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Fourier series With coefficients:. Complex Fourier series Fourier transform (transforms series from time to frequency domain) Discrete Fourier transform.

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Presentation on theme: "Fourier series With coefficients:. Complex Fourier series Fourier transform (transforms series from time to frequency domain) Discrete Fourier transform."— Presentation transcript:

1 Fourier series With coefficients:

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

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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

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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

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27 Probability Degrees of freedom

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29 Includes low frequency N=1512

30 Excludes low frequency N=1512

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