Time-Frequency Localization for the Fractional Fourier Transform in Signal Processing and Uncertainty Principles

The fractional Fourier transform (FrFT) is a generalization of the usual Fourier transform. The aim of this paper is to show the compression of sound signal in FrFT domain and to prove the qualitative and quantitative uncertainty principles for the FrFT. The first of these results consists the Hardy's and an L-p-L-q version of Miyachi's theorems for the FrFT, which estimates of decay of two fractional Fourier transforms F-alpha(f) and F-gamma(f), with gamma - alpha not equal n pi, for all n is an element of Z. The second result consists an extension of Faris's local uncertainty principle which states that if a non zero function F alpha(f) is an element of L-2(R) is highly localized near a single point then F-gamma(f) cannot be concentrated in a set of finite measure with gamma - alpha not equal n pi, for all n is an element of Z. From our results we deduce the usual uncertainty principles for the fractional Fourier transform which states these theorems between a function f and its fractional Fourier transform F-gamma(f).