Spectrum continuation analysis


Spectrum continuation analysis

Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain.

Recall that a Fourier series is only suitable to the analysis of periodic (or finite-domain) functions "f"("x") with period 2π. It can be expressed as an infinite series of sinusoids:

:f(x) = sum_{n=-infty}^{infty} F_n ,e^{inx}

where F_n is the amplitude of the individual harmonics.

In SCA however, one decomposes the spectrum into optimized discrete frequencies. As a consequence, and as the period of the sampled function is supposed to be infinite or not yet known, each of the discrete periodic functions that compose the sampled function fragment can not be considered to be a multiple of the fundamental frequency:

:f(x) = sum_{n=-infty}^{infty} F_n ,e^{i omega_n x}. As such, SCA does not necessarily deliver 2 pi periodic functions, as would have been the case in Fourier analysis.For real-valued functions, the SCA series can be written as:

:f(x) = sum_{n=0}^inftyleft [A_n cos(omega_n x)+B_n sin(omega_n x) ight] + C(x)

where "A""n" and "B""n" are the series amplitudes. The amplitudes can only be solved if the series of values omega_n is previously optimized for a desired objective function (usually least residuals).C(x) is not necessarily the average value over the sampled interval: one might prefer to include predominant information on the behavior of the offset value in the time domain.

Etymology

SCA deals with the prediction problem of continuing a frequency spectrum beyond a sampled (usually stochastic) time series fragment. Unlike ordinary Fourier analysis that infinitely repeats an observed function period or time domain, SCA filters the exact composing frequencies out of the observed spectrum and let them continue (resp. precede) in the time domain. In the scientific terminology, therefore preference is given to the term "continuation" rather than for instance "extrapolation".

Algorithm

An algorithm is required to cope with several problems: detrending, decomposition, frequency resolution optimization, superposition, transformation and computational efficiency.

* Detrending or trend estimation.

* Decomposition.

Since discrete Fourier transform is inherently related to Fourier analysis, this type of spectral analysis is by definition not suitable for spectrum decomposition in SCA. DFT (or FFT) may provide however an initial approximation, which often speeds up the decomposition.

* Improving frequency resolution.

After decomposition of a discrete frequency, it should be filtered for optimal resolution (i.e. varying three parameters: frequency value, amplitude and phase).

* Transformation.

Spectrum dispersion

Compared to DFT (or FFT), which is characterized by perfect spectral resolution, but poor temporal information, SCA favours temporal information, but yields higher spectrum dispersion. This property shows where the analytic strength of SCA is located. For instance, discrete composing frequency resolution is by definition far better in SCA than in DFT.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • List of Fourier analysis topics — This is an alphabetical list of Fourier analysis topics. See also the list of Fourier related transforms, and the list of harmonic analysis topics. Almost periodic function ATS theorem Autocorrelation Autocovariance Banach algebra Bessel function …   Wikipedia

  • List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • SCA — may refer to: US Government * Stored Communications Act * Electronic Communications Privacy Act Retail Stores: *Super Cheap AutoOrganizations: *Scottish Canoe Association *Secular Coalition for America *Sexual Compulsives Anonymous *Société en… …   Wikipedia

  • eye, human — ▪ anatomy Introduction  specialized sense organ capable of receiving visual images, which are then carried to the brain. Anatomy of the visual apparatus Structures auxiliary to the eye The orbit       The eye is protected from mechanical injury… …   Universalium

  • Major depressive disorder — For other depressive disorders, see Mood disorder. Major Depressive Disorder Classification and external resources …   Wikipedia

  • UNITED STATES OF AMERICA — UNITED STATES OF AMERICA, country in N. America. This article is arranged according to the following outline: introduction Colonial Era, 1654–1776 Early National Period, 1776–1820 German Jewish Period, 1820–1880 East European Jewish Period,… …   Encyclopedia of Judaism

  • literature — /lit euhr euh cheuhr, choor , li treuh /, n. 1. writings in which expression and form, in connection with ideas of permanent and universal interest, are characteristic or essential features, as poetry, novels, history, biography, and essays. 2.… …   Universalium

  • Europe, history of — Introduction       history of European peoples and cultures from prehistoric times to the present. Europe is a more ambiguous term than most geographic expressions. Its etymology is doubtful, as is the physical extent of the area it designates.… …   Universalium


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.