Balanced multiwavelet bases based on symmetric FIR filters

This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming we wish to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r² scaling functions φ_i,j. This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using the Grobner basis. The minimal-length K-balanced multiwavelet bases based on even-length symmetric FIR filters are better behaved than those based on odd-length symmetric FIR filters, as illustrated by special relations they satisfy and by examples constructed     

基于MATLAR的时域抽样和频域抽样定理验证

MATLAB应用范围广,包括信号和图像处理,通讯、控制系统设计、测试和测量等众多应用领域,是众多领域不可获缺的工具。本文基于MATLAB,对指数、序列信号进行不同频率的时域,频域抽样。根据不同抽样频率下还原的信号与原信号均方差,时域逼近程度的差别来验证时域和频域抽样定理。     

The origins of the sampling theorem

The publications of Claude E. Shannon brought the sampling theorem to the broad attention of communication engineers. This article demonstrates how practicians, theoreticians, and mathematicians discovered the implications of the sampling theorem almost independent of one another     

Scale periodicity and its sampling theorem

The scalar transform is a new representation for signals, offering a perspective that is different from the Fourier transform. We introduce the notion of a scalar periodic function. These functions are then represented through the discrete scale series. We also define the notion of a strictly scale-limited signal. Analogous to the Shannon interpolation formula, we show that such signals can be exactly reconstructed from exponentially spaced samples of the signal in the time domain. As an interesting, practical application, we show how properties unique to the scale transform make it very useful in computing depth maps of a scene     

A generalized version of the sampling theorem

The sampling theorem states that any strictly band-limited signal can be exactly reconstructed from its sample values. This situation is generalized in two directions: signals are considered which are not strictly band-limited and it is supposed that the samples are subject to a certain distortion before being used for signal reconstruction.     

A note on the multidimensional sampling theorem

The multidimensional sampling theorem is extended to allow nonuniform, but periodic, sampling with explicit and simple reconstruction formulas. The extension is applicable to hexagonal and other nonrectangular lattices; such sampling schemes can be as efficient as rectangular sampling.     

Translation invariance and sampling theorem of wavelet

The sampling theorem for wavelet spaces built by Walter (1992) lacks the translation invariance except for Walter's weak translation invariant wavelet, i.e., Meyer's wavelet. Indeed, we must know a priori the shift offset a in the samples {f(n+a);n∈Z}; otherwise, the waveform cannot be recovered since the interpolation function is dependent on this offset. In this correspondence, we generalize our metric functional to metrize weak shiftability and find a somewhat surprising result that the B spline wavelets of order n⩾3 are degenerate shiftable. Thus, we can recover approximately the waveform by double sampling without any information on shift offset a     

Unified fractional Fourier transform and sampling theorem

The fractional Fourier transform (FRT) is an extension of the ordinary Fourier transform (FT). Applying the language of the unified FT, we develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity. The FRT sampling theorem is derived as an extension of its ordinary counterpart     

带通信号的直接采样理论

用基本的数学方法和信号处理理论研究了带通信号的直接采样问题,得到了一般性的结论,进一步丰富和深化了信号的直接采样理论。     

Synchronous sampling theorem for nonlinear systems

It is shown that the sampled output of a nonlinear system with sampled-data input is given by the discrete equivalent of a Volterra series in which the kernels are the sampled Volterra kernels of the system. This theorem is then extended to the common case in which the sampled-data input is applied to the nonlinear system through a zero-order hold.     

