1. TIME SERIES ANALYSIS
Time series – collection of observations in time: x( ti )
x( ti ) discrete time series with Δt
Deterministic process: Can be predicted exactly for all the values of the independent variable ti
Stochastic process: Basically unpredictable – most geophysical phenomena

2. 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

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

4. Construct y(t) through infinite Fourier series
To obtain coefficients, multiply equation above times cos and sin
and integrate over all frequencies:
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

5. Fourier series can also be expressed in compact form:
Phase between each Fourier coefficient Bn and An, or
Phase displacement at the beginning of the time series for each Cn

6. (j)

7. SUMMARY
Divide by 2 for “mathematical convenience”

8. 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t and time increment j

9. 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 yj cos nj and yj sin nj
3) Increase n and repeat steps 1 and 2.
Requires ~N2 operations (multiplication & addition)
Data from James River

10. Cn An Bn

15. m
radians