiple fractional domains, FRFT based on graph may be very useful in graph signal processing, and discrete FRFT based on quantum computation may greatly reduce the complexity. By summarizing the research history, presenting research focus, and discussing future research directions of the FRFT, we try to provide a relatively comprehensive overview to the research progress in the FRFT to help readers to understand this filed better.
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图 1 线性调频信号的时域、频域及分数域表示
Figure 1. Chirp signal in time, frequency, and fractional domain
表 1 离散分数傅里叶变换特征值及其分配规则[30]
Table 1. Eigenvalues and their assignment rule of the discrete fractional Fourier transform (DFRFT) matrix[30]
N 离散分数傅里叶变换的特征值 4m exp(-ikα), k=0, 1, …, 4m-2, 4m 4m+1 exp(-ikα), k=0, 1, …, 4m-1, 4m 4m+2 exp(-ikα), k=0, 1, …, 4m, 4m+2 4m+3 exp(-ikα), k=0, 1, …, 4m+1, 4m+2 
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表 2 离散分数傅里叶变换算法的性质比较
Table 2. Comparison of the properties of DFRFT algorithms
酉性 阶次可加性 可逆性 逼近连续FRFT 闭合式 计算复杂度 Ozaktas采样型 × × × √ × O(N·log2N) Pei采样型 √ × √ √ √ O(N·log2N) 特征分解型 √ √ √ √ × O(N2)
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