Fourier Transformation
f(x)
Fourier
Transformasjon
F(u)
1
Continuous Fourier Transform
Def
The Fourier transform
of a one-dimentional function f(x)
fˆ (u ) 


f ( x)e  j 2ux dx


The Inverse Fourier Transform
f ( x) 

fˆ (u )e j 2ux du

2
Continuous Fourier Transform
Def - Notation
The Fourier transform
of a one-dimentional function f(x)

fˆ (u )  F (u )  e j 2ux f ( x)  F  f ( x)   e  j 2ux f ( x)dx

The inverse Fourier Transform of F(u)
 

f ( x)  e  j 2ux fˆ (u )  F 1 fˆ (u )   e j 2ux fˆ (u )du

3
Continuous Fourier Transform
Alternative Def
F (u )  F  f ( x) 


f ( x )e
 j 2ux
dx
F ( )  F  f ( x) 
f ( x)  F

F (u)   fˆ (u )e


f ( x)e  jx dx


1

 j 2ux
du
1
f ( x)  F F ( ) 
2
1
1
F ( )  F  f ( x) 
2
1
f ( x)  F F ( ) 
2
1

 fˆ (u)e
jx
d



f ( x)e  jx dx



fˆ (u )e jx d

4
Continuous Fourier Transform
Example - cos(2ft)
5
Continuous Fourier Transform
Example - cos(t)
6
Continuous Fourier Transform
Example - sin(t)
7
Continuous Fourier Transform
Example - Delta-function
8
Continuous Fourier Transform
Example - Gauss function
9
Signals and Fourier Transform
Frequency Information
y1  sin( 1t )
FT
FT
y2  sin(  2t )
FT
y3  sin( 1t )  sin(  2t )
10
Stationary / Non-stationary signals
Stationary
FT
y3  sin( 1t )  sin(  2t )
Non stationary
sin( 1t )
y4  
sin(  2t )
hvis t  60
FT
hvis t  60
The stationary and the non-stationary signal both have the same FT.
FT is not suitable to take care of non-stationary signals to give information about time. 11
Transient Signal
Frequency Information
Constant function in [-3,3].
Dominating frequency  = 0
and some freequency because of edges.
Transient signal
resulting in extra frequencies > 0.
Narrower transient signal
resulting in extra higher frequencies
pushed away from origin.
12
Transient Signal
No Information about Position
Moving the transient part of the signal to
a new position does not result
in any change in the transformed signal.
Conclusion: The Fourier transformation
contains information of a transient part
of a signal, but only the frequency
not the position.
13
fˆ (u ) 

 f ( x )e
 j 2ux
dx


Inverse Fourier Transform [1/3]
f ( x) 
 fˆ (u)e

Theorem:

e
t 2

e

e  jt dt 

Proof:


f ( y )   e t e yt dt   e t
2


y2 
4
2
2

4
 yt
 0

dt   e
 ( t 
y 2 y2
) 
2
4
dt

 4y
t

e  e dt 
e



1
2
2
y   j


e

  4
 jt
e dt 
e

2
t
2
14
j 2ux
du
fˆ (u ) 

 f ( x )e
 j 2ux
dx


Inverse Fourier Transform [2/3]
f ( x) 
 fˆ (u)e

Theorem:



f(y)gˆ(ay)dy 
-
Proof:

fˆ(ay)g(y)dy
f , g  L1 ( R)
-


 j 2ayx
ˆ
f(y)
g
(ay)dy

f(y)
g
(
x
)
e
dx
 dy
-
- 



 

  f ( y ) g ( x )e
 j 2ayx
dxdy
  
 


f ( y ) g ( x)e  j 2ayx dydx
  


    f ( y )e  j 2ayx dy  g ( x)dx
   




 fˆ (ax) g ( x)dx



 fˆ (ay) g ( y)dy

15
j 2ux
du
fˆ (u ) 

 f ( x )e
 j 2ux
dx

Inverse Fourier Transform [3/3]

f ( x) 
 fˆ (u)e
j 2ux
du

g ( x ) 
1
2 
gˆ (u )  e  ( 2u )
e

x2
4
f ( x)  f ( x  0)
2


 lim
 g ( x)dx  1
 0
 f ( y) g ( x  y)dy


 lim  f ( x) * g ( x)
 0

 lim
 0
 y 
ˆ
f
(
y
)
g
 dy

 2 

 lim
 0
1 jtx t 2
g (t ) 
e e
2

1
jtx t 2  j 2yt
gˆ ( y ) 
e
e e
dt
2 
1

2
1

2

t  j ( 2y  x ) t
dt
e e
2

  ( 24y x )
e
 g ( x  2y )

