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EE 2260 Lecture 18
Fourier Transform
Reading: 17.1 – 17.3
HW #18 AP: 17.1, 17.2, 17.3
§17.1 The Derivation of the Fourier Transform
For a periodic function f(t) we have a Fourier series:
or in the exponential form:
If the function f(t) is not periodic, we may think that the period T   . In this situation,

1
d
 jt
dt

, n0   , C n T   f (t )e
T
2

We define the Fourier transform of f(t) as
F ( ) 


f (t )e  jt dt

The Fourier expansion becomes the Inverse Fourier transform:
1
f
(
t
)

jn0t 1
f ( t )   (C n T ) e

2
T
n  


 F ( )e
j t
d

Example1:
Example2: f (t )  e  at u(t ) , a > 0.
F ( ) 
PLOTS



0
 at
 jt
 ( a  j ) t
dt 
 e u(t )e dt   e
e  ( a  j ) t 
1
|0 
 ( a  j )
a  j
.
.
§17.2 The Convergence of the Fourier Integral
We consider some useful but not integrable functions.
Example:

Find the Fourier transform of a constant f(t) = 1:
1 e
 jt

dt 

e

 jt
e  jt 
dt 
|   ?
 j
Let’s consider a different function: f (t )  e  a |t | , a > 0.
F ( ) 

 a|t|  jt
 e e dt 

0


0
at  jt
 at  jt
 e e dt   e e dt 
1
1
2a

 2
a  j a  j a   2
2a
0
a  2
2a
If   0 , let a  0 , 2  
a
If   0 , let a  0 ,
2
Notice that the area under the curve (not shown)

2a
x
2a
is
d  2 tan 1 |  2

2
2
2
2
a
a 
 a  
Therefore, F ( )  2 ( ) when f(t) = 1.
§17.3 Using Laplace Transforms to Find Fourier Transforms
Case #1: f(t) = 0 for t < 0 and let s =j.
Example:
f (t )  e  at cos(0 t )u(t ) ,
Case #2: f(t) = 0 for t > 0 and let s = -j.
F ( ) 



f (t )e  jt dt 
0


f (t )e  jt dt 
Change f(t) to f(-t), and let s = -j.

jt
 f ( t )e dt
0
Example:
f ( t )  e at cos(  0 t )[ 1  u ( t )] , f ( t )  e  at cos(0 t )[1  u( t )]  e at cos(0 t )u(t )
Case #3: For a general f (t )  f  (t )  f  (t ) , where f  (t )  f (t )u(t ) , f  (t )  f (t )[1  u(t )]
Example:
f (t )  e  a|t|  e  at u(t )  e at [1  u(t )] , f  (t )  e  at u(t ) , f  ( t )  e  at u(t )