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Fourier Transforms of Special Functions 主講者:虞台文

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Presentation on theme: "Fourier Transforms of Special Functions 主講者:虞台文"— Presentation transcript:

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 Dirac Delta Function and 0 t 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) Write t as t + t 0

9 Properties of Unit Impulse Function Pf) Write t as t/a Consider a>0Consider 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 Unit Step Function u(t) Define 0 t u(t)u(t)

18 Derivative of the Unit Step Function Show that

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

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

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

22 Show that The integration converges to in the sense of generalized function.

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 Shifted Impulse Function 0  1 |F(j  )| F Use the fact 0 t  (t  t 0 ) t0t0

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

27 Fourier Transform of a Constant

28 F 0 t A A2  (  ) 0  F(j)F(j)

29 Fourier Transform of Exponential Wave

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

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

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 Example: Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T Find the FT of the impulse train.

41 Example: Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T Find the FT of the impulse train. cncn

42 Example: Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T Find the FT of the impulse train. cncn 00

43 Example: Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T 00 0  00 2020 3030  0 2020 3030 2/T2/T F

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

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

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

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


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