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Fourier Transforms of Special Functions

Fourier Transforms of Special Functions. 主講者：虞台文. Content. Introduction More on Impulse Function Fourier Transform Related to Impulse Function Fourier Transform of Some Special Functions Fourier Transform vs. Fourier Series. Introduction.

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Fourier Transforms of Special Functions

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  1. Fourier Transforms of Special Functions 主講者：虞台文

  2. Content • Introduction • More on Impulse Function • Fourier Transform Related to Impulse Function • Fourier Transform of Some Special Functions • Fourier Transform vs. Fourier Series

  3. Introduction • Sufficient condition for the existence of a Fourier transform • That is, f(t) is absolutely integrable. • However, the above condition is not the necessary one.

  4. Some Unabsolutely Integrable Functions • Sinusoidal Functions: cos t, sin t,… • Unit Step Function: u(t). • Generalized Functions: • Impulse Function (t); and • Impulse Train.

  5. Fourier Transforms of Special Functions More on Impulse Function

  6. t 0 Dirac Delta Function and Also called unit impulse function.

  7. Generalized Function • The value of delta function can also be defined in the sense of generalized function: (t): Test Function • We shall never talk about the value of (t). • Instead, we talk about the values of integrals involving (t).

  8. Properties of Unit Impulse Function Pf) Writet as t + t0

  9. Properties of Unit Impulse Function Pf) Writet as t/a Consider a>0 Consider a<0

  10. Properties of Unit Impulse Function Pf)

  11. Properties of Unit Impulse Function Pf)

  12. Properties of Unit Impulse Function

  13. Generalized Derivatives The derivative f’(t) of an arbitrary generalized function f(t) is defined by: Show that this definition is consistent to the ordinary definition for the first derivative of a continuous function. =0

  14. Derivatives of the -Function

  15. Product Rule Pf)

  16. Product Rule Pf)

  17. u(t) t 0 Unit Step Function u(t) • Define

  18. Derivative of the Unit Step Function • Show that

  19. (t) u(t) t t 0 0 Derivative of the Unit Step Function Derivative

  20. Fourier Transforms of Special Functions Fourier Transform Related to Impulse Function

  21. (t) F(j) 1 t 0  0 Fourier Transform for (t) F

  22. The integration converges to in the sense of generalized function. Fourier Transform for (t) Show that

  23. Fourier Transform for (t) Show that Converges to (t) in the sense of generalized function.

  24. Two Identities for (t) These two ordinary integrations themselves are meaningless. They converge to (t) in the sense of generalized function.

  25. |F(j)| (t  t0) 1  t 0 0 t0 Shifted Impulse Function Use the fact F

  26. Fourier Transforms of Special Functions Fourier Transform of a Some Special Functions

  27. Fourier Transform of a Constant

  28. F(j) A2() A  t 0 0 Fourier Transform of a Constant F

  29. Fourier Transform of Exponential Wave

  30. f(t)=cos0t F(j) (+0) (0) t  0 0 0 Fourier Transforms of Sinusoidal Functions F

  31. Fourier Transform of Unit Step Function Let F(j)=? Can you guess it?

  32. Fourier Transform of Unit Step Function Guess 0 B() must be odd

  33. Fourier Transform of Unit Step Function Guess 0

  34. Fourier Transform of Unit Step Function Guess

  35. |F(j)| f(t) 1 ()  t 0 0 Fourier Transform of Unit Step Function F

  36. Fourier Transforms of Special Functions Fourier Transform vs. Fourier Series

  37. Find the FT of a Periodic Function • Sufficient condition --- existence of FT • Any periodic function does not satisfy this condition. • How to find its FT (in the sense of general function)?

  38. Find the FT of a Periodic Function We can express a periodic function f(t) as:

  39. Find the FT of a Periodic Function We can express a periodic function f(t) as: The FT of a periodic function consists of a sequence of equidistant impulses located at the harmonic frequencies of the function.

  40. t T 2T 0 T 2T 3T 3T Example:Impulse Train Find the FT of the impulse train.

  41. t T 2T 0 T 2T 3T 3T Find the FT of the impulse train. Example:Impulse Train cn

  42. t T 2T 0 T 2T 3T 3T Find the FT of the impulse train. cn Example:Impulse Train 0

  43. 0 t T 2T 0 T 2T 3T 3T 2/T  0 0 20 30 20 0 30 Example:Impulse Train F

  44. f(t) t T/2 T/2 fo(t) t T/2 T/2 Find Fourier Series Using Fourier Transform

  45. f(t) t T/2 T/2 fo(t) t T/2 T/2 Sampling the Fourier Transform of fo(t) with period 2/T, we can find the Fourier Series of f(t). Find Fourier Series Using Fourier Transform

  46. f(t) fo(t) 1 1 d t t 0 0 Example:The Fourier Series of a Rectangular Wave

  47. f(t) 1 d t 0 Example:The Fourier Transform of a Rectangular Wave F [f(t)]=?

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