1. Fourier Series
Lecture 14: 27-Aug-12
Dr. P P Das

2. Source
This presentation is lifted from:
http://www.ele.uri.edu/courses/ele436/FourierSeries.ppt

3. Content
Periodic Functions
Fourier Series
Complex Form of the Fourier Series
Impulse Train
Analysis of Periodic Waveforms
Half-Range Expansion
Least Mean-Square Error Approximation

4. Fourier Series
Periodic Functions

5. The Mathematic Formulation
 Any function that satisfies
( ) ( )
f t f t T
 
where T is a constant and is called the period
of the function.

6. Example:
4
cos
3
cos
)
(
t
t
t
f 
 Find its period.
)
(
)
( T
t
f
t
f 
 )
(
4
1
cos
)
(
3
1
cos
4
cos
3
cos T
t
T
t
t
t





Fact: )
2
cos(
cos 



 m

 m
T
2
3

 n
T
2
4

 m
T 6

 n
T 8

24
T smallest T

7. Example:
t
t
t
f 2
1 cos
cos
)
( 


 Find its period.
)
(
)
( T
t
f
t
f 
 )
(
cos
)
(
cos
cos
cos 2
1
2
1 T
t
T
t
t
t 










 m
T 2
1


 n
T 2
2
n
m



2
1
2
1

 must be a
rational number

8. Example:
t
t
t
f )
10
cos(
10
cos
)
( 



Is this function a periodic one?





10
10
2
1 not a rational
number

9. Fourier Series
Fourier Series

10. Introduction
 Decompose a periodic input signal into
primitive periodic components.
A periodic sequence
A periodic sequence
T 2T 3T
t
f(t)

11. Synthesis
T
nt
b
T
nt
a
a
t
f
n
n
n
n




 





2
sin
2
cos
2
)
(
1
1
0
DC Part Even Part Odd Part
T is a period of all the above signals
)
sin(
)
cos(
2
)
( 0
1
0
1
0
t
n
b
t
n
a
a
t
f
n
n
n
n 



 





Let 0=2/T.

12. Orthogonal Functions
Call a set of functions {k} orthogonal
on an interval a < t < b if it satisfies








 n
m
r
n
m
dt
t
t
n
b
a
n
m
0
)
(
)
(

13. Orthogonal set of Sinusoidal
Functions
Define 0=2/T.
0
,
0
)
cos(
2
/
2
/
0 




m
dt
t
m
T
T
0
,
0
)
sin(
2
/
2
/
0 




m
dt
t
m
T
T









 n
m
T
n
m
dt
t
n
t
m
T
T 2
/
0
)
cos(
)
cos(
2
/
2
/
0
0









 n
m
T
n
m
dt
t
n
t
m
T
T 2
/
0
)
sin(
)
sin(
2
/
2
/
0
0
n
m
dt
t
n
t
m
T
T
and
all
for
,
0
)
cos(
)
sin(
2
/
2
/
0
0 




We now prove this one

14. Proof
dt
t
n
t
m
T
T




2
/
2
/
0
0 )
cos(
)
cos(
0
)]
cos(
)
[cos(
2
1
cos
cos 









dt
t
n
m
dt
t
n
m
T
T
T
T 
 







2
/
2
/
0
2
/
2
/
0 ]
)
cos[(
2
1
]
)
cos[(
2
1
2
/
2
/
0
0
2
/
2
/
0
0
]
)
sin[(
)
(
1
2
1
]
)
sin[(
)
(
1
2
1 T
T
T
T
t
n
m
n
m
t
n
m
n
m 











m  n
]
)
sin[(
2
)
(
1
2
1
]
)
sin[(
2
)
(
1
2
1
0
0




n
m
n
m
n
m
n
m






0
0


15. Proof
dt
t
n
t
m
T
T




2
/
2
/
0
0 )
cos(
)
cos(
0
)]
cos(
)
[cos(
2
1
cos
cos 









dt
t
m
T
T




2
/
2
/
0
2
)
(
cos
2
/
2
/
0
0
2
/
2
/
]
2
sin
4
1
2
1
T
T
T
T
t
m
m
t






m = n
2
T

]
2
cos
1
[
2
1
cos2




dt
t
m
T
T





2
/
2
/
0 ]
2
cos
1
[
2
1









 n
m
T
n
m
dt
t
n
t
m
T
T 2
/
0
)
cos(
)
cos(
2
/
2
/
0
0

16. Orthogonal set of Sinusoidal
Functions
Define 0=2/T.
0
,
0
)
cos(
2
/
2
/
0 




m
dt
t
m
T
T
0
,
0
)
sin(
2
/
2
/
0 




m
dt
t
m
T
T









 n
m
T
n
m
dt
t
n
t
m
T
T 2
/
0
)
cos(
)
cos(
2
/
2
/
0
0









 n
m
T
n
m
dt
t
n
t
m
T
T 2
/
0
)
sin(
)
sin(
2
/
2
/
0
0
n
m
dt
t
n
t
m
T
T
and
all
for
,
0
)
cos(
)
sin(
2
/
2
/
0
0 






















,
3
sin
,
2
sin
,
sin
,
3
cos
,
2
cos
,
cos
,
1
0
0
0
0
0
0
t
t
t
t
t
t


















,
3
sin
,
2
sin
,
sin
,
3
cos
,
2
cos
,
cos
,
1
0
0
0
0
0
0
t
t
t
t
t
t
an orthogonal set.
an orthogonal set.

17. Decomposition
dt
t
f
T
a
T
t
t



0
0
)
(
2
0

,
2
,
1
cos
)
(
2
0
0
0


 

n
tdt
n
t
f
T
a
T
t
t
n

,
2
,
1
sin
)
(
2
0
0
0


 

n
tdt
n
t
f
T
b
T
t
t
n
)
sin(
)
cos(
2
)
( 0
1
0
1
0
t
n
b
t
n
a
a
t
f
n
n
n
n 



 






18. Proof
Use the following facts:
0
,
0
)
cos(
2
/
2
/
0 




m
dt
t
m
T
T
0
,
0
)
sin(
2
/
2
/
0 




m
dt
t
m
T
T









 n
m
T
n
m
dt
t
n
t
m
T
T 2
/
0
)
cos(
)
cos(
2
/
2
/
0
0









 n
m
T
n
m
dt
t
n
t
m
T
T 2
/
0
)
sin(
)
sin(
2
/
2
/
0
0
n
m
dt
t
n
t
m
T
T
and
all
for
,
0
)
cos(
)
sin(
2
/
2
/
0
0 





