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Orthonormal Basis Functions

Orthonormal Basis Functions. A set of signals are called orthogonal on an interval ( a , b ) if any two signals and in the set satisfy If the magnitude of each signal is set to one, it is called normalized.

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Orthonormal Basis Functions

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  1. Orthonormal Basis Functions • A set of signals are called orthogonal on an interval (a, b) if any two signals and in the set satisfy • If the magnitude of each signal is set to one, it is called normalized. • A set of normalized orthogonal functions can form an orthonormal basis set。 • The complex exponential signals are orthogonal on any interval over a period . 2.3

  2. Generalized Transformation of Signal • A signal x(t) on any interval over a period T0, i.e., (t0 ,t0+T0 ), can be expressed as following in terms of • Parseval theorem：(Will be discussed later) : Integrateover a period. 2.3

  3. Fourier theory Jean Baptiste Joseph Fourier (1768-1830) proposed that a periodic signal can be represented by a summation of a (possibly infinite) number of sinusoids each with a particular amplitude and phase. Assume that x(t) is a periodic signal with fundamental period T0, then x(t) can be represented as The series is called Trigonometric Fourier series because it is represented in terms of sinusoids. or 4.3

  4. Coefficients of trigonometric Fourier series (1/7) Given a periodic signal with fundamental period T0 that can be represented by the trigonometric Fourier series The problem is how to find the coefficients a0, an and bn. We begin with a0 Integrating the series term by term over one period T0, we obtain Note：Integrating the sinusoidal over one or integer number of periods is 0. 4.3

  5. Coefficients of trigonometric Fourier series (2/7) Find coefficients an Multiplying both sides of the series equation by cos k0t and integrating over one period T0, we obtain a0 is the average value of the waveform (signal). 4.3

  6. Coefficients of trigonometric Fourier series (3/7) The terms I1 and I3 will be discussed in details later 4.3

  7. Coefficients of trigonometric Fourier series (4/7) Find coefficients bn Similarly we can obtain I2 will be discussed in details later 4.3

  8. Coefficients of trigonometric Fourier series (5/7) • In the procedure of finding coefficients an and bn , we used the properties of integrals involving products of sines and cosines. • Consider If If 4.3

  9. Coefficients of trigonometric Fourier series (6/7) • Consider If If 4.3

  10. Coefficients of trigonometric Fourier series (7/7 ) • Consider If If 4.3

  11. Trigonometric Fourier series Given a periodic signal with period T0 that can be represented by the trigonometric Fourier series, where the average value of the signal n  0 n  0 4.3

  12. Harmonic form Fourier series By using the Trigonometric equality where we have Let c0 = a0 and we have the harmonic form Fourier series of x(t) 4.3

  13. Limits of Fourier series at the discontinuities limits of Fourier series Discontinue at t = t1+ nT0 • Limits of Fourier series The Fourier series of waveform converges to the mean of the right- and left-hand limits at the discontinue point. 4.3

  14. Dirichlet conditions for Fourier series • A periodic signal x(t) can be represented as a Fourier series only if it satisfy the following Dirichlet conditions: • x(t) is absolutely integrable over any period, that is • x(t) has a finite number of maxima and minima within any finite interval of t. • x(t) has a finite number of discontinuities within any finite interval of t, andeach of these discontinuities is finite. 4.3

  15. Example 4-9 (1/6) x(t) 1 t 3 2  /2 /2  2 3 • Please find the trigonometric Fourier series for the periodic rectangular pulse train signal with period 2. 4.3

  16. Example 4-9 (2/6) 【Sol.】 • The trigonometric Fourier series with 0 = 2 /T0= 2 /2 = 1 • Find a0 4.3

  17. Example 4-9 (3/6) • Find an 4.3

  18. Example 4-9 (4/6) • Find bn • The signal x(t) is written by 4.3

  19. Example 4-9 (5/6) • The trigonometric Fourier series for the periodic rectangular pulse train signal with period 2 is obtained. 4.3

  20. Example 4-9 (6/6) Note: The phases are either 0 or  , the amplitude and phase are combined in this special case. Plot the series in frequency domain. 4.3

  21. Example 4-10 (1/6) • Please find the trigonometric Fourier series for the periodic triangle pulse train signal with period 2. 4.3

