Stability of Gabor frames with arbitrary sampling points

Summary We study the stability of Gabor frames with arbitrary sampling points in the time-frequency plane, in several aspects. We prove that a Gabor frame generated by a window function in the Segal algebra S₀(R^d) remains a frame even if (possibly) all the sampling points undergo an arbitrary perturbation, as long as this is uniformly small. We give explicit stability bounds when the window function is nice enough, showing that the allowed perturbation depends only on the lower frame bound of the original family and some qualitative parameters of the window under consideration. For the perturbation of window functions we show that a Gabor frame generated by any window function with arbitrary sampling points remains a frame when the window function has a small perturbation in S₀(R^d) sense. We also study the stability of dual frames, which is useful in practice but has not found much attention in the literature. We give some general results on this topic and explain consequences to Gabor frames.     

Gabor (super)frames with Hermite functions

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions H _n . Let h = (H ₀, H ₁, . . . , H _n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices \({\Lambda \subseteq \mathbb{R} ^2}\) such that the Gabor system \({\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}\) is a frame for \({L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}\). As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann–Fock space, and structural results about vector-valued Gabor frames.     

The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

Let K and L be two full-rank lattices in Rd. We give a complete characterization for all the Gabor frames that admit tight dual of the same type. The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time-frequency lattice K×L. Two applications of this characterization were obtained: (i) We are able to prove that every Gabor frame has a tight dual if and only if the volume of K×L is less than or equal to . (ii) We are able to obtain sufficient or necessary conditions for the existence of tight Gabor pseudo-duals for subspace Gabor frames in various cases. In particular, we prove that every subspace Gabor frame has a tight Gabor pseudo-dual if either the volume or v(K×L)?2. Moreover, if K=αZd, L=βZd with αβ=1, then a subspace Gabor frame G(g,L,K) has a tight Gabor pseudo-dual only when G(g,L,K) itself is already tight.     

On Dual Gabor Frame Pairs Generated by Polynomials

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for frames generated by any compactly supported function φ whose integer-translates form a partition of unity, e.g., a B-spline, we construct a class of dual frame generators, formed by linear combinations of translates of φ. This allows us to chose a dual generator with special properties, for example, the one with shortest support, or a symmetric one in case the frame itself is generated by a symmetric function. One of these dual generators has the property of being constant on the support of the frame generator.     

Compactly supported multiwindow dual Gabor frames of rational sampling density

We consider multiwindow Gabor systems (G _N ; a, b) with N compactly supported windows and rational sampling density N/ab. We give another set of necessary and sufficient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski–Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski–Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle (UEP) condition generate a tight Gabor system when restricted on [0, 2] with a?=?1 and b?=?1. As another application, we show that a multiwindow Gabor system (G _N ; 1, 1) forms an orthonormal basis if and only if it has only one window (N?=?1) which is a sum of characteristic functions whose supports ‘essentially’ form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows. (Section 4)     

Irregular Gabor frames and their stability

In this paper we give sufficient conditions for irregular Gabor systems to be frames. We show that for a large class of window functions, every relatively uniformly discrete sequence in with sufficiently high density will generate a Gabor frame. Explicit frame bounds are given. We also study the stability of irregular Gabor frames and show that every Gabor frame with arbitrary time-frequency parameters is stable if the window function is nice enough. Explicit stability bounds are given.

     

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.     

Weighted irregular Gabor tight frames and dual systems using windows in the Schwartz class

We give a characterization for the weighted irregular Gabor tight frames or dual systems in L²(Rn) in terms of the distributional symplectic Fourier transform of a positive Borel measure on R2n naturally associated with the system and the short-time Fourier transform of the windows in the case where the window (or at least one of the windows in the case of dual systems) belongs to S(Rn). This result implies that, for certain classes of windows such as generalized Gaussians or “extreme-value” windows, the only weighted irregular Gabor tight frames (or even dual systems with both windows in the same class) that can be constructed with these windows are the trivial ones, corresponding to the measure μ=1 on R2n. Furthermore, we show that, if a such Gabor system admits a dual which is of Gabor type, then the Beurling density of the associated measure exists and is equal to one.     

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 symmetry group of a finite frame

We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of Vale and Waldron (2005) [12] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and classify some classes of highly symmetric tight frames.     

Variations on a theme of Boole and Stein-Weiss

We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure |E| of such a set. We exploit a rational change of variable of the type used by George Boole in his paper “On the comparison of transcendents, with certain applications to the theory of definite integrals” together with the observation that if two functions f and g have the same Lp norm in a range of exponents p₁<p<p₂ then their distribution functions coincide.     

