Download presentation

Presentation is loading. Please wait.

Published byPierce Watkins Modified about 1 year ago

1
Fourier series With coefficients:

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

3

4
Red Spectrum Wind velocity spectrum

5
Blue Spectrum

6
White Spectrum Noise

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

8

9

10

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

16
Cycles per day m 2 /cpd Spectrum of raw data Spectrum of high-pass filtered data

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

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

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

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

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

22
Cycles per day m 2 /cpd Original from Raw Data with Hanning window with Hamming window Windows reduce noise produced by side-lobe effects Noise reduction is effected at different frequencies

23
Cycles per day m 2 /cpd with Hanning window with Hamming and Kaiser- Bessel (α=3) windows

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

26

27
Probability Degrees of freedom

28

29
Includes low frequency N=1512

30
Excludes low frequency N=1512

31

32

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google