Properties of Fourier transforms (4):
correlation

Correlation of g(t) with f(t): - g(t)h(t+t) dt
which is the Fourier transform of G(f*)H(f)
Closely related to the convolution theorem, relevant for finding
features in data.
The Central Limit Theorem
The Central Limit Theorem says:
The convolution of the convolution of the convolution etc.
approaches a Gaussian.
Mathematically,
f(x) * f(x) * f(x) * f(x) * ... * f(x)  exp[(-x/a)2]
or:
f(x)*n  exp[(-x/a)2]
The Central Limit Theorem is why nearly everything has a Gaussian
distribution.
5. Encontrar la transformada de Fourier de la función:
  t
e a cos( t) , t  0
0
f (t)  

, t 0
0
;
(a  0)
f t   h(t)g(t)
  t
2
 a

2


a
 a , t  0
F.T .
ˆ
h(t)  e
;(a  0) h   2 1   2 a 2  i 1  2 a2 

0 , t  0
gt   cos( 0t)
gˆ( ) 
F.T .

 (   0 )   (   0 )

2
fˆ  hg  hˆ  gˆ
 
hˆ  gˆ ( ) 
1

2 2



2
2
1
2


h(t)g(t)
e
it
dt


ˆ(' )gˆ(   ' )d ' 
h


 a
' a 2  

  (   '  0 )   (   '  0 )d'

2 2 i
2 2 
1   ' a
1   ' a  2

2
 (   )a 2

a
a
(



)a
0
0



2 2 
2 2  i 
2 2 
2 2 

1 (   0 ) a 1  (   0 ) a 
1  (   0 ) a 1  (   0 ) a

=

1
2
 
hˆ  gˆ ( ) 
1
2
fˆ ( )
The Central Limit Theorem for a square function
Note that P(x)*4 already looks like a Gaussian!
The Autocorrelation

The autoconvolution of a function f(x) is given by:
ff 

f (t ) f (t  t ) dt 

Suppose, however, that prior to multiplication and integration we do not reverse one
of the two component factors; then we have the integral: 

f (t ) f (t   t )dt 

which may be denoted by f f. A single value of f f is represented by:
The shaded area is the value of the autocorrelation
for the displacement x. In optics, we often define the
autocorrelation with a complex conjugate:



f (t ) f * (t   t ) dt 
The Autocorrelation Theorem
The Fourier Transform of the autocorrelation is the spectrum!
Proof:


F   f (t ) f *(t   t ) dt   F




F   f (t ) f *(t   t ) dt  
 

















 exp(it ) 
 f (t )
2
f (t ) f *(t   t ) dt  dt

*
f (t )   exp(it ) f (t   t ) dt  dt 
 



*
f (t )   exp(i y ) f (t   y ) dy  dt    f (t )  F ( ) exp(it )  *dt 

 

f (t ) exp(it ) dt  F *( )  F ( ) F *( )  F ( )
2
The Autocorrelation Theorem in action
F
rect(t )  sinc( / 2)
sinc( / 2)
rect(t )
sinc( / 2) 
rect(t ) rect(t ) 

t
(t )
(t )
sinc2(/2)
t
F
(t )  sinc2 ( / 2)

sinc( / 2) 
sinc 2 ( / 2)
The Autocorrelation Theorem for a light wave field
The Autocorrelation Theorem can be applied to a light wave field,
yielding an important result:


2
F   E (t ) E *(t   t dt   F {E (t )}
 

 E ( 
2
= the spectrum!
Remarkably, the Fourier transform of a light-wave field’s autocorrelation
is its spectrum!
This relation yields an alternative technique for measuring a light wave’s
spectrum.
This version of the Autocorrelation Theorem is known as the “WienerKhintchine Theorem.”
The Autocorrelation Theorem for a light
wave intensity
The Autocorrelation Theorem can be applied to a light wave intensity,
yielding a less important, but interesting, result:


F   I (t ) I (t   t ) dt   F I (t )
 

2
Many techniques yield the intensity autocorrelation of an ultrashort
laser pulse in an attempt to measure its intensity vs. time (which is
difficult).
The above result shows that the intensity autocorrelation is not
sufficient to determine the intensity—it yields the magnitude, but not
the phase, of F {I (t )}