seismic data processing lecture 4
DESCRIPTION
Fourier series and Fourier transformTRANSCRIPT
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Seismic Data ProcessingLecture 4
Fourier Series and Fourier TransformPrepared by
Dr. Amin E. KhalilSchool of Physics, USM, Malaysia
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Today's Agenda
• Examples on Fourier Series
• Definition of Fourier transform
•Examples on Fourier transform
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Examples on Fourier Series
Example: 1
Solution:
The function f(x) is an odd function, thus the a- terms vanishes and the transform will be:
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Increasing the number of terms we arrive at better approximation.
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Another Example
The function is even function and thus:
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Fourier-Discrete Functions
iN
xi2
.. the so-defined Fourier polynomial is the unique interpolating function to the function f(xj) with N=2m
it turns out that in this particular case the coefficients are given by
,...3,2,1,)sin()(2
,...2,1,0,)cos()(2
1
*
1
*
kkxxfN
b
kkxxfN
a
N
jjj
N
jjj
k
k
)cos(2
1)sin()cos(
2
1)( *
1
1
****0 kxakxbkxaaxg m
m
km kk
... what happens if we know our function f(x) only at the points
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Fourier Spectrum
)(
)(arctan)(arg)(
)()()()(
)()()()(
22
)(
R
IF
IRFA
eAiIRF i
)(
)(
A Amplitude spectrum
Phase spectrum
In most application it is the amplitude (or the power) spectrum that is of interest.
Remember here that we used the properties of complex numbers.
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When does the Fourier transform work?
Gdttf )(
Conditions that the integral transforms work:
f(t) has a finite number of jumps and the limits exist from both sides
f(t) is integrable, i.e.
Properties of the Fourier transform for special functions:
Function f(t) Fouriertransform F(w)
even even
odd odd
real hermitian
imaginary antihermitian
hermitian real
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Some properties of the Fourier Transform
Defining as the FT: )()( Ftf
Linearity
Symmetry
Time shifting
Time differentiation
)()()()( 2121 bFaFtbftaf
)(2)( Ftf
)()( Fettf ti
)()()( Fi
t
tf nn
n
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Time differentiation )()()( Fi
t
tf nn
n
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Examples on Fourier Transform
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Graphically the spectrum is:
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Important applications of FT
• Convolution and Deconvolution
• Filtering
• Sampling of Seismic time series
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Thank you