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Fourier Transforms of Special Functions. http://www.google.com/search?hl=en&sa=X&oi=spell&resnum=0&ct=result&cd=1&q=unit+step+fourier+transform&spell=1. 主講者:虞台文. Content. Introduction More on Impulse Function Fourier Transform Related to Impulse Function

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fourier transforms of special functions

Fourier Transforms of Special Functions

http://www.google.com/search?hl=en&sa=X&oi=spell&resnum=0&ct=result&cd=1&q=unit+step+fourier+transform&spell=1

主講者:虞台文

content
Content
  • Introduction
  • More on Impulse Function
  • Fourier Transform Related to Impulse Function
  • Fourier Transform of Some Special Functions
  • Fourier Transform vs. Fourier Series
introduction
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
Some Unabsolutely Integrable Functions
  • Sinusoidal Functions: cos t, sin t,…
  • Unit Step Function: u(t).
  • Generalized Functions:
    • Impulse Function (t); and
    • Impulse Train.
dirac delta function

t

0

Dirac Delta Function

and

Also called unit impulse function.

generalized function
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).
properties of unit impulse function9
Properties of Unit Impulse Function

Pf)

Writet as t/a

Consider a>0

Consider a<0

generalized derivatives
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

fourier transforms of special functions20

Fourier Transforms of Special Functions

Fourier Transform Related to

Impulse Function

fourier transform for t22

The integration converges to

in the sense of generalized function.

Fourier Transform for (t)

Show that

fourier transform for t23
Fourier Transform for (t)

Show that

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

two identities for t
Two Identities for (t)

These two ordinary integrations themselves are meaningless.

They converge to (t) in the sense of generalized function.

shifted impulse function

|F(j)|

(t  t0)

1

t

0

0

t0

Shifted Impulse Function

Use the fact

F

fourier transforms of special functions26

Fourier Transforms of Special Functions

Fourier Transform of a Some Special Functions

fourier transforms of sinusoidal functions

f(t)=cos0t

F(j)

(+0)

(0)

t

0

0

0

Fourier Transforms of Sinusoidal Functions

F

fourier transform of unit step function
Fourier Transform of Unit Step Function

Let

F(j)=?

Can you guess it?

fourier transforms of special functions36

Fourier Transforms of Special Functions

Fourier Transform vs. Fourier Series

find the ft of a periodic function
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 function38
Find the FT of a Periodic Function

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

find the ft of a periodic function39
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

0

T

2T

3T

3T

Example:Impulse Train

Find the FT of the impulse train.

example impulse train41

t

T

2T

0

T

2T

3T

3T

Find the FT of the impulse train.

Example:Impulse Train

cn

example impulse train42

t

T

2T

0

T

2T

3T

3T

Find the FT of the impulse train.

cn

Example:Impulse Train

0

example impulse train43

0

t

T

2T

0

T

2T

3T

3T

2/T

0

0

20

30

20

0

30

Example:Impulse Train

F

find fourier series using fourier transform45

f(t)

t

T/2

T/2

fo(t)

t

T/2

T/2

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