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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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