19. Example (Square Wave)
1
1
2
2
0
0 

 

dt
a

,
2
,
1
0
sin
1
cos
2
2
0
0








 n
nt
n
ntdt
an
,
6
,
4
,
2
0
,
5
,
3
,
1
/
2
)
1
cos
(
1
cos
1
sin
2
2
0
0











  

n
n
n
n
n
nt
n
ntdt
bn







 2 3 4 5
-
-2
-3
-4
-5
-6
f(t)
1

20. 1
1
2
2
0
0 

 

dt
a

,
2
,
1
0
sin
1
cos
2
2
0
0








 n
nt
n
ntdt
an
,
6
,
4
,
2
0
,
5
,
3
,
1
/
2
)
1
cos
(
1
cos
1
sin
2
1
0
0



















 

n
n
n
n
n
nt
n
ntdt
bn
 2 3 4 5
-
-2
-3
-4
-5
-6
f(t)
1
Example (Square Wave)











 
t
t
t
t
f 5
sin
5
1
3
sin
3
1
sin
2
2
1
)
( 










 
t
t
t
t
f 5
sin
5
1
3
sin
3
1
sin
2
2
1
)
(

21. 1
1
2
2
0
0 

 

dt
a

,
2
,
1
0
sin
1
cos
2
2
0
0








 n
nt
n
ntdt
an
,
6
,
4
,
2
0
,
5
,
3
,
1
/
2
)
1
cos
(
1
cos
1
sin
2
1
0
0



















 

n
n
n
n
n
nt
n
ntdt
bn
 2 3 4 5
-
-2
-3
-4
-5
-6
f(t)
1
Example (Square Wave)
-0.5
0
0.5
1
1.5











 
t
t
t
t
f 5
sin
5
1
3
sin
3
1
sin
2
2
1
)
( 










 
t
t
t
t
f 5
sin
5
1
3
sin
3
1
sin
2
2
1
)
(

22. Harmonics
T
nt
b
T
nt
a
a
t
f
n
n
n
n




 





2
sin
2
cos
2
)
(
1
1
0
DC Part Even Part Odd Part
T is a period of all the above signals
)
sin(
)
cos(
2
)
( 0
1
0
1
0
t
n
b
t
n
a
a
t
f
n
n
n
n 



 






23. Harmonics
t
n
b
t
n
a
a
t
f
n
n
n
n 0
1
0
1
0
sin
cos
2
)
( 



 





T
f





2
2 0
0
Define , called the fundamental angular frequency.
0


 n
n
Define , called the n-th harmonic of the periodic function.
t
b
t
a
a
t
f n
n
n
n
n
n 



 





sin
cos
2
)
(
1
1
0

24. Harmonics
t
b
t
a
a
t
f n
n
n
n
n
n 



 





sin
cos
2
)
(
1
1
0
)
sin
cos
(
2 1
0
t
b
t
a
a
n
n
n
n
n 



 





















1
2
2
2
2
2
2
0
sin
cos
2 n
n
n
n
n
n
n
n
n
n
n t
b
a
b
t
b
a
a
b
a
a
 











1
2
2
0
sin
sin
cos
cos
2 n
n
n
n
n
n
n t
t
b
a
a
)
cos(
1
0 n
n
n
n t
C
C 



 



25. Amplitudes and Phase Angles
)
cos(
)
(
1
0 n
n
n
n t
C
C
t
f 



 


2
0
0
a
C 
2
2
n
n
n b
a
C 










 
n
n
n
a
b
1
tan
harmonic amplitude phase angle

26. Fourier Series
Complex Form of the
Fourier Series

27. Complex Exponentials
t
n
j
t
n
e t
jn
0
0 sin
cos
0





 
t
jn
t
jn
e
e
t
n 0
0
2
1
cos 0






t
n
j
t
n
e t
jn
0
0 sin
cos
0






   
t
jn
t
jn
t
jn
t
jn
e
e
j
e
e
j
t
n 0
0
0
0
2
2
1
sin 0













28. Complex Form of the Fourier Series
t
n
b
t
n
a
a
t
f
n
n
n
n 0
1
0
1
0
sin
cos
2
)
( 



 





   
t
jn
t
jn
n
n
t
jn
t
jn
n
n e
e
b
j
e
e
a
a 0
0
0
0
1
1
0
2
2
1
2














 


















1
0 0
0
)
(
2
1
)
(
2
1
2 n
t
jn
n
n
t
jn
n
n e
jb
a
e
jb
a
a
 










1
0
0
0
n
t
jn
n
t
jn
n e
c
e
c
c
)
(
2
1
)
(
2
1
2
0
0
n
n
n
n
n
n
jb
a
c
jb
a
c
a
c






)
(
2
1
)
(
2
1
2
0
0
n
n
n
n
n
n
jb
a
c
jb
a
c
a
c







29. Complex Form of the Fourier Series
 










1
0
0
0
)
(
n
t
jn
n
t
jn
n e
c
e
c
c
t
f












1
1
0
0
0
n
t
jn
n
n
t
jn
n e
c
e
c
c






n
t
jn
ne
c 0
)
(
2
1
)
(
2
1
2
0
0
n
n
n
n
n
n
jb
a
c
jb
a
c
a
c






)
(
2
1
)
(
2
1
2
0
0
n
n
n
n
n
n
jb
a
c
jb
a
c
a
c







30. Complex Form of the Fourier Series




2
/
2
/
0
0 )
(
1
2
T
T
dt
t
f
T
a
c
)
(
2
1
n
n
n jb
a
c 






 


 
 

2
/
2
/
0
2
/
2
/
0 sin
)
(
cos
)
(
1 T
T
T
T
tdt
n
t
f
j
tdt
n
t
f
T






2
/
2
/
0
0 )
sin
)(cos
(
1 T
T
dt
t
n
j
t
n
t
f
T





2
/
2
/
0
)
(
1 T
T
t
jn
dt
e
t
f
T



 


2
/
2
/
0
)
(
1
)
(
2
1 T
T
t
jn
n
n
n dt
e
t
f
T
jb
a
c )
(
2
1
)
(
2
1
2
0
0
n
n
n
n
n
n
jb
a
c
jb
a
c
a
c






)
(
2
1
)
(
2
1
2
0
0
n
n
n
n
n
n
jb
a
c
jb
a
c
a
c







31. Complex Form of the Fourier Series






n
t
jn
ne
c
t
f 0
)
( 





n
t
jn
ne
c
t
f 0
)
(
dt
e
t
f
T
c
T
T
t
jn
n 




2
/
2
/
0
)
(
1 dt
e
t
f
T
c
T
T
t
jn
n 




2
/
2
/
0
)
(
1
)
(
2
1
)
(
2
1
2
0
0
n
n
n
n
n
n
jb
a
c
jb
a
c
a
c






)
(
2
1
)
(
2
1
2
0
0
n
n
n
n
n
n
jb
a
c
jb
a
c
a
c






If f(t) is real,
*
n
n c
c 

n
n j
n
n
n
j
n
n e
c
c
c
e
c
c 





 |
|
,
|
| *
2
2
2
1
|
|
|
| n
n
n
n b
a
c
c 

 










 
n
n
n
a
b
1
tan

,
3
,
2
,
1 



n
0
0
2
1
a
c 

32. Complex Frequency Spectra
n
n j
n
n
n
j
n
n e
c
c
c
e
c
c 





 |
|
,
|
| *
2
2
2
1
|
|
|
| n
n
n
n b
a
c
c 

 










 
n
n
n
a
b
1
tan 
,
3
,
2
,
1 



n
0
0
2
1
a
c 
|cn|

amplitude
spectrum
n

phase
spectrum

33. Example
2
T
2
T
 T
T

2
d
t
f(t)
A
2
d

dt
e
T
A
c
d
d
t
jn
n 




2
/
2
/
0
2
/
2
/
0
0
1
d
d
t
jn
e
jn
T
A



















 

 2
/
0
2
/
0
0
0
1
1 d
jn
d
jn
e
jn
e
jn
T
A
)
2
/
sin
2
(
1
0
0
d
n
j
jn
T
A





2
/
sin
1
0
0
2
1
d
n
n
T
A








 





 

T
d
n
T
d
n
T
Ad
sin

34. 




