1. Fourier series
With coefficients:

2. Complex Fourier series
Fourier transform
(transforms series from time to frequency domain)
Discrete Fourier transform

3. Discrete Fourier
transform

4. Wind velocity spectrum
Red Spectrum
Wind velocity spectrum

5. Blue Spectrum

6. White Spectrum
Noise

7. Let’s reproduce this function with Fourier coefficients
Real part of Fourier Series (An)

11. What are the dominant frequencies?
Fourier transforms decompose a data sequence into a set of discrete spectral estimates – separate the variance of a time series as a function of frequency.
A common use of Fourier transforms is to find the frequency components of a signal buried in a time domain signal.

12. FAST FOURIER TRANSFORM (FFT)
In practice, if the time series f(t) is not a power of 2, it should be padded with zeros

13. What is the statistical significance of the peaks?
Each spectral estimate has a confidence limit defined by a chi-squared distribution

14. Spectral Analysis Approach
1. Remove mean and trend of time series
2. Pad series with zeroes to a power of 2
3. To reduce end effect (Gibbs’ phenomenon) use a window
(Hanning, Hamming, Kaiser) to taper the series
4. Compute the Fourier transform of the series, multiplied times the window
5. Rescale Fourier transform by multiplying times 8/3 for the Hanning Window
6. Compute band-averages or block-segmented averages
7. Incorporate confidence intervals to spectral estimates

15. July 1, 2007 (day “0” in the abscissa) to September 1, 2007
1. Remove mean and trend of time series (N = 1512)
2. Pad series with zeroes to a power of 2 (N = 2048) m
Sea level at Mayport, FL
July 1, 2007 (day “0” in the abscissa) to September 1, 2007
Raw data and Low-pass filtered data m
High-pass filtered data

16. Spectrum of high-pass filtered data
Spectrum of raw data
m2/cpd
Spectrum of high-pass filtered data
m2/cpd
Cycles per day

17. Hanning Window Hamming Window
Value of the Window
3. To reduce end effect (Gibbs’ phenomenon) use a window (Hanning, Hamming, Kaiser) to taper the series
Day from July 1, 2007

18. Hanning Window Hamming Window Kaiser-Bessel, α = 2
Value of the Window
Day from July 1, 2007
3. To reduce end effect (Gibbs’ phenomenon) use a window (Hanning, Hamming, Kaiser) to taper the series

19. Raw series x Hanning Window (one to one)
4. Compute the Fourier transform of the series, multiplied times the window
Raw series x Hanning Window
(one to one) m
Raw series x Hamming Window
(one to one) m
To reduce side-lobe effects
Day from July 1, 2007

20. High-pass series x Hanning Window (one to one)
4. Compute the Fourier transform of the series, multiplied times the window
High-pass series x Hanning Window
(one to one) m
High pass series x Hamming Window
(one to one) m
To reduce side-lobe effects
Day from July 1, 2007

21. High pass series x Kaiser-Bessel Window α=3 (one to one)m
4. Compute the Fourier transform of the series, multiplied times the window
Day from July 1, 2007

22. Windows reduce noise produced by side-lobe effects
with Hanning window
m2/cpd
Original from Raw Data
Windows reduce noise produced by side-lobe effects
Noise reduction is effected at different frequencies
with Hamming window
m2/cpd
Cycles per day

23. with Hamming and Kaiser- Bessel (α=3) windows
with Hanning window
m2/cpd
with Hamming and Kaiser-
Bessel (α=3) windows
m2/cpd
Cycles per day

24. 5. Rescale Fourier transform by multiplying:
times 8/3 for the Hanning Window
times for the Hamming Window
times ~8/3 for the Kaiser-Bessel (Depending on alpha)

25. 6. Compute band-averages or block-segmented averages
7. Incorporate confidence intervals to spectral estimates
Upper limit:
Lower limit:
1-alpha is the confidence (or probability)
nu are the degrees of freedom
gamma is the ordinate reference value

27.
Probability 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Degrees of freedom

29. Includes low frequency

30. Excludes low frequency

31. N=1512

32. N=1512