Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Signals and Spectral Methods
in Geoinformatics
Lecture 2:
Fourier Series
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a function
defined in an interval
into Fourier Series
Jean Baptiste Joseph Fourier
2 kt
2 kt
f (t )  ao   [ak cos
 bk sin
]
T
T
k 1

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
REPRESENTING A FUNCTION BY NUMBERS
f (t)
t
Τ
0
function f
known base functions
φ1, φ2, ...
coefficients α1, α2, ...
of the function
f = a1φ1+ a2 φ2 + ...
2 kt
2 kt
f (t )  ao   [ak cos
 bk sin
]
T
T
k 1

A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The base functions of Fourier series
0a (t )  1
+1
0
–1
0
1b (t )  sin
+1
2 t
T
Τ
2b (t )  sin
+1
0
Τ
0
1a (t )  cos
+1
2 t
T
+1
–1
Τ
0
+1
2 t
T /2
Τ
–1
–1
Τ
0
+1
2 t
T /3
Τ
–1
+1
2 t
T /4
–1
Τ
0
4a (t )  cos
+1
2 t
T /4
0
0
0
4b (t )  sin
0
3a (t )  cos
0
0
2 t
T /3
0
2a (t )  cos
0
A. Dermanis
3b (t )  sin
0
–1
–1
2 t
T /2
0
Τ
–1
0
Τ
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:

f (t )  ao   [a k cos
k 1
2kt
2kt
 bk sin
]
T
T

f (t )  ao   [ak cos(2 sk t )  bk sin(2 sk t )]
k 1

f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:

f (t )  ao   [a k cos
k 1
Every base function has:
2kt
2kt
 bk sin
]
T
T
period:
T
k

f (t )  ao   [ak cos(2 sk t )  bk sin(2 sk t )]
k 1

f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:

f (t )  ao   [a k cos
k 1
Every base function has:
2kt
2kt
 bk sin
]
T
T

f (t )  ao   [ak cos(2 sk t )  bk sin(2 sk t )]
k 1
period:
frequency:
T
k
sk 
k
T

f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:

f (t )  ao   [a k cos
k 1
Every base function has:
2kt
2kt
 bk sin
]
T
T

f (t )  ao   [ak cos(2 sk t )  bk sin(2 sk t )]
period:
frequency:
T
k
sk 
k 1

f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
A. Dermanis
k
T
angular frequency:
 k  2sk 
2k
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:

f (t )  ao   [a k cos
k 1
Every base function has:
2kt
2kt
 bk sin
]
T
T

f (t )  ao   [ak cos(2 sk t )  bk sin(2 sk t )]
period:
frequency:
T
k
sk 
k 1

f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
fundamental period
T
A. Dermanis
fundamental frequency
sT  1/ T
k
T
angular frequency:
 k  2sk 
2k
T
fundamental angular frequency
T  2 sT  2 / T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:

f (t )  ao   [a k cos
k 1
Every base function has:
2kt
2kt
 bk sin
]
T
T

f (t )  ao   [ak cos(2 sk t )  bk sin(2 sk t )]
period:
frequency:
T
k
sk 
k 1

f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
fundamental period
fundamental frequency
k
T
angular frequency:
 k  2sk 
2k
T
fundamental angular frequency
sT  1/ T
T  2 sT  2 / T
term periods
term frequencies
term angular frequencies
Tk  T / k
sk  k sT
T
A. Dermanis
k  kT
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series

simplest form:
f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
2
k  k T  k
 k 2 sT
T
2
T 
T
A. Dermanis
1
sT 
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series

simplest form:
f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
2
T 
T
Fourier basis (base functions):
2 0t
0 (t )  1  cos
 cos 0t
T
ka (t )  cos
2
k  k T  k
 k 2 sT
T
1
sT 
T
0  0
2 kt
 cos k t
T
2 kt
 (t )  sin
 sin k t
T
b
k
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Development of a real function f(t) defined in the interval [0,T ] into Fourier series

simplest form:
f (t )  ao   [a k cos k t  bk sin  k t ]
k 1
2
T 
T
Fourier basis (base functions):
2 0t
0 (t )  1  cos
 cos 0t
T
ka (t )  cos
A. Dermanis
1
sT 
T
0  0
2 kt
 cos k t
T
2 kt
 (t )  sin
 sin k t
T
b
k
2
k  k T  k
 k 2 sT
T


f (t )  a00 (t )   a  (t )   bkkb (t )
k 1
a
k k
k 1
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
An example for the development of a function
in Fourier series
Separate analysis of each term for
A. Dermanis
k = 0, 1, 2, 3, 4, …
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
k=0
base function
+1
0
–1
A. Dermanis
0
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
k=0
contribution of term
+1
a0
0
–1
A. Dermanis
0
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
k=1
base functions
+1
+1
0
0
–1
A. Dermanis
0
T
–1
0
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
k=1
contributions of term
a1 cos
+1
2 t
T
b1 sin
+1
2 t
T
b1
0
0
a1
–1
A. Dermanis
0
T
–1
0
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
k=2
base functions
+1
+1
0
0
–1
A. Dermanis
T
2
–1
T
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
contributions of term
a2 cos
+1
2 t
T /2
k=2
b2 sin
+1
a2
b2
0
0
–1
A. Dermanis
T
2
–1
2 t
T /2
T
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
k=3
base functions
+1
+1
0
0
–1
A. Dermanis
T
3
–1
T
3
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
contributions of term
a3 cos
+1
k=3
2 t
T /3
b3 sin
+1
2 t
T /3
a3
0
0
b3
–1
A. Dermanis
T
3
–1
T
3
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
k=4
base functions
+1
+1
0
0
–1
A. Dermanis
T
4
–1
T
4
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
f (t )  a0 1  a1 cos
+1
f (x)
2 t
2 t
 b1 sin

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin

T /4
T /4
0
–1
contributions of term
a4 cos
+1
k=4
2 t
T /4
b4 sin
+1
a4
b4
0
0
–1
A. Dermanis
T
4
–1
2 t
T /4
T
4
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
a0 1 
 a1 cos
2 t
2 t
 b1 cos

