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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 0 t Also called unit impulse function.

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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).

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Properties of Unit Impulse Function Pf) Write t as t + t 0

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

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Fourier Transforms of Special Functions Fourier Transform Related to Impulse Function

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Fourier Transform for (t) 0 t (t)(t) 0 1 F(j)F(j) F

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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 0 1 |F(j )| F Use the fact 0 t (t t 0 ) t0t0

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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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F 0 t A A2 ( ) 0 F(j)F(j)

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Fourier Transform of Exponential Wave

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Fourier Transforms of Sinusoidal Functions F ( + 0 ) 0 F(j)F(j) ( 0 ) 0 00 t f(t)=cos 0 t

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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 0 B ( ) must be odd

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Fourier Transform of Unit Step Function Guess 0

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Fourier Transform of Unit Step Function Guess

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

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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 0 t T2T2T3T3T TT 2T2T 3T3T Find the FT of the impulse train.

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Example: Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T Find the FT of the impulse train. cncn

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Example: Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T Find the FT of the impulse train. cncn 00

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Example: Impulse Train 0 t T2T2T3T3T TT 2T2T 3T3T 00 0 00 2020 3030 0 2020 3030 2/T2/T F

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

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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).

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

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

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