分数傅里叶变换理论及其应用研究进展

Research progress in theories and applications of the fractional Fourier transform

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。

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.