Date post: | 19-Jun-2015 |
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Seismic Data ProcessingLecture 4
Fourier Series and Fourier TransformPrepared by
Dr. Amin E. KhalilSchool of Physics, USM, Malaysia
Today's Agenda
• Examples on Fourier Series
• Definition of Fourier transform
•Examples on Fourier transform
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:
Increasing the number of terms we arrive at better approximation.
Another Example
The function is even function and thus:
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
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.
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
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
Time differentiation )()()( Fi
t
tf nn
n
Examples on Fourier Transform
Graphically the spectrum is:
Important applications of FT
• Convolution and Deconvolution
• Filtering
• Sampling of Seismic time series
Thank you