Meet the editor

Dr. Mohammad Younus Bhat is an assistant professor in the Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir. He obtained a master's degree in Mathematics from the University of Kashmir and a doctorate degree in Mathematics from the Central University of Jammu. He has more than sixty research papers and five book chapters to his credit. His prime research areas are signal and

image processing, harmonic analysis, wavelet analysis, numerical analysis, and differential equations.

## Contents


Preface

Joseph Fourier (1770–1830) first introduced the remarkable idea of expansion of a function in terms of trigonometric series without giving any attention to rigorous mathematical analysis. The integral formulas for the coefficients of the Fourier expansion were already known to Leonardo Euler (1707–1783) and others. In fact, Fourier developed his new idea for finding the solution of heat (or Fourier) equation in terms of Fourier series so that the Fourier series can be used as a practical tool for determining the Fourier series solution of partial differential equations under prescribed

The Fourier transform originated from the Fourier integral theorem that was stated in the Fourier treatise titled *La Théore Analytique de la Chaleur,* and its deep significance has subsequently been recognized by mathematicians and physicists. It is generally believed that the theory of Fourier series and Fourier transforms is one of the most remarkable discoveries in mathematical sciences and it has widespread applications in mathematics, physics, and engineering. Both the Fourier series and Fourier transforms are related in many important ways. Many applications, including the analysis of stationary signals and real-time signal processing, make effective use of the Fourier

In time-frequency analysis, the Fourier transform is one of the oldest tools to dominate signal processing. However, due to its drawbacks in the analysis of non-stationary signals, different alternative transforms have gained much popularity in recent years, including windowed Fourier transform, fractional Fourier transform, linear canonical transform, quadratic-phase Fourier transform, and so on. These transforms are known

The main reason for writing this book is to stimulate interactions among mathematicians, computer scientists, engineers, and economists, as well as biological and physical scientists. The text is suitable for advanced graduate students but is primarily intended for post-graduate students and researchers in wavelets and their applications.

The book begins with an elementary chapter that introduces general Fourier transforms like windowed Fourier transform, fractional Fourier transform, linear canonical

Hybrid transforms are constructed by associating the Wigner-Ville distribution (WVD) with widely known signal processing tools, such as fractional Fourier transform, linear canonical transform, offset linear canonical transform (OLCT), and their quaternion-valued versions. Chapter 2 summarizes research on hybrid transforms by reviewing a computationally efficient type of WVD-OLCT, which has simplicity in

marginal properties compared to classic WVD-OLCT and WVD.

boundary conditions.

transform in time and frequency domains.

as generalizations of the classic Fourier transform.

transform, and quadratic-phase Fourier transform.
