FFT for data filtering The Fourier Transformation Fourier

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 Curso Caracas, 2006

The Fourier. Transformation The FT transformsdata from the time domain x(t) to the frequencydomain. X(f) or from space domainf(x) to wavelengthdomain. 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

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

Fourier. Series. Example 1 Curso Caracas, 2006

Fourier. Series. Example 2 Curso Caracas, 2006

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

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

The trickof 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

FFT: The differencein runtime Curso Caracas, 2006

The trickof Fast FT Curso Caracas, 2006

The trickof Fast FT (sheme) Curso Caracas, 2006

The trickof Fast FT (example) Curso Caracas, 2006

FFT as Matrix. Multiplication Curso Caracas, 2006

Filterdesigns In potentialfield analysisone often wantsto seperatethe regionalfrom the local field • High Pass • Low Pass • Band Pass • Upward Continuation • Downward Continuation Curso Caracas, 2006

How to apply the filter? We multiply. X(f) witha specialfunction (Convolution ) to surpressor emphasis particularfrequency ranges. Curso Caracas, 2006

Unfiltered. Data Curso Caracas, 2006

Frequencydomain Curso Caracas, 2006

Low Pass Curso Caracas, 2006

High Pass Curso Caracas, 2006

Band Pass Curso Caracas, 2006

Upward. Continuation Curso Caracas, 2006

Downwardcontinuation Curso Caracas, 2006

Other Examples#1 Curso Caracas, 2006

Other Examples#2 Curso Caracas, 2006

Other Examples#3 Curso Caracas, 2006

Other Examples#4 How to filterthe diagonalstripes? FFT i. FFT Now simplymask the dominant wavelengthspots. Source: H. W. Lang, FH Flensburg Curso Caracas, 2006

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

Summary • Fourier. Transformationis an importanttool for filtering data. • Potentialfield data can be seperatedin local and regional components • Noisereductioncan be performedon seismic/seismolgical data • SAR processingcan be achived • Just the FFT makesthe transformationquick enoughfor processinghuge data sets • Besidesgeoscience, FFT is used for encoding/compression telephone, internet, image andvideo-streams. Curso Caracas, 2006

References 1. Buttkus: Spectral Analysis and Filter Theory in Applied Geophysics, 2000, Springer. Verlag, Berlin, Germany (ISBN: 3540 -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: Cursode postgrado: Los métodospotencialesen la interpretacióngeológica– geofísica integrada, 1996, Universidad Nacionalde La Plata, Argentina 4. http: //www. iti. fh-flensburg. de/lang/algorithmen/fft. htm Curso Caracas, 2006