2
ˆ  y  g ( y )dy
f
 2 
 
1
 lim
 0 2
1

2

ˆ  y e jyx e y 2 dy
f
 2 
 

ˆ  y
f
 2
 
 jyx
e dy


1

2  fˆ ( y )e j 2yx dy
2

 F [ fˆ ( x)]
1
16
Properties
F[ f  g ]
 F[ f ]  F[ g ]
F [cf ]  cF [ f ]
F 1[ f  g ]
 F 1[ f ]  F 1[ g ]
F 1[cf ]  cF 1[ f ]
F[ f (n) ]
dn
F [ f ]
 j
n
d
n
n d
 ( j )
F 1[ f ]
n
dt
 ( j ) n F [ f ]
F 1[ f ( n ) ]
 ( jt ) n F 1[ f ]
F [ f (t  a )]
 e  ja F [ f (t )]
n
F [t f ]
F 1[ n f ]
n

F [ f (at )]( ) 
F [ f ]( )

1
 
F [ f (t )] 
a
a
 L[ f ]( j )

L[ f ]   f (t )e ts dt
0
17
Fourier Transforms of
Harmonic and Constant Function
f ( x)  F  (u  u0 )   (u  u0 ) 
1

j 2ux


du
e
)
u

u
(


)
u

u
(

0
0






   (u  u0 )e j 2ux du    (u  u0 )e j 2ux du
2 cos( 2u0 x)
 e j 2ux  e  j 2ux  
 2 j sin( 2u0 x)



-

1
 (u   0 )   (u  u0 )
2
j
F sin( 2u0 x)   (u  u0 )   (u  u0 )
2
F cos( 2u0 x) 
F 1   (u )
18
Fourier Transforms of
Some Common Functions
f ( x)
e j 2u0 x
F (u )
1
 (u  u0 )   (u  u0 )
2
j
δ(u  u0 )  δ(u  u0 )
2
 (u  u0 )
δ(x)
1
cos(2u 0 x)
sin (2u 0 x)
 ( x)
( x)
sin 2 (u )
u
sin 2 (u )
(u ) 2
u ( x)
1
j

(
u
)

2 
u 
e x
e u
2
2
19
Even and Odd Functions [1/3]
Def
f even (t ) 
f even (t )
f odd (t )   f odd (t )
Every function can be split
in an even and an odd part
f (t )

f even (t )  f odd (t )
1
 f (t )  f (t )
2
1
 f (t )  f (t )
f odd (t ) 
2
f even (t ) 
Every function can be split
in an even and an odd part
and each of this can in turn be split
in a real and an imaginary part
f (t )

f even (t )  f odd (t )

f even ,real (t )  f odd ,real (t )
 f even ,imag (t )  f odd ,imag (t )
20
Even and Odd Functions [2/3]

F (u ) 
 f ( x )e
 j 2ux
dx






 f ( x) cos(2ux)dx  j  f (t ) sin( 2ux)dx


f
even
( x) cos( 2ux)dx 

f

f
odd








( x) cos( 2ux)dx  j  f even ( x) sin( 2ux)dx  j  f odd ( x) sin( 2ux)dx

even
( x) cos( 2ux)dx  j  f odd ( x) sin( 2ux)dx

 Feven (u )  jFodd (u )
1.
2.
3.
Even component in f produces an even component in F
Odd component in f produces an odd
component in F
Odd component in f produces an coefficient -j
21
Even and Odd Functions [3/3]
f(t)
F(u )
Even
Odd
Real  Even
Even
Odd
Real  Even
Real  Odd
Imag  Odd
Imag  Even
Imag  Even
Complex  Even
Complex  Even
Complex  Odd
Complex  Odd
Real
Hermite
Real  Even plus Imag  Odd
Real  Odd plus Imag  Even
Real
Imag
Hermite
F (u )  F * (u )
22
The Shift Theorem
F  f ( x  a ) 


f ( x  a )e  j 2ux dx




f ( x)e  j 2u ( x  a ) dx

 e  j 2ua


f ( x)e  j 2ux dx

F  f ( x  a)  e  j 2ua F  f ( x)
 e  j 2ua F (u )
 e  j 2ua F  f ( x)
 e  j 2ua F (u )
23
The Similarity Theorem
F  f (ax) 


f (ax)e  j 2ux dx

1

a


 f ( x )e

1 u
F 
a a
 j 2 au x
dx
1 u
F  f (ax)  F  
a a
24
The Convolution Theorem

f (t ) * g (t ) 
 f (u ) g (t  u )du


F  f ( x) * g ( x) 

