Theory, implementation and applications of nonstationary Gabor frames

Abstract

Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.

A Phase Vocoder Based on Nonstationary Gabor Frames

“…In this section we extend the classical Gabor theory to the nonstationary case [15]. Just as for the stationary case, we denote the total number of sampling points in time by N ∈ N, however, we do not assume these points to be uniformly distributed.…”

“…We choose to work with the procedure described in [15] since it is suitable for representing signals, which consist mainly of transient and sinusoidal components. The adaptation procedure is based on the idea that window functions with small support should be used around the onsets of attack transients whereas window functions with longer support should be used between these onsets.…”

“…Transient detection: To perform the transient detection we use a spectral flux (SF) onset detection function as described in [15], [20]. This function is computed with a DGT of redundancy 16, and it measures the sum of (positive) change in magnitude for all frequency channels.…”

“…The construction is performed in a smooth way such that the change from one step to the next corresponds to a window function that is either half as long, twice as long or of the same length. For details see [15].…”

“…We will consider the task of time stretching in the framework of Gabor theory [13], [14]. Applying nonstationary Gabor frames (NSGFs) [15], [16] we extend the theory of the PV to incorporate TF representations with the above-mentioned adaptive TF resolution.…”