Complex Fourier

8/10/2019 Complex Fourier

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University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

Fourier Transforms




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()( 00

0

t nbt nat f n

n

n

)sin()cos(2)( 001

0

t nbt na

a

t f nn

n




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()(20




Fourier series

Alternate forms

where

))(cos(2

))sin()tan()(cos(2

))sin()(cos(2

)(

0

1

0

00

1

0

00

1

0

n

n

n

n

n

n

n

n

n

n

t nca

t nt naa

t na

bt na

at f

n

nnnnn

a

bbac 122 tanand




Complex exponential notation

Euler’s formula )sin()cos( xi xeix

Phasor notation:

x

y

iy xiy x

z z

y x z

e z iy x i

1

22

tanand

))((

where




Euler’s formula

Taylor series expansions

Even function ( f( x) = f(- x) )

Odd function ( f( x) = -f(- x) )

...!4!3!2

1432

x x x

xe x

...!8!6!4!2

1)cos(

8642

x x x x x

...

!9!7!5!3

)sin(9753

x x x x

x x

)sin()cos(

...!7!6!5!4!3!2

1765432

xi x

ix xix xix xixeix




Complex exponential form

Consider the expression

So

Since an and b

n are real, we can let

and get

)sin()()cos()(

)sin()cos()(

00

0

000

t n F F it n F F

t niF t n F e F t f

nnn

n

n

n

n

n

n

t in

n

)(and nnnnnn F F ib F F a

nn F F

2)Im(and

2)Re(

)Im(2and)Re(2

nn

nn

nnnn

b F

a F

F b F a




Complex exponential form

Thus

So you could also write

nin

T t

t

t in

T t

t

T t

t

T t

t

n

e F

dt et f T

dt t nidt t nt f T

dt t nt f idt t nt f T

F

0

0

0

0

0

0

0

0

0

)(1

))sin())(cos((1

)sin()()cos()(1

00

00

n

t ni

nne F t f

)( 0)(




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 t 0=-T/2 and takinglimits as T and n approach

n

t inne F t f 0)(

dt et f T

F

T t

t

t in

n

0

0

0)(1




Fourier transform

dt et f T

e

dt et f T ee F t f

t T

inT

T n

t T

in

t in

T

T n

t in

n

t in

n

22/

2/

2

2/

2/

)(2

12

)(

1

)( 000

d T T

2lim

d nnlim

d F e

d dt et f e

dt et f d et f

t i

t it i

t it i

)(2

1

)(2

1

2

1

)(2

1)(




Fourier transform

So we have (unitary form, angular frequency)

Alternatives (Laplace form, angular frequency)

d e F t f F

dt et f F t f

t i

t i

)(2

1)())((

)(2

1)())((

1-

F

F

d e F t f F

dt et f F t f

t i

t i

)(2

1)())((

)()())((

1-F

F




Fourier transform

Ordinary frequency 2

d e F t f F

dt et f F t f

t i

t i

)()())((

)()())((

1-F

F




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 finite

interval

Square integrable functions (L2 space)

Tempered distributions, like Dirac delta

dt t f )(

dt t f 2

)]([

2

1))(( t F




Fourier transform

Complex form – orthonormal basis functions forspace of tempered distributions

)(

22 21

21

dt ee t it i




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-

FFF

FFF

FFF

FFF

)()(

'')''(')'(

)'(')'(

')'()'()*(

'''

)'('

g f

dt et g dt et f

dt et t g dt et f

dt dt et t g t f g f

t it i

t t it i

t i

FF

F

Complex Fourier

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