 





 

T
d
n
T
d
n
T
Ad
cn
sin
8
2
5
1
T
,
4
1
,
20
1
0 







T
d
T
d
Example
40 80 120
-40 0
-120 -80
A/5
50 100 150
-50
-100
-150

35. 




 





 

T
d
n
T
d
n
T
Ad
cn
sin
4
2
10
1
T
,
2
1
,
20
1
0 

 




T
d
T
d
Example
40 80 120
-40 0
-120 -80
A/10
100 200 300
-100
-200
-300

36. Fourier Series &
Fourier Transform
Lecture 15-16: 28-Aug-12
Dr. P P Das

37. Example
dt
e
T
A
c
d t
jn
n 



0
0
d
t
jn
e
jn
T
A
0
0
0
1 

















 

0
0
1
1 0
jn
e
jn
T
A d
jn
)
1
(
1 0
0
d
jn
e
jn
T
A 




2
/
0
sin
d
jn
e
T
d
n
T
d
n
T
Ad 






 





 

T
T
 d
t
f(t)
A
0
)
(
1 2
/
2
/
2
/
0
0
0
0 d
jn
d
jn
d
jn
e
e
e
jn
T
A 








38. Fourier Series
Analysis of
Periodic Waveforms

39. Waveform Symmetry
Even Functions
Odd Functions
)
(
)
( t
f
t
f 

)
(
)
( t
f
t
f 



40. Decomposition
Any function f(t) can be expressed as the
sum of an even function fe(t) and an odd
function fo(t).
)
(
)
(
)
( t
f
t
f
t
f o
e 

)]
(
)
(
[
)
( 2
1
t
f
t
f
t
fe 


)]
(
)
(
[
)
( 2
1
t
f
t
f
t
fo 


Even Part
Odd Part

41. Example







0
0
0
)
(
t
t
e
t
f
t
Even Part
Odd Part







0
0
)
(
2
1
2
1
t
e
t
e
t
f t
t
e








0
0
)
(
2
1
2
1
t
e
t
e
t
f t
t
o

42. Half-Wave Symmetry
)
(
)
( T
t
f
t
f 
 and  
2
/
)
( T
t
f
t
f 


T
T/2
T/2

43. Quarter-Wave Symmetry
Even Quarter-Wave Symmetry
T
T/2
T/2
Odd Quarter-Wave Symmetry
T
T/2
T/2

44. Hidden Symmetry
 The following is a asymmetry periodic function:
 Adding a constant to get symmetry property.
A
T
T
A/2
A/2
T
T

45. Fourier Coefficients of
Symmetrical Waveforms
 The use of symmetry properties simplifies the
calculation of Fourier coefficients.
– Even Functions
– Odd Functions
– Half-Wave
– Even Quarter-Wave
– Odd Quarter-Wave
– Hidden

46. Fourier Coefficients of Even Functions
)
(
)
( t
f
t
f 

t
n
a
a
t
f
n
n 0
1
0
cos
2
)
( 

 


 

2
/
0
0 )
cos(
)
(
4 T
n dt
t
n
t
f
T
a

47. Fourier Coefficients of Odd Functions
)
(
)
( t
f
t
f 


t
n
b
t
f
n
n 0
1
sin
)
( 



 

2
/
0
0 )
sin(
)
(
4 T
n dt
t
n
t
f
T
b

48. Fourier Coefficients for Half-Wave Symmetry
)
(
)
( T
t
f
t
f 
 and  
2
/
)
( T
t
f
t
f 


T
T/2
T/2
The Fourier series contains only odd harmonics.
The Fourier series contains only odd harmonics.

49. Fourier Coefficients for Half-Wave Symmetry
)
(
)
( T
t
f
t
f 
 and  
2
/
)
( T
t
f
t
f 


)
sin
cos
(
)
(
1
0
0







n
n
n t
n
b
t
n
a
t
f







 odd
for
)
cos(
)
(
4
even
for
0
2
/
0
0 n
dt
t
n
t
f
T
n
a T
n







 odd
for
)
sin(
)
(
4
even
for
0
2
/
0
0 n
dt
t
n
t
f
T
n
b T
n

50. Fourier Coefficients for
Even Quarter-Wave Symmetry
T
T/2
T/2
]
)
1
2
cos[(
)
( 0
1
1
2 t
n
a
t
f
n
n 





 



4
/
0
0
1
2 ]
)
1
2
cos[(
)
(
8 T
n dt
t
n
t
f
T
a

51. Fourier Coefficients for
Odd Quarter-Wave Symmetry
]
)
1
2
sin[(
)
( 0
1
1
2 t
n
b
t
f
n
n 





 



