KEEE 325-02
Signals and Systems
JUNE 2006
Spring 2006
Homework #5 Solution
Fourier Transform
1.
a. x( ) is aperiodic signal.
t  R :
x t  

1
2



1 
             5   e jt d



2
 2 
j 5 t    
1 
2
j  t  2 
 5 
 e j 5t  
1

e

e
,
is not rational number.




2 
2

/
5

X   e jt dw 
1
1  e j t

2
b. y ( ) is periodic signal.
t  R :
y t   x t   h t   


1
2
1

2


1
1  e j  e j 5   u  t     u  t    2  d

2


   u  t     u  t    2    u  t     u t    2   e  u t     u t    2   e d

j
j5


t
t 2
t
t
t 2
t 2
d   e j d   e j 5 d 
1 
1 j t
1 j 5t
 j 2
 j10 
 2  e 1  e   e 1  e  
2 
j
j5

1 
j  j10
1 
j  j10
j 5 t  2 / 5  
j 5t 
 2   e  1 e  
 2   e  1 e
  y  t  2 / 5   y  t  T 
2  5
5
 2 

c. We can see that the convolution of two aperiodic signals is periodic with upper problem.
2.
a. False
Let x( )  jxodd ( ) , where xodd ( ) is an odd and pure real-valued function.

X ( )   x(t )e jt dt













 
jxodd (t )e jt dt
jxodd (t )e  jt  dt 
,where t   t
jxodd (t )e  jt dt
j ( xodd (t ))e  jt dt



xodd ( ) is an odd function 
jxodd (t )e  jt dt
 X ( )
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Signals and Systems
 X ( ) is an odd Fourier Transform.
JUNE 2006
Spring 2006

X  ( )  (  x(t )e  jt dt )


  x (t )(e  jt ) dt



  ( jxodd (t ))e jt dt


  ( jxodd (t ))e  jt  dt 

xodd ( ) is a real-valued function 
,where t   t

  ( jxodd (t ))e  jt dt




jxodd (t ))e  jt dt

xodd ( ) is an odd function 
 X( )
 X ( ) is a real-valued Fourier Transform.
b. True
Let FT [ x1 ( )]  X1 ( ) , where X 1 ( ) is an odd and real-valued function,
FT [ x2 ( )]  X 2 ( ) , where X 2 ( ) is an even an odd and real-valued function.
x( ) x1 ( )  x2 ( )
 IFT  X 1 ( ) X 2 ( ) 
1 
X 1 ( w) X 2 ( w)e jw dw
2 
1 

X 1 ( w) X 2 ( w)(cos( w )  j sin( w )dw
2 
1 

X 1 ( w) X 2 ( w) j sin( w )dw ( X 1 ( ) X 2 ( ) cos( w ) is an odd function)
2 
1 
 j
X 1 ( w) X 2 ( w) sin( w )dw
2 

Here, integration value of real function is real-value.
 x( ) is always a pure imaginary-valued signal.
Korea Univ.
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Signals and Systems
3.
JUNE 2006
Spring 2006
f ( )  (1/  2 )e
F 


2
(  is constant value.)
/ 2 2
 
2
2 
1
f  t  e  j t dt   
e  t / 2   e  j t dt

  2

1

 2
1

 2






e
 t2
 2  j
 2

e


1

e 2
 2
1
t  j 

 2 j


1

e 2
 2




1

e 2
 2



2
e
2
e

e
 2 j
2
1
dt 
 2
2
2 2
2

t 

1
2 2
1
2 2

1
2 2
4
  2

 
2

4





dt

e
1
2 2
t  j 
2
2
dt
, where u  t   2 j
du

du
e
e

1 
2
 t  j
2 2 
2
u2




1
dt 
e 2
 2
u2
2


Cauchy Integral Theorem 
2
2
Two-dimensional Fourier Transform
4. The technique of Fourier analysis can be extended to signals having two independent
variables. The two-dimensional Fourier Transform of x( 1 , 2 ) is defined as X ( 1 , 2 )
such that:
 1 , 2  : X ( 1 , 2 )  





x(t1 , t2 )e j (1t1 2t2 ) dt1dt2 .
a. This two-dimensional Fourier Transform of x( 1 , 2 ) can be performed as two
successive one-dimensional Fourier Transforms.
With t 2 regarded as fixed, we can perform integral with respect to t1 first.
Hence we can write
X ( 1, 2 )  





x(t1 , t2 )e  j ( 1 t1  2 t2 ) dt1dt2


    x(t1 , t2 )e  j 1 t1 dt1  e  j 2 t2 dt2

 
 
