Nonlinear approximation with nonstationary Gabor frames

We consider sparseness properties of adaptive time-frequency representations obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical Gabor frames by allowing for adaptivity in either time or frequency. It is known that the concept of painless nonorthogonal expansions generalizes to the nonstationary case, providing perfect reconstruction and an FFT based implementation for compactly supported window functions sampled at a certain density. It is also known that for some signal classes, NSGFs with flexible time resolution tend to provide sparser expansions than can be obtained with classical Gabor frames. In this article we show, for the continuous case, that sparseness of a nonstationary Gabor expansion is equivalent to smoothness in an associated decomposition space. In this way we characterize signals with sparse expansions relative to NSGFs with flexible time resolution. Based on this characterization we prove an upper bound on the approximation error occurring when thresholding the coefficients of the corresponding frame expansions. We complement the theoretical results with numerical experiments, estimating the rate of approximation obtained from thresholding the coefficients of both stationary and nonstationary Gabor expansions.Comment: 19 pages, 2 figure

A Phase Vocoder based on Nonstationary Gabor Frames

We propose a new algorithm for time stretching music signals based on the theory of nonstationary Gabor frames (NSGFs). The algorithm extends the techniques of the classical phase vocoder (PV) by incorporating adaptive time-frequency (TF) representations and adaptive phase locking. The adaptive TF representations imply good time resolution for the onsets of attack transients and good frequency resolution for the sinusoidal components. We estimate the phase values only at peak channels and the remaining phases are then locked to the values of the peaks in an adaptive manner. During attack transients we keep the stretch factor equal to one and we propose a new strategy for determining which channels are relevant for reinitializing the corresponding phase values. In contrast to previously published algorithms we use a non-uniform NSGF to obtain a low redundancy of the corresponding TF representation. We show that with just three times as many TF coefficients as signal samples, artifacts such as phasiness and transient smearing can be greatly reduced compared to the classical PV. The proposed algorithm is tested on both synthetic and real world signals and compared with state of the art algorithms in a reproducible manner.Comment: 10 pages, 6 figure

Frame Theory for Signal Processing in Psychoacoustics

This review chapter aims to strengthen the link between frame theory and signal processing tasks in psychoacoustics. On the one side, the basic concepts of frame theory are presented and some proofs are provided to explain those concepts in some detail. The goal is to reveal to hearing scientists how this mathematical theory could be relevant for their research. In particular, we focus on frame theory in a filter bank approach, which is probably the most relevant view-point for audio signal processing. On the other side, basic psychoacoustic concepts are presented to stimulate mathematicians to apply their knowledge in this field