4
/
0
0
1
2 ]
)
1
2
sin[(
)
(
8 T
n dt
t
n
t
f
T
b
T
T/2
T/2

52. Example
Even Quarter-Wave Symmetry
 



4
/
0
0
1
2 ]
)
1
2
cos[(
)
(
8 T
n dt
t
n
t
f
T
a  


4
/
0
0 ]
)
1
2
cos[(
8 T
dt
t
n
T
4
/
0
0
0
]
)
1
2
sin[(
)
1
2
(
8
T
t
n
T
n








 
)
1
2
(
4
)
1
( 1
n
n
T
T/2
T/2
1
1
T T/4
T/4

53. Example
Even Quarter-Wave Symmetry
 



4
/
0
0
1
2 ]
)
1
2
cos[(
)
(
8 T
n dt
t
n
t
f
T
a  


4
/
0
0 ]
)
1
2
cos[(
8 T
dt
t
n
T
4
/
0
0
0
]
)
1
2
sin[(
)
1
2
(
8
T
t
n
T
n








 
)
1
2
(
4
)
1
( 1
n
n
T
T/2
T/2
1
1
T T/4
T/4













 
t
t
t
t
f 0
0
0 5
cos
5
1
3
cos
3
1
cos
4
)
( 












 
t
t
t
t
f 0
0
0 5
cos
5
1
3
cos
3
1
cos
4
)
(

54. Example
T
T/2
T/2
1
1
T T/4
T/4
Odd Quarter-Wave Symmetry
 



4
/
0
0
1
2 ]
)
1
2
sin[(
)
(
8 T
n dt
t
n
t
f
T
b  


4
/
0
0 ]
)
1
2
sin[(
8 T
dt
t
n
T
4
/
0
0
0
]
)
1
2
cos[(
)
1
2
(
8
T
t
n
T
n









)
1
2
(
4
n

55. Example
T
T/2
T/2
1
1
T T/4
T/4
Odd Quarter-Wave Symmetry
 



4
/
0
0
1
2 ]
)
1
2
sin[(
)
(
8 T
n dt
t
n
t
f
T
b  


4
/
0
0 ]
)
1
2
sin[(
8 T
dt
t
n
T
4
/
0
0
0
]
)
1
2
cos[(
)
1
2
(
8
T
t
n
T
n









)
1
2
(
4
n













 
t
t
t
t
f 0
0
0 5
sin
5
1
3
sin
3
1
sin
4
)
( 












 
t
t
t
t
f 0
0
0 5
sin
5
1
3
sin
3
1
sin
4
)
(

56. Fourier Series
Half-Range
Expansions

57. Non-Periodic Function Representation
 A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).


58. Without Considering Symmetry
 A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
 T

59. Expansion Into Even Symmetry
 A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
 T=2

60. Expansion Into Odd Symmetry
 A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).

T=2

61. Expansion Into Half-Wave Symmetry
 A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
 T=2

62. Expansion Into
Even Quarter-Wave Symmetry
 A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).

T/2=2
T=4

63. Expansion Into
Odd Quarter-Wave Symmetry
 A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).

T/2=2 T=4

64. Fourier Series
Least Mean-Square
Error Approximation

65. Approximation a function
Use  







k
n
n
n
k t
n
b
t
n
a
a
t
S
1
0
0
0
sin
cos
2
)
(
to represent f(t) on interval T/2 < t < T/2.
Define )
(
)
(
)
( t
S
t
f
t k
k 






2
/
2
/
2
)]
(
[
1 T
T
k
k dt
t
T
E Mean-Square
Error

66. Approximation a function
Show that using Sk(t) to represent f(t) has
least mean-square property.




2
/
2
/
2
)]
(
[
1 T
T
k
k dt
t
T
E
 
 














2
/
2
/
2
1
0
0
0
sin
cos
2
)
(
1 T
T
k
n
n
n dt
t
n
b
t
n
a
a
t
f
T
Proven by setting Ek/ai = 0 and Ek/bi = 0.

67. Approximation a function




2
/
2
/
2
)]
(
[
1 T
T
k
k dt
t
T
E
 
 














2
/
2
/
2
1
0
0
0
sin
cos
2
)
(
1 T
T
k
n
n
n dt
t
n
b
t
n
a
a
t
f
T
0
)
(
1
2
2
/
2
/
0
0







T
T
k
dt
t
f
T
a
a
E 0
)
(
1
2
2
/
2
/
0
0







T
T
k
dt
t
f
T
a
a
E
0
cos
)
(
2 2
/
2
/
0 







T
T
n
n
k
tdt
n
t
f
T
a
a
E 0
cos
)
(
2 2
/
2
/
0 







T
T
n
n
k
tdt
n
t
f
T
a
a
E
0
sin
)
(
2 2
/
2
/
0 







T
T
n
n
k
tdt
n
t
f
T
b
b
E 0
sin
)
(
2 2
/
2
/
0 







T
T
n
n
k
tdt
n
t
f
T
b
b
E

68. Mean-Square Error




2
/
2
/
2
)]
(
[
1 T
T
k
k dt
t
T
E
 
 














2
/
2
/
2
1
0
0
0
sin
cos
2
)
(
1 T
T
k
n
n
n dt
t
n
b
t
n
a
a
t
f
T
 






2
/
2
/
1
2
2
2
0
2
)
(
2
1
4
)]
(
[
1 T
T
k
n
n
n
k b
a
a
dt
t
f
T
E  






2
/
2
/
1
2
2
2
0
2
)
(
2
1
4
)]
(
[
1 T
T
k
n
n
n
k b
a
a
dt
t
f
T
E

69. Mean-Square Error




2
/
2
/
2
)]
(
[
1 T
T
k
k dt
t
T
E
 
 














2
/
2
/
2
1
0
0
0
sin
cos
2
)
(
1 T
T
k
n
n
n dt
t
n
b
t
n
a
a
t
f
T
 





2
/
2
/
1
2
2
2
0
2
)
(
2
1
4
)]
(
[
1 T
T
k
n
n
n b
a
a
dt
t
f
T
 





2
/
2
/
1
2
2
2
0
2
)
(
2
1
4
)]
(
[
1 T
T
k
n
n
n b
a
a
dt
t
f
T

70. Mean-Square Error




2
/
2
/
2
)]
(
[
1 T
T
k
k dt
t
T
E
 
 












2
/
2
/
2
1
0
0
0
sin
cos
2
)
(
1 T
T
k
n
n
n dt
t
n
b
t
n
a
a
t
f
T


 






2
/
2
/
1
2
2
2
0
2
)
(
2
1
4
)]
(
[
1 T
T
n
n
n b
a
a
dt
t
f
T
 






2
/
2
/
1
2
2
2
0
2
)
(
2
1
4
)]
(
[
1 T
T
n
n
n b
a
a
dt
t
f
T

71. Fourier Series
Impulse Train

72. Dirac Delta Function








0
0
0
)
(
t
t
t and 1
)
( 





dt
t
0
t
Also called unit impulse function,
but not actually a function in true
sense.

73. Sifting Property
)
0
(
)
(
)
( 







dt
t
t )
0
(
)
(
)
( 







dt
t
t
)
0
(
)
(
)
0
(
)
0
(
)
(
)
(
)
( 








 











dt
t
dt
t
dt
t
t
(t): Test Function

74. Impulse Train
0
t
T 2T 3T
T
2T
3T








n
T nT
t
t )
(
)
( 







n
T nT
t
t )
(
)
(

75. Fourier Series of the Impulse Train








n
T nT
t
t )
(
)
( 







n
T nT
t
t )
(
)
(
T
dt
t
T
a
T
T
T
2
)
(
2 2
/
2
/
0 

 

T
dt
t
n
t
T
a
T
T
T
n
2
)
cos(
)
(
2 2
/
2
/
0 


 

0
)
sin(
)
(
2 2
/
2
/
0 


 

dt
t
n
t
T
b
T
T
T
n








n
T t
n
T
T
t 0
cos
2
1
)
( 







n
T t
n
T
T
t 0
cos
2
1
)
(
)
0
(
)
(
)
( 







dt
t
t )
0
(
)
(
)
( 







dt
t
t

76. Complex Form
Fourier Series of the Impulse Train
T
dt
t
T
a
c
T
T
T
1
)
(
1
2
2
/
2
/
0
0 


 

T
dt
e
t
T
c
T
T
t
jn
T
n
1
)
(
1 2
/
2
/
0


 










n
t
jn
T e
T
t 0
1
)
( 






n
t
jn
T e
T
t 0
1
)
(








n
T nT
t
t )
(
)
( 







n
T nT
t
t )
(
)
(
)
0
(
)
(
)
( 







dt
t
t )
0
(
)
(
)
( 







dt
t
t

77. Fourier Transform
Impulse Train

78. Fourier Series / Transform
 Fourier Series deals with discrete variable
(harmonics of )
 Fourier Transform deals with continuous
frequency
0
0 2 f

