Quilted Gabor frames – A new concept for adaptive time-frequency representation

Certain signal classes such as audio signals call for signal representations with the ability to adapt to the signalʼs properties. In this article we introduce the new concept of quilted frames, which aim at adaptivity in time-frequency representations. As opposed to Gabor or wavelet frames, this new class of frames allows for the adaptation of the signal analysis to the local requirements of signals under consideration. Quilted frames are constructed directly in the time-frequency domain in a signal-adaptive manner. Validity of the frame property guarantees the possibility to reconstruct the original signal. The frame property is shown for specific situations and the Bessel property is proved for the general setting. Strategies for reconstruction from coefficients obtained with quilted Gabor frames and numerical simulations are provided as well.     

Gabor Frames for Model Sets

Journal of Fourier Analysis and Applications - We generalize three main concepts of Gabor analysis for lattices to the setting of model sets: fundamental identity of Gabor analysis, Janssen’s...     

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.     

Gabor analysis over finite Abelian groups

Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time–frequency plane, are the central topic of this paper. Our generic approach covers both multi-dimensional signals as well as non-separable lattices, and in fact the multi-window case as well. Our generic approach includes most of the fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler–Raz or Tolimieri–Orr, Bastiaans and Van-Leest and others. In our presentation the spreading representation of linear operators between finite-dimensional Hilbert space as well as a symplectic version of Poisson's summation formula over the finite time–frequency plane are essential ingredients. They bring us to the so-called Fundamental Identity of Gabor Analysis. In addition, we highlight projective representations of the time–frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting discussed in this paper these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames, independent of the structure theory theorem for finite Abelian groups.     

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.     

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space.     

Nonstationary Gabor frames for

Recently, continuous‐time nonstationary Gabor (NSG) frames were introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper, we focus on the existence and construction of NSG frames in the discrete‐time setting. We provide existence results for painless NSG frames and for NSG frames with fast decaying window functions. We also construct NSG frames with noncompactly supported window functions from a known painless NSG frame. Some examples are provided to illustrate the general theory.     

Measure functions for frames

This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.     

非均匀Gabor框架的稳定性

研究了当窗函数变化时非均匀Gabor框架的稳定性.对紧支撑Gabor框架,将均匀情况下关于稳定性的结论推广到了非均匀的情况;对一般的Gabor框架,利用W（L^∞,e^1）范数给出了其稳定的一个充分条件.     

Super Gabor frames on discrete periodic sets

Due to its potential applications in multiplexing techniques such as time division multiple access and frequency division multiple access, superframe has interested some mathematicians and engineering specialists. In this paper, we investigate super Gabor systems on discrete periodic sets in terms of a suitable Zak transform matrix, which can model signals to appear periodically but intermittently. Complete super Gabor systems, super Gabor frames and Gabor duals for super Gabor frames on discrete periodic sets are characterized; An explicit expression of Gabor duals is established, and the uniqueness of Gabor duals is characterized. On the other hand, discrete periodic sets admitting complete super Gabor systems, super Gabor frames, super Gabor Riesz bases are also characterized. Some examples are also provided to illustrate the general theory.     

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.     

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.     

Gabor systems on discrete periodic sets

Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χ _E are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory. This work was supported by National Natural Science Foundation of China (Grant No. 10671008), Beijing Natural Science Foundation (Grant No. 1092001), PHR (IHLB) and the project sponsored by SRF for ROCS, SEM of China     

Bessel sequences,wavelet frames,duals and extensions

From the perspectives of duality and extensions, Gabor frames and wavelet frames have contrasting behaviour. Our chief concern here is about duality. Canonical duals of wavelet frames may not be wavelet frames, whereas canonical duals of Gabor frames are Gabor frames. Keeping these in view, we give several constructions of wavelet frames with wavelet canonical duals. For this, a simple characterisation of Bessel sequences and a general commutativity result are given, the former also leading naturally to some extension results.     

Constructing super Gabor frames: the rational time-frequency lattice case

For a time-frequency lattice Λ = A Z d B Z d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det( AB) | = 1 L . The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles R d by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.     

时频分析与算子代数

Gabor分析中几个著名的基本定理(如对偶原理和稠密性定理)与群表示和算子代数理论密切相连.尽管时频分析与算子代数之间的某些联系是Jon von Neumann于1930年代建立的,可是它们在近期得到广泛研究,这主要应归于小波/Gabor理论或更一般的框架理论近二十年的发展.本文将讨论过去几年得到的一些主要结果,同时也给出一些新的结果、解释和问题,我们主要考虑来源于时频分析并能反映与群表示理论存在内在联系的那些结果.特别地,针对群表示的时频分析,将详细说明抽象的对偶原理及其与算子代数理论中几个公开问题的联系.     

Time-domain characterization of multiwindow Gabor systems on discrete periodic sets

This paper addresses multiwindow Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. We give some necessary and/or sufficient conditions for multiwindow Gabor systems to foe frames on discrete periodic sets, and characterize two multiwindow Gabor Bessel sequences to foe dual frames on discrete periodic sets. For a given multiwindow Gabor frame, we derive all its Gabor duals, among which we obtain an explicit expression of the canonical Gabor dual. In addition, we generalize multiwindow Gabor systems to the case of a different sampling rate for each window, and investigate multiwindow Gabor frames and dual frames in this case. We also show the properties of the multiwindow Gabor systems are essentially not changed by replacing the exponential kernel with other kernels.     

Some Results on the Lattice Parameters of Quaternionic Gabor Frames

Gabor frames play a vital role not only in modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture does not hold true in the quaternionic case.     

Deformation of Gabor systems

We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames “without inequalities” from lattices to non-uniform sets.     

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.