T
T
 a2 cos
2 t
2 t
 b2 sin

T /2
T /2
 a3 cos
2 t
2 t
 b3 sin

T /3
T /3
 a4 cos
2 t
2 t
 b4 sin
 f (t )
T /4
T /4
T
+1
f (t)
0
–1
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Exploiting the idea of function othogonality
vector:
e1 , e2 , e3
inner product:
A. Dermanis
v  v1e1  v2e2  v3e3
orthogonal vector basis
u  w  u w cosu ,w
ei  ek ,
ei  ek  0,
ik
ik
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Exploiting the idea of function othogonality
vector:
e1 , e2 , e3
inner product:
v  v1e1  v2e2  v3e3
orthogonal vector basis
u  w  u w cosu ,w
ei  ek ,
ei  ek  0,
ik
ik
Computation of vector components:
v  e1  v1e1  e1  v2e2  e1  v3e3  e1  v1e1  e1  v1 e1
2
v  e2  v1e1  e2  v2e2  e2  v3e3  e2  v2e2  e2  v2 e2
v  e3  v1e1  e3  v2 e2  e3  v3e3  e3  v3e3  e3  v3 e3
A. Dermanis
2
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Exploiting the idea of function othogonality
v  v1e1  v2e2  v3e3
vector:
e1 , e2 , e3
orthogonal vector basis
inner product:
u  w  u w cosu ,w
ei  ek ,
ei  ek  0,
ik
ik
Computation of vector components:
0
0
v  e1  v1e1  e1  v2e2  e1  v3e3  e1  v1e1  e1  v1 e1
0
2
0
v  e2  v1e1  e2  v2e2  e2  v3e3  e2  v2e2  e2  v2 e2
0
0
v  e3  v1e1  e3  v2 e2  e3  v3e3  e3  v3e3  e3  v3 e3
A. Dermanis
2
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Exploiting the idea of function othogonality
v  v1e1  v2e2  v3e3
vector:
e1 , e2 , e3
orthogonal vector basis
inner product:
u  w  u w cosu ,w
ei  ek ,
ei  ek  0,
ik
ik
Computation of vector components:
0
0
v  e1  v1e1  e1  v2e2  e1  v3e3  e1  v1e1  e1  v1 e1
0
2
0
v  e2  v1e1  e2  v2e2  e2  v3e3  e2  v2e2  e2  v2 e2
0
0
v  e3  v1e1  e3  v2 e2  e3  v3e3  e3  v3e3  e3  v3 e3
A. Dermanis
2
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Exploiting the idea of function othogonality
v  v1e1  v2e2  v3e3
vector:
e1 , e2 , e3
orthogonal vector basis
inner product:
u  w  u w cosu ,w
ei  ek ,
ei  ek  0,
ik
ik
Computation of vector components:
0
0
v  e1  v1e1  e1  v2e2  e1  v3e3  e1  v1e1  e1  v1 e1
0
2
0
v  e2  v1e1  e2  v2e2  e2  v3e3  e2  v2e2  e2  v2 e2
0
0
v  e3  v1e1  e3  v2 e2  e3  v3e3  e3  v3e3  e3  v3 e3
A. Dermanis
2
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Exploiting the idea of function othogonality
v  v1e1  v2e2  v3e3
vector:
e1 , e2 , e3
orthogonal vector basis
inner product:
u  w  u w cosu ,w
ei  ek ,
ei  ek  0,
ik
ik
Computation of vector components:
0
0
v  e1  v1e1  e1  v2e2  e1  v3e3  e1  v1e1  e1  v1 e1
0
2
0
v  e2  v1e1  e2  v2e2  e2  v3e3  e2  v2e2  e2  v2 e2
0
v2 
v  e2
v3 
v  e3
0
v  e3  v1e1  e3  v2 e2  e3  v3e3  e3  v3e3  e3  v3 e3
A. Dermanis
2
v1 
v  e1
2
e1
e2
e3
2
2
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Orthogonality of Fourier base functions
T
Inner product of two functions:
 f , g    f (t ) g (t ) dt
0
Fourier basis:
0 (t )  1
ka (t )  cos k t  cos k1t
Orthogonality relations (km):
2 kt
2 mt
 ,     cos
cos
dt  0
T
T
0
kb (t )  sin k t  sin k1t
Norm (length) of a function:
T
a
k
a
m
T
kb , mb    sin
0
2 kt
2 mt
sin
dt  0
T
T
2 mt
2 kt
 ,     sin
cos
dt  0
T
T
0
T
|| f ||   f , f  
0
T
0   12 dt  T
2
0
T
a
k
b
m
2 kt
2 kt
 ,     sin
cos
dt  0
T
T
0
T
a
k
A. Dermanis
b
k

| f (t ) |2 dt
a 2
k

T
  cos 2
0
b 2
k

T
  sin 2
0
2 kt
T
dt 
T
2
2 kt
T
dt 
T
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
Ortjhogonality relations (km):
A. Dermanis
a
k k
k 1
ka , ma   0, kb , mb   0, ka , mb   0, ka , kb   0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
a
k k
k 1
ka , ma   0, kb , mb   0, ka , mb   0, ka , kb   0
Ortjhogonality relations (km):


0 , f   a0 0 , 0    ak 0 ,     bk 0 , kb   a0 0 , 0   a0 || 0 ||2  a0T
a
k
k 1
k 1


 , f   a0  , 0    ak  ,     bk ma , kb   am ma , ma   am || ma ||2  am
a
m
a
m
k 1
a
m
a
k

k 1

 , f   a0  , 0    ak  ,     bk mb , kb   bm mb , mb   bm || mb ||2  bm
b
m
A. Dermanis
b
m
k 1
b
m
a
k
k 1
T
2
T
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
a
k k
k 1
ka , ma   0, kb , mb   0, ka , mb   0, ka , kb   0
Ortjhogonality relations (km):
0

0

0 , f   a0 0 , 0    ak 0 , ka    bk 0 , kb   a0 0 , 0   a0 || 0 ||2  a0T
k 1
0
k 1
0 for km


0
ma , f   a0 ma , 0    ak ma , ka    bk ma , kb   am ma , ma   am || ma ||2  am
k 1
0

k 1
0

0 for km
mb , f   a0 mb , 0    ak mb , ka    bk mb , kb   bm mb , mb   bm || mb ||2  bm
k 1
A. Dermanis
k 1
T
2
T
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
a
k k
k 1
ka , ma   0, kb , mb   0, ka , mb   0, ka , kb   0
Ortjhogonality relations (km):
0

0

0 , f   a0 0 , 0    ak 0 , ka    bk 0 , kb   a0 0 , 0   a0 || 0 ||2  a0T
k 1
0
k 1
0 for km


0
ma , f   a0 ma , 0    ak ma , ka    bk ma , kb   am ma , ma   am || ma ||2  am
k 1
0

k 1
0

0 for km
mb , f   a0 mb , 0    ak mb , ka    bk mb , kb   bm mb , mb   bm || mb ||2  bm
k 1
A. Dermanis
k 1
T
2
T
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
a
k k
k 1
ka , ma   0, kb , mb   0, ka , mb   0, ka , kb   0
Ortjhogonality relations (km):
0

