Efficient design of orthonormal wavelet bases for signal representation

The efficient representation of a signal as a linear combination of elementary "atoms" or building blocks is central to much signal processing theory and many applications. Wavelets provide a powerful, flexible, and efficiently implementable class of such atoms. In this paper, we develop an efficient method for selecting an orthonormal wavelet that is matched to a given signal in the sense that the squared error between the signal and some finite resolution wavelet representation of it is minimized. Since the squared error is not an explicit function of the design parameters, some form of approximation of this objective is required if conventional optimization techniques are to be used. Previous approximations have resulted in nonconvex optimization problems, which require delicate management of local minima. In this paper, we employ an approximation that results in a design problem that can be transformed into a convex optimization problem and efficiently solved. Constraints on the smoothness of the wavelet can be efficiently incorporated into the design. We show that the error incurred in our approximation is bounded by a function that decays to zero as the number of vanishing moments of the wavelet grows. In our examples, we demonstrate that our method provides wavelet bases that yield substantially better performance than members of standard wavelet families and are competitive with those designed by more intricate nonconvex optimization methods.     

Class of nonorthogonal transformations for signal processing

A class of nonorthogonal transformations is described in which the reference waveforms are circularly shifted versions of a single sequence. The sequence must have a 2-level auto-correlation functton: Barker and pseudonoise sequences are examples. The transformation is believed to have applications in impulsive-noise reduction.     

Optimized signal expansions for sparse representation

Traditional signal decompositions such as transforms, filterbanks, and wavelets generate signal expansions using the analysis-synthesis setting: the expansion coefficients are found by taking the inner product of the signal with the corresponding analysis vector. In this paper, we try to free ourselves from the analysis-synthesis paradigm by concentrating on the synthesis or reconstruction part of the signal expansion. Ignoring the analysis issue completely, we construct sets of synthesis vectors, which are denoted waveform dictionaries, for efficient signal representation. Within this framework, we present an algorithm for designing waveform dictionaries that allow sparse representations: the objective is to approximate a training signal using a small number of dictionary vectors. Our algorithm optimizes the dictionary vectors with respect to the average nonlinear approximation error, i.e., the error resulting when keeping a fixed number n of expansion coefficients but not necessarily the first n coefficients. Using signals from a Gaussian, autoregressive process with correlation factor 0.95, it is demonstrated that for established signal expansions like the Karhunen-Loeve transform, the lapped orthogonal transform, and the biorthogonal 7/9 wavelet, it is possible to improve the approximation capabilities by up to 30% by fine tuning of the expansion vectors     

Complete Gabor transformation for signal representation

Properties of the Gabor transformation used for image representation are discussed. The properties can be expressed in matrix notation, and the complete Gabor coefficients can be found by multiplying the inverse of the Gabor (1946) matrix and the signal vector. The Gabor matrix can be decomposed into the product of a sparse constant complex matrix and another sparse matrix that depends only on the window function. A fast algorithm is suggested to compute the inverse of the window function matrix, enabling discrete signals to be transformed into generalized nonorthogonal Gabor representations efficiently. A comparison is made between this method and the analytical method. The relation between the window function matrix and the biorthogonal functions is demonstrated. A numerical computation method for the biorthogonal functions is proposed.     

Optimal biorthogonal wavelet bases for signal decomposition

The selection of scaling functions for optimal signal representation by general multidimensional biorthogonal wavelet bases is investigated. Criterion for optimality is the minimization of the mean-square approximation error at each level of the decomposition. Conditions are given under which the approximation error of the decomposition approaches zero as the level increases. Given arbitrary synthesis filters, the optimal corresponding analysis filters are determined. Globally optimal families of filters are also found, and suboptimal linear and nonlinear-phase filters for the realization of the optimal scaling functions are explicitly determined     

On sparse representation in pairs of bases

In previous work, Elad and Bruckstein (EB) have provided a sufficient condition for replacing an l/sub 0/ optimization by linear programming minimization when searching for the unique sparse representation. We establish here that the EB condition is both sufficient and necessary.     

