Computation and conditioning of the finite-discrete Gabor transform

We present a new approach to analysing the continuous Gabor transform and two critically sampled discretizations of it: the periodic finite-discrete and non-periodic finite-discrete versions. In particular, we distinguish between the analysis and synthesis forms of the transform, and introduce an intermediate operation that decomposes both transforms to collections of independent Toeplitz operators. In the continuous and the periodic finite-discrete case this decomposition allows us to show that for appropriate window functions the analysis and synthesis transforms are inverses of each other. In the nonperiodic finite-discrete case this relation no longer holds, but we are still able to use the decomposition and results on Toeplitz matrices to show that both transforms and their inverses are computable in O(PlogP) operations (after a setup cost of O(Plog²P)) for P discrete samples. Moreover the decomposition also facilitates analysis of the conditioning of finite versions of the transform