离散Gabor展开及其关系

Ｇａｂｏｒ 展开是一种分析非平稳信号的工具。为了能进行数值计算,需要对连续 Ｇａｂｏｒ展开进行离散化和有限化。在过采样的一般情况下,给出了有限离散时域 Ｇａｂｏｒ 展开系数与有限离散频域 Ｇａｂｏｒ 展开系数之间的关系,并给出了一个计算实例。     

连续Gabor展开离散化后的混叠问题

为能进行数值计算，需对连续Ｇａｂｏｒ展开离散化有限化。本文就时域与频域两种情况，给出了在过采样率的一般情况下，有限离散化后的Ｇａｂｏｒ展开系数与原连续Ｇａｂｏｒ展开系数之间的关系，并由此推出了离散Ｇａｂｏｒ展开系数不产生混叠的条件——连续Ｇａｂｏｒ展开离散化的采样定理。最后给出了一个计算实例     

临界抽样Gabor展开的非局部性分析

Ｇａｂｏｒ展开是一在时－频混合空间描述信号的非正交展开。由于展开的非正交性，使得展开系数的计算较为困难。现有的关于Ｇａｂｏｒ展开的文献大都集中在讨论Ｇａｂｏｒ展开的计算，而对临界抽样Ｇａｂｏｒ展开的非局部性问题没有给予足够的重视。本文将证明当临界抽样Ｇａｂｏｒ展开的窗函数为连续或对称函数时，Ｇａｂｏｒ展开不仅存在非局部性问题，而且收敛性也得不到保证。同时我们还将给出临界抽样Ｇａｂｏｒ展开非局部性的     

Gabor expansion for adaptive echo cancellation

A good echo cancellation algorithm should have a fast convergence rate, small steady-state residual echo, and less implementation cost. The normalized least mean square (NLMS) adaptive filtering algorithm may not achieve this goal. We show that using the Gabor expansion is a way to achieve this goal. For direct digital signal processing compatibility the Gabor expansion introduced in this paper is for discrete-time signals, although the Gabor expansion also can be used for continuous-time signals. The Gabor expansion can be defined as a discrete-time signal representation in the joint time-frequency domain of a weighted sum of the collection of functions (known as the synthesis functions). There are several design issues in the echo canceller based on the Gabor expansion: the design of the analysis functions for the far-end speech, the design of the analysis functions for the near-end signal containing the echo plus the near-end speech, the design of the adaptive filters in the subsignal path, and the design of the synthesis functions. All the adaptive filters are designed using identical NLMS adaptive filtering algorithms     

一种新颖的求解离散Gabor展开对偶窗的方法

本文给出了一种新颖的简捷的求解离散Gabor展开最佳对偶窗的方法．首先推导了离散Gabor展开的公式，给出了连续Gabor展开和离散Gabor展开间的关系．最后利用连续展开和离散展开间的关系给出了一求解离散Gabor展开最佳对偶窗的表达式及实例．     

Noise reduction for NMR FID signals via Gabor expansion

The parameters in a nuclear magnetic resonance (NMR) free induction decay (FID) signal contain information that is useful in biological and biomedical applications and research. A real time-sampled FID signal is well modeled as a finite mixture of modulated exponential sequences plus noise. The authors propose to use the generalized Gabor expansion for noise reduction, where the generalized Gabor expansion represents a signal in terms of a collection of time-shifted and frequency-modulated versions of a single sequence (prototype sequence). For FID signal-fitting, the authors choose the exponential sequence as the prototype function. Using the generalized Gabor expansion and exponential prototype sequences for FID model-fitting, an NMR FID signal can be-well represented by the Gabor coefficients distributed in the joint time-frequency domain (JTFD). The Gabor coefficients reflect the weights of modulated exponential sequences in a signal. One of the important features is that the nonzero Gabor coefficients of a modulated exponential sequence will span a very small area in the JTFD, whereas the Gabor coefficients of the noise will not. If the exponent constant of the prototype sequence in the generalized Gabor expansion matches that of a modulated exponential sequence in the signal, then only one of the Gabor coefficients is nonzero in the JTFD. This is a very important property since it can be exploited to separate a signal from noise and to estimate modulated exponential sequence parameters     

离散拟正交GaBor展开

利用一般条件的离散Ｚａｋ变换及连续Ｇｏｂｏｒ展开和离散Ｇａｂｏｒ展开间的关系。本文首次提出了在整数倍过抽样条件下，由Ｗｅｙｌ－Ｈｅｉｓｅｎｂｅｒｇ紧框架构造离散ＧＡＢＯＲ展开的方法。     

用Gabo:展开进行地震数据的滤波

Gabor展开是用一组在时域及频域都局部化且具有能量集中性质的函数来展开信号,这种特征使得Gabor展开适于处理那些时间无关或非平稳的信号。利用框架理论,类似于SVD特征映像滤波方法,本文用Gabor展开滤波方法来进行地震信号的去噪处理。仿真结果显示出Gabor展开滤波方法的优越性。     

A discrete fractional Gabor expansion for multi-component signals

Gabor expansion is widely used to represent the time-varying frequency content of non-stationary signals. Recently, new representations are presented on a general non-rectangular time–frequency grid. In this paper, we present a closed-form, discrete fractional Gabor expansion and show that it can be used to estimate a high resolution time–frequency representation for multi-component signals. The proposed expansion uses the discrete fractional Fourier kernel and generates a parallelogram-shaped time–frequency plane tiling. Completeness and biorthogonality conditions of the new expansion are derived. We also present a search algorithm to obtain optimal analysis fraction orders for the compact representation of multi-component signals.     

