Tight Gabor Frames Research Articles

Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames

In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} (d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document}), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.

Localization of the window functions of dual and tight Gabor frames generated by the Gaussian function

Gabor frames generated by the Gaussian function are considered. The localization of the window functions of dual frames is estimated in terms of the uncertainty constants, it its dependence on the relation between the parameters of the time-frequency window and the degree of overcompleteness. It is shown that localization worsens rapidly with the increasing disproportion in the parameters of the window. On the other hand, the higher the system of functions forming the frame is overdetermined, the better the window function of the dual frame is localized. For a tight frame the localization of the window function with the same set of parameters is much better than that for the dual frame. This problem is closely related to the problem of interpolation by we have uniform shifts of the Gaussian function. Both the nodal interpolation function and the window function of the dual frame are constructed from the same coefficients. These coefficients play an important role also in the derivation of formulae for the uncertainty constants. This is why their properties related to sign alternation and the monotonicity of decrease of the absolute value are considered in the paper. Bibliography: 38 titles.

Localization of window functions of dual and tight gabor frames generated by the Gaussian function

Рассматриваются фреймы Габора, порожденные функцией Гаусса. С помощью констант неопределенности оценивается локализация функций двойственных фреймов в зависимости от соотношения параметров частотно-временного окна и степени переполненности. Общий вывод таков: при увеличении диспропорции окна локализация быстро ухудшается. С другой стороны, чем более переопределена исходная система функций, тем лучше локализованы функции двойственного фрейма. Для жесткого фрейма локализация при одном и том же наборе параметров существенно лучше, чем для двойственного фрейма. Рассматриваемая задача тесно связана с задачей интерполяции по равномерным сдвигам функции Гаусса. Построение узловой функции при интерполяции и функции окна двойственного фрейма осуществляется с помощью одних и тех же коэффициентов. Эти коэффициенты играют важную роль и при выводе формул для констант неопределенности. Поэтому в работе изучаются их свойства, связанные со знакочередуемостью и монотонностью убывания по модулю. Библиография: 38 названий.

Spline function generating Gabor systems on the half real line

Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line is an LCA group under multiplication and the usual topology. This paper addresses spline Gabor frames for , where is the corresponding Haar measure. We introduce the concept of spline functions on by ‐convolution and estimate their Gabor frame sets, that is, lattice sets such that spline generating Gabor systems are frames for . For an arbitrary spline Gabor frame with special lattices, we present its one dual Gabor frame window, which has the same smoothness as the initial window function. For a class of special spline Gabor Bessel sequences, we prove that they can be extended to a tight Gabor frame by adding a new window function, which has compact support and same smoothness as the initial windows. And we also demonstrate that two spline Gabor Bessel sequences can always be extended to a pair of dual Gabor frames with the adding window functions being compactly supported and having the same smoothness as the initial windows.

FTN multicarrier transmission based on tight Gabor frames

A multicarrier signal can be synthesized thanks to a symbol sequence and a Gabor family (i.e., a regularly time–frequency shifted version of a generator pulse). In this article, we consider the case where the signaling density is increased such that inter-pulse interference is unavoidable.Over an additive white Gaussian noise channel, we show that the signal-to-interference-plus-noise ratio is maximized when the transmitter and the receiver use the same tight Gabor frame. What is more, we give practical efficient realization schemes and show how to build tight frames based on usual generators. Theoretical and simulated bit-error probability are given for a non-coded system using quadrature amplitude modulations. Such a characterization is then used to predict the convergence of a coded system using low-density parity-check codes. We also study the robustness of such a system to errors on the received bits in an interference cancellation context.

Constructing Tight Gabor Frames Using CAZAC Sequences

The construction of finite tight Gabor frames plays an important role in many applications. These applications include significant ones in signal and image processing. We explore when constant amplitude zero autocorrelation (CAZAC) sequences can be used to generate tight Gabor frames. The main theorem uses Janssen’s representation and the zeros of the discrete periodic ambiguity function to give necessary and sufficient conditions for determining whether any Gabor frame is tight. The relevance of the theorem depends significantly on the construction of examples. These examples are necessarily intricate, and to a large extent, depend on CAZAC sequences. Finally, we present an alternative method for determining whether a Gabor system yields a tight frame. This alternative method does not prove tightness using the main theorem, but instead uses the Gram matrix of the Gabor system.