Then, we can see that the integration with respect to t1 in the bracket means
performing the one-dimensional Fourier transform of x(t1 , t 2 ) in t1 .
Letting



x(t1 , t2 )e j 1 t1 dt1   ( 1 , t2 ) , we can have

X ( 1 , 2 )    ( 1 , t2 )e j 2 t2 dt2 .

Korea Univ.
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NGITL
KEEE 325-02
JUNE 2006
Signals and Systems
Spring 2006
This equation is also one-dimensional Fourier Transform of  ( 1 , t2 ) in t 2 .
So using the one-dimensional inverse Fourier Transform formula, we can write

 ( 1, 2 ) 
1
2

x( 1 , 2 ) 
1
2


X ( 1 , 2 )e j2 2 d2 .
And also


 (1 , 2 )e j d1 .
1 1
Combining the above two equations, we can have
x( 1 , 2 ) 
1
2


 (1 , 2 )e j d1
1 1

1  1 

X (1 , 2 )e j2 2 d 2  e j1 1 d 1


 2
 2 



1

X (1 , 2 )e j (1 1 2 2 ) d 1d 2
2  
(2 )


X ( 1, 2 )  

b.
i)


x(t 1 , t2 )e  j ( 1 t1  2 t2 ) dt1dt2
 



 

 
2

 1
e  t1  2t2 u (t1  1)u (2  t2 )e  j ( 1 t1  2 t2 ) dt1dt2
e  t1  2t2 e  j ( 1 t1  2 t2 ) dt1dt2
2

  e 2t2   e  t1 e  j 1 t1 dt1  e  j 2 t2 dt2

 1


2
 e
2 t2


e  (1 j 1 )
1 j 1


1
e  (1 j 1 )t1  e  j 2 t2 dt2

 1  j1
1

2

e(2 j 2 )t2 dt2
2

e  (1 j 1 )  1

e(2 j 2 )t2 

1 j 1  2  j 2
 

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(1  j 1 )(2  j 2 )
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Signals and Systems
ii)
X ( 1, 2 )  
JUNE 2006
Spring 2006


t1
 t1
0
e 2t1 e  j ( 1 t1  2 t2 ) dt2 dt1  
0

 t2
 t2
2
2



0
2
1

0

 t1
 t1
2


0
1

e 2t2 e  j ( 1 t1  2 t2 ) dt1dt2
e 2t2 e  j ( 1 t1  2 t2 ) dt1dt2  
e  (2 j 1 )t1 sin 2 t1 dt1 
0

t2
 t2
0



e(2 j 2 )t2 sin 1 t2 dt2 
2

1
 2
2
 (2  j 1 ) 
2
2

e  (2 j 2 )t2 sin 1 t2 dt1

0

2
1
(2  j 2 ) 
2
e 2t1 e  j ( 1 t1  2 t2 ) dt2 dt1
2
1

e(2 j 1 ) t1 sin 2 t1 dt1
1
(2  j 2 ) 
2
2
1



(2  j 1 )  
1
2
t2
 e 2t1
 2t2
e
x(t1 , t2 )   2t
2
e
 e 2t1

t1  t2  0, t1  t2  0
(a )
t1  t2  0, t1  t2  0
(b)
t1  t2  0, t1  t2  0
(c )
t1  t2  0, t1  t2  0
(d )
2
2
t 2  t1
(b)
(d )
(a )
t1
(c)
t 2  t 1
c. The properties of two-dimensional Fourier Transform according to all the properties
of one-dimensional Fourier Transform given in our textbook are:
i) Time Shifting: t1 , t2  :
If FT  x( 1 , 2 )   X ( 1 , 2 ), then


