arXiv subjects

Peter Balazs

Publications and source records attributed to Peter Balazs.

At least 73 records · Page 4Linked to original sources

arXiv2015

The Pole Behaviour of the Phase Derivative of the Short-Time Fourier Transform

The short-time Fourier transform (STFT) is a time-frequency representation widely used in applications, for example in audio signal processing. Recently it has been shown that not only the amplitude, but also the phase of this representation can be successfully exploited for improved analysis and processing. In this paper we describe a rather peculiar pole phenomenon in the phase derivative, a recurring pattern that appears in a characteristic way in the neighborhood around any of the zeros of the STFT, a negative peak followed by a positive one. We describe this phenomenon numerically and provide a complete analytical explanation.

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arXiv2015

Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups

In this paper we introduce and investigate the concept of repro- ducing pairs as a generalization of continuous frames. Reproducing pairs yield a bounded analysis and synthesis process while the frame condition can be omitted at both stages. Moreover, we will investigate certain continuous frames (resp. reproducing pairs) on LCA groups, which can be described as a continuous version of nonstationary Ga- bor systems and state sufficient conditions for these systems to form a continuous frame (resp. reproducing pair). As a byproduct we iden- tify the structure of the frame operator (resp. resolution operator). We will apply our results to systems generated by a unitary action of a subset of the affine Weyl-Heisenberg group in $L^2(\mathbb{R})$. This setup will also serve as a nontrivial example of a system for which, whereas continuous frames exist, no dual system with the same structure exists even if we drop the frame property.

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arXiv2014

Admissibility for $α$-Modulation Spaces

This paper is concerned with frame decompositions of $α$-modulation spaces. These spaces can be obtained as coorbit spaces for square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. The theory yields canonical constructions for Banach frames or atomic decompositions in $α$-modulation spaces. A necessary ingredient in this abstract machinery is the existence of generating functions that are admissible for the representation. For numerical purposes, admissible atoms with compact support are necessary. We show new admissibility conditions that considerably improve upon known results. In particular, we prove the existence of admissible vectors that have compact support in time domain.

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arXiv2014

The continuous nonstationary Gabor transform on LCA groups with applications to representations of the affine Weyl-Heisenberg group

In this paper we introduce and investigate the concept of reproducing pairs which generalizes continuous frames. We will introduce a concept that represents a unifying way to look at certain continuous frames (resp. reproducing pairs) on LCA groups, which can be described as continuous nonstationary Gabor systems and investigate conditions for these systems to form a continuous frame (resp. reproducing pair). As a byproduct we identify the structure of the frame operator (resp. resolution operator). Moreover, we ask the question, whether there always exist mutually dual systems with the same structure such that the resolution operator is given by the identity, i.e. given $A:X\rightarrow B(\mathcal{H})$, if there exist $ψ,φ\in\mathcal{H}$, s.t. \begin{equation*} f=\int_X \langle f,A(x)ψ\rangle A(x)φdμ(x),\ \ \forall f\in \mathcal{H} \end{equation*} and show that the answer is not affirmative. As a counterexample we use a system generated by a unitary action of a subset of the affine Weyl-Heisenberg group in $L^2(\mathbb{R})$.

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arXiv2014

An operator based approach to irregular frames of translates

We consider translates of functions in $L^2(\RRd)$ along an irregular set of points. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel sequence or Riesz sequence. We find a representation of the canonical dual of a frame sequence $\{ϕ( \cdot - λ_k)\}_{k\in \ZZ}$ - where $ϕ$ is a bandlimited function - in terms of its Fourier transform.

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arXiv2013

Representation of the inverse of a frame multiplier

Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. They are not only interesting mathematical objects, but also important for applications, for example for the realization of time-varying filters. In this paper we show a surprising result about the inverse of such operators, if existing, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones. Furthermore, one of those dual frames is uniquely determined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in $\ell^2$. We also investigate invertible Gabor multipliers. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case.

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arXiv2012

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 Calderón- Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as well 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.

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arXiv2011

Frames by Multiplication

In this note we study frame-related properties of a sequence of functions multiplied by another function. In particular we study frame and Riesz basis properties. We apply these results to sets of irregular translates of a bandlimited function $h$ in $L^2(\R^d)$. This is achieved by looking at a set of exponentials restricted to a set $E \subset \R^d$ with frequencies in a countable set $Λ$ and multiplying it by the Fourier transform of a fixed function $h \in L^2(E)$. Using density results due to Beurling, we prove the existence and give ways to construct frames by irregular translates.

