Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Signals and Spectral Methods
in Geoinformatics
Lecture 3:
Fourier Transform
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform
and
inverse Fourier transform
direct
inverse
A. Dermanis
F ()  


1
f (t ) 
2

f (t )e


i t
dt
i t
F ( )e d 
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform
and
inverse Fourier transform
direct
F ()  


f (t )e
i t
dt
from the number domain
to the frequency domain
inverse
A. Dermanis
1
f (t ) 
2



i t
F ( )e d 
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform
and
inverse Fourier transform
direct
F ()  


f (t )e
i t
dt
from the frequency domain
to the number domain
inverse
A. Dermanis
1
f (t ) 
2



i t
F ( )e d
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series
in the interval [ 0, Τ ]
f (t ) 


k 
ck e
i k t
Fourier transform
in the interval (-,+)
1
f (t ) 
2



F ( )ei t d 
inverse
1 T
ck   f (t )e i k t dt
T 0
A. Dermanis
F ()  


f (t )e
i t
dt
direct
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
f (t ) 


k 
A. Dermanis
ck e
i k t
1 T
ck   f (t )e i k t dt
T 0
2
k  k
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
f (t ) 


k 
ck e
i k t
2
k  k
T
1 T
ck   f (t )e i k t dt
T 0
Change of notation
F (k )  ckT
A. Dermanis
k  k 1  k  (k  1)
2
2 2
k

T
T
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
f (t ) 


k 
ck e
i k t
2
k  k
T
1 T
ck   f (t )e i k t dt
T 0
Change of notation
F (k )  ckT
f (t ) 
A. Dermanis
1
2
k  k 1  k  (k  1)
2
1
i k t
F
(

)
e


k
T
2
k
2
2 2
k

T
T
T
i k t
F
(

)
e
 k k
k
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
f (t ) 


k 
ck e
i k t
2
k  k
T
1 T
ck   f (t )e i k t dt
T 0
Change of notation
F (k )  ckT
f (t ) 
1
2
k  k 1  k  (k  1)
2
1
i k t
F
(

)
e


k
T
2
k
2
2 2
k

T
T
T
i k t
F
(

)
e
 k k
k
T
F (k )  ckT   f (t )ei k t dt
0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
f (t ) 
1
2
2
1
i k t
F
(

)
e


k
T
2
k
i k t
F
(

)
e
 k k
k
T
F (k )  ckT   f (t )ei k t dt
0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
1
f (t ) 
2
A. Dermanis
i k t
F
(

)
e
 k k
k
T
F (k )   f (t )ei k t dt
0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
1
f (t ) 
2
i k t
F
(

)
e
 k k
T
F (k )   f (t )ei k t dt
k
0
[0, T ]  (, )
k  0
k  
G( )


k
k
  G( )d

k
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
1
f (t ) 
2
i k t
F
(

)
e
 k k
k
T
F (k )   f (t )ei k t dt
0
G(k )
[0, T ]  (, )
k  0
k  
G( )


k
k
  G( )d

k
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
From Fourier series to Fourier transform
1
f (t ) 
2
i k t
F
(

)
e
 k k
T
F (k )   f (t )ei k t dt
k
0
[0, T ]  (, )
k  0
k  
G( )


k
k
  G( )d

k
1
f (t ) 
2
inverse
A. Dermanis



i t
F ( )e d 
F ()  


f (t )ei t dt
direct
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in a continuously increasing interval Τ ∞
f (t )
t
 

k  a  b
2
k
2
k
L
2

t
 
L
2
T L

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in a continuously increasing interval Τ ∞
f (t )
2
k  k
T
k  k 1  k 
2 2


 
T
L
 
k  a  b
2
k
2
k
t

L
2

t
 
L
2
T L


A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in a continuously increasing interval Τ ∞
f (t )
2
k  k
T
k  k 1  k 
2 2


