Feichtinger H.G., Strohmer T. (eds.) Gabor Analysis and Algorithms. Theory and Applications

Birkhäuser, 1998. — 507 p.In his paper Theory of Communication, D. Gabor proposed the use of a family of functions obtained from one Gaussian by time- and frequency-shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coefficients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal- as far as I know there were no serious attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions (it even has one function to spare) but it does not constitute what we now call a frame, leading to numerical instabilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density.
Interestingly, the same time-frequency lattice of functions was also proposed in an entirely different context by von Neumann, and became subsequently known as the von Neumann lattice, and lived an essential parallel life among quantum physicists. In addition, there is also a very clear connection to the short-time Fourier transform or windowed Fourier transform, used extensively in electrical engineering. Here too, the need to go to overcritical sampling, corresponding to the higher time-frequency density mentioned above, was discovered, independently.
Of course, in order to be useful practically, a transform must not only have good mathematical properties; it must also go hand-in-hand with efficient discrete algorithms, and for the Gabor transform these were developed extensively in the last decade.The duality condition for Weyl-Heisenberg frames
Gabor systems and the Balian-Low Theorem
A Banach space of t est functions for Gabor analysis
Pseudodifferential operators, Gabor frames, and local trigonometric bases
Perturbation of frames and applications to Gabor frames
Aspects of Gabor analysis on locally compact abelian groups
Quantization of TF lattice-invariant operators on elementary LCA groups
Numerical algorithms for discrete Gabor expansions
Oversampled modulated filter banks
Adaptation of Weyl-Heisenberg frames to underspread environments
Gabor representation and signal detection
Multi-window Gabor schemes in signal and image representations
Gabor kernels for affine-invariant object recognition
Gabor's signal expansion in optics