This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefitted from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon's sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets. Practically, this allows for simpler-and possibly more realistic-interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.

/pdf/sampling-50-years-after-shannon-1i3rgczukl.pdf

Sampling-50 years after Shannon

The wavelet transform has emerged over recent years as a powerful time-frequency analysis and signal coding tool favoured for the interrogation of complex nonstationary signals. Its application to biosignal processing has been at the forefront of these developments where it has been found particularly useful in the study of these, often problematic, signals: none more so than the ECG. In this review, the emerging role of the wavelet transform in the interrogation of the ECG is discussed in detail, where both the continuous and the discrete transform are considered in turn.

/pdf/wavelet-transforms-and-the-ecg-a-review-36a6zg0xg1.pdf

Wavelet transforms and the ECG: a review.

This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift-invariant subspaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p-spaces.

/pdf/nonuniform-sampling-and-reconstruction-in-shift-invariant-3fggp0zfbi.pdf

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Digital coding of waveforms: Principles and applications to speech and video

Current research in indexing and mining time series data has produced many interesting algorithms and representations. However, the algorithms and the size of data considered have generally not been representative of the increasingly massive datasets encountered in science, engineering, and business domains. In this work, we show how a novel multi-resolution symbolic representation can be used to index datasets which are several orders of magnitude larger than anything else considered in the literature. Our approach allows both fast exact search and ultra fast approximate search. We show how to exploit the combination of both types of search as sub-routines in data mining algorithms, allowing for the exact mining of truly massive real world datasets, containing millions of time series.

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iSAX: indexing and mining terabyte sized time series

This paper presents a new electrocardiogram (ECG) compression method based on orthonormal wavelet transform and an adaptive quantization strategy, by which a predetermined percent root mean square difference (PRD) can be guaranteed with high compression ratio and low implementation complexity.

A wavelet transform-based ECG compression method guaranteeing desired signal quality

From the Paley-Wiener 1/4-theorem, the finite energy signal f(t) can be reconstructed from its irregularly sampled values f(k+/spl delta//sub /spl kappa//) if f(t) is band-limited and sup/sub /spl kappa//|/spl delta//sub /spl kappa//|<1/4. We consider the signals in wavelet subspaces and wish to recover the signals from its irregular samples by using scaling functions. Then the method of estimating the upper bound of sup/sub /spl kappa//|/spl delta//sub /spl kappa//| such that the irregularly sampled signals can be recovered is very important. Following the work done by Liu and Walter (see J. Fourier Anal. Appl., vol.2, no.2, p.181-9, 1995), we present an algorithm which can estimate a proper upper bound of sup/sub /spl kappa//|/spl delta//sub /spl kappa//|. Compared to Paley-Wiener 1/4-theorem, this theorem can relax the upper bound for sampling in some wavelet subspaces.

/pdf/irregular-sampling-theorems-for-wavelet-subspaces-pbrmfnijk4.pdf

Irregular sampling theorems for wavelet subspaces

A necessary and sufficient condition for sampling in the general framework of shift invariant spaces is derived. Then this result is applied, respectively, to the regular sampling and the perturbation of regular sampling in shift invariant spaces. A simple necessary and sufficient condition for regular sampling in shift invariant spaces is attained. Furthermore, an improved estimate for the perturbation is derived for the perturbation of regular sampling in shift invariant spaces. The derived estimate is easy to calculate, and shown to be optimal in some shift invariant spaces. The algorithm to calculate the reconstruction frame is also presented.

Shuichi Itoh

Papers

A wavelet transform-based ECG compression method guaranteeing desired signal quality

Irregular sampling theorems for wavelet subspaces

On sampling in shift invariant spaces

ECG Data Compression by Using Wavelet Transform

A hierarchical group key management scheme for secure multicast increasing efficiency of key distribution in leave operation