 
T
L
 
k  a  b
2
k

2
k
t

L
2

t
 
L
2
T L
2 2
k 

 
T
L

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in a continuously increasing interval Τ ∞
f (t )
t
 

k  a  b
2
k

2
k
L
2

t
 
L
2
T L
2 2
k 

 
T
L

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in a continuously increasing interval Τ ∞
f (t )
t
 

k  a  b
2
k
2
k
2L
2

t
 
2L
2
T  2L
2 2 
k 


T
2L
2

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in a continuously increasing interval Τ ∞
f (t )
3
t
 

k  a  b
2
k
2
k
4L
2

t
 
4L
2
T  4L
2 2 
k 


T
4L
4

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in a continuously increasing interval Τ ∞
f (t )
t
 

8L
2

t
k  a  b
2
k
2
k
8L
2
 
T  8L
2 2 
k 


T
8L
8

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in a continuously increasing interval Τ ∞
The Fourier series expansion of a function in a continuously larger
interval Τ, provides coefficients for continuously denser frequencies ωk.
As the length of the interval Τ tends to infinity, the frequencies ωk tend to
cover more and more from the set of the real values frequencies ()
For an infinite interval Τ, i.e. for (  t  ) the total real set of
frequencies () is required and from the Fourier series expansion
we pass to the inverse Fourier transform
f (t ) 


k 
ck e
i k t
discrete frequencies ωk
wit step Δω = 2π / Τ
A. Dermanis
1
f (t ) 
2



F ( )ei t d 
continuous frequencies - all
possible values (    )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a complex function
F ( )  R()  i I ()  


f (t )  f R (t )  i f I (t )
A. Dermanis
f (t )ei t dt
ei t  cos t  i sin t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a complex function
F ( )  R()  i I ()  


f (t )  f R (t )  i f I (t )
f (t )ei t dt
ei t  cos t  i sin t
f (t )eit   f R (t )  i f I (t )cos t  i sin t  
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a complex function
F ( )  R()  i I ()  


f (t )  f R (t )  i f I (t )
f (t )ei t dt
ei t  cos t  i sin t
f (t )eit   f R (t )  i f I (t )cos t  i sin t  
 f R (t ) cos t  i f R (t ) sin t  i f I (t ) cos t  i 2 f I (t ) sin t 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a complex function
F ( )  R()  i I ()  


f (t )  f R (t )  i f I (t )
f (t )ei t dt
ei t  cos t  i sin t
f (t )eit   f R (t )  i f I (t )cos t  i sin t  
 f R (t ) cos t  i f R (t ) sin t  i f I (t ) cos t  i 2 f I (t ) sin t 
  f R (t )cos t  f I (t )sin t   i  f I (t )cos t  f R (t )sin t 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a complex function
F ( )  R()  i I ()  


f (t )  f R (t )  i f I (t )
f (t )ei t dt
ei t  cos t  i sin t
f (t )eit   f R (t )  i f I (t )cos t  i sin t  
 f R (t ) cos t  i f R (t ) sin t  i f I (t ) cos t  i 2 f I (t ) sin t 
  f R (t )cos t  f I (t )sin t   i  f I (t )cos t  f R (t )sin t 

f R (t ) cos t  f I (t )sin t  dt


R( )  



I ( )  
A. Dermanis
f I (t ) cos t  f R (t )sin t  dt
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a complex function
F ()  


f (t )ei t dt
complex
form
f (t )  f R (t )  i f I (t )
F ( )  R( )  i I ( )
R( )  


I ( )  


A. Dermanis
 f R (t ) cos t  f I (t )sin t  dt

f I (t ) cos t  f R (t )sin t  dt
real
form
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a complex function
direct
F ()  


f (t )e
inverse
i t
1
f (t ) 
2
dt



F ( )ei t d 
Notation
F

F
f 

F 1
f F
F F (f)
f F
1
(F )
Usual (mathematically incorect) notation
f (t )  F ( )
A. Dermanis
F ()  F
 f (t ) 
f (t )  F
1
 F () 
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a real function
Complex function:
R( )  


I ( )  


A. Dermanis
 f R (t ) cos t  f I (t )sin t  dt

f I (t ) cos t  f R (t )sin t  dt
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a real function
Complex function:
R( )  


I ( )  


 f R (t ) cos t  f I (t )sin t  dt

f I (t ) cos t  f R (t )sin t  dt
Real function:
f (t )  f R (t )  i f I (t )  real
A. Dermanis
f R (t )  f (t )
f I (t )  0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a real function
Complex function:
R( )  


