FOURIER ANALYSIS OF DETERMINISTIC PROCESS

1. 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

2. Any time series y(t) can be reproduced with a summation of cosines and sines: Fourier series 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 density Both 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.

3. Construct y(t) through infinite Fourier series To obtain coefficients: An and Bn provide a measure of the relative importance of each frequency to the overall signal variability. e.g. if there is much more spectral energy at frequency 1 than at 2

4. Fourier series expressed in compact form:

5. (j)

7. To obtain coefficients: Multiplying data times sin and cos functions picks out frequency components specific to their trigonometric arguments Orthogonality requires that arguments be integer multiples of total record length T = Nt, otherwise original series cannot be replicated correctly Arguments2nj/N, are based on hierarchy of equally spaced frequencies n=2n/Ntand time increment j

8. Steps for computing Fourier coefficients: 1) Calculate arguments nj= 2nj/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 yjcosnj and yjsinnj 3) Increase n and repeat steps 1 and 2. Requires ~N2 operations (multiplication & addition)

9. Cn An Bn

14. m radians