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arXiv2011

Sound Analysis and Synthesis Adaptive in Time and Two Frequency Bands

We present an algorithm for sound analysis and resynthesis with local automatic adaptation of time-frequency resolution. There exists several algorithms allowing to adapt the analysis window depending on its time or frequency location; in what follows we propose a method which select the optimal resolution depending on both time and frequency. We consider an approach that we denote as analysis-weighting, from the point of view of Gabor frame theory. We analyze in particular the case of different adaptive time-varying resolutions within two complementary frequency bands; this is a typical case where perfect signal reconstruction cannot in general be achieved with fast algorithms, causing a certain error to be minimized. We provide examples of adaptive analyses of a music sound, and outline several possibilities that this work opens.

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arXiv2011

Can any unconditionally convergent multiplier be transformed to have the symbol (1) and Bessel sequences by shifting weights?

Multipliers are operators that combine (frame-like) analysis, a multiplication with a fixed sequence, called the symbol, and synthesis. They are very interesting mathematical objects that also have a lot of applications for example in acoustical signal processing. It is known that bounded symbols and Bessel sequences guarantee unconditional convergence. In this paper we investigate necessary and equivalent conditions for the unconditional convergence of multipliers. In particular we show that, under mild conditions, unconditionally convergent multipliers can be transformed by shifting weights between symbol and sequence, into multipliers with symbol (1) and Bessel sequences.

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arXiv2011

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.

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arXiv2010

Classification of General Sequences by Frame-Related Operators

This note is a survey and collection of results, as well as presenting some original research. For Bessel sequences and frames, the analysis, synthesis and frame operators as well as the Gram matrix are well-known, bounded operators. We investigate these operators for arbitrary sequences, which in general lead to possibly unbounded operators. We characterize various classes of sequences in terms of these operators and vice-versa. Finally, we classify these sequences by operators applied on orthonormal bases.

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arXiv2010

Detailed characterization of unconditional convergence and invertibility of multipliers

In this paper we investigate the possibility of unconditional convergence and invertibility of multipliers $M_{m,Φ,Ψ}$ depending on the properties of the sequences $Ψ$,$Φ$ and $m$. We characterize a complete set of conditions for the invertibility and the unconditional convergence of multipliers, and collect those results in tables. We either prove that unconditional convergence and invertibility is not possible, that one or both of these conditions are always the case for the given parameters, or we give examples for the feasible combinations. We give a full list of examples for all conditions.

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arXiv2010

Multipliers for p-Bessel sequences in Banach spaces

Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters.

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arXiv2009

Weighted and controlled frames

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally we analyze weighted frames numerically. First we investigate three possibilities for finding weights in order to tighten a given frame, i.e., decrease the frame bound ratio. Then we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame.

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arXiv2006

Matrix Representation of Operators Using Frames

In this paper it is investigated how to find a matrix representation of operators on a Hilbert space with Bessel sequences, frames and Riesz bases. In many applications these sequences are often preferable to orthonormal bases (ONBs). Therefore it is useful to extend the known method of matrix representation by using these sequences instead of ONBs for these application areas. We will give basic definitions of the functions connecting infinite matrices defining bounded operators on l2 and operators on the Hilbert space. We will show some structural results and give some examples. Furthermore in the case of Riesz bases we prove that those functions are isomorphisms. Finally we are going to apply this idea to the connection of Hilbert-Schmidt operators and Frobenius matrices. Keywords: frames, discrete expansion, operators, matrix representation, Hilbert-Schmidt operators, Frobenius matrices, Riesz bases.

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arXiv2006

Hilbert-Schmidt Operators and Frames - Classification, Approximation by Multipliers and Algorithms

In this paper we deal with the connection of frames with the class of Hilbert Schmidt operators. First we give an easy criteria for operators being in this class using frames. It is the equivalent to the criteria using orthonormal bases. Then we construct Bessel sequences frames and Riesz bases for the class of Hilbert Schmidt operators using the tensor product of such sequences in the original Hilbert space. We investigate how the Hilbert Schmidt inner product of an arbitrary operator and such a rank one operator can be calculated in an efficient way. Finally we give an algorithm to find the best approximation, in the Hilbert Schmidt sense, of arbitrary matrices by operators called frame multipliers, which multiply the analysis coefficents by a fixed symbol followed by a synthesis.

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arXiv2006

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.

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