0

0 , f   a0 0 , 0    ak 0 , ka    bk 0 , kb   a0 0 , 0   a0 || 0 ||2  a0T
k 1
0
k 1
0 for km


0
ma , f   a0 ma , 0    ak ma , ka    bk ma , kb   am ma , ma   am || ma ||2  am
k 1
0

k 1
0

0 for km
mb , f   a0 mb , 0    ak mb , ka    bk mb , kb   bm mb , mb   bm || mb ||2  bm
k 1
A. Dermanis
k 1
T
2
T
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
a
k k
k 1
ka , ma   0, kb , mb   0, ka , mb   0, ka , kb   0
Ortjhogonality relations (km):
0

0

0 , f   a0 0 , 0    ak 0 , ka    bk 0 , kb   a0 0 , 0   a0 || 0 ||2  a0T
k 1
0
k 1
0 for km


0
ma , f   a0 ma , 0    ak ma , ka    bk ma , kb   am ma , ma   am || ma ||2  am
k 1
0

k 1
0

0 for km
mb , f   a0 mb , 0    ak mb , ka    bk mb , kb   bm mb , mb   bm || mb ||2  bm
k 1
A. Dermanis
k 1
T
2
T
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
0 , f   a0T
T
2
T
mb , f   bm
2
ma , f   am
a
k k
k 1
1
0 , f 
T
2
am  ma , f 
T
2
bm  mb , f 
T
a0 
Computation of Fourier series coefficients
of a known function:
T
1
a0   f (t ) dt
T 0
T
2
ak   f (t ) cos k t dt
T 0
A. Dermanis
T
2
bk   f (t ) sin k t dt
T 0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
a
k k
k 1
T
1
a0   f (t ) dt
T 0
T
2
ak   f (t ) cos k t dt
T 0
k  1, 2,
T
2
bk   f (t ) sin k t dt
T 0
k  1, 2,
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients


f  a00   a    bkkb
k 1
a
k k
k 1
T
1
a0   f (t ) dt
T 0
T
2
ak   f (t ) cos k t dt
T 0
k  1, 2,
change of
notation
a0  2a0
ak  ak
bk  bk
T
T
bk 
2
f (t ) sin k t dt

T 0
k  1, 2,
A. Dermanis
2
a0  2a0   f (t ) dt
T 0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients




a0
a
f  0   akk   bkkb
2
k 1
k 1
f  a00   a    b 
k 1
a
k k
k 1
T
1
a0   f (t ) dt
T 0
T
2
ak   f (t ) cos k t dt
T 0
k  1, 2,
b
k k
change of
notation
a0  2a0
ak  ak
bk  bk
T
2
ak   f (t ) cos k t dt
T 0
k  0,1, 2,
T
T
2
bk   f (t ) sin k t dt
T 0
k  1, 2,
A. Dermanis
2
a0  2a0   f (t ) dt
T 0
T
2
bk   f (t ) sin k t dt
T 0
k  1, 2,
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients
ao 
2 kt
2 kt
f (t )    [ak cos
 bk sin
]
2 k 1
T
T
T
2
ak   f (t ) cos k t dt
T 0
T
k  1, 2,
2
bk   f (t ) sin k t dt
T 0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Computation of Fourier series coefficients
2 kt
2 kt
f (t )  ao   [ak cos
 bk sin
]
T
T
k 1

T
1
ak   f (t ) cos k t dt
T 0
A. Dermanis
T
2
ak   f (t ) cos k t dt
T 0
k  1, 2,
T
2
bk   f (t ) sin k t dt
T 0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Alternative forms of Fourier series (polar forms)
Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk !
bk
ρk = «length»
k
k
k
ak
A. Dermanis
θk = «azimuth»
φk = «direction angle»
φk + θk = 90
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Alternative forms of Fourier series (polar forms)
Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk !
bk
ρk = «length»
k
k
k
ak
A. Dermanis
θk = «azimuth»
φk = «direction angle»
φk + θk = 90
ak  k sin k
ak  k cos k
bk  k cos k
bk  k sin k
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Alternative forms of Fourier series (polar forms)
Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk !
bk
ρk = «length»
k
k
k
ak
θk = «azimuth»
φk = «direction angle»
φk + θk = 90
ak  k sin k
ak  k cos k
bk  k cos k
bk  k sin k
k  ak2  bk2
ak
tan  k 
bk
A. Dermanis
bk

tan k 
 cot  k  tan(   k )
ak
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Alternative forms of Fourier series (polar forms)

f  a0   (ak cos k t  bk sin k t)
k 1
ak  k sin k
k  ak2  bk2
tan  k 
A. Dermanis
ak
bk
bk  k cos k
ak  k cos k
bk  k sin k
k  ak2  bk2
tan k 
bk

 cot  k  tan(   k )
ak
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Alternative forms of Fourier series (polar forms)

f  a0   (ak cos k t  bk sin k t)
k 1
ak  k sin k
bk  k cos k
k  ak2  bk2
tan  k 
ak
bk
ak  k cos k
bk  k sin k
k  ak2  bk2
tan k 
bk

 cot  k  tan(   k )
ak
2
ak cos k t  bk sin k t 
 k sin k cos k t  k cos k sin k t 
 k sin(k t  k )
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Alternative forms of Fourier series (polar forms)

f  a0   (ak cos k t  bk sin k t)
k 1
ak  k sin k
bk  k cos k
k  ak2  bk2
tan  k 
ak  k cos k
bk  k sin k
k  ak2  bk2
ak
bk
tan k 
bk

 cot  k  tan(   k )
ak
2
ak cos k t  bk sin k t 
 k sin k cos k t  k cos k sin k t 
 k sin(k t  k )

f (t )    k sin(k t   k )
k 0
θk = phase (sin)
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Alternative forms of Fourier series (polar forms)

f  a0   (ak cos k t  bk sin k t)
k 1
ak  k sin k
bk  k cos k
k  ak2  bk2
tan  k 
ak  k cos k
bk  k sin k
k  ak2  bk2
ak
bk
tan k 
ak cos k t  bk sin k t 
bk

 cot  k  tan(   k )
ak
2
ak cos k t  bk sin k t 
 k sin k cos k t  k cos k sin k t 
 k cos k cos k t  k sin k sin k t 
 k sin(k t  k )
 k cos(k t  k )

f (t )    k sin(k t   k )
k 0
θk = phase (sin)
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Alternative forms of Fourier series (polar forms)

f  a0   (ak cos k t  bk sin k t)
k 1
ak  k sin k
bk  k cos k
k  ak2  bk2
tan  k 
bk  k sin k
k  ak2  bk2
ak
bk
tan k 
ak cos k t  bk sin k t 
bk