I ( )  


 f R (t ) cos t  f I (t )sin t  dt

f I (t ) cos t  f R (t )sin t  dt
Real function:
f (t )  f R (t )  i f I (t )  real
A. Dermanis
f R (t )  f (t )
f I (t )  0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a real function
Complex function:
R( )  


I ( )  


 f R (t ) cos t  f I (t )sin t  dt

f I (t ) cos t  f R (t )sin t  dt
Real function:
f (t )  f R (t )  i f I (t )  real
R( )  


f (t ) cos t dt
cosine transform
A. Dermanis
f R (t )  f (t )
I ( )  


f I (t )  0
f (t )sin t dt
sine transform
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a real function
Complex function:
R( )  


I ( )  


 f R (t ) cos t  f I (t )sin t  dt

f I (t ) cos t  f R (t )sin t  dt
Real function:
f (t )  f R (t )  i f I (t )  real
R( )  


f (t ) cos t dt
cosine transform
f R (t )  f (t )
I ( )  


f I (t )  0
f (t )sin t dt
sine transform





F ( )  R( )  i I ( )   f (t ) cos t dt  i   f (t )sin t dt 
 
  

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform of a real function
R()  


I ( )  
f (t ) cos(t ) dt  



f (t )sin(t ) dt  


R( )
f (t ) cos t dt  R()
f (t )sin t dt  I ( )
even
function
odd
function
I ( )

A. Dermanis


Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform in polar form
F ( )  R( )  i I ( )
R()  F () cos  ()
I ()  F () sin  ()
F ( )  R( )2  I ( )2
I ( )
 ( )
I ( )
 ( )  arctan
R( )
R( )
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform in polar form
F ( )  R( )  i I ( )
R()  F () cos  ()
I ()  F () sin  ()
amplitude spectrum
phase spectrum
F ( )  R( )  I ( )
2
2
I ( )
 ( )  arctan
R( )
F ()  R()  i I ()  F () cos  ()  i sin  ()  F () ei ( )
polar form:
A. Dermanis
F ()  F () ei ( )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier transform in polar form
R( )  R( )
even
I ( )   I ( )
odd
F ( )  R( ) 2  I ( ) 2  R( ) 2  [ I ( )]2  R( ) 2  I ( ) 2 
 F ( )
 ( )  arctan
A. Dermanis
I ( )
 I ( )
I ( )
 arctan
  arctan
  ( )
R( )
R( )
R( )
amplitude spectrum
F ()  F ()
even function
phase spectrum
 ( )   ( )
odd function
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
f (t )  F ( )
Linearity
A. Dermanis
g (t )  G ( )
af (t )  bg (t )  aF ( )  bG ( )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
f (t )  F ( )
g (t )  G ( )
Linearity
af (t )  bg (t )  aF ( )  bG ( )
Symmetry
F (t )  2 f ( )
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
f (t )  F ( )
g (t )  G ( )
Linearity
af (t )  bg (t )  aF ( )  bG ( )
Symmetry
F (t )  2 f ( )
Time translation
f (t  t0 )  F ()eit0  F () ei[F ( )t0 ]
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
g (t )  G ( )
f (t )  F ( )
Linearity
af (t )  bg (t )  aF ( )  bG ( )
Symmetry
F (t )  2 f ( )
Time translation
f (t  t0 )  F ()eit0  F () ei[F ( )t0 ]
g (t )  f (t  t0 )  G ( )  | G ( ) | eiG ( )
f (t )
g (t )
t0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
g (t )  G ( )
f (t )  F ( )
Linearity
af (t )  bg (t )  aF ( )  bG ( )
Symmetry
F (t )  2 f ( )
Time translation
f (t  t0 )  F ()eit0  F () ei[F ( )t0 ]
g (t )  f (t  t0 )  G ( )  | G ( ) | eiG ( )
f (t )
g (t )
G()  F ()
t0
A. Dermanis
G ( )  F ( )  t0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
f (t )  F ( )
Phase translation
g (t )  G ( )
f (t )ei 0t  F (  0 )
f (t )e i 0t  F (  0 )
ei 0t  ei 0t  (cos 0t  i sin 0t )  (cos 0t  i sin 0t )  2 cos 0t
ei 0t  e i 0t  (cos 0t  i sin 0t )  (cos 0t  i sin 0t )  2i sin 0t
1
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
1
f (t ) sin 0t   F (  0 )  F (  0 ) 
2i
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
f (t )  F ( )
Phase translation
g (t )  G ( )
f (t )ei 0t  F (  0 )
f (t )e i 0t  F (  0 )
ei 0t  ei 0t  (cos 0t  i sin 0t )  (cos 0t  i sin 0t )  2 cos 0t
ei 0t  e i 0t  (cos 0t  i sin 0t )  (cos 0t  i sin 0t )  2i sin 0t
1
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
1
f (t ) sin 0t   F (  0 )  F (  0 ) 
2i
A. Dermanis
Modulation theorem
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
Modulation theorem
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
Proof:
A. Dermanis
g (t )  f (t ) cos 0t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
Modulation theorem
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
g (t )  f (t ) cos 0t
Proof:
G()  


