DREAM

1. DREAM IDEA PLAN IMPLEMENTATION

3. Introduction to Image Processing Present to:Amirkabir University of Technology (Tehran Polytechnic) & Semnan University Dr. Kourosh Kiani Email: kkiani2004@yahoo.com Email: Kourosh.kiani@aut.ac.ir Email: Kourosh.kiani@semnan.ac.ir Web: www.kouroshkiani.com

4. Lecture 07 FOURIER SERIES

5. Periodic Functions • Periodic functions play a very important role in the study of dynamical systems • Definition: A function f (t) is said to be periodic of period T, if for all t D(f) , we have • f (t + T) = f (t)

7. Periodic Signals • A periodic signal f(t) Unchanged when time-shifted by one period Two-sided: extent is t (-, ) May be generated by periodically extending one period Area under f(t) over any interval of duration equal to the period is the same; e.g., integrating from 0 to T0 would give the same value as integrating from –T0/2 to T0 /2

8. A Amplitude (volts) Sine Wave 0 -A T=1/f A Square Wave Amplitude (volts) 0 -A T=1/f A Amplitude (volts) Triangular Wave 0 -A T=1/f Samples of periodic signals

10. Examples

11. Examples

12. Common Period Consider two periodic function f1(t) and f2(t) with respectively periods T1 and T2. If the ratio T1/T2 is a rational number, then T=n1T1=n2T2 is a common period of f1(t) and f2(t) If T1/T2 is a rational number, the cos(w1t) and cos(w2t) have a common period. If T1/T2 is irrational, they do not have a common period.

13. Example What is period of the function

14. Example What is period of the function The f(t) is not periodic.

15. Example What is period of the function

16. Example What is period of the function

17. Analytic description of a periodic function

18. Analytic description of a periodic function

19. Analytic description of a periodic function

20. Analytic description of a periodic function

21. Analytic description of a periodic function

22. Orthogonal functions If two different functions f(x) and g(x) are defined on the interval a ≤x≤ b and Then we say that the two functions are orthogonal to each other on the interval a ≤x≤ b. The trigonometric functions sin(nx) and cos(nx) where n=0, 1, 2,…. Form an infinite collection of periodic functions that are mutually orthogonal on the interval , indeed on any interval of width That is

23. Fourier series We turn now to our main objective: the study of properties of linear combinations of the functions These functions are periodic with common period . Therefore, any linear combination is also periodic with period T. The most general function that can be so formed is an infinite sum: This is the Fourier series expansion of f(x) where the an and bn are constants called the Fourier coefficients. But how do we find the values of these constants?

24. Fourier coefficients

25. Fourier coefficients

26. Summary Fourier series

27. Dirichlet Conditions • If a function f(t) is such that: • f(t) is defined, single-valued and periodic • f(t) and f ’(t) have at most a finite number of finite discontinuities over a single period – that is they are piecewise continuous. • Then the series • Where • Converges to f(t)

28. Example

31. 1,0 0.0 -1.0 1.75T 1.50T 0.75T 2.00T 0.25T 1.00T 1.25T 0.0 0.50T 1,0 0.0 -1.0 1.75T 1.50T 0.75T 2.00T 0.25T 1.00T 1.25T 0.0 0.50T Fourier components of square wave

32. 1.5 1.0 0.5 0.0 0 f 2f 3f 4f 1,0 0.0 -1.0 . 1.00T 0.50T 1.50T 1.75T 0.0 0.25T 0.75T 1.25T 2.00T Fourier components and spectra of square wave The spectrumof a signal is a graphical representation of the relative amounts of signal as a function of frequency. Modified from Stallings (2000) fig. 3.5

33. 1,0 0.0 -1.0 1.75T 1.50T 0.75T 2.00T 0.25T 1.00T 1.25T 0.0 0.50T 1,0 0.0 -1.0 0.0 0.25T 0.50T 0.75T 1.00T 1.25T 1.50T 1.75T 2.00T Fourier components of a square wave

34. 1.5 1.0 0.5 0.0 0 1f 2f 3f 4f 5f 6f 7f 8f 9f 1,0 0.0 -1.0 1.75T 1.50T 0.75T 2.00T 0.25T 1.00T 1.25T 0.0 0.50T Fourier components and spectra of square wave

35. Spectrum 9f 7f 3f 5f f 9f 7f 3f 5f f Fourier components and spectra of wave

36. 1.5 1.0 0.5 0.0 0 1f 2f 3f 4f 5f 6f 7f 8f 9f Bandwidth = 7f –f = 6f 1,0 0.0 -1.0 1.75T 1.50T 0.75T 2.00T 0.25T 1.00T 1.25T 0.0 0.50T The absolute bandwidth of a signal is the width of the spectrum, that is the width of the frequency range containing any signal The effective bandwidth of a signal is the width of the portion of the spectrum containing ‘significant’ energy

41. Ideally need infinite terms.

42. Fourier series with n = 20 Fourier series with n = 100

44. Example. Find the Fourier series of the function Answer. Since f(x) is odd, then an = 0, for n≥0. We turn our attention to the coefficients bn. For any n≥0 , we have

45. Example. Find the Fourier series of the function

46. Example. Find the Fourier series of the function Since this function is the function of the example above minus the constant   . So Therefore, the Fourier series of f(x) is

47. Example. Find the Fourier series of the function

48. f(t) 1 e-t/2 -p 0 p Example • Fundamental period T0 = p • Fundamental frequency f0 = 1/T0 = 1/ p Hz w0 = 2 p /T0 = 2 rad/s

49. f(t) A -1 0 1 -A Example • Fundamental period T0 = 2 • Fundamental frequency f0 = 1/T0 = 1/2 Hz w0 = 2p/T0 = p rad/s

50. First observed by Michelson, 1898. Explained by Gibbs. • Max overshoot pk-to-pk = 8.95% of discontinuity magnitude. • Just a minor annoyance. • FS converges to (-1+1)/2 = 0 @ discontinuities, in this case. Gibbs phenomenon Overshoot exist @ each discontinuity