 cot  k  tan(   k )
ak
2
ak cos k t  bk sin k t 
 k sin k cos k t  k cos k sin k t 
 k cos k cos k t  k sin k sin k t 
 k sin(k t  k )
 k cos(k t  k )

f (t )    k sin(k t   k )
k 0
θk = phase (sin)
A. Dermanis
ak  k cos k

f (t )    k cos(k t  k )
k 0
φk = phase (cosine)
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series for a complex function
f (t )  f1 (t )  if 2 (t )
i  1
Fourier series of real functions:
«imaginary» part
«real» part

f1 (t )  a   [a cos k t  b sin k t ]
1
0
A. Dermanis
k 1
1
k
1
k

f 2 (t )  a   [ak2 cos k t  bk2 sin k t ]
2
0
k 1
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series for a complex function
f (t )  f1 (t )  if 2 (t )
i  1
Fourier series of real functions:
«imaginary» part
«real» part

f1 (t )  a   [a cos k t  b sin k t ]
1
0
k 1
1
k
1
k

f 2 (t )  a   [ak2 cos k t  bk2 sin k t ]
2
0
k 1
setting
a0  a01  i a02
ak  a1k  i ak2
bk  bk1  i bk2
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series for a complex function
f (t )  f1 (t )  if 2 (t )
i  1
Fourier series of real functions:
«imaginary» part
«real» part

f1 (t )  a   [a cos k t  b sin k t ]
1
0
1
k
k 1
1
k

f 2 (t )  a   [ak2 cos k t  bk2 sin k t ]
2
0
k 1
setting
a0  a01  i a02

ak  a  i a
1
k
2
k
f (t )  a0   [ak cos k t  bk sin k t ]
k 1
bk  bk1  i bk2
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series for a complex function
Implementation of complex symbolism:
cos  
A. Dermanis
ei  cos  i sin 
ei  cos  i sin 
1 i
(e  ei )
2
i
sin   (ei  ei )
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series for a complex function
cos  

f (t )  a0   [ak cos k t  bk sin k t ]  a0 
k 1
1 i
(e  ei )
2

[ a (e

2
1
k 1
ik t
k
 a0 
A. Dermanis
ei  cos  i sin 
ei  cos  i sin 
Implementation of complex symbolism:
1
2

e
ik t
)  i bk (e
 (ak  ibk )e
k 1
i
sin   (ei  ei )
2
ik t

ik t
1
2

e
ik t
)] 
 (ak  ibk )e
ik t
k 1
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series for a complex function
cos  

f (t )  a0   [ak cos k t  bk sin k t ]  a0 
k 1
c0  a0  a  i a
1
0
2
0
ei  cos  i sin 
ei  cos  i sin 
Implementation of complex symbolism:
1 i
(e  ei )
2

[ a (e

2
1
k 1
ik t
k
 a0 
1
2

e
ik t
)  i bk (e
 (ak  ibk )e
k 1
i
sin   (ei  ei )
2
ik t

ik t
1
2

e
ik t
)] 
 (ak  ibk )e
ik t
k 1
k  0:
 k  kT  k
ck  12 (ak  ibk ) 
 12 [(a1k  bk2 )  i (ak2  bk1 )]
c k  12 (ak  ibk ) 
 12 [(a1k  bk2 )  i (ak2  bk1 )]
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series for a complex function
cos  

f (t )  a0   [ak cos k t  bk sin k t ]  a0 
k 1
c0  a0  a  i a
1
0
2
0
ei  cos  i sin 
ei  cos  i sin 
Implementation of complex symbolism:
1 i
(e  ei )
2

[ a (e

2
1
k 1
ik t
k
 a0 
1
2
e
ik t
i
sin   (ei  ei )
2
)  i bk (e

 (ak  ibk )e
ik t

k 1
ik t
1
2
e
ik t
)] 

 (ak  ibk )e
ik t
k 1
k  0:
 k  kT  k
ck  12 (ak  ibk ) 

f (t )  c0   ck e
k 1
ik t

  c k e
i k t
k 1
 12 [(a1k  bk2 )  i (ak2  bk1 )]
c k  12 (ak  ibk ) 
 12 [(a1k  bk2 )  i (ak2  bk1 )]
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series for a complex function
cos  

f (t )  a0   [ak cos k t  bk sin k t ]  a0 
k 1
c0  a0  a  i a
1
0
2
0
ei  cos  i sin 
ei  cos  i sin 
Implementation of complex symbolism:
1 i
(e  ei )
2

[ a (e

2
1
k 1
ik t
k
 a0 
1
2
e
ik t
i
sin   (ei  ei )
2
)  i bk (e

 (ak  ibk )e
ik t

k 1
ik t
1
2
e
ik t
)] 

 (ak  ibk )e
ik t
k 1
k  0:
 k  kT  k
ck  12 (ak  ibk ) 

f (t )  c0   ck e
ik t
k 1

  c k e
i k t
k 1
 12 [(a1k  bk2 )  i (ak2  bk1 )]
c k  12 (ak  ibk ) 
 12 [(a1k  bk2 )  i (ak2  bk1 )]
A. Dermanis
f (t ) 

i k t
c
e
 k
k 
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Complex form of Fourier series
Development of a complex function
into a Fourier series
with complex base functions
and complex coefficients
f (t ) 

ce
k 
ik t
k
k  k
2
T
T
Computation of complex coefficients
for a known complex function
A. Dermanis
1
 ik t
ck   f (t ) e
dt
T 0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Ortjhogonality of the complex basis
Conjugateς z* of a complex number z :
ek (t )  e
z *  ( z1  iz2 )*  z1  iz2
ik t
e
ik
2 t
T
| z |2  zz *  z12  z22
inner product:
T
 f , g    f (t ) g * (t ) dt
0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
ek (t )  e
Ortjhogonality of the complex basis
z *  ( z1  iz2 )*  z1  iz2
Conjugateς z* of a complex number z :
T
ek , em    ek (t )e (t ) dt   e
inner product:
0
T
 f , g    f (t ) g * (t ) dt
0
T
*
m
A. Dermanis
e
T
2 t
T
T
dt   ei 2 ( k m)t / T dt  0
0
T
| ek |  ek , ek    ek (t )e (t ) dt   e
2
e
ik
| z |2  zz *  z12  z22
0
*
k
0
ek , em   0
i 2 kt / T i 2 mt / T
ik t
0
i 2 kt / T i 2 kt / T
e
T
T
dt   e dt   dt  T
0
0
0
| ek |2   ek , ek   T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
ek (t )  e
Ortjhogonality of the complex basis
z *  ( z1  iz2 )*  z1  iz2
Conjugateς z* of a complex number z :
T
T
ek , em    ek (t )e (t ) dt   e
*
m
inner product:
0
T
 f , g    f (t ) g * (t ) dt
0
T
T
dt   ei 2 ( k m)t / T dt  0
0
T
| ek |  ek , ek    ek (t )e (t ) dt   e
2
2 t
T
*
k
i 2 kt / T i 2 kt / T
0
e
T
T
dt   e dt   dt  T
0
0
0
| ek |2   ek , ek   T
ek , em   0
A. Dermanis
e
e
ik
| z |2  zz *  z12  z22
0
0
 f , ek  
i 2 kt / T i 2 mt / T
ik t

