Fractional Fourier analysis — a new perspective of signal processing

Signal contains information. Signal processing deals with the representation, transformation, and manipulation of signals and the information the signal contains, hence it is widely used in engineering. As one of the most important signal processing tools, Fourier transform paves the way for the analysis of stationary signals and linear time invariant systems. However, nonstationary signal and linear time-variant system are commonly used in modern information systems, such as air defense, missile defense and TT&C, radar detection and identification of high speed/high acceleration targets, etc. Fourier analysis cannot meet the demand for high precision and real-time, thus new signal processing theories and methods are required. Fractional Fourier transform is a generalization of Fourier transform. As a novel signal processing tool, fractional Fourier transform can be interpreted as decomposition of a signal into chirps which are typical non-stationary signals, thus it is very suitable for non-stationary signal analysis. Meanwhile, fractional Fourier transform can provide characteristics of signals in multiple fractional domains, including time domain and frequency domain, thus it provides multiple perspectives of signal processing. There are many theoretical research directions associated with fractional Fourier transform, including sampling, filter and parameter estimation, multi-domains analysis, etc. Meanwhile, fractional Fourier transform has been widely used in the field of radar, communication, medical, and information system, etc.
The research team of Professor Tao Ran is from Beijing Institute of Technology. They have been engaged in the research of theories and applications of fractional Fourier analysis for over twenty years. The main achievements are as follows:
1) the research team obtained a series of sampling theorems to solving the sampling problem of non-stationary signals, including uniform sampling, non-uniform sampling, and sampling rate conversion theorem, etc.  All of these results can help us to effective and accurate sampling the non-stationary signals, which is one of the most important problems in sampling theory.
2) The research team extended the concept of power spectrum to the concept of fractional power spectrum, revealed the mechanism of multiplicative filtering in the fractional domain, found that the time-frequency resolution cells reflect not only the first order frequency information but also the second-order frequency information, and proposed effective signal detection and parameter estimation methods. These achievements provided the effectively methods for the signal detection and parameter estimation.
3) The research team revealed the mechanism of signal processing in multiple fractional domains. Based on these mechanism, signals can be analyzed not only in any single fractional domain, but also in multiple and joint fractional domains. These results can help us to analysis and utilize the detailed characteristics of signals in these domains, which are very important issue in modern signal processing community.

The research team of Professor Tao Ran reviews recent developments of the FRFT in theory and summarizes progress in several application areas, including radar and communication, image encryption and optical measurement, medicine, and mechanical instrument and so on. This review paper also discusses the challenges of the basic theory, such as how to improve the computational complexity of the discrete fractional Fourier transform algorithm and reduce the sampling rate, and indicates the future research directions of the FRFT.

Periodic nonuniform sampling model

Chirp signal in time, frequency, and fractional domains

Multi-domains analysis

About Team
Fractional signals and systems research team is the innovative research group of the National Natural Science Foundation of China, innovation team of the Ministry of Education of the People’s Republic of China, innovation team of National Defense Science and Technology, Huang Da-Nian Teacher team. The research team consists of 3 professors, 5 associated professors, and 7 lecturers. One of the researchers was awarded by Academician of China Engineering Academy and Academic Divisions of the Chinese Academy of Science. One of the researchers was awarded by the Distinguished Young Scholars of National Science Foundation of China, Yangtze River Scholar Distinguished Professor, New Century Hundred-Thousand-Ten Thousand Talents Project. One of the researchers was awarded by Yangtze River Scholar Chaired Professor, Foundation for Distinguished Young Scholars, the Recruitment Program of Global Experts, IEEE Fellow. The research team has published 156 SCI papers and authorized 84 patents. It also undertook many important projects, including Innovative Research Groups of the National Natural Science Foundation of China, the National Science Fund for Distinguished Young Scholars, Key Project of National Natural Science Foundation, etc. The research team was awarded by the first prize of the National Science & Technology Awards, the first prize of the Natural Science Award of the Ministry of Education of the People’s Republic of China, the first prize of the Science & Technology Awards by Ministry of Education of the People's Republic of China, special prize of the National Teaching Achievement Award, and National Excellent Doctoral Thesis nomination Award, etc.

Article
Ma J M, Miao H X, Su X H, et al. Research progress in theories and applications of the fractional Fourier transform[J]. Opto-Electronic Engineering, 2018, 45(6): 170747.
DOI: 10.12086/oee.2018.170747