Fourier Transforms of
Special Functions
主講者：虞台文
Content
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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





| f (t ) |dt  
That is, f(t) is absolutely integrable.
However, the above condition is not the
necessary one.
Some Unabsolutely Integrable Functions
Functions: cos t, sin t,…
 Unit Step Function: u(t).
 Sinusoidal
 Generalized
–
–
Functions:
Impulse Function (t); and
Impulse Train.
Fourier Transforms of
Special Functions
More on
Impulse Function
Dirac Delta Function
0 t  0
(t )  
 t  0
and



(t )dt  1
Also called unit impulse function.
0
t
Generalized Function

The value of delta function can also be defined
in the sense of generalized function:





(t )(t )dt  (0)
(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



(t  t0 )(t )dt  (t0 )
Pf)
Write t as t + t0




(t  t0 )(t )dt   (t )(t  t0 )dt

 (t0 )
Properties of Unit Impulse Function

1
 (at )(t )dt  | a | (0)
Pf) Write t as t/a
Consider a>0



(at )(t )dt
1 
t
  (t ) dt
a 
a
1

(0)
|a|
Consider a<0



(at )(t )dt
1 
t
   (t ) dt
a 
a
1

(0)
|a|
Properties of Unit Impulse Function
f (t )(t )  f (0)(t )
Pf)




[ f (t )(t )](t )dt   (t )[ f (t )(t )]dt

 f (0)(0)

 f (0)  (t )(t )dt


  [ f (0)(t )](t )dt

Properties of Unit Impulse Function
f (t )(t )  f (0)(t )
Pf)
1
(at ) 
(t )
|a|
1
1 
 (at )(t )dt  | a | (0)  | a |  (t )(t )dt


1

(t )(t )dt
 | a |
Properties of Unit Impulse Function
f (t )(t )  f (0)(t )
1
(at ) 
(t )
|a|
t(t )  0
(t )  (t )
Generalized Derivatives
The derivative f’(t) of an arbitrary
generalized function f(t) is defined by:




f ' (t )(t )dt    f (t )' (t )dt

Show that this definition is consistent to the ordinary
definition for the first derivative of a continuous function.





f ' (t )(t )dt  f (t )(t )    f (t )' (t )dt

=0
Derivatives of the -Function




' (t )(t )dt    (t )' (t )dt  ' (0)

d(t )
' (t ) 
,
dt



d(t )
' (0) 
dt t 0
 (t )(t )dt (1)  (0)
( n)
n
d
(t )
(n)
 (t ) 
,
n
dt
n
( n)
n
d
(t )
(n)
 (0) 
dt n t 0
Product Rule
[ f (t )(t )]'  f ' (t )(t )  f (t )' (t )
Pf)




[ f (t )(t )]' (t )dt   [ f (t )(t )]' (t )dt   (t )[ f (t )' (t )]dt



  (t ){[ f (t )(t )]' f ' (t )(t )}dt





  (t )[ f (t )(t )]' dt   (t )[ f (t )' (t )]dt




  ' (t )[ f (t )(t )]dt   (t )[ f (t )' (t )]dt

  [' (t ) f (t )  (t ) f ' (t )](t )dt

Product Rule
f (t )' (t )  f (0)' (t )  f ' (0)(t )
Pf)
f (t )' (t )  [ f (t )(t )]' f (t )' (t )
 [ f (0)(t )]'
 f (0)' (t )
 f ' (0)(t )
Unit Step Function u(t)
 Define




u(t )(t )dt   (t )dt
0
u(t)
0
t
1 t  0
u (t )  
0 t  0
Derivative of the Unit Step Function
 Show



that u ' (t )  (t )

u' (t )(t )dt   u(t )' (t )dt


  ' (t )dt
0
 [()  (0)]  (0)

  (t )(t )dt

Derivative of the Unit Step Function
(t)
u(t)
Derivative
0
t
0
t
Fourier Transforms of
Special Functions
Fourier Transform
Related to
Impulse Function
Fourier Transform for (t)
(t ) 
1
F

F [(t )]   (t )e
 jt

 jt

e
dt
t 0
1
F(j)
(t)
F
0
t
1
0

Fourier Transform for (t)
Show that
1  jt
(t ) 
e d

2  
1  jt
1  jt
e d
1e d 
(t )  F [1] 


2  
2  
1
1  j t
e d converges to
The integration

2  
in the sense of generalized function.
(t )
Fourier Transform for (t)
1 
Show that (t )   cos td
 0
1 
1  jt
(cos t  j sin t )d
(t ) 
e d 


2  
2  
1 
j 

cos td 
sin td


2  
2  
1 
Converges to (t) in the sense of
  cos td generalized function.
 0
Two Identities for (t)
1  jxy
( y ) 
e dx

2  
1 
( y )   cos xydx
 0
These two ordinary integrations themselves are meaningless.
They converge to (t) in the sense of generalized function.
Shifted Impulse Function
(t  t0 )  e
F
 jt 0
Use the fact F [ f (t  t0 )]  F ( j)e
 jt 0
(t  t0)
|F(j)|
F
0
t0
t
1
0