 

79. Fourier Series / Transform







 dt
e
t
f
F
t
f t
j 
 2
)
(
)
(
)}
(
{







 

 
d
e
F
F
t
f t
j2
1
)
(
)}
(
{
)
(





n
t
f
jn
ne
c
t
f 0
2
)
( 




2
/
2
/
2 0
)
(
1 T
T
t
f
jn
n dt
e
t
f
T
c 
0
0 2 f

 

80. Example

81. Example
dt
e
t
f
F t
j





 
 2
)
(
)
(
2
/
2
/
2
2
1
W
W
t
j
e
j
A



 

 
W
j
W
j
e
e
j
A 





2
 
 
W
W
AW


sin


82. Example

83. Example: Unit Impulse
dt
e
t
F t
j





 

 2
)
(
)
(
1
0
0
2


 
e
e j 
dt
e
t
t
F t
j






 

 2
0 )
(
)
(
0
2 t
j
e 



84. Fourier Series of the Impulse Train
T
dt
e
t
s
T
c
T
T
t
j
T
n
T
n



 






1
)
(
1 2
/
2
/
2








n
t
T
n
j
T e
T
t
s

2
1
)
( 







n
t
T
n
j
T e
T
t
s

2
1
)
(




 


n
T T
n
t
t
s )
(
)
( 




 


n
T T
n
t
t
s )
(
)
( 

85. Fourier Transform of the Impulse Train



















n
n
t
T
n
j
T
T
n
T
e
T
t
s
S
)
(
1
}
1
{
)}
(
{
)
(
2























n
n
t
T
n
j
T
T
n
T
e
T
t
s
S
)
(
1
}
1
{
)}
(
{
)
(
2




)
(
}
{
2
T
n
e
t
T
n
j



 



)
(
}
{
2
T
n
e
t
T
n
j



 




86. Fourier Transform
Convolution

87. Convolution
 A mathematical operator which computes the
“amount of overlap” between two functions.
Can be thought of as a general moving
average
 Discrete domain:
 Continuous domain:

88. Discrete domain
 Basic steps
1. Flip (reverse) one of the digital functions.
2. Shift it along the time axis by one sample.
3. Multiply the corresponding values of the two digital functions.
4. Summate the products from step 3 to get one point of the
digital convolution.
5. Repeat steps 1-4 to obtain the digital convolution at all times
that the functions overlap.
 Example

89. Continuous domain example

90. Continuous domain example

91. Convolution Theorem
)
(
)
(
)}
(
)
(
{ 
 G
F
t
g
t
f 


)
(
)
(
)}
(
)
(
{ t
g
t
f
G
F 

 


92. Fourier Transform
Sampling

93. Convolution Theorem

94. Sampling









n
T
T
n
t
t
f
t
s
t
f
t
f
)
(
)
(
)
(
)
(
)
(
~





 


n
T T
n
t
t
s )
(
)
( 




 


n
T T
n
t
t
s )
(
)
( 

95. Sampling & FT















n
T
T
n
F
T
S
F
t
s
t
f
t
f
F
)
(
1
)
(
)
(
)}
(
)
(
{
)}
(
~
{
)
(
~







 



n T
n
T
S )
(
1
)
( 

 


 



n T
n
T
S )
(
1
)
( 



96. Sampling & FT

97. Nyquist Rate

98. Reconstruction Filter

99. Aliasing

100. Discrete & Fast
Fourier Transform
Lecture 17: 03-Sep-12
Dr. P P Das

101. Sampling & FT















n
T
T
n
F
T
S
F
t
s
t
f
t
f
F
)
(
1
)
(
)
(
)}
(
)
(
{
)}
(
~
{
)
(
~







 



n T
n
T
S )
(
1
)
( 

 


 



n T
n
T
S )
(
1
)
( 



102. Sampling & FT

103. DFT: How to compute FT?
T
n
j
n
n
n
t
j
t
j
n
t
j
e
f
dt
e
T
n
t
t
f
dt
e
T
n
t
t
f
dt
e
t
f
F
























 
 






















2
2
2
2
)
(
)
(
)
(
)
(
)
(
~
)
(
~
)
(
)
(
)
(
T
n
f
dt
T
n
t
t
f
fn




 





104. Discrete Fourier Transform
T
n
j
n
ne
f
F 





 
 2
)
(
~ )
( T
n
f
fn 

1
,
2
,
1
,
0
,
T
M
m
T
1/
to
0
period
over the
)
(
~
of
samples
spaced
equally
Obtain
DFT
for the
basis
the
is
period
one
Sampling
period
one
over
zed
characteri
be
to
need
)
(
~
T
1
period
with
periodic
infinitely
and
continuous
is
)
(
~
function
discrete
a
is














M
m
F
M
F
F
fn








105. Discrete Fourier Transform
T
n
j
n
ne
f
F 





 
 2
)
(
~
)
( T
n
f
fn 

1
,
2
,
1
,
0
,
T
M
m



 M
m 

1
,
2
,
1
,
0
,
DFT
1
0
/
2






M
m
e
f
F
M
n
M
mn
j
n
m 

1
,
2
,
1
,
0
,
1
IDFT
1
0
/
2


 


M
n
e
F
M
f
M
m
M
mn
j
m
n 


106. DFT: Notations
D
-
2
in
variables
transform
Discrete
:
,
D
-
2
in
variables
function
Discrete
:
,
D
-
2
in
variables
transform
Continuous
:
,
D
-
2
in
variables
function
Continuous
:
,
D
-
1
in
variable
)
(frequency
Continuous
:
D
-
1
in
variable
(time)
Continuous
:
v
u
y
x
z
t
t




107. DFT Pair
1
,
2
,
1
,
0
,
)
(
)
(
DFT
1
0
/
2






M
u
e
x
f
u
F
M
x
M
ux
j


1
,
2
,
1
,
0
,
)
(
1
)
(
IDFT
1
0
/
2


 


M
x
e
u
F
M
x
f
M
u
M
ux
j



108. Discrete Fourier Transform Pair





1
0
/
2
)
(
)
(
M
x
M
ux
j
e
x
f
u
F 




1
0
/
2
)
(
1
)
(
M
u
M
ux
j
e
u
F
M
x
f 
uniformly
taken
samples
discrete
of
set
finite
any
to
applicable
Is
/
1
Interval
Frequency
of
t
Independen
Interval
Sampling
of
t
Independen
:
pair
DFT