ce
m 
m m
, ek 

c
m 
m
em , ek   ck ek , ek   ck | ek |2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
ek (t )  e
Ortjhogonality of the complex basis
z *  ( z1  iz2 )*  z1  iz2
Conjugateς z* of a complex number z :
T
T
ek , em    ek (t )e (t ) dt   e
*
m
inner product:
0
T
 f , g    f (t ) g * (t ) dt
e
T
2 t
T
T
dt   ei 2 ( k m)t / T dt  0
0
T
| ek |  ek , ek    ek (t )e (t ) dt   e
*
k
0
e
ik
| z |2  zz *  z12  z22
0
2
0
i 2 kt / T i 2 kt / T
0
e
T
T
dt   e dt   dt  T
0
0
0
| ek |2   ek , ek   T
ek , em   0
 f , ek  
i 2 kt / T i 2 mt / T
ik t

ce
m 
m m
, ek 
T

c
m 
m
em , ek   ck ek , ek   ck | ek |2
T
1
1
1
ik t
*
ck 

f
,
e


f
(
t
)
e
(
t
)
dt

f
(
t
)
e
dt
k
k
| ek |2
T 0
T 0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
ek (t )  e
Ortjhogonality of the complex basis
z *  ( z1  iz2 )*  z1  iz2
Conjugateς z* of a complex number z :
T
T
ek , em    ek (t )e (t ) dt   e
*
m
inner product:
0
T
 f , g    f (t ) g * (t ) dt
T
2 t
T
T
dt   ei 2 ( k m)t / T dt  0
0
T
| ek |  ek , ek    ek (t )e (t ) dt   e
*
k
0
i 2 kt / T i 2 kt / T
e
T
T
dt   e dt   dt  T
0
0
0
0
| ek |2   ek , ek   T
ek , em   0

ce
m 
m m
, ek 
T

c
m 
m
em , ek   ck ek , ek   ck | ek |2
T
1
1
1
ik t
*
ck 

f
,
e


f
(
t
)
e
(
t
)
dt

f
(
t
)
e
dt
k
k
| ek |2
T 0
T 0
A. Dermanis
e
e
ik
| z |2  zz *  z12  z22
0
2
0
 f , ek  
i 2 kt / T i 2 mt / T
ik t
T
1
ck   f (t ) e  ik t dt
T 0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
Implementation of complex symbolism:
cos  
A. Dermanis
ei  cos  i sin 
ei  cos  i sin 
1 i
(e  ei )
2
i
sin   (ei  ei )
2
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
cos  

f (t )  a0   [ak cos k t  bk sin k t ]  a0 
k 1
1 i
(e  ei )
2

[ a (e

2
1
k 1
ik t
k
 a0 
A. Dermanis
ei  cos  i sin 
ei  cos  i sin 
Implementation of complex symbolism:
1
2

e
ik t
)  i bk (e
 (ak  ibk )e
k 1
i
sin   (ei  ei )
2
ik t

ik t
1
2

e
ik t
)] 
 (ak  ibk )e
ik t
k 1
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
cos  

f (t )  a0   [ak cos k t  bk sin k t ]  a0 
k 1
c0  a0
ei  cos  i sin 
ei  cos  i sin 
Implementation of complex symbolism:
1 i
(e  ei )
2

[ a (e

2
1
k 1
ik t
k
 a0 
1
2

e
ik t
)  i bk (e
 (ak  ibk )e
k 1
i
sin   (ei  ei )
2
ik t

ik t
1
2

e
ik t
)] 
 (ak  ibk )e
ik t
k 1
k  0:
 k  kT  k
ck  12 (ak  ibk )
c k  12 (ak  ibk )
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
cos  

f (t )  a0   [ak cos k t  bk sin k t ]  a0 
k 1
c0  a0
ei  cos  i sin 
ei  cos  i sin 
Implementation of complex symbolism:
1 i
(e  ei )
2

[ a (e

2
1
k 1
ik t
k
 a0 
1
2
e
ik t
i
sin   (ei  ei )
2
)  i bk (e

 (ak  ibk )e
ik t

k 1
ik t
1
2
e
ik t
)] 

 (ak  ibk )e
ik t
k 1
k  0:
 k  kT  k
ck  12 (ak  ibk )

f (t )  c0   ck e
k 1
ik t

  c k e
i k t
k 1
c k  12 (ak  ibk )
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
cos  

f (t )  a0   [ak cos k t  bk sin k t ]  a0 
k 1
c0  a0
ei  cos  i sin 
ei  cos  i sin 
Implementation of complex symbolism:
1 i
(e  ei )
2

[ a (e

2
1
k 1
ik t
k
 a0 
1
2
e
ik t
i
sin   (ei  ei )
2
)  i bk (e

 (ak  ibk )e
ik t

k 1
ik t
1
2
e
ik t
)] 

 (ak  ibk )e
ik t
k 1
k  0:
 k  kT  k
ck  12 (ak  ibk )

f (t )  c0   ck e
ik t
k 1
c k  12 (ak  ibk )
f (t ) 

  c k e
i k t
k 1

i k t
c
e
 k
k 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
f  f1  i f 2  f1
1
f1  f
a1k  ak
f2  0
a b 0
2
k
2
k
bk1  bk
ck  (ak  ibk )
2
c0  a0
1
c k  (ak  ibk )  ck
2
f (t ) 

i k t
c
e
 k
k 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
f  f1  i f 2  f1
1
f1  f
a1k  ak
f2  0
a b 0
2
k
ck  (ak  ibk )
2
bk1  bk
c0  a0
2
k
1
c k  (ak  ibk )  ck
2
f (t ) 

i k t
c
e
 k
k 
A. Dermanis
f (t) = real function
c k  ck*
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
f  f1  i f 2  f1
1
f1  f
a1k  ak
f2  0
a b 0
2
k
ck  (ak  ibk )
2
bk1  bk
c0  a0
2
k
1
c k  (ak  ibk )  ck
2
f (t ) 