FT  x( 1 T1 , 2 T2 )    x(t1  T1 , t2  T2 )e j (




 
1 t1  2 t2 )
dt1dt2
x(t1 ', t2 ')e  j ( 1 (t1 ' T1 )  2 (t2 ' T2 )) dt1 ' dt2 '
 e j ( 1 T1  2 T2 ) 



 
x(t1 ', t2 ')e j ( 1 t1 ' 2 t2 ') dt1 ' dt2 '
 e j ( 1 T1  2 T2 ) X ( 1 , 2 )
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Signals and Systems
ii) Time Scaling: t1 , t2 
JUNE 2006
Spring 2006
:
If FT  x( 1 , 2)   X ( ,1 ),2 then


FT  x(a 1 , b 2 )    x(at1 , bt2 )e j (
1 t 1
t2 )
2
dt
1dt2
1 1  
 (( 1 / a ) 1 ( /2b ) ) 2
d 1d 2
 a b   x( 1 , 2 )e

 1 1  
 (( 1 / a ) 1 ( /2b ) ) 2
d 1d 2
( a ) b   x( 1 , 2 )e

 1 ( 1 )   x( , )e (( 1 / a )1  ( 2 / b ) 2 ) d d
1
2
 a b   1 2

 
( 1 )( 1 )
x( 1 , 2 )e  (( 1 / a )1  ( 2 / b ) 2 ) d 1d 2




 a
b
1 1

X( 1 , 2)
a b
a b
iii) Linearity: t1 , t2 
a  0, b  0
a  0, b  0
a  0, b  0
a  0, b  0
:
If FT  x1 ( 1 , 2 )   X 1 ( 1 , 2 ) & FT  x2 ( 1 , 2 )   X 2 ( 1 , 2 ), then


FT  ax1 ( 1 , 2 )  bx2 ( 1 , 2 )     ax1 (t1 , t2 )  bx2 (t1 , t2 ) e  j (


  x (t , t ) e
 b    x (t , t ) e
 a
 

 j ( 1 t1  2 t2 )
1
1
2

 
2
1
dt1dt2
dt1dt2
 j ( 1 t1  2 t2 )
2
1 t1  2 t2 )
dt1dt2
 aX 1 ( 1 , 2 )  bX 2 ( 1 , 2 )
iv) Symmetry: t1 , t2 
:
If x( 1 , 2 ) is a real valued signal, then


X * ( 1 , 2 )     x(t1 , t2 )e  j ( 1 t1  2 t2 ) dt1dt2 
  

*


     x(t1 , t2 )e  j 1 t1 dt1  e  j 2 t2 dt2 

   

*


    x(t1 , t2 )e j 1 t1 dt1  e j 2 t2 dt2
 
 



    x(t1 , t2 )e  j (  1 ) t1 dt1  e  j (  2 ) t2 dt2

 
 
 X ( 1 ,  2 )
Korea Univ.
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Signals and Systems
v) Differentiation: t1 , t2 , 1 , 2  :
JUNE 2006
Spring 2006
If FT  x( 1 , 2 )   X ( 1 , 2 ), then
 2 x(t1 , t2 )
2  1


t1t2
t1t2  4 2


  1

t1  4 2

1
4
2
 







X (1 , 2 ) j2 e j (1t1 2t2 ) d 1d 2 



 
 

X (1 , 2 )e j (1t1 2t2 ) d 1d 2 


 
X (1 , 2 ) j1 j2 e j (1t1 2t2 ) d 1d 2
  x(t , t ) 
2
1 2
FT 
  j1 j2 X (1 , 2 )
 t1t2 
vi) Energy of Aperiodic Signals: t1 , t2 , 1 , 2  :
E


 

 


 
x(t1 , t2 ) x* (t1 , t2 )dt1dt2
 1
   x(t1 , t2 ) 
2
 
 (2 )
 