A. Dermanis
g (t )e
 it
dt  


f (t ) cos 0t eit dt 
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
Modulation theorem
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
g (t )  f (t ) cos 0t
Proof:
G()  


A. Dermanis
g (t )e
 it
dt  


f (t ) cos 0t eit dt 

1 i 0t
cos 0t  e  e i 0t
2

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
Modulation theorem
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
g (t )  f (t ) cos 0t
Proof:
G()  





A. Dermanis
g (t )e
f (t )
 it
dt  



f (t ) cos 0t eit dt 

1 i 0t
e  e i 0t e it dt 
2

1 i 0t
cos 0t  e  e i 0t
2

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
Modulation theorem
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
g (t )  f (t ) cos 0t
Proof:
G()  





g (t )e
f (t )
 it
dt  



f (t ) cos 0t eit dt 

1 i 0t
e  e i 0t e it dt 
2

1 i 0t
cos 0t  e  e i 0t
2

1 
1 
i 0t  it
  f (t )e e dt   f (t )e  i 0t e it dt 
2 
2 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
Modulation theorem
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
g (t )  f (t ) cos 0t
Proof:
G()  





g (t )e
f (t )
 it
dt  



f (t ) cos 0t eit dt 

1 i 0t
e  e i 0t e it dt 
2

1 i 0t
cos 0t  e  e i 0t
2

1 
1 
i 0t  it
  f (t )e e dt   f (t )e  i 0t e it dt 
2 
2 
1 
1 
 i ( 0 ) t
  f (t )e
dt   f (t )e i ( 0 )t dt 
2 
2 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
Modulation theorem
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
g (t )  f (t ) cos 0t
Proof:
G()  





g (t )e
f (t )
 it
dt  



f (t ) cos 0t eit dt 

1 i 0t
e  e i 0t e it dt 
2

1 i 0t
cos 0t  e  e i 0t
2

1 
1 
i 0t  it
  f (t )e e dt   f (t )e  i 0t e it dt 
2 
2 
1 
1 
 i ( 0 ) t
  f (t )e
dt   f (t )e i ( 0 )t dt 
2 
2 
1
1
 F (  0 )  F (  0 )
2
2
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
f (t )
Modulation theorem
signal
cos 0t
carrier frequency
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
f (t )
Modulation theorem
signal
g (t )  f (t ) cos 0t
cos 0t
carrier frequency
modulated signal
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
g (t )  f (t ) cos 0t  G( )
f (t )  F ( )
(amplitude) spectrum of modulated signal
(amplitude) spectrum of signal
| F ( ) |
| G ( ) |

A. Dermanis
Modulation theorem

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
Modulation theorem
g (t )  f (t ) cos 0t  G( )
f (t )  F ( )
(amplitude) spectrum of modulated signal
(amplitude) spectrum of signal
| F ( ) |
| G ( ) |
1
| F ( ) |
2

A. Dermanis

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
Modulation theorem
g (t )  f (t ) cos 0t  G( )
f (t )  F ( )
(amplitude) spectrum of modulated signal
(amplitude) spectrum of signal
| F ( ) |
| G ( ) |
1
| F ( ) |
2
1
| F (  0 ) |
2


