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