Nicki Holighaus

Theory, implementation and applications of nonstationary Gabor frames

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.

The ERBlet transform: An auditory-based time-frequency representation with perfect reconstruction

This paper describes a method for obtaining a perceptually motivated and perfectly invertible time-frequency representation of a sound signal. Based on frame theory and the recent non-stationary Gabor transform, a linear representation with resolution evolving across frequency is formulated and implemented as a non-uniform filterbank. To match the human auditory time-frequency resolution, the transform uses Gaussian windows equidistantly spaced on the psychoacoustic “ERB” frequency scale. Additionally, the transform features adaptable resolution and redundancy. Simulations showed that perfect reconstruction can be achieved using fast iterative methods and preconditioning even using one filter per ERB and a very low redundancy (1.08). Comparison with a linear gammatone filterbank showed that the ERBlet approximates well the auditory time-frequency resolution.

A Framework for Invertible, Real-Time Constant-Q Transforms

Audio signal processing frequently requires time-frequency representations and in many applications, a non-linear spacing of frequency bands is preferable. This paper introduces a framework for efficient implementation of invertible signal transforms allowing for non-uniform frequency resolution. Non-uniformity in frequency is realized by applying nonstationary Gabor frames with adaptivity in the frequency domain. The realization of a perfectly invertible constant-Q transform is described in detail. To achieve real-time processing, independent of signal length, slice-wise processing of the full input signal is proposed and referred to as sliCQ transform. By applying frame theory and FFT-based processing, the presented approach overcomes computational inefficiency and lack of invertibility of classical constant-Q transform implementations. Numerical simulations evaluate the efficiency of the proposed algorithm and the methods applicability is illustrated by experiments on real-life audio signals .

The Large Time-Frequency Analysis Toolbox 2.0

The Large Time Frequency Analysis Toolbox (LTFAT) is a modern Octave/Matlab toolbox for time-frequency analysis, synthesis, coefficient manipulation and visualization. It’s purpose is to serve as a tool for achieving new scientific developments as well as an educational tool. The present paper introduces main features of the second major release of the toolbox which includes: generalizations of the Gabor transform, the wavelets module, the frames framework and the real-time block processing framework.

Structure of nonstationary Gabor frames and their dual systems

Abstract We investigate the structural properties of dual systems for nonstationary Gabor frames. In particular, we prove that some inverse nonstationary Gabor frame operators admit a Walnut-like representation, i.e. the operator acting on a function can be described by weighted translates of that function. In this case, which only occurs when compactly supported window functions are used, the canonical dual frame partially inherits the structure of the original frame, with differences that we describe in detail. Moreover, we determine a necessary and sufficient condition for a pair of nonstationary Gabor frames to form dual frames, valid under mild restrictions. This condition is then applied in a simple setup, to prove the existence of dual pairs of nonstationary Gabor systems with coarser frequency sampling than allowed by previous results [3] . We also explore a connection to recent work of Christensen, Kim and Kim on Gabor frames with compactly supported window function.

Representing and Counting the Subgroups of the Group

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group , where and are arbitrary positive integers. We obtain formulas for the total number of subgroups and the number of subgroups of a given order.

Efficient Algorithms for Discrete Gabor Transforms on a Nonseparable Lattice

The Discrete Gabor Transform (DGT) is the most commonly used transform for signal analysis and synthesis using a linear frequency scale. It turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup. Most of the literature focuses on separable subgroups, in this paper we will survey existing methods for a generalization to arbitrary groups, as well as present an improvement on existing methods. Comparisons are made with respect to the computational complexity, and the running time of optimized implementations in the C programming language. The new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download. By summarizing general background information on the state of the art, this article can also be seen as a research survey, sharing with the readers experience in the numerical work in Gabor analysis.

Reassignment and synchrosqueezing for general time-frequency filter banks, subsampling and processing

In this contribution, we extend the reassignment method (RM) and synchrosqueezing transform (SST) to arbitrary time-frequency localized filters and, in the first case, arbitrary decimation factors. A sufficient condition for the invertibility of the SST is provided. RM and SST are techniques for deconvolution of short-time Fourier and wavelet representations. In both methods, the partial phase derivatives of a complex-valued time-frequency representation are used to determine the instantaneous frequency and group delay associated to the individual representation coefficients. Subsequently, the coefficient energy is reassigned to the determined position. Combining RM and frame theory, we propose a processing scheme that benefits both from the improved localization of the reassigned representation and the frame properties of the underlying complex-valued representation. This scheme is particularly interesting for applications that require low redundancy not achievable by an invertible SST.

Phase vocoder done right

The phase vocoder (PV) is a widely spread technique for processing audio signals. It employs a short-time Fourier transform (STFT) analysis-modify-synthesis loop and is typically used for time-scaling of signals by means of using different time steps for STFT analysis and synthesis. The main challenge of PV used for that purpose is the correction of the STFT phase. In this paper, we introduce a novel method for phase correction based on phase gradient estimation and its integration. The method does not require explicit peak picking and tracking nor does it require detection of transients and their separate treatment. Yet, the method does not suffer from the typical phase vocoder artifacts even for extreme time stretching factors.

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.