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# Seismic data processing lecture 4

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Fourier series and Fourier transform

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### Seismic data processing lecture 4

1. 1. Seismic Data Processing Lecture 4 Fourier Series and Fourier Transform Prepared by Dr. Amin E. Khalil School of Physics, USM, Malaysia
2. 2. Today's Agenda • Examples on Fourier Series • Definition of Fourier transform •Examples on Fourier transform
3. 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:
4. 4. Increasing the number of terms we arrive at better approximation.
5. 5. Another Example The function is even function and thus:
6. 6. 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 )
7. 7. 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.
8. 8. 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
9. 9. 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
10. 10.  f (t ) n  Time differentiation t n  (  i  ) F ( ) n
11. 11. Examples on Fourier Transform
12. 12. Graphically the spectrum is:
13. 13. Important applications of FT • Convolution and Deconvolution • Sampling of Seismic time series • Filtering
14. 14. Thank you