Discrete multi-Gabor expansions

A discrete multi-Gabor expansion (DMGE) is developed to meet the requirements of localized and refined time-frequency (TF) representation of signals. The DMGE uses multiple windows and their translations and complex modulations as synthesis (or analysis) waveforms. It includes and generalizes the metaplectic (translation, modulation, and dilation) representations which are useful in signal analysis. Uniform, nonuniform, and proportional time sampling schemes are analyzed. The fundamental features and the importance of the DMGE are discussed. We focus on the construction of DMGE and deriving fast algorithms for the computation of related multi-analysis sequences. With matrix algebra, the algorithms derived apply to both multi-Gabor expansions and uni-(window) Gabor expansions. Another useful feature of the DMGE lies in the fact that the multi-Gabor transform can be realized in a parallel FFT-based implementation structure. Examples of DMGE and their applications to TF analysis are also discussed.