On denoising and best signal representation

We propose a best basis algorithm for signal enhancement in white Gaussian noise. The best basis search is performed in families of orthonormal bases constructed with wavelet packets or local cosine bases. We base our search for the “best” basis on a criterion of minimal reconstruction error of the underlying signal. This approach is intuitively appealing, because the enhanced or estimated signal has an associated measure of performance, namely, the resulting mean-square error. Previous approaches in this framework have focused on obtaining the most “compact” signal representations, which consequently contribute to effective denoising. These approaches, however, do not possess the inherent measure of performance which our algorithm provides. We first propose an estimator of the mean-square error, based on a heuristic argument and subsequently compare the reconstruction performance based upon it to that based on the Stein (1981) unbiased risk estimator. We compare the two proposed estimators by providing both qualitative and quantitative analyses of the bias term. Having two estimators of the mean-square error, we incorporate these cost functions into the search for the “best” basis, and subsequently provide a substantiating example to demonstrate their performance     

Optimized nonorthogonal transforms for image compression

The transform coding of images is analyzed from a common standpoint in order to generate a framework for the design of optimal transforms. It is argued that all transform coders are alike in the way they manipulate the data structure formed by transform coefficients. A general energy compaction measure is proposed to generate optimized transforms with desirable characteristics particularly suited to the simple transform coding operation of scalar quantization and entropy coding. It is shown that the optimal linear decoder (inverse transform) must be an optimal linear estimator, independent of the structure of the transform generating the coefficients. A formulation that sequentially optimizes the transforms is presented, and design equations and algorithms for its computation provided. The properties of the resulting transform systems are investigated. In particular, it is shown that the resulting basis are nonorthogonal and complete, producing energy compaction optimized, decorrelated transform coefficients. Quantization issues related to nonorthogonal expansion coefficients are addressed with a simple, efficient algorithm. Two implementations are discussed, and image coding examples are given. It is shown that the proposed design framework results in systems with superior energy compaction properties and excellent coding results.     

Discrete inverses for nonorthogonal wavelet transforms

Discrete nonorthogonal wavelet transforms play an important role in signal processing by offering finer resolution in time and scale than their orthogonal counterparts. The standard inversion procedure for such transforms is a finite expansion in terms of the analyzing wavelet. While this approximation works quite well for many signals, it fails to achieve good accuracy or requires an excessive number of scales for others. This paper proposes several algorithms that provide more adequate inversion and compares them in the case of Morlet wavelets. In the process, both practical and theoretical issues for the inversion of nonorthogonal wavelet transforms are discussed     

Electromagnetic pulse signal representation using orthonormalLaguerre sequences

The notion of representing discrete-time electromagnetic pulse (EMP) signals using orthonormal Laguerre sequences is introduced. The motivation for doing so is that EMP signals and Laguerre sequences are both characterized by decaying exponentials. It is shown a that very efficient Laguerre representation of EMP signals is possible using this approach     

Loading the bases: a new number representation with applications

A number system has recently been introduced that uses two orthogonal bases (Double-base Number System-DBNS). In its direct form the system provides a very sparse two-dimensional number representation which appears, initially, to be a curiosity. After some research by our group, however, the number system has proved to have some interesting and potentially far-reaching applications. The number system has been extended to more than 2 bases and a logarithmic version, which we refer to as the Multi-dimensional Logarithmic Number System (MDLNS), has also proved useful for implementing digital filters. An important property of the MDLNS, that the computational complexity associated with each base reduces both as the number of bases and as the number of digits (or logarithmic components) increase, gives rise to some simple implementation procedures. In this article we will explore some of the theory of the linear and logarithmic systems associated with this new representation, and provide examples of applications in cryptography and digital filter implementation.     

A generalized uncertainty principle and sparse representation in pairs of bases

An elementary proof of a basic uncertainty principle concerning pairs of representations of R/sup N/ vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pairs of orthonormal bases as overcomplete dictionaries. The main contribution in this paper is the improvement of an important result due to Donoho and Huo (2001) concerning the replacement of the l/sub 0/ optimization problem by a linear programming (LP) minimization when searching for the unique sparse representation.     

A system representation for control, digital signal processing, and estimation

In a recent review article, Willsky describes certain limitations of state-space models for relating digital signal processing research to control research. Here generalized system representations of current interest in algebraic control and estimation research are outlined. These representations also have potential as models of signal processing algorithms.     

