Fourier series and Fourier transform

1. Seismic Data Processing
Lecture 4
Fourier Series and Fourier Transform
Prepared by
Dr. Amin E. Khalil
School of Physics, USM, Malaysia

2. Today's Agenda
• Examples on Fourier Series
• Definition of Fourier transform
•Examples on Fourier transform

3. 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:

6. Increasing the number of terms we arrive at
better approximation.

7. Another Example
The function is even function and thus:

10. Fourier-Discrete Functions
... what happens if we know our function f(x) only at the points
xi 
2
i
N
it turns out that in this particular case the coefficients are given by
ak 
*
bk 
*
2
N
2
N
N

f ( x j ) cos( kx j ) ,
k  0 ,1, 2 ,...
f ( x j ) sin( kx j ) ,
k  1, 2 , 3 ,...
j 1
N

j 1
.. the so-defined Fourier polynomial is the unique interpolating function to the
function f(xj) with N=2m
g ( x) 
*
m
1
2
m 1
a0 
*
 a
k 1
*
k

cos( kx )  b k sin( kx ) 
*
1
2
*
a m cos( kx )

11. Fourier Spectrum
F ( )  R ( )  iI ( )  A ( ) e
A ( )  F ( ) 
R ( )  I ( )
2
 ( )  arg F ( )  arctan
A ( )
 ( )
i (  )
2
I ( )
R ( )
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.

12. 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
sides

 f(t) is integrable, i.e.

f ( t ) dt  G  

Properties of the Fourier transform for special functions:
Function f(t)
Fouriertransform F()
even
even
odd
odd
real
hermitian
imaginary
antihermitian
hermitian
real

13. Some properties of the Fourier Transform
Defining as the FT:
 Linearity
 Symmetry
 Time shifting
f ( t )  F ( )
af 1 ( t )  bf 2 ( t )  aF1 ( )  bF 2 ( )
f (  t )  2 F (   )
f (t   t )  e
 f (t )
i  t
F ( )
n
 Time differentiation
t
n
 (  i  ) F ( )
n

14.  f (t )
n
 Time differentiation
t
n
 (  i  ) F ( )
n

15. Examples on Fourier Transform

17. Graphically the spectrum
is:

21. Important applications of FT
• Convolution and Deconvolution
• Sampling of Seismic time series
• Filtering

22. Thank you