The centered discrete fractional Fourier transform and linear chirp signals

The basis functions for the fractional Fourier transform are chirp signals where a precise relationship between the fractional parameter and the chirp angle can be established. The recently introduced centered discrete fractional Fourier transform, based on the Gr/spl uml/nbaum commuting matrix, has basis functions that have a sigmoidal instantaneous frequency and produces a transform that is approximately an impulse for discrete chirps. However, no such precise relation between the fractional parameter and the chirp rate of the basis functions exists in the discrete case. We study the relationship between the chirp rate and the fractional parameter in the discrete case and specifically look at two approximate expressions that relate the chirp rate and the angle for which one obtains an impulse-like transform. We study the efficacy of these estimates by applying them to the analysis of monocomponent and two component chirp signals.