On Wilson Bases in $L^2(\mathbb{R}^d)$

A Wilson system is a collection of finite linear combinations of time frequency shifts of a square integrable function. It is well known that, starting from a tight Gabor frame for $L^{2}(\mathbb{R})$ with redundancy 2, one can construct an orthonormal Wilson basis for $L^2(\mathbb{R})$ whose generator is well localized in the time-frequency plane. In this paper we use the fact that a Wilson system is a shift-invariant system to explore its relationship with Gabor systems. Specifically, we show that one can construct $d$-dimensional orthonormal Wilson bases starting from tight Gabor frames of redundancy $2^k$, where $k=1, 2, \hdots, d$. These results generalize most of the known results about the existence of orthonormal Wilson bases.

Relating tribological stimuli to somatosensory electroencephalographic responses.

The present study deals with the extraction of neural correlates evoked by tactile stimulation of the human fingertip. A reciprocal sliding procedure was performed using a home-built tribometer while simultaneously electroencephalographic (EEG) data from the somatosensory cortex was recorded. The tactile stimuli were delivered by a sliding block with equidistant, perpendicular ridges. The experiments were designed and performed in a fully passive way to prevent attentional locked influences from the subjects. In order to improve the signal-to-noise ratio (SNR) of event related single-trials (ERPs), nonlocal means in addition to 2D-anisotropic denoising schemes based on tight Gabor frames were applied. This novel approach allowed for an easier extraction of ERP alternations. A negative correlation between the latency of the P100 component of the resulting brain responses and the intensity of the underlying lateral forces was found. These findings lead to the conclusion that an increasing stimulus intensity results in a decreasing latency of the brain responses.

Certain Characterizations of Tight Gabor Frames on Local Fields

Certain Characterizations of Tight Gabor Frames on Local Fields

Asymptotic Boundary Forms for Tight Gabor Frames and Lattice Localization Domains

We consider Gabor localization operators defined by two parameters, the generating function of a tight Gabor frame , indexed by a lattice , and a domain whose boundary consists of line segments connecting certain points of . We provide an explicit formula for the boundary form , the normalized limit of the projection functional , where are the eigenvalues of the localization operators applied to dilated domains , R is an integer and is the area of the fundamental domain. The boundary form expresses quantitatively the asymptotic interactions between the generating function and the oriented boundary from the point of view of the projection functional, which measures to what degree a given trace class operator fails to be an orthogonal projection. Keeping the area of the localization domain bounded above corresponds to controlling the relative dimensionality of the localization problem.

On R-Duals and the Duality Principle in Gabor Analysis

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et el., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames.

From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa

We discuss an elementary procedure that allows us to construct dual pairs of wavelet frames based on certain dual pairs of Gabor frames and vice versa. The construction preserves tightness of the involved frames. Starting with Gabor frames generated by characteristic functions the construction leads to a class of tight wavelet frames that include the Shannon (orthonormal) wavelet, and applying the construction to Gabor frames generated by certain exponential B-splines yields wavelet frames generated by functions whose Fourier transforms are compactly supported splines with geometrically distributed knot sequences. On the other hand, the pendant of the Meyer wavelet turns out to be a tight Gabor frame generated by a C∞(R) function with compact support. As an application of our results we show that for each given pair of bandlimited dual wavelet frames it is possible to construct dual wavelet frames for any desired scaling and translation parameters.

Density results for subspace multiwindow Gabor systems in the rational case

Let \(\mathbb{S}\) be a periodic set in ℝ and \(L^2 \left( \mathbb{S} \right)\) be a subspace of L2(ℝ). This paper investigates the density problem for multiwindow Gabor systems in \(L^2 \left( \mathbb{S} \right)\) for the case that the product of timefrequency shift parameters is a rational number. We derive the density conditions for a multiwindow Gabor system to be complete (a frame) in \(L^2 \left( \mathbb{S} \right)\). Under such conditions, we construct a multiwindow tight Gabor frame for \(L^2 \left( \mathbb{S} \right)\) with window functions being characteristic functions. We also provide a characterization of a multiwindow Gabor frame to be a Riesz basis for \(L^2 \left( \mathbb{S} \right)\), and obtain the density condition for a multiwindow Gabor Riesz basis for \(L^2 \left( \mathbb{S} \right)\).

