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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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