i k t
c
e
 k
k 
ik t
]k 0  c0e
ik t
 c k e
[ck e
ck e
A. Dermanis
i0t
ik t
f (t) = real function
c k  ck*
 c0e0  c0  a0
1
1
2
2
 (ak  ibk )(cos k t  i sin k t )  (ak  ibk )(cos k t  i sin k t ) 
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
f  f1  i f 2  f1
1
f1  f
a1k  ak
f2  0
a b 0
2
k
ck  (ak  ibk )
2
bk1  bk
c0  a0
2
k
1
c k  (ak  ibk )  ck
2
f (t ) 

i k t
c
e
 k
k 
ik t
]k 0  c0e
ik t
 c k e
[ck e
ck e
i0t
ik t
c k  ck*
f (t) = real function
 c0e0  c0  a0
1
1
2
2
 (ak  ibk )(cos k t  i sin k t )  (ak  ibk )(cos k t  i sin k t ) 
1
 (ak cos k t  ibk cos k t  iak sin k t  bk sin k t 
2
ak cos k t  ibk cos k t  iak sin k t  bk sin k t ) 
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
f  f1  i f 2  f1
1
f1  f
a1k  ak
f2  0
a b 0
2
k
ck  (ak  ibk )
2
bk1  bk
c0  a0
2
k
1
c k  (ak  ibk )  ck
2
f (t ) 

i k t
c
e
 k
k 
ik t
]k 0  c0e
ik t
 c k e
[ck e
ck e
i0t
ik t
c k  ck*
f (t) = real function
 c0e0  c0  a0
1
1
2
2
 (ak  ibk )(cos k t  i sin k t )  (ak  ibk )(cos k t  i sin k t ) 
1
 (ak cos k t  ibk cos k t  iak sin k t  bk sin k t 
2
ak cos k t  ibk cos k t  iak sin k t  bk sin k t ) 
 ak cos k t  bk sin k t
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function using complex notation
f  f1  i f 2  f1
1
f1  f
a1k  ak
f2  0
a b 0
2
k
ck  (ak  ibk )
2
bk1  bk
c0  a0
2
k
1
c k  (ak  ibk )  ck
2
f (t ) 

i k t
c
e
 k
k 
ik t
]k 0  c0e
ik t
 c k e
[ck e
ck e
i0t
ik t
c k  ck*
f (t) = real function
 c0e0  c0  a0
1
1
2
2
 (ak  ibk )(cos k t  i sin k t )  (ak  ibk )(cos k t  i sin k t ) 
1
 (ak cos k t  ibk cos k t  iak sin k t  bk sin k t 
2
ak cos k t  ibk cos k t  iak sin k t  bk sin k t ) 
 ak cos k t  bk sin k t
A. Dermanis
The imaginary part disappears !
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function: Real and complex form
a0 
2 kt
2 kt
f (t )    [ak cos
 bk sin
]
2 k 1
T
T
2
2 kt
ak   f (t ) cos
dt
T 0
T
T
2
2 kt
bk   f (t ) sin
dt
T 0
T
T
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function: Real and complex form
a0 
2 kt
2 kt
f (t )    [ak cos
 bk sin
]
2 k 1
T
T
2
2 kt
ak   f (t ) cos
dt
T 0
T
T
1
ck  (ak  ibk )
2
2
2 kt
bk   f (t ) sin
dt
T 0
T
T
1
c k  (ak  ibk )  ck
2
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function: Real and complex form
a0 
2 kt
2 kt
f (t )    [ak cos
 bk sin
]
2 k 1
T
T
2
2 kt
ak   f (t ) cos
dt
T 0
T
T
1
ck  (ak  ibk )
2
2
2 kt
bk   f (t ) sin
dt
T 0
T
T
1
c k  (ak  ibk )  ck
2
e
A. Dermanis
ik t
e
ik
2 t
T
2 t
2 t
 cos k
 i sin k
T
T
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function: Real and complex form
a0 
2 kt
2 kt
f (t )    [ak cos
 bk sin
]
2 k 1
T
T
2
2 kt
ak   f (t ) cos
dt
T 0
T
T
1
ck  (ak  ibk )
2
2
2 kt
bk   f (t ) sin
dt
T 0
T
T
1
c k  (ak  ibk )  ck
2
e
f (t ) 
ik t


k 
A. Dermanis
e
ik
2 t
T
2 t
2 t
 cos k
 i sin k
T
T
ck eik t
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series of a real function: Real and complex form
a0 
2 kt
2 kt
f (t )    [ak cos
 bk sin
]
2 k 1
T
T
2
2 kt
ak   f (t ) cos
dt
T 0
T
T
1
ck  (ak  ibk )
2
2
2 kt
bk   f (t ) sin
dt
T 0
T
T
1
c k  (ak  ibk )  ck
2
e
f (t ) 
ik t


k 
A. Dermanis
e
ik
2 t
T
2 t
2 t
 cos k
 i sin k
T
T
T
ck eik t
1
ck   f (t ) e  ik t dt
T 0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Extension of the function
f (t ) 

ik t
 ck e
k 
f (t) outside the interval [0, T ]
1 T
 i t
i 
    f ( )e k d  e k
k   T 0


f (t )  f (t )
t  [0, T ]
f (t )  f (t )
t  [0, T ]
f (t )
f (t )
–2T
–T
0
T
2T
3T
The extension f (t ) is a periodic function, with period Τ
f (t  nT )  f (t ) t  [0, T ]
for every integer n
CAUSE OF USUAL MISCONCEPTION:
A. Dermanis
“Fourier series expansion
deals with periodic functions»
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the circle
(naturally periodic domain)
θ
T  2
k 
2 k 2 k

k
T
2
t 


k 1
k 1
(angle)
f ( )  a0   [ak cos k  bk sin k ]  a0   [ak cos(k )  bk sin(k )]
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the circle
(naturally periodic domain)
θ
T  2
k 
2 k 2 k

k
T
2
t 


k 1
k 1
(angle)
f ( )  a0   [ak cos k  bk sin k ]  a0   [ak cos(k )  bk sin(k )]

f ( )  ao   [ak cos(k )  bk sin(k )]
k 1
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the circle
(naturally periodic domain)
θ
T  2
k 
2 k 2 k

k
T
2
t 


k 1
k 1
(angle)
f ( )  a0   [ak cos k  bk sin k ]  a0   [ak cos(k )  bk sin(k )]
ao 