Fourier Transforms of
Special Functions
Fourier Transform of a
Some Special Functions
Fourier Transform of a Constant
f (t )  A 
 F ( j)  A2()
F

F ( j)  F [ A]   Ae jt d

 1  j (  ) t 
 2A  e
dt 
 2  

 2A()
Fourier Transform of a Constant
f (t )  A 
 F ( j)  A2()
F
F(j)
0
A2()
F
A
t
0

Fourier Transform of Exponential Wave
f (t )  e
F [ f (t )e
j0t
j0t
 F ( j)  2(  0 )
F
]  F [ j (  0 )]
F [1]  2()
F [e
j0t
]  2(  0 )
Fourier Transforms of Sinusoidal Functions
cos 0t  (  0 )  (  0 )
F
sin 0t   j(  0 )  j(  0 )
F
F(j)
(+0) (0)
f(t)=cos0t
t
F
0
0
0

Fourier Transform of Unit Step Function
Let F [u (t )]  F ( j)
F [u (t )]  F ( j)
u (t )  u (t )  1 (except for t  0)
F [u (t )  u (t )]  F [1]
F [u (t )]  F [u (t )]  2()
F ( j)  F ( j)  2()
F(j)=?
Can you guess it?
Fourier Transform of Unit Step Function
Guess F ( j)  k()  B()
k 
F ( j)  F ( j)  k()  k()  B()  B()
 2k()  B()  B()
0
B() must be odd
F ( j)  F ( j)  2()
Fourier Transform of Unit Step Function
k 
Guess F ( j)  k()  B()
u ' (t )  (t )
F [u (t )]  F ( j)
1
B() 
j
F [u ' (t )]  F [(t )]  1
F [u ' (t )]  jF ( j)
 j[()  B()]
 j()  jB()
0
Fourier Transform of Unit Step Function
Guess F ( j)  k()  B()
1
u (t ) 
 () 
j
F
k 
1
B() 
j
Fourier Transform of Unit Step Function
|F(j)|
f(t)
F
1
t
0
1
u (t ) 
 () 
j
F
()
0

Fourier Transforms of
Special Functions
Fourier Transform vs.
Fourier Series
Find the FT of a Periodic Function

Sufficient condition --- existence of FT





| f (t ) |dt  
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:
f (t ) 

c e
n  
n
jn0t
,
2
0 
T

 

F ( j)  F [ f (t )]  F   cn e jn0t    cnF [e jn0t ]
n 
 n  


 c 2(  n )
n  
n
0

 2   c n  (   n 0 )
n  
Find the FT of a Periodic Function
We can express a periodic function f(t) as:
f (t ) 

c e
n  
jn0t
n
,
2
0 
T

F ( j)  2  cn (  n0 )
n  
The FT of a periodic function consists of a sequence of
equidistant impulses located at the harmonic frequencies
of the function.
Example:
Impulse Train
3T 2T T
T (t ) 
0
T
2T
3T
t

 (t  nT )
n  
Find the FT of the
impulse train.
Example:
Impulse Train
3T 2T T
0
T
2T
3T
t


1
jn0t
Find
the
FT
of
the
T (t )   (t  nT ) T (t )   e
impulse T
train.
n  
n  
cn
2
F [T (t )] 
(  n0 )

Example:
T n  

0
Impulse Train
3T 2T T
0
T
2T
3T
t


1
jn0t
Find
the
FT
of
the
T (t )   (t  nT ) T (t )   e
impulse T
train.
n  
n  
cn
2
F [T (t )] 
(  n0 )

Example:
T n  

0
Impulse Train
3T 2T T
0
T
2T
3T
0
20 30
t
F
2/T
30 20 0
0

Find Fourier Series Using
Fourier Transform
f(t)
t
T/2
f (t ) 

c e
n  
jn0t
n
1
cn 
T
Fo ( j)   f o (t )e  jt

T / 2

T /2
T / 2
f (t )e
 jn0t
1
cn  Fo ( jn0 )
T


T /2
T/2
f (t )e  jt
T/2
fo(t)
t
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
f(t)
t
T/2
f (t ) 

c e
n  
jn0t
n
1
cn 
T
Fo ( j)   f o (t )e  jt

T / 2

T /2
T / 2
f (t )e
 jn0t
1
cn  Fo ( jn0 )
T


T /2
T/2
f (t )e  jt
T/2
fo(t)
t
T/2
Example:
The Fourier Series of a Rectangular Wave
f(t)
1
1
d
f (t ) 

0
jn0t
c
e
n
n  
fo(t)
t
t
0
Fo ( j)  
d /2
e jt dt
d / 2
2  d 
1
 sin 

cn  Fo ( jn0 )
  2 
T
2
1
 n0 d 
 n0 d 

sin 
sin 
 

Tn0  2  n  2 

F ( j)  2  cn (  n0 )
Example:
n  
The Fourier Transform of a Rectangular Wave
f(t)
1
d
f (t ) 

t
0
jn0t
c
e
n
n  
F [f(t)]=?
2  n0 d 
F ( j)   sin 
(  n0 )
 2 
n   n

1
cn  Fo ( jn0 )
T
2
1
 n0 d 
 n0 d 

sin 
sin 
 

Tn0  2  n  2 