Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform

Conventional Fourier analysis has many schemes for different types of signals. They are Fourier transform (FT), Fourier series (FS), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT). The goal of this article is to develop two absent schemes of fractional Fourier analysis methods. The proposed methods are fractional Fourier series (FRFS) and discrete-time fractional Fourier transform (DTFRFT), and they are the generalizations of Fourier series (FS) and discrete-time Fourier transform (DTFT), respectively.