On the Grunbaum commuter based discrete fractional Fourier transform

The basis functions of the continuous fractional Fourier transform (FRFT) are linear chirp signals that are suitable for time-frequency analysis of signals with chirping time-frequency content. Efforts to develop a discrete computable version of the fractional Fourier transform (DFRFT) have focussed on furnishing an orthogonal set of eigenvectors for the DFT that serve as discrete versions of the Gauss-Hermite functions. Analysis of the DFRFT obtained from Grunbaum´s tridiagonal commuter and the kernel associated with it reveals the presence of both amplitude and frequency modulation in contrast to just frequency modulation seen in the continuous case. Furthermore the instantaneous frequencies of the basis functions of the DFRFT are sigmoidal rather than linear.