Peter Balazs

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 Linear Time Frequency Analysis Toolbox

The Linear Time Frequency Analysis Toolbox is a MATLAB/Octave toolbox for computational time-frequency analysis. It is intended both as an educational and computational tool. The toolbox provides the basic Gabor, Wilson and MDCT transform along with routines for constructing windows (filter prototypes) and routines for manipulating coefficients. It also provides a bunch of demo scripts devoted either to demonstrating the main functions of the toolbox, or to exemplify their use in specific signal processing applications. In this paper we describe the used algorithms, their mathematical background as well as some signal processing applications.

Basic definition and properties of Bessel multipliers

This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a special case the multipliers for Riesz bases are examined and it is shown that multipliers in this case can be easily composed and inverted. Finally the continuous dependence of a Bessel multiplier on the parameters (i.e., the involved sequences and the symbol in use) is verified, using a special measure of similarity of sequences.

Time–Frequency Sparsity by Removing Perceptually Irrelevant Components Using a Simple Model of Simultaneous Masking

We present an algorithm for removing time-frequency components, found by a standard Gabor transform, of a ldquoreal-worldrdquo sound while causing no audible difference to the original sound after resynthesis. Thus, this representation is made sparser. The selection of removable components is based on a simple model of simultaneous masking in the auditory system. Important goals were the applicability to any real-world music and speech sound, integrating mutual masking effects between time-frequency components, coping with the time-frequency spread of such an operation, and computational efficiency. The proposed algorithm first determines an estimation of the masked threshold within an analysis window. The masked threshold function is then shifted in level by an amount determined experimentally, and all components falling below this function (the irrelevance threshold) are removed. This shift gives a conservative way to deal with uncertainty effects resulting from removing time-frequency components and with inaccuracies in the masking model. The removal of components is described as an adaptive Gabor multiplier. Thirty-six normal hearing subjects participated in an experiment to determine the maximum shift value for which they could not discriminate the irrelevance filtered signal from the original signal. On average across the test stimuli, 32 percent of the time-frequency components fell below the irrelevance threshold.

Invertibility of multipliers

Abstract In the present paper the invertibility of multipliers is investigated in detail. Multipliers are operators created by (frame-like) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or necessary conditions for invertibility are determined depending on the properties of the analysis and synthesis sequences, as well as the symbol. Examples are given, showing that the established bounds are sharp. If a multiplier is invertible, a formula for the inverse operator is determined and n-term error bounds are given. The case when one of the sequences is a Riesz basis is completely characterized.

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.

MULTIPLIERS FOR CONTINUOUS FRAMES IN HILBERT SPACES

In this paper, we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include anti-Wick operators, STFT multipliers or Calder?n?Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten-class properties. Furthermore, the concepts of controlled and weighted frames are transferred to the continuous setting.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Coherent states: mathematical and physical aspects?.

HILBERT–SCHMIDT OPERATORS AND FRAMES — CLASSIFICATION, BEST APPROXIMATION BY MULTIPLIERS AND ALGORITHMS

In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.

Frames and semi-frames

Loosely speaking, a semi-frame is a generalized frame for which one of the frame bounds is absent. More precisely, given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator, with bounded inverse. We study mostly upper semi-frames, both in the continuous case and in the discrete case, and give some remarks for the dual situation. In particular, we show that reconstruction is still possible in certain cases.

Adapted and Adaptive Linear Time-Frequency Representations: A Synthesis Point of View

A large variety of techniques exist to display the time and frequency content of a given signal. In this article, we give an overview of linear time-frequency representations, focusing mainly on two fundamental aspects. The first is the introduction of flexibility, more precisely, the construction of time-frequency waveform systems that can be adapted to specific signals or specific signal processing problems. To do this, we base the constructions on frame theory, which allows many options while still ensuring perfect reconstruction. The second aspect is the choice of the synthesis framework rather than the usual analysis framework. Instead of considering the correlation, i.e. the inner product, of the signal with the chosen waveforms, we find appropriate coefficients in a linear combination of those waveforms to synthesize the given signal. We show how this point of view allows the easy introduction of prior information into the representation. We give an overview of methods for transform domain modeling, in particular those based on sparsity and structured sparsity. Finally, we present an illustrative application for these concepts: a denoising scheme.