Generalized Gabor expansion associated with linear canonical transform series

The Gabor expansion (GE), which maps the time domain signal into the joint time and frequency domain, has been recognized as very useful for signal processing. However, sinusoidal analysis used in the traditional GE is not appropriate for a compact representation for chirp-type signals. In this paper, a generalized Gabor expansion (GGE) is proposed in order to rectify the limitations of the GE, the proposed expansion not only inherits the advantage of GE, but also has the capability of signal representations in the linear canonical transform (LCT) domain which is similar to the LCT. Basis functions of the proposed expansion are obtained via LCT basis. Compared with the traditional GE, the GGE can offer signal representations on a general, non-rectangular time–frequency plane tiling. Besides, the completeness and biorthogonality conditions of the GGE are derived.