T
T


109. DFT: Periodicity

,
2
,
1
,
0
),
(
)
(
DFT


 k
kM
u
F
u
F

,
2
,
1
,
0
),
(
)
(
IDFT


 k
kM
x
f
x
f

110. Sampling & Frequency Intervals
T
interval
sampling
the
on
depends
DFT
the
by
spanned
s
frequencie
of
Range
sampled.
is
)
(
over which
duration
the
on
depends
frequency
in
Resolution
1
;
1
1
;















t
f
T
u
T
u
M
T
T
M
u
T
M
T

111. DFT: Example
4
)
3
(
;
4
)
2
(
;
2
)
1
(
;
1
)
0
( 


 f
f
f
f

112. DFT: Example
 
)
2
3
(
)
3
(
);
0
1
(
)
2
(
2
3
)
4
0
(
)
0
4
(
)
2
0
(
1
4
4
2
1
)
(
)
1
(
11
4
4
2
1
)
3
(
)
2
(
)
1
(
)
0
(
)
(
)
0
(
2
/
3
2
/
0
3
0
4
/
)
1
(
2
3
0
j
F
j
F
j
j
j
j
e
e
e
e
e
x
f
F
f
f
f
f
x
f
F
j
j
j
x
x
j
x













































113. DFT: Example
    1
4
4
1
2
3
1
2
3
11
4
1
)
(
4
1
)
(
4
1
)
0
(
3
0
3
0
)
0
(
2














j
j
u
F
e
u
F
f
u
u
u
j 

114. M. Wu: ENEE631 Digital Image Processing (Spring'09)
1-D FFT in Matrix/Vector
 { z(n) }  { Z(k) }
n, k = 0, 1, …, N-1, WN = exp{ - j2 / N }
~ complex conjugate of primitive Nth root of unity
 

















































































)
1
(
)
1
(
)
0
(
)
1
(
)
2
(
)
1
(
)
0
(
1
1
1
1
1
1
1
)
1
(
)
2
(
)
1
(
)
0
(
1
1
0
/
)
1
(
2
/
)
1
(
2
2
/
)
1
(
2
/
)
1
(
2
2
/
4
2
/
2
2
/
)
1
(
2
/
2
2
/
2
2 N
Z
Z
Z
a
a
a
N
Z
Z
Z
Z
e
e
e
e
e
e
e
e
e
N
z
z
z
z
N
N
N
j
N
N
j
N
N
j
N
N
j
N
j
N
j
N
N
j
N
j
N
j






































1
0
1
0
)
(
1
)
(
)
(
1
)
(
N
k
nk
N
N
n
nk
N
W
k
Z
N
n
z
W
n
z
N
k
Z
UMCP
ENEE631
Slides
(created
by
M.Wu
©
2001/2004)
z
a
a
a
N
z
z
z
z
e
e
e
e
e
e
e
e
e
N
Z
Z
Z
Z
T
N
T
T
N
N
j
N
N
j
N
N
j
N
N
j
N
j
N
j
N
N
j
N
j
N
j



































































 






















*
1
*
1
*
0
/
)
1
(
2
/
)
1
(
2
2
/
)
1
(
2
/
)
1
(
2
2
/
4
2
/
2
2
/
)
1
(
2
/
2
2
/
2
)
1
(
)
2
(
)
1
(
)
0
(
1
1
1
1
1
1
1
)
1
(
)
2
(
)
1
(
)
0
(
2




















115. DFT: Example
4
)
3
(
;
4
)
2
(
;
2
)
1
(
;
1
)
0
( 


 f
f
f
f

























4
4
2
1
)
3
(
)
2
(
)
1
(
)
0
(
f
f
f
f

116. DFT: By Matrix















































)
3
(
)
2
(
)
1
(
)
0
(
1
1
1
1
1
1
1
)
3
(
)
2
(
)
1
(
)
0
(
2
/
9
3
2
/
3
3
2
2
/
3
2
/
f
f
f
f
e
e
e
e
e
e
e
e
e
F
F
F
F
j
j
j
j
j
j
j
j
j




























































)
3
(
)
2
(
)
1
(
)
0
(
0
0
1
0
1
0
1
0j
1
0
1
1
0
0j
1
0
1
1
1
1
1
)
3
(
)
2
(
)
1
(
)
0
(
f
f
f
f
j
j
j
j
j
j
j
F
F
F
F

117. DFT: By Matrix

































































































































j
j
j
j
j
j
j
j
j
j
f
f
f
f
j
j
j
j
j
j
j
F
F
F
F
2
3
1
2
3
11
4
4
2
1
4
4
2
1
4
4
2
1
4
4
2
1
4
4
2
1
1
1
1
1
1
1
1
1
1
1
1
1
)
3
(
)
2
(
)
1
(
0
(
0
0
1
0
1
0
1
0j
1
0
1
1
0
0j
1
0
1
1
1
1
1
)
3
(
)
2
(
)
1
(
)
0
(

























4
4
2
1
)
3
(
)
2
(
)
1
(
)
0
(
f
f
f
f

118. IDFT: By Matrix






































)
3
(
)
2
(
)
1
(
)
0
(
1
1
1
1
1
1
1
)
3
(
)
2
(
)
1
(
)
0
(
4
2
/
9
3
2
/
3
3
2
2
/
3
2
/
F
F
F
F
e
e
e
e
e
e
e
e
e
f
f
f
f
j
j
j
j
j
j
j
j
j





























































)
3
(
)
2
(
)
1
(
)
0
(
0
0
1
0
1
0
1
0j
1
0
1
1
0
0j
1
0
1
1
1
1
1
)
3
(
)
2
(
)
1
(
)
0
(
4
F
F
F
F
j
j
j
j
j
j
j
f
f
f
f

119. IDFT: By Matrix










































































































































16
16
8
4
2
3
1
2
3
11
2
3
1
2
3
11
2
3
1
2
3
11
2
3
1
2
3
11
2
3
1
2
3
11
1
1
1
1
1
1
1
1
1
1
1
1
)
3
(
)
2
(
)
1
(
0
(
0
0
1
0
1
0
1
0j
1
0
1
1
0
0j
1
0
1
1
1
1
1
)
3
(
)
2
(
)
1
(
)
0
(
4
j
j
j
j
j
j
j
j
j
j
j
j
j
j
F
F
F
F
j
j
j
j
j
j
j
f
f
f
f






























j
j
F
F
F
F
2
3
1
2
3
11
)
3
(
)
2
(
)
1
(
)
0
(

120. Inverse Discrete Fourier Transform
1
,
2
,
1
,
0
,
)
(
)
(
:
DFT
1
0
/
2






M
u
e
x
f
u
F
M
x
M
ux
j


))
(
(
1
)
(
))
(
(
1
)
(
1
)
(
:
conjugate
complex
Taking
1
,
2
,
1
,
0
,
)
(
1
)
(
:
IDFT
*
*
*
1
0
/
2
*
*
1
0
/
2
u
F
DFT
M
x
f
u
F
DFT
M
e
u
F
M
x
f
M
x
e
u
F
M
x
f
M
u
M
ux
j
M
u
M
ux
j

















121. DFT Complexity
point DFT
an
in
additions
complex
of
Number
:
)
(
point DFT
an
in
tions
multiplica
complex
of
Number
:
)
(
M
M
Add
M
M
Mult