0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
1
f (t ) cos 0t   F (  0 )  F (  0 ) 
2
Modulation theorem
g (t )  f (t ) cos 0t  G( )
f (t )  F ( )
(amplitude) spectrum of modulated signal
(amplitude) spectrum of signal
| F ( ) |
| G ( ) |
1
| F ( ) |
2
1
| F (  0 ) |
2
1
| F (  0 ) |
2


0
A. Dermanis
0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
f (t )  F ( )
Change of time scale:
A. Dermanis
g (t )  G ( )
1  
f (at ) 
F 
|a|  a 
f (t )  F ( )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
f (t )  F ( )
Change of time scale:
1  
f (at ) 
F 
|a|  a 
Differentiation theorem with respect to time:
A. Dermanis
g (t )  G ( )
f (t )  F ( )
df
(t )  i  F ( )
dt
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
f (t )  F ( )
1  
f (at ) 
F 
|a|  a 
Change of time scale:
Differentiation theorem with respect to time:
g (t ) 
df
(t )
dt
f (t )  F ( )
df
(t )  i  F ( )
dt
G( ) | G( ) | eiG ( )  i  F ( )  i  | F ( ) | eiF ( ) 
 ei / 2  | F ( ) | eiF ( )   | F ( ) | ei[F ( ) / 2]
| G ( ) |   | F ( ) |
A. Dermanis
g (t )  G ( )
G ( )  F ( )   / 2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
g (t )  G ( )
f (t )  F ( )
1  
f (at ) 
F 
|a|  a 
Change of time scale:
df
(t )  i  F ( )
dt
Differentiation theorem with respect to time:
g (t ) 
df
(t )
dt
G( ) | G( ) | eiG ( )  i  F ( )  i  | F ( ) | eiF ( ) 
 ei / 2  | F ( ) | eiF ( )   | F ( ) | ei[F ( ) / 2]
| G ( ) |   | F ( ) |
G ( )  F ( )   / 2
Differentiation theorem with respect to frequency:
A. Dermanis
f (t )  F ( )
dF
it f (t ) 
( )
d
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)
  (t )
 0
 (t )  lim   (t )
 0
1


A. Dermanis

2


2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)
  (t )
1

 (t )  lim   (t )
 0
area =1

A. Dermanis
 0

2


2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)
  (t )
area =1
1


A. Dermanis

2


2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)
  (t )
1

area =1

A. Dermanis



2
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)
  (t )
1

area =1

A. Dermanis



2
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)
  (t )
 0
 (t )  lim   (t )
 0
1


A. Dermanis

2


2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)
  (t )
 0
 (t )  lim   (t )
 0
1






2


2
 (t   ) ( )d 

1
 / 2

 /2
 (t   )d
= average value of φ in the interval

 
t

,
t

 2
2 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)
  (t )
 (t )  lim   (t )
 0
 0
1






2


2
 (t   ) ( )d 

1
 / 2

 /2
 0
 (t   )d
= average value of φ in the interval



 (t   ) ( )d   (t )

 
t

,
t

 2
2 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)



A. Dermanis

 ( ) (t  )d    (t  ) ( )d   (t )

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)



 ( ) (t  )d    (t  ) ( )d   (t )
du
  (t ) 
(t )
dt
1

 / 2  / 2
A. Dermanis


 0
1 t

u (t )   
2 
 0
t   / 2
 / 2  t   / 2
t   / 2
u (t )
1
 / 2  / 2
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)



 ( ) (t  )d    (t  ) ( )d   (t )
du
  (t ) 
(t )
dt
1

 / 2  / 2
A. Dermanis


 0
 0
1 t

u (t )   
2 
 0
t   / 2
 / 2  t   / 2
t   / 2
u (t )
1
 / 2  / 2
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The Dirac delta function δ(t)




 ( ) (t  )d    (t  ) ( )d   (t )

u (t )
0 t  0
u (t )  
1 t  0
du
 (t )  (t )
dt
1
t
Dirac delta function
du
  (t ) 
(t )
dt
1

