Studies on convolution and correlation theorems for the linear canonical transform and their applications in signal processing

Abstract

The main aim of the proposed work is to provide an inclusive approach towards the introduction to the principles and applications of the product and convolution theorem in the linear canonical transform (LCT) domain. As a generalization of fractional Fourier transform (FRFT), Fresnel transform (FST) and Fourier transform (FT), the LCT is a three variable class of integral transform and has been used in many fields of optics and signal processing. The LCT has proved to be a powerful tool for the analysis of time-varying signals by representing rotation of a signal in the time-frequency plane. In the applications, where FT and fractional domain concepts are used, the performance can be enhanced through the use of LCT because of its three extra degrees of freedom as compared to one degree of freedom for FRFT and no degree of freedom for FT. Many properties of the LCT are currently well known, including sampling, uncertainty principle, product and convolution theorems, which are generalization of the corresponding properties of the FT and FRFT. The product and convolution theorems for the LCT available in the literature, however these do not generalize very nicely to the classical result for the FT and FRFT. The proposed work can be divided into two broader segments. The first segment includes, the efforts made in establishing the LCT a complete integral transform by developing and deriving the weighted convolution and correlation identities. Also the proposed definitions of these theorems are compared with existing ones and their superiority has been determined with the help of some newly devised performance metrics. The second segment comprises of the applications of proposed identities along with some new application areas of the LCT. In the first phase, a comprehensive closed-form analytical expression of the behavior of Dirichlet, Generalized Hamming and triangular window functions is established, utilizing various special mathematical functions in the LCT domain.