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