A fractional Gabor transform

We present a fractional Gabor expansion on a general, non-rectangular time-frequency lattice. The traditional Gabor expansion represents a signal in terms of time- and frequency-shifted basis functions, called Gabor logons. This constant-bandwidth analysis results in a fixed, rectangular time frequency plane tiling. Many of the practical signals require a more flexible, non-rectangular time-frequency lattice for a compact representation. The proposed fractional Gabor expansion uses a set of basis functions that are related to the fractional Fourier basis and generate a non-rectangular tiling. The completeness and bi-orthogonality conditions of the new Gabor basis are discussed.