 / 2  / 2
A. Dermanis
Heaviside step function
 0
 0
1 t

u (t )   
2 
 0
t   / 2
 / 2  t   / 2
t   / 2
u (t )
1
 / 2  / 2
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
involving the Dirac delta function δ(t)
1  2  ( )
 (t )  1
 (t  t0 )  e it
0
A. Dermanis
ei0t  2  (  0 )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Properties of the Fourier transform
involving the Dirac delta function δ(t)
ei0t  2  (  0 )
e
 i0t
 2  (  0 )
ei0t  ei0t
cos(0t ) 
2
ei0t  ei0t
sin(0t )  i
2
| F ( ) |
cos(0t )    (  0 )    (  0 )
sin(0t )  i   (  0 )  i   (  0 )
A. Dermanis


0
0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution

definition:
notation:
A. Dermanis
g (t )  (h * f )(t )   h(s) f (t  s)ds

g  h* f
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution

definition:
g (t )  (h * f )(t )   h(s) f (t  s)ds

g  h* f
notation:
s  t  s
( f * h)(t )  


A. Dermanis

f (s)h(t  s)ds   h(s) f (t  s)ds  g (t )

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution

definition:
g (t )  (h * f )(t )   h(s) f (t  s)ds

g  h* f
notation:
s  t  s
( f * h)(t )  



f (s)h(t  s)ds   h(s) f (t  s)ds  g (t )

property:
g  h* f  f *h




g (t )   h(s) f (t  s)ds   h(t  s) f (s)ds
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution
g  h* f
Mathematical mapping:
A. Dermanis

g (t )   h(t  s) f (s)ds

g
h*
f
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution
g  h* f
Mathematical mapping:

g (t )   h(t  s) f (s)ds

g
h*
f
The value g(t) of the function g for a particular t follows by
multiplying each value f(s) of the function f
with a factor (weight) h(t-s)
which depends on the “distance” t-s
between the particular t and the varying s (-∞<s<+∞).
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution
g  h* f
Mathematical mapping:

g (t )   h(t  s) f (s)ds

g
h*
f
The value g(t) of the function g for a particular t follows by
multiplying each value f(s) of the function f
with a factor (weight) h(t-s)
which depends on the “distance” t-s
between the particular t and the varying s (-∞<s<+∞).
Thus every value g(t) of the function g is a
“weighted mean” of the function f(s)
with weights h(t-s) determined by the function h(t).
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying

Convolution
g (t )   h(t  s) f (s)ds

h( s )
s
0
*
g (t )
f (s)
s
0
A. Dermanis
t
0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying

Convolution
g (t )   h(t  s) f (s)ds

h(  s )
h( s )
s
s
0
0
g (t )
f (s)
s
0
A. Dermanis
t
0
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying

Convolution
g (t )   h(t  s) f (s)ds

h(  s )
h( s )
h(t  s )
s
s
0
s
0
0
t
g (t )
f (s)
s
0
A. Dermanis
t
0
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying

Convolution
g (t )   h(t  s) f (s)ds

h(  s )
h( s )
h(t  s )
s
s
0
s
0
0
t
g (t )
h(t  s )
f (s)
1
s
0
A. Dermanis
t
t
0
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying

Convolution
g (t )   h(t  s) f (s)ds

h(  s )
h( s )
h(t  s )
s
s
0
s
0
h(t  s )
0
t
g (t )
f ( s )h(t  s )
f (s)
1
s
0
A. Dermanis
t
t
0
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying

Convolution
g (t )   h(t  s) f (s)ds

h(  s )
h( s )
h(t  s )
s
s
0
s
0
h(t  s )
0
t
g (t )
f ( s )h(t  s )
f (s)
1
s
t
0
t
0
t
area



A. Dermanis
h(t  s) f (s)ds  g (t )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying

Convolution
g (t )   h(t  s) f (s)ds

h(  s )
h( s )
h(t  s )
s
s
0
s
0
h(t  s )
0
t
g (t )
f ( s )h(t  s )
f (s)
g (t )
1
s
t
0
t
0
t
area



A. Dermanis
h(t  s) f (s)ds  g (t )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
h(t  s )
1
s
1
0
s
1
0
s
0
t
g (t )
f (s)
h(t  s )
s
t
0
t
g (t )
0
t
area



