Editorial, 2018 Issue 6: Theories and methods of signal processing in fractional Fourier domain (2018)

The theory and method of fractional Fourier domain signal processing (including fractional Fourier transform (FRFT), linear canonical transformation and fractional calculus, etc.) are the results of continuous improvement, enrichment and development of Fourier analysis. Since the theory and method of fractional Fourier domain signal processing have the unique characteristics of fractional domain, their related researches have been rapidly infiltrated from the early fields of mathematics , quantum mechanics and optical domain (optical image compression) to encryption, digital watermarking, communication, information security, medical imaging, radar and sonar and so on. With the continuous development and the in-depth research of the theory and method of fractional Fourier domain signal processing in optics and engineering applications fields, they have become one of the most important topics in information processing fields, and will have a more extensive application prospect.  According to the increasing demand of fractional Fourier theory in various fields, this special issue further studies the basic theory and application of fractional domain signal processing based on the existing theoretical results. The experts and scholars in this field are invited to elaborate on the latest research results and development trends to further promoting the synergistic development of relevant theories, technologies and applications. Through this topic, the aim is to grasp the research opportunities together, face the challenges of research and promote the progress and development of this field.  The articles published include two reviews and twelve research papers. “Research Progress in Theories and Applications of the Fractional Fourier Transform” 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. The other review paper –“Research Progress on the Discretization of Linear Canonical Transform” summarizes the discretization and fast algorithm of a class of generalized linear integral transform (linear canonical transform) in fractional field. It classifies the existing discrete algorithms, compares and analyzes the properties (unitary, additive, etc.) and computational complexity of the algorithms. In addition, it discusses the technical difficulties and the applicable restrictive conditions of each kind of discrete algorithm and further forecasts the future research direction.  This special research papers introduce the latest theoretical research results of theory and method of fractional Fourier domain signal processing, including sampling theorems, uncertainty principles, convolution theorems, sparse signal processing theories and methods. At the same time, these research papers discuss the applications of theory and method of fractional Fourier domain signal processing in radar and image processing, such as radar moving target detection, magnetic resonance imaging, optical encryption and face recognition. These reviews and research papers will give readers a better understanding of the current research status, trends and application prospects of theory and method of fractional Fourier domain signal processing, as well as provides the help for researchers in related fields.