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Fourier Analysis

It is the study of the way generalfunctionsmay be represented or approximated by sums of simplertrigonometric functions.

In the sciences and engineering, the process of decomposing a function into simpler pieces is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known asFourier synthesis

Ref: http://www.frca.co.uk/images/fourier-1.gifIn mathematics, the termFourier analysisoften refers to the study of both operations. The decomposition process itself is called a Fourier transform.1Ref: http://www.dspguide.com/graphics/F_8_2.gif

Variants of Fourier TransformFourier analysis has many scientific applications in physics, partial differential equations, number theory, signal processing, probability theory, statistics, numerical analysis, acoustics, optics, and other areas.

Applications in signal processing:When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection and/or removal. Some examples include:Equalization of audio recordings with a series of bandpass filtersDigital radio reception with no superheterodyne circuit, as in a modern cell phone or radio scannerApplications:3Application in Reservoir SimulationFourier analysis is proposed as an alternative nonparametric method to simulate streamflows.An observed series is decomposed into its components at various resolutions and then recombined randomly to generate synthetic series.Autocorrelation coefficients and the dependence structure are better preserved when Fourier analysis is used, but the mean and variance remain constant when the simulated and observed series have the same length.Therefore, simulations reflect only the different flow sequences in the case of the Fourier analysis.Fourier analysis is not successful in generating streamflow series that require a wide range of required reservoir capacity, an important statistic in water resources studies.

Autocorrelation: Correlation between the elements of a series and others from the same series separated from them by a given interval. Thecorrelation coefficientcan be used to measure the linear dependence between two random variables. Unlike parametric statistics, nonparametric statistics make no assumptions about the probability distributions of the variables being assessed. The difference between parametric model and non-parametric model is that the former has a fixed number of parameters, while the latter grows the number of parameters with the amount of training data4