Publication

An Error Analysis for the Sampling of Periodic Signals

Michaël Unser, Thierry Blu, Mathews Jacob
2001
Conference Papers

Abstract

We analyze the representation of periodic signals in a scaling function basis. This representation is sufficiently general to include the widely used approximation schemes like wavelets, splines and Fourier series representation. We derive a closed form expression for the approximation error in the scaling function representation. The error formula takes the simple form of a Parseval like sum, weighted by an appropriate error kernel. This formula may be useful in choosing the right representation for a class of signals. We also experimentally verify the theory in the particular case of description of closed curves.

Official source

https://infoscience.epfl.ch/entities/publication/d8cb8465-f452-4088-9b70-3d782089d347

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Michaël Unser, Thierry Blu, Mathews Jacob
2001
Conference Papers

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/d8cb8465-f452-4088-9b70-3d782089d347

About this result

Related concepts (32)

Approximation error

The approximation error in a data value is the discrepancy between an exact value and some approximation to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value). An approximation error can occur for a variety of reasons, among them a computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.

Fourier series

A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.

Wavelet

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second.

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An Error Analysis for the Sampling of Periodic Signals

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