FOURIER ANALYSIS OF DETERMINISTIC PROCESS

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FOURIER ANALYSIS OF DETERMINISTIC PROCESS. Fourier Analysis is concerned with orthogonal functions:. Any time series y(t) can be reproduced with a summation of cosines and sines :. Fourier series. Average. Constants – Fourier Coefficients. - PowerPoint PPT Presentation

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TIME SERIES ANALYSISTime series collection of observations in time: x( ti ) x( ti ) discrete time series with tDeterministic process: Can be predicted exactly for all the values of the independent varriable ti Stochastic process: Basically unpredictable most geophysical phenomenaFOURIER ANALYSIS OF DETERMINISTIC PROCESSFourier Analysis is concerned with orthogonal functions:

Any time series y(t) can be reproduced with a summation of cosines and sines:

Fourier seriesAverageConstants Fourier Coefficients

Fourier seriesAny time series y(t) can be reproduced with a summation of cosines and sines:Collection of Fourier coefficients An and Bn forms a periodogram

defines contribution from each oscillatory component n to the total energy of the observed signal power spectral densityBoth An and Bn need to be specified to build a power spectrum periodogram. Therefore, there are 2 dof per spectral estimate for the raw periodogram.

Construct y(t) through infinite Fourier series

An and Bn provide a measure of the relative importance of each frequency to the overall signal variability.

e.g. ifthere is much more spectral energy at frequency 1 than at 2

To obtain coefficients:

Fourier series can also be expressed in compact form:

(j)

SUMMARY

To obtain coefficients:

Multiplying data times sin and cos functions picks out frequency components specific to their trigonometric argumentsOrthogonality requires that arguments be integer multiples of total record length T = Nt, otherwise original series cannot be replicated correctlyArguments2nj/N, are based on hierarchy of equally spaced frequencies n=2n/Nt and time increment j Steps for computing Fourier coefficients:1) Calculate arguments nj = 2nj/N, for each integer j and n = 1.2) For each j = 1, 2, , N evaluate the corresponding cos nj and sin nj ; effect sums of yj cos nj and yj sin nj 3) Increase n and repeat steps 1 and 2.Requires ~N2 operations (multiplication & addition)

AnBnCn

mradians