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Fourier Methods

Fourier Methods. 신상민 jjindda@korea.ac.kr 윤상기 lockdown99@korea.ac.kr 최현상 realchs@korea.ac.kr. Fourier Series. For periodic data, it is more appropriate to use sine and cosine functions for the approximation or interpolation Fourier Series

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Fourier Methods

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  1. Fourier Methods 신상민 jjindda@korea.ac.kr 윤상기 lockdown99@korea.ac.kr 최현상 realchs@korea.ac.kr

  2. Fourier Series • For periodic data, it is more appropriate to use sine and cosine functions for the approximation or interpolation • Fourier Series • An investigation into data approximation and interpolation using trigonometric polynomials • The formulas for the coefficients are found by using the appropriate orthogonality results for the sine and cosine function. • Derivation from ‘Euler formulas’

  3. Fourier Series • Example n = 1 n = 2 n = 10 n = 10000

  4. contents • Fourier Approximation and Interpolation • Fast Fourier Transforms for n=2r • Fast Fourier Transforms for General n

  5. Fourier Approximation and Interpolation • Discrete Fourier Approximation • Formulas • If number of samples per period are odd, • If number of samples per period are even,  Derivation of the formulas for the coefficients??

  6. Fourier Approximation and Interpolation • Discrete Fourier Approximation

  7. Fourier Approximation and Interpolation • Example 10.1 Trigonometric Interpolation

  8. Fourier Approximation and Interpolation • Example 10.2 Trigonometric Approximation

  9. Fourier Approximation and Interpolation • 10.1.1 Matlab function for Fourier Interpolation or Approximation

  10. Fourier Approximation and Interpolation • Example 10.3 A step function z = [ 1 1 1 1 0 0 0 0] m=4 [a,b]=Trig_poly(z,m) a = [ 0.5 0.25 0 0.25 0] b = [0 0.6036 0 0.1036 0] m=4 m=3

  11. Fourier Approximation and Interpolation • Example 10.4 Geometric Figures

  12. Fast Fourier Transform

  13. Discrete Fourier Transform • discrete-time Fourier transform • The discrete-time Fourier transform X(ejw) of a sequence x[n] is defined by • discrete Fourier transform • uniformly sampling X(ejω) on the ω-axis between 0 ≤ω≤ 2πat ωk=2πk/N, 0 ≤ k ≤ N-1

  14. Discrete Fourier Transform • Commonly used notation • We can rewrite DFT equation as • Inverse discrete Fourier transform (IDFT)

  15. Discrete Fourier Transform • Matrix Relations • The DFT samples can be expressed in matrix form as • DFT can be computed in O(N2) operations. • FFT can reduce the computational complexity to about O(Nlog2N) operations

  16. contents • Fourier Approximation and Interpolation • Fast Fourier Transforms for n=2r • Fast Fourier Transforms for General n

  17. Fast Fourier Transforms for n = 2r • begin by considering the FFT when n is power of 2, i.e., n=2r • Example of n = 4 • Each value of j can be written in binary form as j=2r-1jr+…22j3+2j2+j1. • We can also write k in binary form, but as k = 2k1+k2

  18. Fast Fourier Transforms for n = 2r • begin by writing out the linear system of equations for the Fourier transform components for the case n=4: • w4 = w0 = 1, and interchanging the order of the second and third equations

  19. Fast Fourier Transforms for n = 2r • We now factor the coefficient matrix • Substituting the factored form of the coefficient matrix into DFT eq.

  20. Fast Fourier Transforms for n = 2r • First we find the product • Then we form the second product

  21. z0 z1 z2 z3 g0 g1 g2 g3 s0 s1 s2 s3 w2 w2 w w3 w2 1st stage 2nd stage Fast Fourier Transforms for n = 2r • Pathways with powers of w on them indicate that the quantity on the left is multiplied by that amount.

  22. Algebraic Form of FFT • Example of n = 4 • to calculate the discrete Fourier transform of the data zk, i.e., • using binary factorization of j and k, we have

  23. Algebraic Form of FFT • We first compute the inner summation for each value of j • Writing the digits so that j is in natural order, we have k = k2+2k1 and j=j1+2j2; the first stage produces the values of s(j1+2k2)

  24. Algebraic Form of FFT • We now compute the outer summation

  25. MATLAB Function for FFT with n = 4

  26. Example – FFT with Four Points

  27. contents • Fourier Approximation and Interpolation • Fast Fourier Transforms for n=2r • Fast Fourier Transforms for General n

  28. FFT for General n • The general FFT does not require the factorization of n • example n=6, r1 = 2 and r2 = 3

  29. FFT for General n • Using preceding factorization of j and k, we have

  30. Example - FFT for Six Data Points • z = [ 0 1 2 3 2 1 ] • compute the inner sum for each pair of values of j1 and k2:

  31. Example - FFT for Six Data Points • compute the outer sum

  32. MATLAB Function for FFT with n=rs

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