Fourier Transform Lecture

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Advanced Engineering

Mathematics

Fourier Transforms

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The Fourier Transform

The Fourier Transform of a nonperiodicx(t) signal

is given by( ) ( ) ( )exp 2X f x t j ft dt

=

The Inverse Fourier Transform ofX(f) is

where

( ) ( ) ( )exp 2x t X f j ft df

=

2 f =

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Dirichlets ConditionsWithin a finite time interval, x(t) is single-

valued.x(t) is absolutely integrable, meaning

Within a finite time interval, x(t) has a finitenumber of minima and maxima.

Within a finite time interval, x(t) has a finitenumber of discontinuities and thesediscontinuities are finite.

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Example 1

Find the Fourier Transforms of (t) and (ta).

( ) ( ) ( )

( )

1

1

exp 2

1

X f t j ft dt

X f

=

=

( ) ( ) ( )( ) ( )

2

2

exp 2

exp 2

X f t a j ft dt

X f j fa

=

=

Therefore,

( ) 1t

( ) ( )exp 2t a j fa

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The Sinc Function

sin

sinc

t

t t

=

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Example 2

Find the Fourier Transform of the function:

( )x t

A

As a function x(t):

t

2b 2

b

( ) 2 2 b bx t A t=

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Example 2

Solution:

( ) ( )

( )

2

2

2

2

exp 2

exp 22

b

b

b

b

X f A j ft dt

Aj ft

j f

=

=

( )

( )

2

2sin

sinc

j fb j fb

j fb j fb

A e ej f

A e e

f jfb

b Ab fbfb

=

=

= =

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Example 3

Find the Fourier Transform of the function:( )x t

2

As a function x(t):

t

1 1

( )2 2 1 0

2 2 0 1

t tx t

t t

+

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Solution:

Example 3

( )2 2

2 2 2 2 2 2 2 2

2 2

1 1 1 1

2 2 2 2

j f j f

j f j f

e eX f

j f f f j f f f

= + + +

2 2 2 2

2 2

22

2

2 2

2

1 cos 2

sin sin 2 2 2sinc

f f

f

f

f f ff f

=

=

= = =

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Example 4

Find the Fourier Transform of

( ) ( )cos3tx t e tu t=

Solution:Using the properties of the unit step function, the

Fourier Transform is given by

( ) ( ) 20

cos3t j ft X f e t e dt

=

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Example 4

Solution:

( ) 2 21 2

cos3t j f

e tu t +

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