Gabor Frames Research Articles

Semi-classical Time-frequency Analysis and Applications

This work represents a first systematic attempt to create a common ground for semi-classical and time-frequency analysis. These two different areas combined together provide interesting outcomes in terms of Schr\"odinger type equations. Indeed, continuity results of both Schr\"odinger propagators and their asymptotic solutions are obtained on $\hbar$-dependent Banach spaces, the semi-classical version of the well-known modulation spaces. Moreover, their operator norm is controlled by a constant independent of the Planck's constant $\hbar$. The main tool in our investigation is the joint application of standard approximation techniques from semi-classical analysis and a generalized version of Gabor frames, dependent of the parameter $\hbar$. Continuity properties of more general Fourier integral operators (FIOs) and their sparsity are also investigated.

A Phase Vocoder Based on Nonstationary Gabor Frames

We propose a new algorithm for time stretching music signals based on the theory of nonstationary Gabor frames (NSGFs). The algorithm extends the techniques of the classical phase vocoder (PV) by incorporating adaptive time-frequency (TF) representations and adaptive phase locking. The adaptive TF representations imply good time resolution for the onsets of attack transients and good frequency resolution for the sinusoidal components. We estimate the phase values only at peak channels and the remaining phases are then locked to the values of the peaks in an adaptive manner. During attack transients we keep the stretch factor equal to one and we propose a new strategy for determining which channels are relevant for reinitializing the corresponding phase values. In contrast to previously published algorithms we use a non-uniform NSGF to obtain a low redundancy of the corresponding TF representation. We show that with just three times as many TF coefficients as signal samples, artifacts such as phasiness and transient smearing can be greatly reduced compared to the classical PV. The proposed algorithm is tested on both synthetic and real world signals and compared with state of the art algorithms in a reproducible manner.

Gabor Frames in $${\ell }^2({\mathbb {Z}})$$ ℓ 2 ( Z ) and Linear Dependence

We prove that an overcomplete Gabor frame in $${\ell }^2({\mathbb {Z}})$$ generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in $${\ell }^2({\mathbb {Z}})$$ with modulation parameter 1 / M and translation parameter N for some $$M,N\in {\mathbb {N}},$$ and generated by a finite sequence g in $${\ell }^2({\mathbb {Z}})$$ with K nonzero entries.

Kernel Theorems for Modulation Spaces

We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces $M^p$ for every $1\leq p\leq\infty$, by the membership of its kernel to (mixed) modulation spaces. Whereas Feichtinger's kernel theorem (which we recapture as a special case) is the modulation space counterpart of Schwartz' kernel theorem for temperate distributions, our results do not have a couterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism upon distribution theory, as already emphasized in Feichtinger's manifesto for a post-modern harmonic analysis, tailored to the needs of mathematical signal processing. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. For operators acting on $M^{p,q}$ a similar characterization is not expected, but sufficient conditions for boundedness can be sated in the same spirit.

Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

Stability results for Gabor frames and the p-order hold models

We prove stability results for a class of Gabor frames in L2(R). We consider window functions in the Sobolev spaces H01(R) and B-splines of order p≥1. Our results can be used to describe the effect of the timing jitters in the p-order hold models of signal reconstruction.

B-Spline Approximations of the Gaussian, their Gabor Frame Properties, and Approximately Dual Frames

We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters $$a,b>0$$ with $$ab<1$$ , the scaled version of $$B_N$$ generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to “almost perfect reconstruction” within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the $$L^p$$ -spaces, as well in the time-domain as in the frequency domain.

Subspace dual super wavelet and Gabor frames

Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L2(R;C L ) setting.

Digital Gabor filters do generate MRA-based wavelet tight frames

Gabor frames, especially digital Gabor filters, have long been known as indispensable tools for local time–frequency analysis of discrete signals. With strong orientation selectivity, tensor products discrete (tight) Gabor frames also see their applications in image analysis and restoration. However, the lack of multi-scale structures existing in MRA-based wavelet (tight) frames makes discrete Gabor frames less effective on modeling local structures of signals with varying sizes. Indeed, historically speaking, it was the motivation of studying wavelet systems. By applying the unitary extension principle on some most often seen digital Gabor filters (e.g. local discrete Fourier transform and discrete Cosine transform), we are surprised to find out that these digital filter banks generate MRA-based wavelet tight frames in square integrable function space, and the corresponding refinable functions and wavelets can be explicitly given. In other words, the discrete tight frames associated with these digital Gabor filters can be used as the filter banks of MRA wavelet tight frames, which introduce both multi-scale structures and fast cascade implementation of discrete signal decomposition/reconstruction. Discrete tight frames generated by such filters with both wavelet and Gabor structures can see their potential applications in image processing and recovery.

