Fractional Fourier Transform, Signal Processing and Uncertainty Principles

Abstract

The fractional Fourier transform (FrFT) is one of the generalizations of the Fourier transform (FT). This paper is centered on the compression of different forms of signal in FrFT domain in order to extract some properties of each one with a comparison between the FrFT and the usual FT. Also, our focus here will be on two qualitative uncertainty principles for the fractional Fourier transform: The Cowling–Price’s theorem and the Lp- Lq version of Morgan’s theorem for the FrFT. These two results estimate the decay of two fractional Fourier transforms Fα(f) and Fγ(f) , with γ- α≠ nπ, ∀ n∈ Z, which allows us to deduce the usual uncertainty principles between a function f and its fractional Fourier transform Fγ(f).