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.

/pdf/isax-indexing-and-mining-terabyte-sized-time-series-20pdh1hhla.pdf

iSAX: indexing and mining terabyte sized time series

XML (Extensible Markup Language) is the global standardized form for representing digital data. It is flexible and extensible, but on the other hand uses vast amounts of main memory for data processing, thus representing a burden. In order to resolve this problem when processing data in large XML documents, in this paper the authors propose the “CSV compaction for XML” method to improve data processing performance by using a format conversion for an XML document with a record structure. In this method, redundant elements are packed in CSV (Comma Separated Values) format from the perspective of accessing the XML document prior to data processing. The results of evaluating this method confirmed that memory usage and deployment time were improved in proportion to the percentage of elements that could be packed and eliminated using the CSV format. This method is a reversible conversion for an XML document. Since the authors created software to execute the method as XSL (Extensible Stylesheet Language) Transformation, it can be executed using various programming languages. In this method, specifications that designate elements to be packed in CSV format in a record are needed, and creating them is troublesome. Thus, the authors created a support tool to create the specifications simply using a GUI (Graphical User Interface). As a result of these developments, complex documents with a hierarchical structure and documents with a large number of elements can be readily handled in this method. © 2007 Wiley Periodicals, Inc. Syst Comp Jpn, 38(5): 1– 13, 2007; Published online in Wiley InterScience (). DOI 10.1002sscj.20736

Development of an XML-document compaction method to improve data-processing performance

An Adaptive Noiseless Coding for Sources with Big Alphabet Size

SUMMARY Following our former works on regular sampling in wavelet subspaces, the paper provides two algorithms to estimate the truncation error and aliasing error respectively when the theorem is applied to calculate concrete signals. Furthermore the shift sampling case is also discussed. Finally some important examples are calculated to show the algorithm.

http://wnt.sjtu.edu.cn/papers/ieice_error_estimation.pdf

The Error Estimation of Sampling in Wavelet Subspaces

On the Internet, the prevalent communication style in recent years has been to distribute voice and video information simultaneously to large numbers of receivers by IP multicasting. With the use of broadband Internet, it is likely that a still larger amount of multicast traffic will be transmitted. Thus, monitoring techniques that allow measurement of traffic levels and communications quality for multicast traffic will become important. However, the performance evaluation tools for the multicast traffic hitherto proposed have only the function of acquiring the transmitted datagram traffic, and cannot provide detailed analyses such as the reasons for the generation or termination of datagram transmission. In order to provide a detailed analysis of multicast traffic transmission, it is necessary to analyze the multicast routing protocol for transmission control of the traffic. This study focuses on PIM-SM (protocol-independent multicast-sparse model) multicast routing protocol and establishes the correspondence between PIM-SM messages exchanged between routers and the transmitted multicast traffic. The design and implementation of an analyzer which provides detailed performance evaluation of multicast communications, and the results of multicast analysis using the proposed analyzer, are described. © 2005 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 88(8): 19–29, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.20198

A performance analyzer for multicast traffic behavior analysis based on observing PIM‐SM messages

We propose a method of predicting the number of requests for video titles in an on-demand video delivery system considering their time dependency. To handle many heterogeneous video requests in a video streaming delivery network system using video servers, it is effective to introduce video caching systems in the network system. For the cache system design in video delivery network systems, an effective cache algorithm for predicting how many requests will occur in the future is needed. Since almost all such proposed algorithms are for WWW systems, they do not consider the time dependency of the number of requests, so they are not applicable to a video system with time-dependent requests because a customer in the system watches one video title just one time. In our method, the number of requests is represented by a function of time by considering how a customer obtains information about a video and requests it and by considering that a customer requests each video only once. We have developed a method of calculating the coefficients of the function from the series of the number of requests and for estimating the number of requests in the future. We evaluated the convergence and calculation time. The results show that the calculations converge and the calculation time is 31 ms. They also show that the calculation cost is acceptable for a general video delivery system.

Shuichi Itoh

Papers

Development of an XML-document compaction method to improve data-processing performance

An Adaptive Noiseless Coding for Sources with Big Alphabet Size

The Error Estimation of Sampling in Wavelet Subspaces

A performance analyzer for multicast traffic behavior analysis based on observing PIM‐SM messages

Method of predicting number of on-demand video requests using time series data for video cache system