f ( )  ao   [ak cos(k )  bk sin(k )]
k 1
ak 
bk 
A. Dermanis
1

1

1
2
2

2

f ( )d
0
f ( ) cos(k )d
0
2

f ( ) sin(k )d
0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Expansion of function
f (x,y)
inside an orthogonal parallelogram (0  x  Tx, 0 
ka ( x, y)  cos
Base functions:
y  Ty)
2 k
2 k
x cos
y  cos(xk x) cos( yk y)
Tx
Ty
Ty
0
0
A. Dermanis
Tx
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Expansion of function
f (x,y)
inside an orthogonal parallelogram (0  x  Tx, 0 
ka ( x, y)  cos
Base functions:
 xk  uk 
2 k
Tx
 yk  vk 
y  Ty)
2 k
2 k
x cos
y  cos(xk x) cos( yk y)
Tx
Ty
2 k
Ty
(angular frequencies along x and y )
Ty
0
0
A. Dermanis
Tx
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Expansion of function
f (x,y)
inside an orthogonal parallelogram (0  x  Tx, 0 
ka ( x, y)  cos
Base functions:
 xk  uk 
2 k
Tx
Ty
0
0
A. Dermanis
Tx
 yk  vk 
y  Ty)
2 k
2 k
x cos
y  cos(xk x) cos( yk y)
Tx
Ty
2 k
Ty
(angular frequencies along x and y )
a
km
( x, y )  cos(uk x) cos(vm y )
 akm
b
km
( x, y )  cos(uk x) sin(vm y )
 bkm
c
km
( x, y )  sin(uk x) cos(vm y )
 ckm
d
km
( x, y )  sin(uk x)sin(vm y)
 d km
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Expansion of function
f (x,y)
inside an orthogonal parallelogram (0  x  Tx, 0 
ka ( x, y)  cos
Base functions:
 xk  uk 
2 k
Tx
Ty
0
Tx
0

f ( x, y )  

 [a
k 0 m 0
A. Dermanis
 yk  vk 
y  Ty)
2 k
2 k
x cos
y  cos(xk x) cos( yk y)
Tx
Ty
2 k
Ty
(angular frequencies along x and y )
a
km
( x, y )  cos(uk x) cos(vm y )
 akm
b
km
( x, y )  cos(uk x) sin(vm y )
 bkm
c
km
( x, y )  sin(uk x) cos(vm y )
 ckm
d
km
( x, y )  sin(uk x)sin(vm y)
 d km
b
c
d
 ( x, y )  bkmkm
( x, y )  ckmkm
( x, y )  d kmkm
( x, y )]
a
km km
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Expansion of function

f ( x, y )  
f (x,y)

 [a
k 0 m 0
A. Dermanis
inside an orthogonal parallelogram (0  x  Tx, 0 
y  Ty)
b
c
d
 ( x, y )  bkmkm
( x, y )  ckmkm
( x, y )  d kmkm
( x, y )]
a
km km
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Expansion of function

f ( x, y )  
f (x,y)

 [a
inside an orthogonal parallelogram (0  x  Tx, 0 
y  Ty)
b
c
d
 ( x, y )  bkmkm
( x, y )  ckmkm
( x, y )  d kmkm
( x, y )]
a
km km
k 0 m 0
Equivalent to double Fourier series: First along x and then along y (or vice-versa)

f ( x, y )  f ( x)   [ak ( y ) cos(uk x)  bk ( y ) sin(uk x)
y
k 0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Expansion of function

f ( x, y )  
f (x,y)

 [a
inside an orthogonal parallelogram (0  x  Tx, 0 
y  Ty)
b
c
d
 ( x, y )  bkmkm
( x, y )  ckmkm
( x, y )  d kmkm
( x, y )]
a
km km
k 0 m 0
Equivalent to double Fourier series: First along x and then along y (or vice-versa)

f ( x, y )  f ( x)   [ak ( y ) cos(uk x)  bk ( y ) sin(uk x)
y
k 0

ak ( y )   [akm cos(vm y )  bkm sin(vm y )]
m0

bk ( y )   [ckm cos(vm y )  d km sin(vm y )]
m0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Expansion of function

f ( x, y )  
f (x,y)

 [a
inside an orthogonal parallelogram (0  x  Tx, 0 
y  Ty)
b
c
d
 ( x, y )  bkmkm
( x, y )  ckmkm
( x, y )  d kmkm
( x, y )]
a
km km
k 0 m 0
Equivalent to double Fourier series: First along x and then along y (or vice-versa)

f ( x, y )  f ( x)   [ak ( y ) cos(uk x)  bk ( y ) sin(uk x)
y
k 0

ak ( y )   [akm cos(vm y )  bkm sin(vm y )]
m0

bk ( y )   [ckm cos(vm y )  d km sin(vm y )]
m0
f ( x, y )  f y ( x) 


  [akm cos(uk x) cos(vm y)  bkm cos(uk x)sin(vm y)  ckm sin(uk x) cos(vm y)  d km sin(uk x)sin(vm y )]
k 0 m 0
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Tx
Inner product:
 f , g  
0
A. Dermanis

f ( x, y ) g ( x, y ) dx dy
0
A
B
km
,  pq
0
m  q for every Α = a,b,c,d
and B = a,b,c,d
T Ty
|| kka ||2 || kkb ||2 || kkc ||2 || kkd ||2  x
2 2
Orthogonal Fourier basis !
|| 00A ||2  TxTy
Ty
kp
ή
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Tx
 f , g  
Inner product:
0

f ( x, y ) g ( x, y ) dx dy
0
A
B
km
,  pq
0
m  q for every Α = a,b,c,d
and B = a,b,c,d
T Ty
|| kka ||2 || kkb ||2 || kkc ||2 || kkd ||2  x
2 2
Orthogonal Fourier basis !
|| 00A ||2  TxTy
Ty
kp
ή
Computation of coefficients:
2 2
4
a
akm 
 f , km

Tx Ty
TxTy
2 2
4
b
bkm 
 f , km

Tx Ty
TxTy
2 2
4
c
ckm 
 f , km

Tx Ty
TxTy
d km 
A. Dermanis
2 2
4
d
 f , km

Tx Ty
TxTy
Tx Ty
  f ( x, y) cos(u x) cos(v
k
m
y ) dx dy
0 0
Tx Ty
  f ( x, y) cos(u x)sin(v
k
m
y ) dx dy
0 0
Tx Ty
  f ( x, y)sin(u x) cos(v
m
y ) dx dy
m
y ) dx dy
k
0 0
Tx Ty
  f ( x, y)sin(u x)sin(v
k
0 0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
Complex form:
f ( x, y ) 


 
k 
1 1
ckm 
Tx Ty
A. Dermanis
m 
ckm e
i (uk x vm y )
Tx Ty
 
0
f ( x, y ) e
i (uk x vm y )
uk 
2 k
Tx
vm 
2 m
Ty
dxdy
0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on the plane
f ( x, y ) 
Complex form:


 
k 
1 1
ckm 
Tx Ty
m 
ckm e
i (uk x vm y )
Tx Ty
 
0
f ( x, y ) e
i (uk x vm y )
uk 
2 k
Tx
vm 
2 m
Ty
dxdy
0
Fourier series in n dimensions
, xn ) 
f ( x1 , x2 ,
ck k
1 2
A. Dermanis
kn



 
 ck k
k1  k2 
T1
1
TT
1 2

Tn
T2
 
0
0
kn 
1 2
kn e
i (1x1 2 x2  n xn )
Tn

f ( x1 , x2 ,
, xn ) e
k 
i (1x1 2 x2  n xn )
dx1dx2
2 k
Tk
dxn
0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in n dimensions
, xn ) 
f ( x1 , x2 ,
ck k
1 2
A. Dermanis
kn



