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Fourier Analysis

Fourier Analysis. PSCI 702 November 2, 2005. Even and Odd Functions. Even Functions. Odd Functions. Even and Odd Functions. Kronecker’s Rule. Periodic Functions. Trigonometric System. Trigonometric System of period 2a. Fourier Series.

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Fourier Analysis

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  1. Fourier Analysis PSCI 702 November 2, 2005

  2. Even and Odd Functions

  3. Even Functions

  4. Odd Functions

  5. Even and Odd Functions

  6. Kronecker’s Rule

  7. Periodic Functions

  8. Trigonometric System

  9. Trigonometric System of period 2a

  10. Fourier Series • The basic Idea behind Fourier series is to express a periodic function in terms of trigonometric system using the orthogonality relations.

  11. Fouriser Coefficient

  12. Fouriser Coefficient

  13. Example for Fourier series source: f(x) =|x|, -8 <= x <= 8 n= 5 n= 1 n= 3 y Successive approximations of f(x) f(x) x

  14. Examples

  15. Even & Odd Extensions • Let the function f be defined on (0,a). The even extension feand odd extension fo of f are the following functions

  16. Even periodic extension

  17. Odd periodic extension

  18. Square Wave Example 1 + 2 + 3 1 + 2 1 1 0 -  2 3 1 + 2 + 3 + 4 + 5 1 + 2 + 3 + 4 Square wave: Y = 0 for - < x < 0 and Y=1 for 0 < x <  Y = 1/2 + 2/pi( sinx + sin3x/3 + sin5x/5 + sin7x/7 … + sin(2m+1)x/(2m+1) + …) 1 2 3 4 5 May do with sum of cosines too.

  19. The Euler identity: The inverse equations: , Using the formulas above and some properties of exponential function, the Fourier series can also be written as an expansion in terms of complex exponentials as: , Complex version of the Fourier expansion

  20. Fourier Transform • Let f is a piecewise smooth function defined over R. Since f may not be periodic, we define fL as the peridic extension of f over (-L,L). fL can be expressed as:

  21. Fourier Transform

  22. Fourier Transform

  23. Fourier Transform: Properties #1 Linearity Nth Derivative

  24. Fourier Transform: Properties #2 Convolution Translation: x-shift & -shift

  25. Convolution illustrated

  26. Table of Fourier Transforms

  27. Table of Fourier Transforms

  28. F(w) f(t) t F 1 w t -t/2 0 t/2 -6p -4p -2p 2p 4p 6p 0 t t t t t t Fourier Transform Pairs

  29. F Fourier Transform Pairs F(w) f(t) (p) (p) t w 0 -w0 w0 0

  30. t F(w) w -6p -4p -2p 2p 4p 6p t t t t t t 0 Duality • Forward/inverse transforms are similar • Example: rect(t/t)  t sinc(wt / 2) • Apply duality t sinc(t t/2)  2 p rect(-w/t) • rect(·) is even t sinc(t t /2)  2 p rect(w/t) f(t) 1 t -t/2 0 t/2

  31. Parseval’s Equality • Suppose that the f is piecewise continous function then:

  32. Famous Fourier Transforms Sine wave Delta function

  33. Famous Fourier Transforms Sinc function Square wave

  34. Famous Fourier Transforms Exponential Lorentzian

  35. Famous Fourier Transforms Gaussian Gaussian

  36. FFT of DHM

  37. FFT of DHM

  38. FFT of DHM

  39. Measuring multiple frequencies

  40. Measuring multiple frequencies

  41. Example problem Find the Fourier transform of

  42. Example problem: Answer. Find the Fourier transform of f(x) = Π(x /4) – Λ(x /2) + .5Λ(x) Using the Fourier transforms of Π and Λ and the linearity and scaling properties, F(u) = 4sinc(4u) - 2sinc2(2u) + .5sinc2(u)

  43. Example problem: Alternative Answer. * –1 -.5 0 .5 1 –2 1 0 1 2 Find the Fourier transform of f(x) = Π(x /4) – 0.5((Π(x /3) * Π(x)) - Using the Fourier transforms of Π and Λ and the linearity and scaling and convolution properties , F(u) = 4sinc(4u) – 1.5sinc(3u)sinc(u)

  44. The discrete Fourier transform Motivation: computer applications of the Fourier transform require that all of the definitions and properties of Fourier transforms be translated into analogous statements appropriate to functions represented by a discrete set of sampling points rather than by continuous functions. Let f(x) be a function.Let {fk = f(xk)} be a set of N function values, k = 0, 1, …, N-1.Let be the separation of the equidistant sampling points.Assumption: N is even. The discrete Fourier transform is: The inverse discrete transform is:

  45. The discrete Fourier transform(2) Let’s examine more closely the formula of the discrete Fourier transform: We know that (it’s called n-th root of unity), so the formula above can be rewritten as:

  46. DFT Example -1 Say we want to perform a 8 point DFT on a discretized version of a continuous input signal having frequency components 1KHz and 2KHz Calculation of Ts : Suppose Period of x(t) = 1/1Khz = 1/1000 8 samples per period => sample time (Ts) = 1/8000 sec Or sample rate = 8000 samples/s t = nTs so n = 0,1,…,7

  47. DFT Example -1 (Contd…) Therefore : DC Component And so on... Where X(k) = (k*8Khz)/8 Actually evaluating we get the values: X(0) = 0 + i 0 (dc) X(2) = 1.414 + i1.414 (2Khz) X(4) = 0 + i0 (4Khz) X(6) = 1.414 – i 1.414 (6Khz) X(1) = 0 – i 4 (1KHz) X(3) = 0 + i 0 (3Khz) X(5) = 0 + i 0 (5Khz) X(7) = 0 + i 4 (7KHz)

  48. Fast Fourier Transform • A direct calculation of N-point DFT requires (N-1)2 multiplications and N(N-1) addition. • The FFT is begun by noting that W0.5N=-1 and splitting the DFT in to two sums

  49. Fast Fourier Transform • Now we halve {fk} into two subsequences, according to whether k is even or odd

  50. FFT

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