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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

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Curso Caracas, 2006 Definitions

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Curso Caracas, 2006 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.

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

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Curso Caracas, 2006 Fourier Series Example 1

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Curso Caracas, 2006 Fourier Series Example 2

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

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Curso Caracas, 2006 Discrete FT - problems Nonperiodic functions can be better handled using window functions, bringing the function down to 0 at both ends. 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.

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Curso Caracas, 2006 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

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Curso Caracas, 2006 FFT: The difference in runtime

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Curso Caracas, 2006 The trick of Fast FT

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

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

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Curso Caracas, 2006 FFT as Matrix Multiplication

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

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

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Curso Caracas, 2006 Unfiltered Data

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Curso Caracas, 2006 Frequency domain

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Curso Caracas, 2006 Low Pass

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Curso Caracas, 2006 High Pass

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Curso Caracas, 2006 Band Pass

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Curso Caracas, 2006 Upward Continuation

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Curso Caracas, 2006 Downward continuation

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Curso Caracas, 2006 Other Examples #1

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Curso Caracas, 2006 Other Examples #2

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Curso Caracas, 2006 Other Examples #3

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

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

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Curso Caracas, 2006 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.

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Curso Caracas, 2006 References 1.Buttkus: Spectral Analysis and Filter Theory in Applied Geophysics, 2000, Springer-Verlag, Berlin, Germany (ISBN: 3-540-62674-3) 2.Brigham: FFT – Schnelle Fourier-Tranformation, 1985, R. Oldenbourg Verlag, Munich, Germany (ISBN: 3-486- 25862-1) 3.Götze, Barrio-Alvers, Schmidt, Alvers: Curso de postgrado: Los métodos potenciales en la interpretación geológica – geofísica integrada, 1996, Universidad Nacional de La Plata, Argentina 4.http://www.iti.fh- flensburg.de/lang/algorithmen/fft/fft.htm

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