Frames and orthonormal bases for variable windowed Fourier transforms

The Gabor transform (or the windowed Fourier transform) is a widely used tool in signal processing. We generalize the windowed Fourier transform to the variable-windowed Fourier transform. This generalization brings the Gabor transform and the wavelet transform under the same framework. Using frame theory we characterize frames and orthonormal bases for the variable windowed Fourier series (VWFS). These characterizations are formulated explicitly in terms of window functions. Therefore they can serve as guidelines for designing windows for the VWFS. We introduce the notion of “complete orthogonal support” and, with the help of this notion, we construct a class of orthonormal VWFS bases for L²(R⁺)