Fourier Transforms of Special Functions

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# Fourier Transforms of Special Functions - PowerPoint PPT Presentation

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## 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

t

0

Dirac Delta Function

and

Also called unit impulse 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).
Properties of Unit Impulse Function

Pf)

Writet as t/a

Consider a>0

Consider a<0

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 Functions

Fourier Transform Related to

Impulse Function

The integration converges to

in the sense of generalized function.

Fourier Transform for (t)

Show that

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.

|F(j)|

(t  t0)

1

t

0

0

t0

Shifted Impulse Function

Use the fact

F

### Fourier Transforms of Special Functions

Fourier Transform of a Some Special 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

Let

F(j)=?

Can you guess it?

### 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.

t

T

2T

0

T

2T

3T

3T

Example:Impulse Train

Find the FT of the impulse train.

t

T

2T

0

T

2T

3T

3T

Find the FT of the impulse train.

Example:Impulse Train

cn

t

T

2T

0

T

2T

3T

3T

Find the FT of the impulse train.

cn

Example:Impulse Train

0

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

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