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Seismic Data Processing
Fourier Series and Fourier Transform
Dr. Amin E. Khalil
School of Physics, USM, Malaysia
• Examples on Fourier Series
• Definition of Fourier transform
•Examples on Fourier transform
Examples on Fourier Series
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
The function is even function and thus:
... what happens if we know our function f(x) only at the points
it turns out that in this particular case the coefficients are given by
f ( x j ) cos( kx j ) ,
k 0 ,1, 2 ,...
f ( x j ) sin( kx j ) ,
k 1, 2 , 3 ,...
.. the so-defined Fourier polynomial is the unique interpolating function to the
function f(xj) with N=2m
g ( x)
cos( kx ) b k sin( kx )
a m cos( kx )
F ( ) R ( ) iI ( ) A ( ) e
A ( ) F ( )
R ( ) I ( )
( ) arg F ( ) arctan
A ( )
i ( )
I ( )
R ( )
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?
Conditions that the integral transforms work:
f(t) has a finite number of jumps and the limits exist from both
f(t) is integrable, i.e.
f ( t ) dt G
Properties of the Fourier transform for special functions:
Some properties of the Fourier Transform
Defining as the FT:
f ( t ) F ( )
af 1 ( t ) bf 2 ( t ) aF1 ( ) bF 2 ( )
f ( t ) 2 F ( )
f (t t ) e
f (t )
F ( )
( i ) F ( )
f (t )
( i ) F ( )