利用改进的稀疏表示处理FBG传感信号

基于 压缩感知(CS)的正交匹配追 踪 ( OMP ) 算法,须以稀疏度 确定 为先验条件, 在 实际 应用 中稀疏度 不 易 确定 的情况下, 本文 提出了 稀疏度确 定方法和 二次正交匹配追踪 (TOMP)算法。 先 引入熵权法 采用 多指标融合并结合饱和值点法确定稀疏度 , 然后利用所提 方 法 对实验信号进行重构 。 实验 仿真结果表明: 与同类算法相比,本文所提 TOMP 算法增加 0.1s 运行时间降低了 12～ 22% 的重构误差,更好折中处理了重构误差和时间;与不同类算法相比,本文 所提方法重构的信号信噪比(SNR)最大可提升 22 dB ,且均方根误差(RMSE)降低 0.7,因此去噪效果更优。     

Discrete signal representation by the method of kernel expansion

In digital processing of continuous-time signals it is necessary to represent the signal by a sequence. In this letter a number of transforms are presented, generalizing the Poisson transform, which in addition to generating the sequences they also map continuous time-invariant convolution to discrete shift-invariant convolution.     

Fourier-series representation for phase density of sum of signal, noise and interference

A new Fourier-series expansion of the phase probability density function for the sum of signal, Gaussian noise and co-channel interference is derived. One of several possible applications of the derived expression is given.     

Advanced acoustic microimaging using sparse signal representation for the evaluation of microelectronic packages

Acoustic microimaging (AMI) has been widely used to nondestructively evaluate microelectronic packages for the presence of internal defects. To detect defects in small devices such as /spl mu/BGA, flip-chip, and chip-scale packages, high acoustic frequencies are required for the conventional AMI systems. The acoustic frequency used in practice, however, is limited by its penetration through materials. In this paper, a novel acoustic microimaging technique, which utilizes nonlinear signal processing techniques to improve the resolution and robustness of conventional AMI, is proposed and investigated. The technique is based on the concept of sparse signal representations in overcomplete time-frequency dictionaries. Simulation and experimental results show its super resolution and high robustness.     

Construction of sparse signal representations with adaptive multiscale orthogonal bases

We propose a new approach to construct adaptive multiscale orthonormal (AMO) bases of R^N that provide highly sparse signal representations. Our new multilayer AMO basis design produces a high proportion of small scale vectors. The basis vectors are built from small scale to large scales, layer by layer. For each layer, the basis vector maximizes a p-norm measure of sparsity. We compare the sparsity ratios SR (i.e. the percentage of negligibly small coefficients) obtained with AMO and Daubechies wavelet bases for seven families of piecewise smooth signals with randomly located discontinuities. The signals are composed of polynomial, sinusoidal and exponential pieces. In all cases, AMO bases produce a SR increase ranging from 6% to 37%. AMO bases have three main advantages over wavelets. First, they are found automatically by solving a sequence of optimization problems, which eliminates the problem of selecting a wavelet for a given signal. Second, they can provide a significantly sparser representation. Finally, they have the ability to produce zero coefficients for a larger family of piecewise smooth signals. The drawbacks of AMO bases are computational: the basis computation is more expensive, the basis vectors require storage space and no fast transform is known.     

MSM激光距离成像雷达中线性调频信号的产生

MSM激光距离成像雷达将微波雷达中调频测距原理应用到激光雷达,采用一个线性调频的射频副载波调制激光输出强度。本文提出了雷达中调频信号发生器产生宽带线性调频信号的设计思路,并着重阐述DDS+PLL实现宽带线性调频的过程。首先建立四阶二型锁相环架构,分析其对DDS产生的线性扫频信号实现频率稳态跟踪的可能性,然后进行锁相环锁频的初步仿真并得到环路基本参数,最后在输入DDS产生的线性扫频信号前提下仿真环路的频率跟踪性能。仿真结果表明四阶二型锁相环实现了对DDS产生的线性扫频信号的频率跟踪及频带扩展,满足了雷达距离精度对线性调频信号的宽频带要求。     

Numerical stability of nonorthogonal FDTD methods

In this paper, a sufficient test for the numerical stability of generalized grid finite-difference time-domain (FDTD) schemes is presented. It is shown that the projection operators of such schemes must be symmetric positive definite. Without this property, such schemes can exhibit late-time instabilities. The origin and the characteristics of these late-time instabilities are also uncovered. Based on this study, nonorthogonal grid FDTD schemes (NFDTD) and the generalized Yee (GY) methods are proposed that are numerically stable in the late time for quadrilateral prism elements, allowing these methods to be extended to problems requiring very long-time simulations. The study of numerical stability that is presented is very general and can be applied to most solutions of Maxwell's equations based on explicit time-domain schemes