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FFT for data filtering

FFT for data filtering. The Fourier Transformation Fourier Series Discrete FT The trick of Fast FT Filter designs Examples. Timo Damm, CAU Kiel, tdamm@geophysik.uni-kiel.de. Definitions. The Fourier Transformation.

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FFT for data filtering

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  1. FFT for data filtering The Fourier Transformation Fourier Series Discrete FT The trick of Fast FT Filter designs Examples Timo Damm, CAU Kiel, tdamm@geophysik.uni-kiel.de

  2. Definitions Curso Caracas, 2006

  3. The Fourier Transformation The FT transforms data from the time domain x(t) to the frequency domain X(f) or from space domain f(x) to wavelength domain F(λ). Normally the FT calculation is carried out using complex numbers. We usually consider the amplitude and phase or the real and imaginary part. Curso Caracas, 2006

  4. Fourier Series Most functions have an approximated Fourier Series representation: Curso Caracas, 2006

  5. Fourier Series Example 1 Curso Caracas, 2006

  6. Fourier Series Example 2 Curso Caracas, 2006

  7. Discrete FT Amplitude and Phase diagram of a Fourier Transformed cosine-function Amplitude and Phase diagram of a Fourier transformed sin-function Curso Caracas, 2006

  8. Discrete FT - problems The Nyquist frequency is the limit for the highest transformable sampling frequency. Higher frequencies will be mapped back into the spectrum beginning with small frequencies! If the Nyquist frequency is 5Hz, 8Hz appears like 2Hz and 13Hz as 3Hz. Nonperiodic functions can be better handled using window functions, bringing the function down to 0 at both ends. Curso Caracas, 2006

  9. The trick of Fast FT 1965 published by Cooley & Tukey 1805 Mr. Gauss used already a special shape of the algorithm for calculation asteroid motion! Classical “divide & conquer”-style O(n log(n)) instead of O(n^2) Using the symmetries of the trigonometric functions Curso Caracas, 2006

  10. FFT: The difference in runtime Curso Caracas, 2006

  11. The trick of Fast FT Curso Caracas, 2006

  12. The trick of Fast FT (sheme) Curso Caracas, 2006

  13. The trick of Fast FT (example) Curso Caracas, 2006

  14. FFT as Matrix Multiplication Curso Caracas, 2006

  15. Filter designs In potential field analysis one often wants to seperate the regional from the local field • High Pass • Low Pass • Band Pass • Upward Continuation • Downward Continuation Curso Caracas, 2006

  16. How to apply the filter? We multiply X(f) with a special function (Convolution) to surpress or emphasis particular frequency ranges. Curso Caracas, 2006

  17. Unfiltered Data Curso Caracas, 2006

  18. Frequency domain Curso Caracas, 2006

  19. Low Pass Curso Caracas, 2006

  20. High Pass Curso Caracas, 2006

  21. Band Pass Curso Caracas, 2006

  22. Upward Continuation Curso Caracas, 2006

  23. Downward continuation Curso Caracas, 2006

  24. Other Examples #1 Curso Caracas, 2006

  25. Other Examples #2 Curso Caracas, 2006

  26. Other Examples #3 Curso Caracas, 2006

  27. Other Examples #4 FFT iFFT Now simply mask the dominant wavelength spots. How to filter the diagonal stripes? Source: H.W. Lang, FH Flensburg Curso Caracas, 2006

  28. JAVA FFT-Lab from Dave Hale, Stanford (http://sepwww.stanford.edu/oldsep/hale/FftLab.html) Curso Caracas, 2006

  29. Summary • Fourier Transformation is an important tool for filtering data. • Potential field data can be seperated in local and regional components • Noise reduction can be performed on seismic/seismolgical data • SAR processing can be achived • Just the FFT makes the transformation quick enough for processing huge data sets • Besides geoscience, FFT is used for encoding/compression telephone, internet, image and video-streams. Curso Caracas, 2006

  30. References • Buttkus: Spectral Analysis and Filter Theory in Applied Geophysics, 2000, Springer-Verlag, Berlin, Germany (ISBN: 3-540-62674-3) • Brigham: FFT – Schnelle Fourier-Tranformation, 1985, R. Oldenbourg Verlag, Munich, Germany (ISBN: 3-486-25862-1) • Götze, Barrio-Alvers, Schmidt, Alvers: Curso de postgrado: Los métodos potenciales en la interpretación geológica – geofísica integrada, 1996, Universidad Nacional de La Plata, Argentina • http://www.iti.fh-flensburg.de/lang/algorithmen/fft/fft.htm Curso Caracas, 2006

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