F  f * g   fˆ  gˆ
 j 2ux


f
(
x
)
*
g
(
x
)
e
dx







f ( y )  g ( x  y )e  j 2ux dtdy




f ( y )e
 j 2uy

G (u )dy 


 
F 1 fˆ  gˆ  f * g
f ( y )e  j 2uy dyG (u )

 F (u )G (u )  fˆ  gˆ
25
Convolution
Edge detection
26
The Adjoint of the Fourier Transform
Theorem:
Suppose f and g er are square integrable. Then:
F f  g
Proof:
F f  g
2
L
 f F 1g 
L2

L2

 fˆ (u ) g (u )du

 

  f ( x )e
 j 2ux
dt g (u )du
  


  f ( x)   g (u )e j 2ux du  dx






 f ( x) F g ( x)dx
1

 f F 1 g 
L2
27
Plancherel Formel - The Parselval’s Theorem
Theorem:
Suppose f and g are square integrable. Then:
F  f  F g  L2
F 1  f  F 1 g 
Proof:
L2
F  f  F g  L2
F 1  f  F 1 g 
2
L

f g

f g

f F 1 F g 

F F 1  f  g

In paricular
L2
F[f] L2

f
L2

L2
L2
 f g
 f g
L2
L2
28
L2
The Rayleigh’s Theorem
Conservation of Energy
F  f  F g  L2 

F[f] L2
f

energy 



f g
2
f ( x) dx

L2

f ( x) dx 
2

2
L




2
F (u ) du


f ( x) dx 
2
 f ( x) f
*
( x)dx

The energy of a signal in the time domain
is the same as the energy in the frequency domain
f
fˆ
L2
L2
29
The Fourier Series Expansion
u a discrete variable - Forward transform
Suppose f(t) is a transient function that is zero outside the interval [-T/2,T/2]
or is considered to be one cycle of a periodic function.
We can obtain a sequence of coefficients by making a discrete variable
and integrating only over the interval.
fˆ (u ) 


f ( x)e  j 2ux dx 

fˆn  fˆn (n  u ) 
T /2

f ( x)e  j 2ux dx
T / 2
T /2

f ( x)e  jn2u dx
T / 2
u 
1
T
30
The Fourier Series Expansion
u a discrete variable - Inverse transform
fˆn  fˆn (n  u ) 
T /2
 f ( x )e
 jn 2u
1
u 
T
dx
T / 2
The inverse transform becomes:

f ( x) 

fˆ ( x)e j 2ux du




n  
fˆn (n  u )e jn2ux u 


n  
fˆn e
jn
2
x
T
2
1 1  ˆ jn T
  f ne
T T n  
x
31
The Fourier Series Expansion
cn coefficients
fˆn  fˆn (n  u ) 
T /2
 f ( x )e
T / 2
 jn 2u
dx
1
u 
T
2
2


jn x
jn x
1
f ( x)   fˆ ( x)e j 2ux du   fˆn e T   cn e T
T n 
n  


f ( x) 

c e
n  
jn
2
x
T
n
2 n
j
x
1
T
cn 
f ( x )e
dx

T T / 2
T /2
32
The Fourier Series Expansion
zn, an, bn coefficients
f ( x) 

c e
n  
jn
2
x
T
n
2 n
j
x
1
T
cn 
f
(
x
)
e
T T/ 2
T /2
f ( x) 

c e
n  
jn
n
2
x
T
2 n
2 n
j
x
j
x
1
T
T
 
f
(
x
)
e
dx

e

n   T T / 2

T /2
2 n
2 n

j
x
j
x
a0
1
T
T
 
f
(
x
)
e
dx

e
2 n   T T/ 2
T /2
n0
T /2
T /2
2 n
2 n
2 n
2 n
j
x
j
x
j
x
j
x
a0  1 
a0 
T
T
T
T
     f ( x )e
dx  e
  f ( x )e
dx  e
    zn
2 n 1 T T / 2
T / 2
 2 n 1
T /2
T /2
2 n
2 n
2 n
2 n
2 nx
2 nx
j
x
j
x
j
t
j
x
j
j


1
1
z n    f ( x)e T dx  e T   f ( x)e T dx  e T   (an  ibn )e T  (an  ibn )e T 
T T / 2