A. Dermanis
h(t  s) f (s)ds  g (t )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution - Example
h(  s )
h( s )
s
h(t  s )
s
s
t
g (t )
f (s)
s
t
A. Dermanis
t
t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution properties
h f  f h
(h  f )  g  h  ( f  g )  h  f  g
h  ( f  g)  h  f  h  g
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Convolution properties
h f  f h
(h  f )  g  h  ( f  g )  h  f  g
h  ( f  g)  h  f  h  g
f   f
f (t )   (t  s)  f (t  s)
  s (t )   (t  s)
A. Dermanis
f  s (t )  f (t  s)
f  s  fs
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The convolution theorem
f F
g  h* f
hH
G  HF
g G
F
(h  g )  F (h)F ( g )
Convolution is replaced by a simple multiplication in the frequency domain !
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The convolution theorem
Proof:
A. Dermanis
F
(h  f )  

t 
(h  f )(t )eit dt 
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The convolution theorem
Proof:
F

(h * f )  

t 
A. Dermanis

t 
(h * f )(t )eit dt 
  h(t  s) g ( s)ds  e it dt 
 s 

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The convolution theorem
Proof:
F
(h  f )  




t 
s 
A. Dermanis

t 
(h  f )(t )eit dt 
  h(t  s) g ( s)ds  e it dt 
 s 

  h(t  s) eit dt  g (s)ds 
 t 

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The convolution theorem
Proof:
F
(h  f )  




t 
t  t  s
s 


s 
A. Dermanis

t 
(h  f )(t )eit dt 
  h(t  s) g ( s)ds  e it dt 
 s 

  h(t  s) eit dt  g (s)ds 
 t 

t  t  s
  h(t ) ei (t  s ) dt  g (s)ds 
 t 

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The convolution theorem
F
Proof:
(h  f )  




t 
t  t  s
s 


s 


s 
A. Dermanis

t 
(h  f )(t )eit dt 
  h(t  s) g ( s)ds  e it dt 
 s 

  h(t  s) eit dt  g (s)ds 
 t 

t  t  s
  h(t ) ei (t  s ) dt  g (s)ds 
 t 

  h(t ) eit  dt   ei s g (s)ds 
 t 

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The convolution theorem
F
Proof:
(h  f )  




t 
t  t  s
s 


s 


s 

t 
(h  f )(t )eit dt 
  h(t  s) g ( s)ds  e it dt 
 s 

  h(t  s) eit dt  g (s)ds 
 t 

t  t  s
  h(t ) ei (t  s ) dt  g (s)ds 
 t 

  h(t ) eit  dt   ei s g (s)ds 
 t 



 it 



 
h(t ) e dt   g (s) e is ds   F (h)F ( g )
 t 
  s 

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
CONVOLUTION THEOREM
for frequencies
f F
G  H F
g  2 hf
hH
g G
F 1 F
A. Dermanis
(h)  F ( g )  2 hg
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
CONVOLUTION THEOREM
for frequencies
f F
1
G
H F
2
g G
F
A. Dermanis
g  hf
hH
1
(hg ) 
2
F
( h)  F ( g )
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
PARSEVAL THEOREM
f F
A. Dermanis

1
 | f (t ) | dt  2
2



| F ( ) |2 d 
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
PARSEVAL THEOREM

1
 | f (t ) | dt  2
f F
2



| F ( ) |2 d 
Similar relation for Fourier series
f (t ) 

ce
k 
k
i k t
a0 
   [ak cos k t  bk sin k t ]
2 k 1

T
0
A. Dermanis
| f (t ) | dt 
2


2
2
|
c
|

a

(
a

b
 k
 k k)
2
k 
2
0
k 1
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
PARSEVAL THEOREM

1
 | f (t ) | dt  2
f F
2



| F ( ) |2 d 
Similar relation for Fourier series
f (t ) 

ce
k 
i k t
k
a0 
   [ak cos k t  bk sin k t ]
2 k 1

T
0
| f (t ) | dt 
2


2
2
|
c
|

a

(
a

b
 k
 k k)
2
2
0
k 
k 1
Comparison with
3
x  x1e1  x2e2  x3e3   xk ek
k 1
A. Dermanis
3
| x |  x  x  x   xk2
2
2
1
2
2
2
3
k 1
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
END
A. Dermanis
Signals and Spectral Methods in Geoinformatics