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**Fourier Transforms of Special Functions**

主講者：虞台文

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**Content Introduction More on Impulse Function**

Fourier Transform Related to Impulse Function Fourier Transform of Some Special Functions Fourier Transform vs. Fourier Series

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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.

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**Some Unabsolutely Integrable Functions**

Sinusoidal Functions: cos t, sin t,… Unit Step Function: u(t). Generalized Functions: Impulse Function (t); and Impulse Train.

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**Fourier Transforms of Special Functions**

More on Impulse Function

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Dirac Delta Function and t Also called unit impulse function.

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**Generalized Function (t): Test 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).

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**Properties of Unit Impulse Function**

Pf) Write t as t + t0

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**Properties of Unit Impulse Function**

Pf) Write t as t/a Consider a>0 Consider a<0

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**Properties of Unit Impulse Function**

Pf)

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**Properties of Unit Impulse Function**

Pf)

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**Properties of Unit Impulse Function**

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**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

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**Derivatives of the -Function**

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Product Rule Pf)

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Product Rule Pf)

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**Unit Step Function u(t)**

Define t u(t)

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**Derivative of the Unit Step Function**

Show that

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**Derivative of the Unit Step Function**

t (t) t u(t) Derivative

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**Fourier Transforms of Special Functions**

Fourier Transform Related to Impulse Function

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**Fourier Transform for (t)**

1 F(j) t (t) F

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**Fourier Transform for (t)**

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

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**Fourier Transform for (t)**

Show that Converges to (t) in the sense of generalized function.

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**Two Identities for (t)**

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

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**Shifted Impulse Function**

Use the fact 1 |F(j)| t (t t0) t0 F

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**Fourier Transforms of Special Functions**

Fourier Transform of a Some Special Functions

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**Fourier Transform of a Constant**

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**Fourier Transform of a Constant**

F(j) t A F

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**Fourier Transform of Exponential Wave**

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**Fourier Transforms of Sinusoidal Functions**

(+0) F(j) (0) 0 0 t f(t)=cos0t F

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**Fourier Transform of Unit Step Function**

Let F(j)=? Can you guess it?

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**Fourier Transform of Unit Step Function**

Guess B() must be odd

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**Fourier Transform of Unit Step Function**

Guess

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**Fourier Transform of Unit Step Function**

Guess

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**Fourier Transform of Unit Step Function**

() |F(j)| t 1 f(t) F

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**Fourier Transforms of Special Functions**

Fourier Transform vs. Fourier Series

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**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)?

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**Find the FT of a Periodic Function**

We can express a periodic function f(t) as:

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**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.

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**Example: Impulse Train**

t T 2T 3T T 2T 3T Find the FT of the impulse train.

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**Example: Impulse Train**

t T 2T 3T T 2T 3T Find the FT of the impulse train. cn

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**Example: Impulse Train**

0 t T 2T 3T T 2T 3T Find the FT of the impulse train. cn

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**Example: Impulse Train**

0 Example: Impulse Train t T 2T 3T T 2T 3T F 0 20 30 0 20 30 2/T

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**Find Fourier Series Using Fourier Transform**

f(t) t T/2 T/2 fo(t) t

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**Find Fourier Series Using Fourier Transform**

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 T/2 T/2 f(t) t fo(t)

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**Example: The Fourier Series of a Rectangular Wave**

f(t) d 1 t t fo(t) 1

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**Example: The Fourier Transform of a Rectangular Wave**

f(t) d 1 t F [f(t)]=?

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