Introduction to coupled fractional fourier transform

Abstract

This thesis includes an introduction to the Coupled Fractional Fourier Trans form (CFrFT) and its various properties. The CFrFT is a generalized version of the Fractional Fourier transform (FrFT) that extends FrFT to two-dimension. The thesis begins with a comprehensive review of the Fourier transforms (FT) and fractional Fourier transforms, providing a solid foundation for the study of the CFrFT. The mathematical framework of the CFrFT is then introduced, including the definition, properties, and some important theorems with exam ples. The main focus of the thesis is to investigate the properties of the CFrFT, including its linearity, symmetry, conjugation, time-frequency shift, etc. Addi tionally, certain properties related to its convolution operator, like commuta tivity, associativity have been explored. Moreover, relation between convolu tion of the product of CFrFTs of two functions with CFrFT of the product of these functions has also been investigated. The results of this thesis contribute to a deeper understanding of the CFrFT and its properties. The findings may be of interest to researchers and practitioners in the fields of electrical engi neering and mathematics. This thesis contains four chapters. Chapter 1 contains a brief introduction to integral transforms and Fourier transform. The definition and inversion formula for FT, as well as some ex amples has been included. A brief overview of features of FT such as linearity, time shift, and so on has been included. This chapter concludes with the limitations of FT.