A sampling theorem for shift-invariant subspace

A sampling theorem for regular sampling in shift-invariant subspaces is established. The sufficient-necessary condition for which it holds is found. Then, the theorem is modified to the shift sampling in shift-invariant subspaces by using the Zak transform. Finally, some examples are presented to show the generality of the theorem     

Extension of the sampling theorem to exponentially decaying causalfunctions

A simple extension of the sampling theorem to nonbandlimited exponentially decaying functions is reported. After an immediate transformation, logarithmically spaced samples are used to reconstruct the original function. One of the most interesting properties of this method is that it enables measuring the time delay introduced by narrowband systems very accurately     

A sampling theorem for EEG electrode configuration

An analytical tool to help in selecting the number of electrodes required for recording electroencephalogram (EEG) signals is presented. The main assumption made is that the scalp can be modeled as a hemispherical surface. The number of sensors required to sample a surface is derived by using a mean square error (MSE) measure to approximate the continuous potential functions on the hemispherical surface. An algorithm for selecting the number of electrodes for arbitrary head geometries is also proposed. A sampling theorem is then derived with conditions on the sampling points for electrode placement     

On sampling theorem, wavelets, and wavelet transforms

The classical Shannon sampling theorem has resulted in many applications and generalizations. From a multiresolution point of view, it provides the sine scaling function. In this case, for a band-limited signal, its wavelet series transform (WST) coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the Nyquist rate. The authors study the properties of cardinal orthogonal scaling functions (COSF), which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants. They show that COSF with compact support have and only have one possibility which is the Haar pulse. They present a family of COSF with exponential decay, which are generalizations of the Haar function. With these COSF, an application is the computation of WST coefficients of a signal by the Mallat (1989) algorithm. They present some numerical comparisons for different scaling functions to illustrate the advantage of COSF. For signals which are not in multiresolution spaces, they estimate the aliasing error in the sampling theorem by using uniform samples     

基于Simulink的采样定理建模与仿真

Simulink是系统建模和仿真广泛使用的集成环境.本文以Simulink环境为平台,建立采样系统的框图模型,对采样定理进行仿真,得到了仿真波形,并对仿真结果进行了分析.     

Optimizing the multiwavelet shrinkage denoising

Denoising methods based on wavelet domain thresholding or shrinkage have been found to be effective. Recent studies reveal that multivariate shrinkage on multiwavelet transform coefficients further improves the traditional wavelet methods. It is because multiwavelet transform, with appropriate initialization, provides better representation of signals so that their difference from noise can be clearly identified. We consider the multiwavelet denoising by using multivariate shrinkage function. We first suggest a simple second-order orthogonal prefilter design method for applying multiwavelet of higher multiplicities. We then study the corresponding thresholds selection using Stein's unbiased risk estimator (SURE) for each resolution level provided that we know the noise structure. Simulation results show that higher multiplicity wavelets usually give better denoising results and the proposed threshold estimator suggests good indication for optimal thresholds.     

Further results on a series representation related to the sampling theorem

During the period 1968–1971 there were several papers dealing with a class of series representations for stochastic processes on an infinite time interval. In a special case, these series reduce to the sampling theorem for bandlimited functions. The present paper continues the study of these series. Series representations of functions are studied, but most of the results should extend to stochastic processes. Sufficient conditions for convergence of the series are given and a bound on the truncation error of the series is obtained. A Kullback-Leibler measure of the interdependence of the coefficients in the series is found. Some partial results are obtained on the degree to which the coefficients depend on measurements which are localized in time. Some additional special properties of the coefficients associated with bandlimited functions are also discussed.     

On quantization,truncation and jitter errors in the sampling theorem and its generalizations

Any band-limited signal f(t) can, according to the sampling theorem, be exactly reconstructed from its sampled values. If the signal is not necessarily band-limited, an alternative model states that it can be approximately reconstructed from its samples. Such signals can also be approximated by generalized sampling sums which can be interpreted as discretized convolution integrals of Fejér's type. In all three cases the physically realized signal is often only roughly equal to f(t) due to errors caused e.g. by the sampling mechanism.In this paper the following types of errors are treated: (1) round-off error arising when quantized sampled values are used, (2) truncation error, arising when a truncated sum is used for representation, (3) time jitter error, a result of sampling at instants slightly different from the sample values. All proofs employ deterministic methods.     

A sampling theorem for nonstationary random processes (Corresp.)

The well-known sampling theorem for wide-sense stationary random processes is generalized to the class of nonstationary random processes.