fourier transforms - cs.utexas.edu
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University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier Transforms
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier series
To go from f(θ ) to f(t) substitute
To deal with the first basis vector being oflength 2π instead of π, rewrite as
ttT
0
2!
"# ==
)sin()cos()( 00
0
tnbtnatf n
n
n !! +="#
=
)sin()cos(2
)( 00
1
0 tnbtnaa
tf n
n
n !! ++= "#
=
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier series
The coefficients become
dttktfT
a
Tt
t
k !+
=0
0
)cos()(2
0"
dttktfT
b
Tt
t
k !+
=0
0
)sin()(2
0"
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier series
Alternate forms
where
!
f (t) =a0
2+ an
n=1
"
# (cos(n$0t) +bn
ansin(n$0t))
=a0
2+ an
n=1
"
# (cos(n$0t) % tan(&n )sin(n$0t))
=a0
2+ cn
n=1
"
# cos(n$0t +&n )
!!"
#$$%
&'=+=
'
n
n
nnnn
a
bbac
122tanand (
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Complex exponential notation
Euler’s formula )sin()cos( xixeix
+=
Phasor notation:
!"
#$%
&=
'+=
=
+=
=+
'
x
y
iyxiyx
zz
yxz
eziyxi
1
22
tanand
))((
where
(
(
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Euler’s formula
Taylor series expansions
Even function ( f(x) = f(-x) )
Odd function ( f(x) = -f(-x) )
...!4!3!2
1
432
+++++=xxx
xex
...!8!6!4!2
1)cos(8642
!+!+!=xxxx
x
...!9!7!5!3
)sin(9753
!+!+!=xxxx
xx
)sin()cos(
...!7!6!5!4!3!2
1765432
xix
ixxixxixxixe
ix
+=
+!!++!!+=
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Complex exponential form
Consider the expression
SoSince an and bn are real, we can letand get
)sin()()cos()(
)sin()cos()(
00
0
000
tnFFitnFF
tniFtnFeFtf
nnn
n
n
n
n
n
n
tin
n
!!
!!!
""
#
=
#
"#=
#
"#=
"++=
+==
$
$$
)(andnnnnnn
FFibFFa !! !=+=
nnFF =!
2)Im(and
2)Re(
)Im(2and)Re(2
n
n
n
n
nnnn
bF
aF
FbFa
!==
!==
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Complex exponential form
Thus
So you could also write
ni
n
Tt
t
tin
Tt
t
Tt
t
Tt
t
n
eF
dtetfT
dttnidttntfT
dttntfidttntfT
F
!
"
""
""
=
=
#=
$$
%
&
''
(
)#=
*
*
**
+
#
+
++
0
0
0
0
0
0
0
0
0
)(1
))sin())(cos((1
)sin()()cos()(1
00
00
!"
#"=
+=n
tni
nneFtf)( 0)(
$%
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier transform
We now have
Let’s not use just discrete frequencies, nω0 ,we’ll allow them to vary continuously too
We’ll get there by setting t0=-T/2 and takinglimits as T and n approach ∞
!"
#"=
=n
tin
neFtf 0)($
dtetfT
F
Tt
t
tin
n !+
"=
0
0
0)(1 #
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier transform
dtetfT
e
dtetfT
eeFtf
tT
inT
Tn
tT
in
tin
T
Tn
tin
n
tin
n
!!
"""
!
!22/
2/
2
2/
2/
)(2
12
)(1
)( 000
#$
#$=
#$
#$=
$
#$=
%&
%&&
=
==
!"
dTT
=#$
%&'
()*
2lim !! =
"#dn
n
lim
!!"
!""
"!
!
!!
!!
dFe
ddtetfe
dtetfdetf
ti
titi
titi
#
# #
##
$
$%
$
$%
%$
$%
%$
$%
$
$%
=
&'
()*
+=
=
)(2
1
)(2
1
2
1
)(2
1)(
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier transform
So we have (unitary form, angular frequency)
Alternatives (Laplace form, angular frequency)
!!"
!
"!
!
!
deFtfF
dtetfFtf
ti
ti
#
#$
$%
%
$
$%
==
==
)(2
1)())((
)(2
1)())((
1-F
F
!!"
!
!
!
!
deFtfF
dtetfFtf
ti
ti
#
#$
$%
%
$
$%
==
==
)(2
1)())((
)()())((
1-F
F
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier transform
Ordinary frequency
!
s ="
2#
!
F( f (t)) = F(s) = f (t)"#
#
$ e" i 2% s t
dt
F-1(F(s)) = f (t) = F(&)ei 2% st
"#
#
$ ds
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier transform
Some sufficient conditions for applicationDirichlet conditions
f(t) has finite maxima and minima within any finite interval
f(t) has finite number of discontinuities within any finiteinterval
Square integrable functions (L2 space)
Tempered distributions, like Dirac delta
!<"!
!#dttf )(
!<"!
!#dttf 2)]([
!"
2
1))(( =tF
University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Fourier transform
Complex form – orthonormal basis functions forspace of tempered distributions
)(22
21
21
!!"##
!!
$=$
%
%$& dtee
titi
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