Gabor Family Research Articles

Totally positive functions and Gabor frames over rational lattices

We show that for an arbitrary totally positive function g∈L1(R) and αβ rational, the Gabor family {e2πiβltg(t−αk):k,l∈Z} is a frame for L2(R), if and only if αβ<1.

Névföldrajzi kutatások – egy családnév földrajzi és történelmi háttere

Language geography studies the spatiality of language. The examination of the Hungarian (geographic) place and personal names, tribe names as well as the family names, looks back about one and a half centuries. However, today’s name geography has only emerged through digital databases and computer-aided cartography. Geographers, however, have generally not been very concerned with the spatial nature of linguistic phenomena. The following is an attempt to address this gap in the scholarship by focusing on the example of the Gabor family name and its name variants in geographic and historical contexts. Amongst Hungarian family names which have been inherited from the 15th and 16th centuries, Gabor is very common, and the oldest use of which was derived from the father’s personal name. Such is the surname of Gabor, whose spatial distribution and temporal changes are presented in the study, with special attention to historical and political influences. The research was based on three databases: the first census of the country after it was completely liberated from Ottoman rule for the purpose of recording the taxpayer (1720); the Internet database of the First World War Loss List (RadixIndex), and data from Hungarian, Slovak, and Austrian web telephone books (no data from Romania and Serbia).

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.

Analysis of a FTN Multicarrier System: Interference Mitigation Based on Tight Gabor Frames

Cognitive radio applications require flexible waveforms to overcome several challenges such as opportunistic spectrum allocation and white spaces utilization. In this context, multicarrier modulations generalizing traditional cyclic-prefix orthogonal frequency-division multiplexing are particularly justified to fit time-frequency characteristics of the channel while improving spectral efficiency.In our theoretical framework, a multicarrier signal is described as a Gabor family the coefficients of which are the symbols to be transmitted and the generators are the time-frequency shifted pulse shapes to be used. In this article, we consider the case where non-rectangular pulse shapes are used with a signaling density increased such that inter-pulse interference is unavoidable. Such an interference is minimized when the Gabor family used is a tight frame. We show that, in this case, interference can be approximated as an additive Gaussian noise. This allows us to compute theoretical and simulated bit-error-probability for a non-coded system using a quadrature phase-shift keying constellation. Such a characterization is then used in order 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.

Matching Pursuit with Asymmetric Functions for Signal Decomposition and Parameterization.

The method of adaptive approximations by Matching Pursuit makes it possible to decompose signals into basic components (called atoms). The approach relies on fitting, in an iterative way, functions from a large predefined set (called dictionary) to an analyzed signal. Usually, symmetric functions coming from the Gabor family (sine modulated Gaussian) are used. However Gabor functions may not be optimal in describing waveforms present in physiological and medical signals. Many biomedical signals contain asymmetric components, usually with a steep rise and slower decay. For the decomposition of this kind of signal we introduce a dictionary of functions of various degrees of asymmetry – from symmetric Gabor atoms to highly asymmetric waveforms. The application of this enriched dictionary to Otoacoustic Emissions and Steady-State Visually Evoked Potentials demonstrated the advantages of the proposed method. The approach provides more sparse representation, allows for correct determination of the latencies of the components and removes the "energy leakage" effect generated by symmetric waveforms that do not sufficiently match the structures of the analyzed signal. Additionally, we introduced a time-frequency-amplitude distribution that is more adequate for representation of asymmetric atoms than the conventional time-frequency-energy distribution.

Zak transforms and Gabor frames of totally positive functions and exponential B-splines

We study totally positive (TP) functions of finite type and exponential B-splines as window functions for Gabor frames. We establish the connection of the Zak transform of these two classes of functions and prove that the Zak transforms have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on the variation-diminishing property of shifts of exponential B-splines. For the exponential B-spline Bm of order m, we determine a set of lattice parameters α,β>0 such that the Gabor family G(Bm,α,β) of time–frequency shifts e2πilβBm(⋅−kα), k,l∈Z, is a frame for L2(R). By the connection of its Zak transform to the Zak transform of TP functions of finite type, our result provides an alternative proof that TP functions of finite type provide Gabor frames for all lattice parameters with αβ<1. For even two-sided exponentials g(x)=λ2e−λ∣x∣ we find lower frame-bounds A, which show the asymptotically linear decay A∼(1−αβ) as the density αβ of the time–frequency lattice tends to the critical density αβ=1.

CONSTRUCTING, COMMODIFYING, AND CONSUMING INVENTED ETHNIC PROVENANCE AMONG ROMANIAN ROMA

AbstractThis article deals with contemporary examples of fraud in the trade of silver beakers and tankards that are defined as valuable prestige goods by two Roma groups living in central Romania―the Gabor Roma and the Cărhar Roma. It demonstrates how some Gabor Roma construct an invented ethnic provenance for silver objects purchased on the European antiques market to sell them for profit to wealthy Cărhar Roma as precious Gabor family heirlooms. I discuss the techniques of fabricating a history of ownership for these pieces (symbolic patina), aging their surfaces (material patina), and manipulating the context of these transactions. The analysis reveals how the politics of object authenticity function and how the artificially created ethnic provenance attributed to commodities has been converted into financial gain in socialist and postsocialist Romania.

Dilations and Completions for Gabor Systems

Let \(\Lambda=\mathcal{K}\times\mathcal{L}\) be a full rank time-frequency lattice in ℝd×ℝd. In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L2(ℝd) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪j=1NG(gj,Λ)) for L2(ℝd). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L2(ℝd). Related results for affine systems are also discussed.

The undersampled discrete Gabor transform

Conventional studies on discrete Gabor transforms have generally been confined to the cases of critical sampling and oversampling in which the Gabor families span the whole signal space. In this paper, we investigate undersampled discrete Gabor transforms. For an undersampled Gabor triple (g,a,b), i.e. a/spl middot/b>N, we show that the associated generalized dual Gabor window (GDGW) function is the same as the one associated with the oversampled (g,N/b,N/a), except for the constant factor (ab/N). Computations of undersampled Gabor transforms are made possible. By applying the methods (algorithms) developed in oversampled settings, the undersampled GDGW is determined. Then, we are able to obtain the best approximation of a signal x by linear combinations of vectors in the Gabor family.

Generalized dual Gabor atoms and best approximations by Gabor family

Let g be a Gabor window of length N and ( a, b) be a pair of lattice constants, a and b are considered as time and frequency gaps for a TF-lattice with N 2 ab elements. The corresponding time-frequency shifts of g form a so-called Gabor family. In this paper, we investigate the structural properties of the discrete Gabor transforms. We address the problem of finding the best approximation of a signal x ϵ C n by linear combinations of a Gabor family. We consider critical sampling, oversampling and undersampling, and do not assume that the Gabor family is a frame. For the task we determine the (generalized) dual Gabor atom (GDGA). This amounts to determining the pseudoinverse of the Gabor matrix and can be solved by the conjugate-gradient (CG) algorithm with O( N) complexity for fixed lattice constants ( a, b). We provide an easy practical criterion for checking whether a Gabor triple ( g, a, b) generates a Gabor frame or not. We propose an efficient algorithm for estimating the Gabor frame bounds and an algorithm for determining tight Gabor atoms.