# Fourier Transforms of Special Functions

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Fourier Transforms of Special Functions

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

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.

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

Fourier Transforms of Special Functions
More on Impulse Function

Dirac Delta Function and t Also called unit impulse function.

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).

Properties of Unit Impulse Function
Pf) Write t as t + t0

Properties of Unit Impulse Function
Pf) Write t as t/a Consider a>0 Consider a<0

Properties of Unit Impulse Function
Pf)

Properties of Unit Impulse Function
Pf)

Properties of Unit Impulse Function

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

Derivatives of the -Function

Product Rule Pf)

Product Rule Pf)

Unit Step Function u(t)
Define t u(t)

Derivative of the Unit Step Function
Show that

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

Fourier Transforms of Special Functions
Fourier Transform Related to Impulse Function

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

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

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

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

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

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

Fourier Transform of a Constant

Fourier Transform of a Constant
F(j) t A F

Fourier Transform of Exponential Wave

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

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

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

Fourier Transform of Unit Step Function
Guess

Fourier Transform of Unit Step Function
Guess

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

Fourier Transforms of Special Functions
Fourier Transform vs. Fourier Series

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)?

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

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.

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

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

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

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

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

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)

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

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