fft for data filtering the fourier transformation fourier series discrete ft the trick of fast ft...
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FFT for data filteringFFT for data filteringFFT for data filteringFFT for data filtering
•The Fourier Transformation
•Fourier Series
•Discrete FT
•The trick of Fast FT
•Filter designs
•Examples
Timo Damm, CAU Kiel, [email protected]
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Curso Caracas, 2006
DefinitionsDefinitions
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Curso Caracas, 2006
The Fourier TransformationThe Fourier Transformation
dfefXtx
dtetxfX
ift
ift
2
2
2
1
)(
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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Fourier SeriesFourier Series
Most functions have an approximated Fourier Series representation:
T
n
T
n
nnn
dttnftxT
b
dttnftxT
a
Tftnfbtnfa
atx
0
0
0
0
1000
0
2sin2
2cos2
1,2sin2cos
2
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Fourier Series Example 1Fourier Series Example 1
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Fourier Series Example 2Fourier Series Example 2
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Discrete FTDiscrete FT
1,...,1,0,1
1,...,1,0,
1
0
2
1
0
2
NneXN
x
NnexX
N
j
N
jni
jn
N
j
N
jni
jn
Amplitude and Phase diagram of a Fourier transformed sin-function
Amplitude and Phase diagram of a Fourier Transformed cosine-function
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Discrete FT - problemsDiscrete FT - problems
WelchT
Txxg
HannT
xxg
BartlettT
Txxg
,
21
21
1
,2
cos12
1
,
21
21
1
2
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.
tff Nyquist
2
1max
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The trick of Fast FTThe 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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FFT: The difference in runtimeFFT: The difference in runtime
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The trick of Fast FTThe trick of Fast FT
2
2
2
2
Nn
N
ni
Nn
n
nN
ni
nn
ZeYX
ZeYX
1
0
2.1,...,2,1,0,
N
j
N
nji
jn NnexX
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The trick of Fast FT (sheme)The trick of Fast FT (sheme)
6420 ,,, xxxxy
62 , xxz
76543210 ,,,,,,, xxxxxxxx
40, xxy
7531 ,,, xxxxz
6
2
xz
xy
7
3
xz
xy
5
1
xz
xy
73, xxz 51, xxy
4
0
xz
xy
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The trick of Fast FT (example)The trick of Fast FT (example)
0,1y
0,0,0,1
0,0z
0
0
z
y
0
1
z
y
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FFT as Matrix MultiplicationFFT as Matrix Multiplication
3
2
1
0
010
001
010
001
100
100
001
001
3
1
2
0
3
2
1
0
1
1
1
1111
3
2
1
0
3
2
1
0
3
2
1
0
0
0
0
0
2
2
0
0
3
1
2
0
0
0
0
0
123
202
321
0
0
0
0
9630
6420
3210
0000
x
x
x
x
W
W
W
W
W
W
W
W
X
X
X
X
x
x
x
x
WWW
WWW
WWW
X
X
X
X
x
x
x
x
WWWW
WWWW
WWWW
WWWW
X
X
X
X
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Filter designsFilter 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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How to apply the filter?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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Unfiltered DataUnfiltered Data
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Frequency domainFrequency domain
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Low PassLow Pass
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High PassHigh Pass
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Band PassBand Pass
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Upward ContinuationUpward Continuation
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Downward continuationDownward continuation
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Other Examples #1Other Examples #1
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Other Examples #2Other Examples #2
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Other Examples #3Other Examples #3
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Other Examples #4Other Examples #4
Source: H.W. Lang, FH Flensburg
How to filter the diagonal stripes?
Now simply mask the dominant wavelength spots.
FFT
iFFT
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JAVA FFT-Lab from Dave Hale, Stanford
(http://sepwww.stanford.edu/oldsep/hale/FftLab.html)
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SummarySummary•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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ReferencesReferences
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