  22. Example 4-10 (2/6) 【Sol.】 • The trigonometric Fourier series with 0 = 2 /T0= 2 /2 =  . • Set the period as [1/2, 3/2], then x(t) in this period is written as • Obtain a0 = 0 . (the average of x(t) is 0) 4.3

  23. Example 4-10 (3/6) • Find an Note: use the odd/even properties 4.3

  24. Example 4-10 (4/6) • Find bn 4.3

  25. Example 4-10 (5/6) • The signal x(t) is written by Using , we rewrite x(t) as 4.3

  26. Example 4-10 (6/6) Only odd order harmonic (n times of fundamental frequency) components exist. The signal is expressed as harmonic form Fourier series and plotted in frequency domain. 4.3

  27. Exponential Fourier series Assume that x(t) is a periodic signal with fundamental period T0, then x(t) can be represented as the exponential Fourier series, 4.4

  28. Coefficients of exponential Fourier series (1/6) Review the trigonometric Fourier series Review the Euler’s equality 4.4

  29. Coefficients of exponential Fourier series (2/6) By using Euler’s equality, we rewrite the trigonometric Fourier series and obtain 4.4

  30. Coefficients of exponential Fourier series (3/6) The relationship between the coefficients of exponential Fourier series and those of trigonometric Fourier series 4.4

  31. Coefficients of exponential Fourier series (4/6) Review the harmonic form Fourier series By using Euler’s equality, we rewrite the trigonometric Fourier series and obtain 4.4

  32. Coefficients of exponential Fourier series (5/6) The relationship between the coefficients of exponential Fourier series and those of trigonometric Fourier series: The exponential Fourier series and trigonometric Fourier series are equivalent. 4.4

  33. Coefficients of exponential Fourier series (6/6) An alternative way to find the coefficients of exponential Fourier series. Multiplying both sides of the series equation by and integrating over one period T0, we obtain 4.4

  34. Example 4-11 (1/6) T0 T0 • Please find the exponential Fourier series for theperiodic rectangular pulse train signal with period T0. . 4.4

  35. Example 4-11 (2/6) 【Sol.】 • The exponential Fourier series with 0 = 2 /T0= 2f0. • Find X0 4.4

  36. Example 4-11 (4/6) • Find Xn 4.4

  37. Example 4-11 (5/6) • Rewrite the coefficients • The exponential Fourier series or the trigonometric Fourier series 4.4

  38. Example 4-11 (6/6) Double-sided amplitude spectrum Double-sided phase spectrum Double-sided spectrums of the periodic rectangular pulse train signal. 4.4

  39. Example 4-12 (1/2) • Please find the exponential Fourier series for the signal 【Sol.】 • The signal consists of 3 complex exponential components with frequencies 1000, 2000 and 3000, respectively. The fundamental frequency of xc(t) is given by GCD(1000, 2000, 3000) = 1000. • The expression of xc(t) is in the form of exponential Fourier series. 4.4

  40. Example 4-12 (2/2) • Rewrite xc(t) Note that xc(t) is a complex signal and thus the symmetry property of Fourier series discussed later is not held. 4.4

  41. Example 4-13 (1/5) • Please find the exponential Fourier series for theperiodic signal with period T0. 4.4

  42. Example 4-13 (2/5) 【Sol.】 • The exponential Fourier series with 0 = 2 /T0. • Find Xn 4.4

  43. Example 4-13 (3/5) 【Sol.】 • The trigonometric Fourier series with 0 = 2 /T0. • Find Xn 4.4

  44. Example 4-13 (4/5) Using we obtain the coefficients 4.4

  45. Example 4-13 (5/5) Similarly 4.4

  46. Example 4-14 (1/4) 1 … … t 3T0 2T0 T0 T0 2T0 3T0 0 • Please find the exponential Fourier series for the periodic impulse train with period T0. 4.4

  47. Example 4-14 (2/4) 【Sol.】 • The exponential Fourier series with 0 = 2 /T0. • Find Xn 4.4

  48. Example 4-14 (3/4) • Double-sided amplitude spectrum of the periodic pulse train . Phase is 0, thus the phase spectrum is not shown 4.4

  49. Example 4-14 (4/4) • Using we obtain • Single-sided amplitude spectrum of the periodic pulse train . 4.4

  50. Exponential Fourier series of common used signals 50 4.4

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