A Geometric Construction of Tight Multivariate Gabor Frames with Compactly Supported Smooth Windows

Fundamental domains of pairs of lattices were used by Han and Wang to construct multivariate Gabor frames for separable lattices. We build upon their results to obtain Gabor frames with smooth and compactly supported window functions. Our results are applicable, for example, if certain pairs of lattices with equal density allow for a common compact and star-shaped fundamental domain.

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)

On the Duality Principle by Casazza, Kutyniok, and Lammers

The R-dual sequences of a frame {fi}i∈I, introduced by Casazza, Kutyniok and Lammers in (J. Fourier Anal. Appl. 10(4):383–408, 2004), provide a powerful tool in the analysis of duality relations in general frame theory. In this paper we derive conditions for a sequence {ωj}j∈I to be an R-dual of a given frame {fi}i∈I. In particular we show that the R-duals {ωj}j∈I can be characterized in terms of frame properties of an associated sequence {ni}i∈I. We also derive the duality results obtained for tight Gabor frames in (Casazza et al. in J. Fourier Anal. Appl. 10(4):383–408, 2004) as a special case of a general statement for R-duals of frames in Hilbert spaces. Finally we consider a relaxation of the R-dual setup of independent interest. Several examples illustrate the results.

Projections in noncommutative tori and Gabor frames

We describe a connection between two seemingly different problems: (a) the construction of projections in noncommutative tori and (b) the construction of tight Gabor frames for . The present investigation relies on interpretation of projective modules over noncommutative tori in terms of Gabor analysis. The main result demonstrates that Rieffel's condition on the existence of projections in noncommutative tori is equivalent to the Wexler-Raz biorthogonality relations for tight Gabor frames. Therefore we are able to invoke results on the existence of Gabor frames in the construction of projections in noncommutative tori. In particular, the projection associated with a Gabor frame generated by a Gaussian turns out to be Boca's projection. Our approach to Boca's projection allows us to characterize the range of existence of Boca's projection. The presentation of our main result provides a natural approach to the Wexler-Raz biorthogonality relations in terms of Hilbert -modules over noncommutative tori.

Optimized Paraunitary Filter Banks for Time-Frequency Channel Diagonalization

We adopt the concept of channel diagonalization to time-frequency signal expansions obtained by DFT filter banks. As a generalization of the frequency domain channel representation used by conventional orthogonal frequency-division multiplexing receivers, the time-frequency domain channel diagonalization can be applied to time-variant channels and aperiodic signals. An inherent error in the case of doubly dispersive channels can be limited by choosing adequate windows underlying the filter banks. We derive a formula for the mean-squared sample error in the case of wide-sense stationary uncorrelated scattering (WSSUS) channels, which serves as objective function in the window optimization. Furthermore, an enhanced scheme for the parameterization of tight Gabor frames enables us to constrain the window in order to define paraunitary filter banks. We show that the design of windows optimized for WSSUS channels with known statistical properties can be formulated as a convex optimization problem. The performance of the resulting windows is investigated under different channel conditions, for different oversampling factors, and compared against the performance of alternative windows. Finally, a generic matched filter receiver incorporating the proposed channel diagonalization is discussed which may be essential for future reconfigurable radio systems.

Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant

We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L 2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in ℝ2 which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on ℝ2n with the property that the short-time Fourier transform defines an isometric embedding from L 2(ℝ n ) to L μ 2 (ℝ2n ) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point.

Support size restrictions on time‐frequency representations of functions on finite abelian groups

AbstractWe obtain uncertainty principles for finite abelian groups that relate the cardinality of the support of a function to the cardinality of the support of its short–time Fourier transform. These uncertainty principles are based on well–established uncertainty principles for the Fourier transform. In terms of applications, the uncertainty principle for the short–time Fourier transform implies the existence of a class of equal norm tight Gabor frames that are maximally robust to erasures. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)