A parametric class of discrete Gabor expansions

The Gabor expansion and its discretization have been widely studied, and many potential applications have been suggested in various signal processing problems. A new approach to the study of the discrete Gabor expansion (DGE) is introduced and analyzed in detail using the theory of pseudoframe decompositions. A parametric and analytical formula for a class of different Gabor analysis sequences is derived. It is a simple algebraic formula rather than another abstract system of equations. For the first time, the structure of analysis sequences is questioned. We show that while there is a class of infinite analysis sequences that possess the Gabor (translation and complex modulation) structure, there are also infinite analysis sequences of arbitrary forms. Simulation results are provided to demonstrate the proposed algorithms. The study of the DGE by means of the theory of pseudoframe decompositions reveals a much broader mathematical perspective on the DGE. The general algorithm derived provides a feasible platform for optimizations in discrete Gabor expansions arising from various applications. This is an area that can surely be exploited as algorithms of DGEs become known and applications become more and more intensive     

Discrete Gabor transform

A feasible algorithm for implementing the Gabor expansion, the coefficients of which are computed by the discrete Gabor transform (DGT), is presented. For a given synthesis window and sampling pattern, computing the auxiliary biorthogonal function of the DGT is nothing more than solving a linear system. The DGT presented applies for both finite as well as infinite sequences. By exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like DGT is obtained. As the discrete Fourier transform (DFT) is a discrete realization of the continuous-time Fourier transform, similarly, the DGT introduced provides a feasible vehicle to implement the useful Gabor expansion     

An efficient algorithm to compute the complete set of discreteGabor coefficients

The discrete Gabor (1946) transform algorithm is introduced that provides an efficient method of calculating the complete set of discrete Gabor coefficients of a finite-duration discrete signal from finite summations and to reconstruct the original signal exactly from the computed expansion coefficients. The similarity of the formulas between the discrete Gabor transform and the discrete Fourier transform enables one to employ the FFT algorithms in the computation. The discrete 1-D Gabor transform algorithm can be extended to 2-D as well.     

Alternative to Gabor's representation of plane aperture radiation

An exact procedure based on frame discretisation is proposed as an alternative for overcoming the inherent limitations of the well-known Gabor expansion. Numerical comparisons of these two phase-space methods are presented in the context of aperture radiation to demonstrate the stability, efficiency and accuracy of the frame decomposition, thereby emphasising its advantages over the Gabor representation     

Frame analysis of the discrete Gabor-scheme

The properties of the discrete Gabor scheme are considered in the context of oversampling. The approach is based on the concept of frames and utilizes the piecewise finite Zak transform (PFZT). The frame operator is represented as a matrix-valued function in the PFZT domain, and its properties are examined in relation to this function. The frame bounds are calculated by means of the eigenvalues of the matrix-valued function, and the dual frame, which is used in calculation of the expansion coefficients, is expressed by means of the inverse matrix. DFT-based algorithms for computation of the expansion coefficients, and for the reconstruction of signals from these coefficients, are generalized for the case of oversampling of the Gabor space. The algorithms are implemented in an example of representation of a nonstationary signal     

Efficient Parallel Computing-Based Implementation Methods of DCT-Kernel-Based Real-Valued Discrete Gabor Transform and Expansion

Journal of Signal Processing Systems - The existing researches of fast parallel algorithms for DCT-based real-valued discrete Gabor transform and expansion are limited to theoretical analysis. In...     

Adaptive Gabor transformation for image processing

An algorithm that computes the Gabor coefficients of an image is presented. An adaptive filter that uses the complex least mean-square algorithm for their computation is proposed, and its numerical characteristics are discussed. It is shown that the filter is stable under certain conditions. Because the Gabor transformation seems to be an excellent tool for image compression, the efficiency of information coding using the Gabor coefficients is investigated and compared with coding that uses the coefficients of the discrete cosine transformation (DCT). Properties of both transformations are discussed.     

An introduction to discrete finite frames

The frame concept was first introduced by Duffin and Schaeffer (1952), and it is widely used today to describe the behavior of vectors for signal representation. The Gabor (1946) expansion and wavelet transform are two special well-known cases. The goal of this article is to describe the frame theory and introduce a simple tutorial method to find discrete finite frame operators and their frame bounds. An easily implementable method for finding the discrete finite frame and subframe operators has been presented by Kaiser (1994). We introduce the method of Kaiser to compute the discrete finite frame operator. Using subframe operators, the biorthogonal basis and projection vectors in a subspace can be easily calculated. Gabor and wavelet analysis are two popular tools for signal processing, and they can reveal time-frequency distribution for a nonstationary signal. Both schemes can be regarded as signal decompositions onto a set of basis functions, and their basis functions are derived from a single prototype function through simple operations. Therefore, the basis functions used in Gabor and wavelet analysis can be regarded as special frames. For completeness we also make some simple introductions on the results of special frames such as discrete Gabor and wavelet analysis     

Discrete Zak transforms, polyphase transforms, and applications

We consider three different versions of the Zak (1967) transform (ZT) for discrete-time signals, namely, the discrete-time ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex z-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss fundamental properties, relations, and transform pairs of the three discrete ZT versions, and we summarize applications of these transforms. In particular, the discrete-time ZT and the cyclic discrete ZT are important for discrete-time Gabor (1946) expansion (Weyl-Heisenberg frame) theory since they diagonalize the Weyl-Heisenberg frame operator for critical sampling and integer oversampling. The polyphase representation plays a fundamental role in the theory of filter banks, especially DFT filter banks. Simulation results are presented to demonstrate the application of the discrete ZT to the efficient calculation of dual Gabor windows, tight Gabor windows, and frame bounds