T / 2
 2
2 nx
j
2
an  ibn 
f (t )e T

T T / 2
T /2
33
The Fourier Series Expansion
an,bn coefficients
a0 
f (t )    z n
2 n 1
j
1
z n  (an  ibn )e
2
T /2
an  ibn 
2 nx
T
2
f (t )e
T T/ 2
j
 (an  ibn )e
2 nx
T
j
2 nx
T



a0  
2nx
2nx 
f ( x)    an cos
 bn sin
2 n 1 
T
T 
2
2nx
an 
f
(
x
)
cos
dx

T T / 2
T
T /2
2
2nx
bn 
f
(
x
)
sin
dx

T T / 2
T
T /2
34
f ( x) 
4
N
1

sin (2i  1)


2i  1 
i 1
Fourier Series
2 
x
 
Pulse train
Pulse train
approximated by Fourier Serie
N=1
N=2
N=5
N = 10
35
Fourier Series
Pulse train – Java program
36
Pulse Train approximated by Fourier Serie
f(x) square wave (T=2)
a0  
2nx
2nx 
  an cos
 bn sin
2 n 1 
T
T 
4  1
 
sin[( 2n  1)x]
 n 1 2n  1
f ( x) 
f ( x) 
4
N
1
 2n  1 sin[( 2n  1)x]

n 1
N=1
N=2
N=10
37
2 N
1
f ( x)   (1)i 1 sin( ikx)
k i 1
i
Fourier Series
k
Zig tag
Zig tag
approximated by Fourier Serie
N=1
N=2
N=5
N = 10
38
2

N
1 
(-1)i
f ( x )     4
cos(ikx)
2
3 2 
i 1 (ik )
2
Fourier Series
k
Negative sinus function
N=1
Negative sinus function
approximated by Fourier Serie
N=2
N=5
N = 10
39
2

1
2 N
1
f ( x)   sin( kx)  
cos(2ikx)
2
 2
 i 1 (2i )  1
1
Fourier Series
k
Truncated sinus function
Truncated sinus function
approximated by Fourier Serie
N=1
N=2
N=5
N = 10
40
2

Fourier Series
Line
Line
approximated by Fourier Serie
N=1
N
a0 N
f ( x)    a j cos( jkx)   b j sin( jkx)
2 j 0
j 0
k
L
N=2
1
a j   f ( x) cos( jkx)dx
L L
L
N=5
41
N = 50
L
L 
1
b j   f ( x) sin( jkx)dx
L L
N = 10

Fourier Series
Java program for approximating Fourier coefficients
Approximate functions by adjusting Fourier coefficients (Java program)
42
The Discrete Fourier Transform - DFT
Discrete Fourier Transform - Discretize both time and frequency
Continuous
Fourier transform
Discrete frequency
Fourier Serie
u  n  u
fˆ (u ) 
T /2
 f ( x )e
 j 2ux
T / 2

f ( x) 
 fˆ (u)e

j 2ux
dx
u 
fˆn  fˆ (nu ) 
Discrete frequency and time
Discrete Fourier Transform
1
T
t  i  t t 
T /2
 f ( x )e
T / 2
2
du
1  ˆ jn T
f ( x)   f n e
T n
x
T
N
n
 jn 2u
dx
N /2
 j 2 i
T
N
ˆf  fˆ (nu ) 
fi e

n
N i  N / 2
1  ˆ j 2 N n
f i  f (ix)   f n e
T n
i
43
The Discrete Fourier Transform - DFT
Discrete Fourier Transform - Discretize both time and frequency
{ fi } sequence of length N,
taking samples of a continuous function at equal intervals
n
N /2
 j 2 i
T
N
ˆf  fˆ (nu ) 
fi e

n
N i  N / 2
ˆf  1
n
N
1  ˆ j 2 N n
f i  f (ix)   f n e
T n
1
fi 
N
i
N 1
fe
i 0
j 2
i
N 1
 fˆ e
n 0
n
i
N
j 2
i
n
N
n
44
Continuous Fourier Transform in two Dimensions
Def
The Fourier transform
of a two-dimentional function f(x,y)
fˆ (u, v) 
 

f ( x, y )e  j 2 (uxvy ) dxdy
  
 
The Inverse Fourier Transform
f ( x, y ) 

fˆ (u, v)e j 2 (ux vy ) dudv
  
45
The Two-Dimensional DFT and Its Inverse
1
fˆ (u, v) 
MN
M 1 N 1
 f ( x, y)e
 j 2 (
1
f ( x, y ) 
MN
M 1 N 1
j 2 (
u
v
x y )
M
N
x 0 y 0
 fˆ (u, v)e
u
v
x y )
M
N
x 0 y 0
46
Fourier Transform in Two Dimensions
Example 1
47
Fourier Transform in Two Dimensions
Example 2
48
End
49