 1
x(t1 , t2 ) 
2

 (2 )
 









1
(2 )
1

(2 ) 2
1

(2 ) 2
Korea Univ.
| x(t1 , t2 ) |2 dt1dt2
2
 


 



 
 
 
 


 


X * (1 , 2 )
*
X (1 , 2 )e
j (1t1 2t2 )

d 1d 2  dt1dt2


X * (1 , 2 )e  j (1t1 2t2 ) d 1d 2  dt1dt2







 

x(t1 , t2 )e  j (1t1 2t2 ) dt1dt2 d 1d 2
X * (1 , 2 ) X (1 , 2 )d 1d 2
| X (1 , 2 ) |2 d 1d 2
-7/9-
NGITL
KEEE 325-02
Signals and Systems
vii) Convolution:
JUNE 2006
Spring 2006
If FT  x( 1 , 2 )   X ( 1 , 2 ) & FT  h( 1 , 2 )   H ( 1 , 2 ), then
FT  x( 1 , 2 )  h( 1 , 2 ) 











  x(
 








 
 
 
 

1

, 2 )  h( 1 , 2 ) e  j (1t1 2t2 ) dt1dt2


 

h( , )  
h( 1 , 2 )
1

x(t1   1 , t2   2 )h( 1 , 2 )d 1d 2 e  j ( 1 t1  2 t2 ) dt1dt2





 
2

 

dt dt d d
x(t1   1 , t2   2 )e  j ( 1 t1  2 t2 ) dt1dt2 d 1d 2
x(t1   1 , t2   2 )e  j ( 1 t1  2 t2 )
1
2
1
2
h( 1 , 2 )  X ( 1 , 2 )e  j ( 11  2  2 ) d 1d 2
 X ( 1, 2 )



 
h( 1 , 2 )e  j ( 11  2  2 ) d 1d 2
 X ( 1, 2 )H ( 1, 2 )
viii) Duality:
If FT  x( 1 , 2 )   X ( 1 , 2 ), then
x( 1 , 2 ) 

1
(2 )
 x( 1 , 2 ) 
2
 

 

1
(2 )
X (1 , 2 )e j (1 1 2 2 ) d 1d 2
2
 

 
 (2 ) 2 x( 1 ,  2 )  

X (t1 , t2 )e j ( 1 t1  2 t2 ) dt1dt2


 
X (t1 , t2 )e  j ( 1 t1  2 t2 ) dt1dt2
FT  X ( 1 , 2 )  (2 ) 2 x( 1 ,  2 )
ix) Modulation:
If FT  x( 1 , 2)   X ( ,1 )2& FT  m( , 1 )  2 M ( , ),1 then
2
using duality property,
FT  X ( 1 , 2)  (2 ) 2 x( ,1 )2
FT  M ( 1 , 2)   (2 ) 2 m( ,1 )2
using convolution property,
FT  X ( 1 , 2 )  M ( 1 , 2 )   (2 ) 2 x( 1 ,  2 )(2 ) 2 m( 1 ,  2 )
using duality property again,
FT (2 ) 2 x( 1 ,  2 )(2 ) 2 m( 1 ,  2 )   (2 ) 2 X ( 1 ,  2 )  M ( 1 ,  2 )
FT  x( 1 , 2 )m( 1 , 2 )  
1
(2 ) 2
X ( 1, 2 )  M ( 1, 2 )
Having given above benign treatment of 2-D Fourier Transform, I must now ask you
Korea Univ.
-8/9-
NGITL
KEEE 325-02
JUNE 2006
Signals and Systems
Spring 2006
this question: “Do YOU have a vision of what is going on? For instance, do you
see in your mind orthogonal vectors? Do you have the MICRO, MACRO,
EQUATION, PRIOR ART, GEOMETRICAL, e.t.c. vision of what is going on? In
particular, can you propose the notion of LTI System for which the basic signal
e j (1 1 2 2 ) becomes an eigen signal?
Korea Univ.
-9/9-
NGITL