1 / 32

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

ciara
Download Presentation

Fourier Methods

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related