Gabor analysis as contraction of wavelets analysis

We use the method of group contractions to relate wavelet analysis and Gabor analysis. Wavelet analysis is associated with unitary irreducible representations of the affine group while the Gabor analysis is associated with unitary irreducible representations of the Heisenberg group. We obtain unitary irreducible representations of the Heisenberg group as contractions of representations of the extended affine group. Furthermore, we use these contractions to relate the two analyses, namely, we contract coherent states, resolutions of the identity, and tight frames. In order to obtain the standard Gabor frame, we construct a family of time localized wavelet frames that contract to that Gabor frame. Starting from a standard wavelet frame, we construct a family of frequency localized wavelet frames that contract to a nonstandard Gabor frame. In particular, we deform Gabor frames to wavelet frames.

2D TM scattering problem for finite dielectric objects in a dielectric stratified medium employing Gabor frames in a domain integral equation.

We present a method to simulate two-dimensional scattering by dielectric objects embedded in a dielectric layered medium with transverse magnetic polarization through a domain integral equation formulation. A mixed spatial-spectral discretization is employed with both a spatial and a spectral representation along the direction of the layer interfaces. In the spectral domain, a discretization on a path through the complex plane is used on which the Green function is well behaved. To calculate the field-material interaction in the spatial domain, an auxiliary field is employed similar to the Li factorization rules. Numerical results show that this auxiliary-field formulation significantly improves accuracy, compared to a formulation that directly employs the electric field.

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.

A domain integral equation approach for simulating two dimensional transverse electric scattering in a layered medium with a Gabor frame discretization

We solve the 2D transverse-electrically polarized domain-integral equation in a layered background medium by applying a Gabor frame as a projection method. This algorithm employs both a spatial and a spectral discretization of the electric field and the contrast current in the direction of the layer extent. In the spectral domain we use a representation on the complex plane that avoids the poles and branchcuts found in the Green function. Because of the special choice of the complex-plane path in the spectral domain and because of the choice to use a Gabor frame to represent functions on this path, fast algorithms based on FFTs are available to transform to and from the spectral domain, yielding an O(Nlog⁡N) scaling in computation time.

A Characterization of Sparse Nonstationary Gabor Expansions

We investigate the problem of constructing sparse time-frequency representations with flexible frequency resolution, studying the theory of nonstationary Gabor frames in the framework of decomposition spaces. Given a painless nonstationary Gabor frame, we construct a compatible decomposition space and prove that the nonstationary Gabor frame forms a Banach frame for the decomposition space. Furthermore, we show that the decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients and we prove an upper bound on the approximation error occurring when thresholding the frame coefficients for signals belonging to the decomposition space.

Density of Gabor Systems Via the Short Time Fourier Transform

We apply a new approach to the study of the density of Gabor systems, and obtain a simple and straightforward proof of Ramanathan and Steger's well known result regarding the density of Gabor frames and Gabor Riesz sequences. Moreover, this point of view allows us to extend this result in several directions. The approach we use was first observed by A. Olevskii and the third author in their study of exponential systems, here we develop and simplify it further.

A study on Gabor frame for estimating instantaneous dynamic characteristics of structures

A structure under damage normally exhibits nonlinear dynamic behaviors and time-dependent stiffness and damping. By estimating instantaneous dynamic characteristics of a structures is generally very useful when assessing structural damage in real application. This study presents a novel and effective approach that accurately estimating instantaneous dynamic characteristic of a structure using time-varying autoregressive exogenous (TVARX) model on Gabor frame. To evaluate the time-varying coefficient matrices of a time-series model, the proposed procedure applies Gaussian-based weighted function and Fourier transform on time series model, then the Gabor-transform-based time series formula could be obtained. The dynamic characteristics of a structure are determined from the coefficient matrices. Numerical analyses demonstrate that the proposed approach is superior to traditional time-frequency analysis in accurately estimating instantaneous dynamic characteristics of a structure. Finally, the proposed approach is applied to process measured data for a steel frame specimen subjected to a series of base excitations in shaking table tests.

Minimal Frame Operator Norms Via Minimal Theta Functions

We investigate sharp frame bounds of Gabor frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper frame bound using a result by Montgomery on minimal theta functions.

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.

Space-variant Gabor decomposition for filtering 3D medical images.

This is an experimental paper in which we introduce the possibility to analyze and to synthesize 3D medical images by using multi-variate Gabor frames with Gaussian windows. Our purpose is to apply a space-variant filter-like operation in the space-frequency domain to correct medical images corrupted by different types of acquisitions errors. The Gabor frames are constructed with Gaussian windows sampled on non-separable lattices for a better packing of the space-frequency plane. An implementable solution for 3D-Gabor frames with non-separable lattice is given and numerical tests on simulated data are presented.

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.