Fourier Class Lessons

1. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier Transforms

2. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier series
To go from f( ) to f(t) substitute
To deal with the first basis vector being of
length 2 instead of , rewrite as
t
t
T
0
2


 

)
sin(
)
cos(
)
( 0
0
0
t
n
b
t
n
a
t
f n
n
n 
 
 


)
sin(
)
cos(
2
)
( 0
0
1
0
t
n
b
t
n
a
a
t
f n
n
n 
 

 



3. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier series
The coefficients become
dt
t
k
t
f
T
a
T
t
t
k 


0
0
)
cos(
)
(
2
0

dt
t
k
t
f
T
b
T
t
t
k 


0
0
)
sin(
)
(
2
0


4. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier series
Alternate forms
where
)
)
(cos(
2
))
sin(
)
tan(
)
(cos(
2
))
sin(
)
(cos(
2
)
(
0
1
0
0
0
1
0
0
0
1
0
n
n
n
n
n
n
n
n
n
n
t
n
c
a
t
n
t
n
a
a
t
n
a
b
t
n
a
a
t
f




































 
n
n
n
n
n
n
a
b
b
a
c 1
2
2
tan
and 

5. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Complex exponential notation
Euler’s formula )
sin(
)
cos( x
i
x
eix


Phasor notation:
















x
y
iy
x
iy
x
z
z
y
x
z
e
z
iy
x i
1
2
2
tan
and
)
)(
(
where



6. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Euler’s formula
Taylor series expansions
Even function ( f(x) = f(-x) )
Odd function ( f(x) = -f(-x) )
...
!
4
!
3
!
2
1
4
3
2






x
x
x
x
ex
...
!
8
!
6
!
4
!
2
1
)
cos(
8
6
4
2






x
x
x
x
x
...
!
9
!
7
!
5
!
3
)
sin(
9
7
5
3






x
x
x
x
x
x
)
sin(
)
cos(
...
!
7
!
6
!
5
!
4
!
3
!
2
1
7
6
5
4
3
2
x
i
x
ix
x
ix
x
ix
x
ix
eix












7. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Complex exponential form
Consider the expression
So
Since an and bn are real, we can let
and get
)
sin(
)
(
)
cos(
)
(
)
sin(
)
cos(
)
(
0
0
0
0
0
0
t
n
F
F
i
t
n
F
F
t
n
iF
t
n
F
e
F
t
f
n
n
n
n
n
n
n
n
n
t
in
n

























)
(
and n
n
n
n
n
n F
F
i
b
F
F
a 
 



n
n F
F 

2
)
Im(
and
2
)
Re(
)
Im(
2
and
)
Re(
2
n
n
n
n
n
n
n
n
b
F
a
F
F
b
F
a







8. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Complex exponential form
Thus
So you could also write
n
i
n
T
t
t
t
in
T
t
t
T
t
t
T
t
t
n
e
F
dt
e
t
f
T
dt
t
n
i
dt
t
n
t
f
T
dt
t
n
t
f
i
dt
t
n
t
f
T
F





























0
0
0
0
0
0
0
0
0
)
(
1
))
sin(
)
)(cos(
(
1
)
sin(
)
(
)
cos(
)
(
1
0
0
0
0






n
t
n
i
n
n
e
F
t
f )
( 0
)
( 


9. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier transform
We now have
Let’s not use just discrete frequencies, n0 ,
we’ll allow them to vary continuously too
We’ll get there by setting t0=-T/2 and taking
limits as T and n approach 





n
t
in
ne
F
t
f 0
)
( 
dt
e
t
f
T
F
T
t
t
t
in
n 



0
0
0
)
(
1 

10. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier transform
dt
e
t
f
T
e
dt
e
t
f
T
e
e
F
t
f
t
T
in
T
T
n
t
T
in
t
in
T
T
n
t
in
n
t
in
n






 2
2
/
2
/
2
2
/
2
/
)
(
2
1
2
)
(
1
)
( 0
0
0





















d
T
T









2
lim 
 


d
n
n
lim













d
F
e
d
dt
e
t
f
e
dt
e
t
f
d
e
t
f
t
i
t
i
t
i
t
i
t
i

 




























)
(
2
1
)
(
2
1
2
1
)
(
2
1
)
(

11. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier transform
So we have (unitary form, angular frequency)
Alternatives (Laplace form, angular frequency)








d
e
F
t
f
F
dt
e
t
f
F
t
f
t
i
t
i













)
(
2
1
)
(
))
(
(
)
(
2
1
)
(
))
(
(
1
-
F
F







d
e
F
t
f
F
dt
e
t
f
F
t
f
t
i
t
i













)
(
2
1
)
(
))
(
(
)
(
)
(
))
(
(
1
-
F
F

12. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier transform
Ordinary frequency



2







d
e
F
t
f
F
dt
e
t
f
F
t
f
t
i
t
i













)
(
)
(
))
(
(
)
(
)
(
))
(
(
1
-
F
F

13. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier transform
Some sufficient conditions for application
Dirichlet conditions
f(t) has finite maxima and minima within any finite interval
f(t) has finite number of discontinuities within any finite
interval
Square integrable functions (L2 space)
Tempered distributions, like Dirac delta






dt
t
f )
(






dt
t
f 2
)]
(
[


2
1
))
(
( 
t
F

14. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Fourier transform
Complex form – orthonormal basis functions for
space of tempered distributions
)
(
2
2
2
1
2
1













 dt
e
e t
i
t
i

15. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell
Convolution theorem
Theorem
Proof (1)
)
(
*
)
(
)
(
)
(
)
(
)
*
(
)
(
*
)
(
)
(
)
(
)
(
)
*
(
G
F
FG
G
F
G
F
g
f
fg
g
f
g
f
1
-
1
-
1
-
1
-
1
-
1
-
F
F
F
F
F
F
F
F
F
F
F
F




)
(
)
(
'
'
)
'
'
(
'
)
'
(
)
'
(
'
)
'
(
'
)
'
(
)
'
(
)
*
(
'
'
'
)
'
(
'
g
f
dt
e
t
g
dt
e
t
f
dt
e
t
t
g
dt
e
t
f
dt
dt
e
t
t
g
t
f
g
f
t
i
t
i
t
t
i
t
i
t
i
F
F
F






 
 
 




























