ABSTRACT
The French scientist Joseph Fourier was the first to introduce the remarkable idea of expansion of a function in terms of trigonometric series. Fourier had not paid much heed to the underlying rigorous mathematical analysis. Despite the success and profound impact of Fourier transforms, the theory is being continuously revisited to cater to the needs of scientific and engineering communities, which has resulted in the advent of fractional Fourier transform, linear canonical transform, special affine Fourier transform and quadratic-phase Fourier transform. The exponential relaxation process results in Lorentzian-shaped spectral lines as is the case with the nuclear magnetic resonance. The notion of convolution is one of the widely applied concepts of mathematics with application areas ranging from functional analysis to different fields of signal and image processing. The celebrated Shannon's sampling theorem in the Fourier domain is one of the remarkable, profound and elegant concepts of digital signal processing which serves as a bridge between the analog and digital signals.
