Modern tools for Weyl-Heisenberg (Gabor) frame theory

D. Gabor formulated a fundamental approach to signal decomposition in terms of elementary signals. He decided to represent a one-dimensional signal in two dimensions, with time and frequency as coordinates. Gabor's idea required a tiling of the time-frequency domain by nonoverlapping rectangles. Gabor reasoned that certain optimal elementary signals should provide an efficient decomposition of the information plane. This decomposition should determine countably many components of the information plane with each component sufficiently localized in time and frequency that a coefficient c associated with a component R would characterize the amount of information from R in the signal. So Gabor needed to find a countable collection of optimal elementary signals with small associated areas in the information plane. He decided to generate his elementary signals from a single building block by using modulation and translation.