Gabor frames, unimodularity, and window decay

We study time-continuous Gabor frame generating window functions g satisfying decay properties in time and/or frequency with particular emphasis on rational time-frequency lattices. Specifically, we show under what conditions these decay properties of g are inherited by its minimal dual γ⁰ and by generalized duals γ. We consider compactly supported, exponentially decaying, and faster than exponentially decaying (i.e., decay like |g(t)|≤Ce^{−α|t| 1/α} for some 1/2≤α<1) window functions. Particularly, we find that g and γ⁰ have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e., its determinant is a constant. In the case of integer oversampling, unimodularity of the Zibulski-Zeevi matrix is equivalent to tightness of the underlying Gabor frame. For arbitrary oversampling, we furthermore consider tight Gabor frames canonically associated to window functions g satisfying certain decay properties. Here, we show under what conditions and to what extent the canonically associated tight frame inherits decay properties of g. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Gabor frame operator, on a result by Jaffard, on a functional calculus for Gabor frame operators, on results from the theory of entire functions, and on the theory of polynomial matrices.     

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. In this paper we show a surprising result about the inverse of such operators, if any, 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 ?²${?}^2$. We also introduce a class of frames (called pseudo-coherent frames), which includes Gabor frames and coherent frames, and investigate invertible pseudo-coherent frame multipliers, allowing a classification for frame-type operators for these frames. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case.     

Smooth Functions Associated with Wavelet Sets on ℝ d , d≥1, and Frame Bound Gaps

The theme is to smooth characteristic functions of Parseval frame wavelet sets by convolution in order to obtain implementable, computationally viable, smooth wavelet frames. We introduce the following: a new method to improve frame bound estimation; a shrinking technique to construct frames; and a nascent theory concerning frame bound gaps. The phenomenon of a frame bound gap occurs when certain sequences of functions, converging in L ² to a Parseval frame wavelet, generate systems with frame bounds that are uniformly bounded away from 1. We prove that smoothing a Parseval frame wavelet set wavelet on the frequency domain by convolution with elements of an approximate identity produces a frame bound gap. Furthermore, the frame bound gap for such frame wavelets in L ²(? ^d ) increases and converges as d increases.     

Density, Overcompleteness, and Localization of Frames. II. Gabor Systems

This work develops a quantitative framework for describing the overcompleteness of a large class of frames. A previous article introduced notions of localization and approximation between two frames F = {f_i}_i∈I and E = {e_j}_j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a : I → G. This article shows that those abstract results yield an array of new implications for irregular Gabor frames. Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane. The notions of localization and related approximation properties are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this article, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with most implications shown to be sharp.     

Functional Gabor frame multipliers

A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional Gabor frame multipliers. We prove that a L^∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular and is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which there is a function ∈ L^∞(ℝ) such that {wg_mn} (resp. ωψ_k,ℝ) is a normalized tight frame.     

Generalized super Gabor duals with bounded invertible operators

In this paper, we introduce generalized super Gabor duals with bounded invertible operators by combining ideas concerning super Gabor frames with the idea of g-duals as proposed by Dehgham and Fard in 2013. Given a super Gabor frame and a bounded invertible operator A, we characterize its generalized super Gabor duals with A, and derive a parametric expression of all its generalized super Gabor duals with A. The perturbation of generalized super Gabor duals is considered as well.     

Constructing and evaluating alternative frames of discernment

We construct alternative frames of discernment from input belief functions. We assume that the core of each belief function is a subset of a so far unconstructed frame of discernment. The alternative frames are constructed as different cross products of unions of different cores. With the frames constructed the belief functions are combined for each alternative frame. The appropriateness of each frame is evaluated in two ways: (i) we measure the aggregated uncertainty (an entropy measure) of the combined belief functions for that frame to find if the belief functions are interacting in interesting ways, (ii) we measure the conflict in Dempster’s rule when combining the belief functions to make sure they do not exhibit too much internal conflict. A small frame typically yields a small aggregated uncertainty but a large conflict, and vice versa. The most appropriate frame of discernment is that which minimizes a probabilistic sum of the conflict and a normalized aggregated uncertainty of all combined belief functions for that frame of discernment.     

On some Hermite series identities and their applications to Gabor analysis

We prove some infinite series identities for the Hermite functions. From these identities we disprove the Gabor frame set conjecture for Hermite functions of order \(4m+2\) and \(4m+3\) for \(m\in \{0\} \cup \mathbb {N}\). The results hold not only for Hermite functions, but for two large classes of eigenfunctions of the Fourier transform associated with the eigenvalues \(-1\) and i, and the results indicate that the Gabor frame set of all such functions must have a rather complicated structure.