122. DFT / IDFT Complexity
)
(
)
1
(
)
(
)
(
)
(
computed.
-
pre
are
that
Assume
1
,
2
,
1
,
0
,
)
(
)
(
2
2
2
/
2
1
0
/
2
M
O
M
M
M
Add
M
O
M
M
Mult
e
M
u
e
x
f
u
F
M
ux
j
M
x
M
ux
j

















123. Fast Fourier Transform
K
M
e
W
M
u
W
x
f
u
F
n
M
j
M
M
x
ux
M
2
2
1
,
2
,
1
,
0
,
)
(
)
(
/
2
1
0












u
M
M
u
M
u
M
M
u
M
ux
K
ux
K W
W
W
W
W
W 2
2
2
2 ;
;
:
Properties



 

1
,
2
,
1
,
0
,
)
(
)
(
1
0
/
2






M
u
e
x
f
u
F
M
x
M
ux
j



124. Fast Fourier Transform























1
0
2
1
0
1
0
)
1
2
(
2
1
0
)
2
(
2
1
2
0
)
1
2
(
)
2
(
)
1
2
(
)
2
(
)
(
)
(
K
x
u
K
ux
K
K
x
ux
K
K
x
x
u
K
K
x
x
u
K
K
x
ux
M
W
W
x
f
W
x
f
W
x
f
W
x
f
W
x
f
u
F
K
M
e
W n
M
j
M 2
2
/
2


   ux
K
ux
K W
W 
2
2

125. Fast Fourier Transform
u
K
odd
even
u
K
odd
even
K
x
ux
K
odd
K
x
ux
K
even
W
u
F
u
F
K
u
F
W
u
F
u
F
u
F
W
x
f
u
F
W
x
f
u
F
2
2
1
0
1
0
)
(
)
(
)
(
)
(
)
(
)
(
)
1
2
(
)
(
)
2
(
)
(














K
M
e
W
n
M
j
M
2
2
/
2


   u
M
M
u
M
u
M
M
u
M W
W
W
W 2
2
; 

 


126. FFT Complexity
point FFT
2
an
in
additions
complex
of
Number
:
)
(
)
(
point FFT
2
an
in
tions
multiplica
complex
of
Number
:
)
(
)
(
n
n
M
n
Add
M
Add
M
n
Mult
M
Mult





127. FFT









1
0
1
0
)
1
2
(
)
(
)
2
(
)
(
K
x
ux
K
odd
K
x
ux
K
even
W
x
f
u
F
W
x
f
u
F
0
2
0
2
1
)
0
(
)
0
(
)
1
(
)
0
(
)
0
(
)
0
(
)
1
(
)
0
(
)
0
(
)
0
(
1
;
2
2
;
1
W
F
F
F
W
F
F
F
f
F
f
F
K
M
n
odd
even
odd
even
odd
even










u
K
odd
even
u
K
odd
even
W
u
F
u
F
K
u
F
W
u
F
u
F
u
F
2
2
)
(
)
(
)
(
)
(
)
(
)
(





2
)
1
(
1
)
1
(


Add
Mult

128. FFT









1
0
1
0
)
1
2
(
)
(
)
2
(
)
(
K
x
ux
K
odd
K
x
ux
K
even
W
x
f
u
F
W
x
f
u
F
1
4
0
4
1
4
0
4
2
)
1
(
)
1
(
)
2
1
(
)
3
(
)
0
(
)
0
(
)
2
0
(
)
2
(
)
1
(
)
1
(
)
1
(
)
0
(
)
0
(
)
0
(
)
3
(
&
)
1
(
from
point FFT
-
2
)
(
)
2
(
&
)
0
(
from
point FFT
-
2
)
(
2
;
4
2
;
2
W
F
F
F
F
W
F
F
F
F
W
F
F
F
W
F
F
F
f
f
u
F
f
f
u
F
K
M
n
odd
even
odd
even
odd
even
odd
even
odd
even


















u
K
odd
even
u
K
odd
even
W
u
F
u
F
K
u
F
W
u
F
u
F
u
F
2
2
)
(
)
(
)
(
)
(
)
(
)
(





4
)
1
(
2
)
2
(
2
)
1
(
2
)
2
(




Add
Add
Mult
Mult

129. FFT Complexity
)
log
(
log
)
(
)
(
)
log
(
log
2
1
)
(
)
(
0
;
;
0
1
;
2
)
1
(
2
)
(
0
;
;
0
1
;
2
)
1
(
2
)
(
2
2
2
2
1
M
M
O
M
M
n
Add
M
Add
M
M
O
M
M
n
Mult
M
Mult
n
n
n
Add
n
Add
n
n
n
Mult
n
Mult
n
n

















 

131. 2-D DFT & Images
Lecture 18-19: 04-Sep-12
Dr. P P Das
Kinect Demo in Lecture 19

132. 2-D Impulse









0
0
,
0
0
)
,
(
z
t
z
t
z
t

and
1
)
,
( 
 






dz
dt
z
t


133. 2-D Sifting Property
)
0
,
0
(
)
,
(
)
,
( f
dz
dt
z
t
z
t
f 
 







)
,
(
)
,
(
)
,
( 0
0
0
0 z
t
f
dz
dt
z
z
t
t
z
t
f 


 








134. 2-D Discrete Impulse








0
1
0
,
0
0
)
,
(
y
x
y
x
y
x


135. 2-D Discrete Sifting Property
)
0
,
0
(
)
,
(
)
,
( f
y
x
y
x
f
x y

 







)
,
(
)
,
(
)
,
( 0
0
0
0 y
x
f
y
y
x
x
y
x
f
x y



 








136. 2-D Impulse

137. 2-D Continuous Fourier
Transform Pair
 








 dz
dt
e
z
t
f
F z
t
j )
(
2
)
,
(
)
,
( 




 







 


 


d
d
e
F
z
t
f z
t
j )
(
2
)
,
(
)
,
(

138. 2-D FT: Example

139. 2-D FT: Example















 
 
 










)
(
)
sin(
)
(
)
sin(
)
,
(
)
,
(
2
/
2
/
2
/
2
/
)
(
2
)
(
2
Z
Z
T
T
ATZ
dz
dt
Ae
dz
dt
e
z
t
f
F
T
T
Z
Z
z
t
j
z
t
j













140. 2-D Sampling
max
max
max
max
2
1
2
1
and
for
0
)
,
(
limited
-
band
is
)
,
(















Z
T
F
z
t
f
 







 




m n
Z
T Z
n
z
T
m
t
z
t
s )
,
(
)
,
( 

141. 2-D Sampling

142. Frequency Aliasing
 Signals sampled below Nyquist rate (under-
sampled) have overlapped periods.
 In aliasing, high frequency components
masquerade as low frequency components in
sampled function. Hence, ‘aliasing’ or ‘false
identity’.

