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Fourier series

Fourier series. With coefficients:. Complex Fourier series. Fourier transform (transforms series from time to frequency domain). Discrete Fourier transform. Discrete Fourier transform. Red Spectrum. Wind velocity spectrum. http://www.acoustics.org/press/154th/webster.html.

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Fourier series

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  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. Red Spectrum Wind velocity spectrum http://www.acoustics.org/press/154th/webster.html

  5. Blue Spectrum www.ifm.zmaw.de/research/remote-sensing-assimilation/research-in-the-lab/gas-transfer/

  6. White Spectrum Noise http://clas.mq.edu.au/speech/perception/workshop_masking/introduction.html

  7. violet blue white pink red By Mwchalmers - Created using Cnoise (a set of realtime noise generation algorithms written for scientific purposes), CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=41544754

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

  9. 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.

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

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

  12. 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

  13. 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

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

  15. 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

  16. Hanning Window Hamming Window Kaiser-Bessel, α = 2 Kaiser-Bessel, α = 3 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

  17. 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

  18. 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

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

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

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

  22. 5. Rescale Fourier transform by multiplying: times 8/3 for the Hanning Window times 2.5164 for the Hamming Window times ~8/3 for the Kaiser-Bessel (Depending on alpha)

  23. 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

  24. 0.995 0.990 0.975 0.950 0.900 0.750 0.500 0.250 0.100 0.050 0.025 0.010 0.005 7.88 6.63 5.02 3.84 2.71 1.32 0.45 0.10 0.02 0.00 0.00 0.00 0.00 10.60 9.21 7.38 5.99 4.61 2.77 1.39 0.58 0.21 0.10 0.05 0.02 0.01 12.84 11.34 9.35 7.81 6.25 4.11 2.37 1.21 0.58 0.35 0.22 0.11 0.07 14.86 13.28 11.14 9.49 7.78 5.39 3.36 1.92 1.06 0.71 0.48 0.30 0.21 16.75 15.09 12.83 11.07 9.24 6.63 4.35 2.67 1.61 1.15 0.83 0.55 0.41 18.55 16.81 14.45 12.59 10.64 7.84 5.35 3.45 2.20 1.64 1.24 0.87 0.68 20.28 18.48 16.01 14.07 12.02 9.04 6.35 4.25 2.83 2.17 1.69 1.24 0.99 21.95 20.09 17.53 15.51 13.36 10.22 7.34 5.07 3.49 2.73 2.18 1.65 1.34 23.59 21.67 19.02 16.92 14.68 11.39 8.34 5.90 4.17 3.33 2.70 2.09 1.73 25.19 23.21 20.48 18.31 15.99 12.55 9.34 6.74 4.87 3.94 3.25 2.56 2.16 26.76 24.72 21.92 19.68 17.28 13.70 10.34 7.58 5.58 4.57 3.82 3.05 2.60 28.30 26.22 23.34 21.03 18.55 14.85 11.34 8.44 6.30 5.23 4.40 3.57 3.07 29.82 27.69 24.74 22.36 19.81 15.98 12.34 9.30 7.04 5.89 5.01 4.11 3.57 31.32 29.14 26.12 23.68 21.06 17.12 13.34 10.17 7.79 6.57 5.63 4.66 4.07 32.80 30.58 27.49 25.00 22.31 18.25 14.34 11.04 8.55 7.26 6.26 5.23 4.60 34.27 32.00 28.85 26.30 23.54 19.37 15.34 11.91 9.31 7.96 6.91 5.81 5.14 35.72 33.41 30.19 27.59 24.77 20.49 16.34 12.79 10.09 8.67 7.56 6.41 5.70 37.16 34.81 31.53 28.87 25.99 21.60 17.34 13.68 10.86 9.39 8.23 7.01 6.26 38.58 36.19 32.85 30.14 27.20 22.72 18.34 14.56 11.65 10.12 8.91 7.63 6.84 40.00 37.57 34.17 31.41 28.41 23.83 19.34 15.45 12.44 10.85 9.59 8.26 7.43 41.40 38.93 35.48 32.67 29.62 24.93 20.34 16.34 13.24 11.59 10.28 8.90 8.03 42.80 40.29 36.78 33.92 30.81 26.04 21.34 17.24 14.04 12.34 10.98 9.54 8.64 44.18 41.64 38.08 35.17 32.01 27.14 22.34 18.14 14.85 13.09 11.69 10.20 9.26 45.56 42.98 39.36 36.42 33.20 28.24 23.34 19.04 15.66 13.85 12.40 10.86 9.89 46.93 44.31 40.65 37.65 34.38 29.34 24.34 19.94 16.47 14.61 13.12 11.52 10.52 48.29 45.64 41.92 38.89 35.56 30.43 25.34 20.84 17.29 15.38 13.84 12.20 11.16 49.64 46.96 43.19 40.11 36.74 31.53 26.34 21.75 18.11 16.15 14.57 12.88 11.81 50.99 48.28 44.46 41.34 37.92 32.62 27.34 22.66 18.94 16.93 15.31 13.56 12.46 52.34 49.59 45.72 42.56 39.09 33.71 28.34 23.57 19.77 17.71 16.05 14.26 13.12 53.67 50.89 46.98 43.77 40.26 34.80 29.34 24.48 20.60 18.49 16.79 14.95 13.79 55.00 52.19 48.23 44.99 41.42 35.89 30.34 25.39 21.43 19.28 17.54 15.66 14.46 56.33 53.49 49.48 46.19 42.58 36.97 31.34 26.30 22.27 20.07 18.29 16.36 15.13 57.65 54.78 50.73 47.40 43.75 38.06 32.34 27.22 23.11 20.87 19.05 17.07 15.82 58.96 56.06 51.97 48.60 44.90 39.14 33.34 28.14 23.95 21.66 19.81 17.79 16.50 60.27 57.34 53.20 49.80 46.06 40.22 34.34 29.05 24.80 22.47 20.57 18.51 17.19 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

  25. N=1512 Includes low frequency

  26. N=1512 Excludes low frequency

  27. N=1512

  28. N=1512

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