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Seismic data processing lecture 4

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Fourier series and Fourier transform

Fourier series and Fourier transform

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  • 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:
  • 4. Increasing the number of terms we arrive at better approximation.
  • 5. Another Example The function is even function and thus:
  • 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. 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. 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. 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.  f (t ) n  Time differentiation t n  (  i  ) F ( ) n
  • 11. Examples on Fourier Transform
  • 12. Graphically the spectrum is:
  • 13. Important applications of FT • Convolution and Deconvolution • Sampling of Seismic time series • Filtering
  • 14. Thank you

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