143. Aliasing

144. Aliasing

145. Frequency Aliasing
 Most band-limited signals have infinite
frequency components in sampled signal due
to finite duration of sampling.
 

















n
T
n
T
n
t
c
T
n
f
t
f
t
h
H
F
F
t
f
/
)
(
sin
)
(
)
(
~
)
(
)}
(
)
(
~
{
)}
(
{
)
( 1
1





 


otherwise
T
t
t
h
0
0
1
)
(

146. Frequency Aliasing
 Aliasing is inevitable while working with
sampled records of finite length
 Aliasing can be reduced by smoothing the
input function to attenuate higher frequencies
(defocusing images)
 This is known as ‘anti-aliasing’ and has to be
done before the function is sampled

147. Aliasing & Interpolation
 









n
T
n
T
n
t
c
T
n
f
t
f /
)
(
sin
)
(
)
(











,
function
sinc
of
sum
of
ion
interpolat
0
)
(
sin
and
1
)
0
(
sin
as
),
(
)
(
T
k
t
m
c
c
T
k
t
T
k
f
t
f

148. Aliasing in Images
 Spatial Aliasing
– Due to under-sampling in space
 Temporal Aliasing
– Due to slow time interval between frames in video
– Wagon-Wheel Effect

149. Aliasing in Spectrum

150. Sampling: 96X96 Pixels

151. Interpolation Techniques
 Nearest Neighbor Interpolation
 Bi-Linear Interpolation
 Bi-Cubic Interpolation

152. Interpolation & Re-sampling
 Zooming
– Over-sampling
– Pixel Replication / Row-Column Duplication
 Shrinking
– Under-sampling
– Row-Column Deletion
– Blur slightly before shrinking

154. Aliasing Artifact: Jaggies
 Block-like image component
 Common in images with string edge content

157. Aliasing Artifact: Moiré Patterns
 Beat patterns produced between two gratings
of approximately equal spacing
 Common in images with periodic or nearly
periodic content

161. 2-D DFT Properties
Lecture 20: 10-Sep-12
Dr. P P Das

162. 2-D DFT Pair








1
0
1
0
)
/
/
(
2
)
,
(
)
,
(
DFT
M
x
N
y
N
vy
M
ux
j
e
y
x
f
v
u
F 







1
0
1
0
)
/
/
(
2
)
,
(
1
)
(
IDFT
M
u
N
v
N
vy
M
ux
j
e
v
u
F
MN
x
f 

163. 2-D: Separability


 























1
0
/
2
1
0
/
2
1
0
1
0
/
2
/
2
1
0
1
0
)
/
/
(
2
)
,
(
)
,
(
where
)
,
(
)
,
(
)
,
(
)
,
(
N
y
N
vy
j
M
x
M
ux
j
M
x
N
y
N
vy
j
M
ux
j
M
x
N
y
N
vy
M
ux
j
e
y
x
f
v
x
F
e
v
x
F
e
y
x
f
e
e
y
x
f
v
u
F






164. Properties of 2-D DFT:
Spatial & Frequency Intervals
Z
N
v
T
M
u






1
1

165. Properties of 2-D DFT:
Translation & Rotation
)
,
(
)
,
(
sin
cos
sin
cos
:
s
coordinate
polar
In
)
,
(
)
,
(
)
,
(
)
,
(
0
0
)
/
/
(
2
0
0
)
/
/
(
2
0
0
0
0
0
0




























F
r
f
v
u
r
y
r
x
e
v
u
F
y
y
x
x
f
e
y
x
f
v
u
u
F
N
v
y
M
u
x
j
N
y
v
M
x
u
j

166. Properties of 2-D DFT:
Periodicity
)
,
(
)
,
(
)
,
(
)
,
(
2
1
2
1
N
k
v
M
k
u
F
N
k
v
u
F
v
M
k
u
F
v
u
F







)
,
(
)
,
(
)
,
(
)
,
(
2
1
2
1
N
k
y
M
k
x
f
N
k
y
x
f
y
M
k
x
f
y
x
f








167. Periodicity: Centering the
Transform
)
2
/
,
2
/
(
)
1
(
,
)
2
/
(
)
1
(
2
/
For
)
(
D
-
1
In
0
0
/
)
(
2 0
N
v
M
u
F
y)
f(x
M
u
F
f(x)
M
u
u
u
F
f(x)e
y
x
x
M
x
u
jπ












169. Properties of 2-D DFT:
Symmetry
)
,
(
2
)
,
(
)
,
(
)
,
(
)
,
(
2
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
y
x
w
y
x
w
y
x
w
y
x
w
y
x
w
y
x
w
y
x
w
y
x
w
y
x
w
y
x
w
y
x
w
o
o
e
e
o
e


















170. Properties of 2-D DFT:
Symmetry
)
,
(
)
,
(
)
,
(
)
,
(
0
)
,
(
)
,
(
1
0
1
0
y
N
x
M
w
y
x
w
y
N
x
M
w
y
x
w
y
x
w
y
x
w
o
o
e
e
o
M
x
N
y
e














171. Symmetry
)
,
(
)
,
(
symmetric
-
anti
conjugate
is
FT
,
)
,
(
imaginary
For
)
,
(
)
,
(
symmetric
conjugate
is
FT
,
)
,
(
real
For
*
*
v
u
F
v
u
F
y
x
f
v
u
F
v
u
F
y
x
f








173. Fourier Spectrum & Phase Angle
 
)
,
(
)
,
(
)
,
(
)
,
(
:
Spectrum
Power
)
,
(
)
,
(
arctan
:
Angle
Phase
)
,
(
)
,
(
)
,
(
:
Spectrum
Fourier
)
,
(
)
,
(
)
,
(
)
,
(
2
2
2
)
,
(
2
/
1
2
2
)
,
(
v
u
I
v
u
R
v
u
F
v
u
P
v
u
R
v
u
I
e
v
u
I
v
u
R
v
u
F
e
v
u
F
v
u
jI
v
u
R
v
u
F
v
u
j
v
u
j


















174. Fourier Spectrum & Phase Angle
)
,
(
)
,
(
)
0
,
0
(
)
,
(
)
,
(
symmetry
odd
has
Angle
Phase
)
,
(
)
,
(
symmetry
even
has
Spectrum
symmetric
conjugate
is
FT
,
)
,
(
real
For
1
0
1
0
y
x
f
MN
y
x
f
F
v
u
v
u
v
u
F
v
u
F
y
x
f
M
x
N
y




















175. y
x
y
x
f 
 )
1
)(
,
( )
,
(
log
1 v
u
F


179. 2-D Convolution
)
(
)
(
)
,
(
)
,
(
)
(
)
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
1
0
1
0




H
F
y
x
h
y
x
f
H
F
y
x
h
y
x
f
n
y
m
x
h
n
m
f
y
x
h
y
x
f
M
m
N
n














180. Effect of DFT on Convolution





399
0
)
(
)
(
)
(
)
(
m
m
x
h
m
f
x
h
x
f

181. Wraparound
Error needs
Zero Padding

182. 2-D: Separability

 



















1
0
/
2
1
0
1
0
/
2
/
2
1
0
1
0
)
/
/
(
2
)
,
(
)
,
(
)
,
(
)
,
(
DFT
M
x
M
ux
j
M
x
N
y
N
vy
j
M
ux
j
M
x
N
y
N
vy
M
ux
j
e
v
x
F
e
y
x
f
e
e
y
x
f
v
u
F





183. Thank you