 

k1  k2 
T1
1
TT
1 2

Tn
T2
 
0
0
kn 
ck k
1 2
kn e
i (1x1 2 x2  n xn )
Tn

f ( x1 , x2 ,
, xn ) e
k 
i (1x1 2 x2  n xn )
dx1dx2
2 k
Tk
dxn
0
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in n dimensions
, xn ) 
f ( x1 , x2 ,
ck k
1 2
kn



 

k1  k2 
T1
1
TT
1 2

Tn
T2
 
0
0
kn 
ck k
1 2
kn e
i (1x1 2 x2  n xn )
Tn

f ( x1 , x2 ,
, xn ) e
2 k
Tk
k 
i (1x1 2 x2  n xn )
dx1dx2
dxn
0
In matrix notation:
x  [ x1 x2
dx  dx1 dx2
xn ]T
dxn
ω  [1 2
n ]T
c[ k ]  ck k
1 2
kn


[k ]
domain of definition:
 n  {x | 0  x1  T1 ,0  x2  T2 ,
Vn |  n | TT
1 2
A. Dermanis
Tn
,0  xn  Tn }


 
k1  k2 


kn 
(orthogonal hyper-parallelepiped)
(parallelepiped volume)
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series in n dimensions
, xn ) 
f ( x1 , x2 ,
ck k
1 2
kn



 

k1  k2 
T1
1
TT
1 2

Tn
T2
 
0
0
kn 
ck k
1 2
kn e
i (1x1 2 x2  n xn )
Tn

f ( x1 , x2 ,
, xn ) e
2 k
Tk
k 
i (1x1 2 x2  n xn )
dx1dx2
dxn
0
In matrix notation:
x  [ x1 x2
dx  dx1 dx2
xn ]T
dxn
ω  [1 2
n ]T
c[ k ]  ck k
1 2

kn

[k ]
domain of definition:
 n  {x | 0  x1  T1 ,0  x2  T2 ,
Vn |  n | TT
1 2
,0  xn  Tn }
Tn

 
k1  k2 


kn 
(orthogonal hyper-parallelepiped)
(parallelepiped volume)
f (x)   c[ k ]e
i (ωT x)
[k ]
A. Dermanis

c[ k ] 
1
Vn

f ( x )e  i ( ω
T x)
dx
n
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Fourier series on any interval [Α, Β]
T  B A
B  AT

f (t )  ao   [ak cos k t  bk sin k t ]
k 
k 1
B
1
a0 
f (t ) dt

B A A
A  0, B  2
A   , B  
T  2 , k  k
T  2 , k  k
1
a0 
2
B
2
ak 
f (t ) cos k t dt
B  A A
ak 
B
2
bk 
f (t )sin k t dt
B  A A
A. Dermanis
bk 
1

1

2

f (t ) dt
0
1
a0 
2
2

f (t ) cos(kt ) dt
ak 
0
2

0
f (t ) sin(kt ) dt
bk 
1


 f (t ) dt


1

2 k
T
 f (t ) cos(kt ) dt


 f (t ) sin(kt ) dt

Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Approximating a function by a finite Fourier series expansion

f (t )  a0   [ak cos
k 1
2 kt
2 kt
 bk sin
]
T
T

Question : What is the meaning of the symbol

in the Fourier series expansion?
k 1
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Approximating a function by a finite Fourier series expansion

f (t )  a0   [ak cos
k 1
2 kt
2 kt
 bk sin
]
T
T

Question : What is the meaning of the symbol

in the Fourier series expansion?
k 1
Certainly not that infinite terms must be summed! This is impossible!
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Approximating a function by a finite Fourier series expansion

f (t )  a0   [ak cos
k 1
2 kt
2 kt
 bk sin
]
T
T

Question : What is the meaning of the symbol

in the Fourier series expansion?
k 1
Certainly not that infinite terms must be summed! This is impossible!
In practice we can use only
a finite sum
N
f N (t )  a0   [ak cos
k 1
2 kt
2 kt
 bk sin
]
T
T
with a «sufficiently large» integer Ν
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Approximating a function by a finite Fourier series expansion
Sufficiently large Ν means:
For whatever small ε > 0 there exists an integer Ν such that
|| f(t) – fN(t)|| < ε
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Approximating a function by a finite Fourier series expansion
Sufficiently large Ν means:
For whatever small ε > 0 there exists an integer Ν such that
|| f(t) – fN(t)|| < ε
Attention:
|| f(t) – fN(t)|| < ε does not necessarily mean that the
difference
A. Dermanis
| f(t) – fN(t)|
is small for every
t !!!
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Approximating a function by a finite Fourier series expansion
Sufficiently large Ν means:
For whatever small ε > 0 there exists an integer Ν such that
|| f(t) – fN(t)|| < ε
Attention:
|| f(t) – fN(t)|| < ε does not necessarily mean that the
difference
| f(t) – fN(t)|
is small for every
t !!!
It would be desirable (though not plausible) that
max | f(t) – fN(t)| < ε
A. Dermanis
in the interval [0,Τ]
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Characteristics of the Fourier series expansion
 The coefficients ak, bk become generally smaller as k increases
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Characteristics of the Fourier series expansion
 The coefficients ak, bk become generally smaller as k increases
 The base functions cosωkt, sinωkt have larger frequency ωk = kωT
and smaller period Τk = T/k (i.e. more detail) as k increases
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Characteristics of the Fourier series expansion
 The coefficients ak, bk become generally smaller as k increases
 The base functions cosωkt, sinωkt have larger frequency ωk = kωT
and smaller period Τk = T/k (i.e. more detail) as k increases
 The terms [ak cosωkt + bk sinωkt]
have a more detailed contribution to fN(t) a k increases
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Characteristics of the Fourier series expansion
 The coefficients ak, bk become generally smaller as k increases
 The base functions cosωkt, sinωkt have larger frequency ωk = kωT
and smaller period Τk = T/k (i.e. more detail) as k increases
 The terms [ak cosωkt + bk sinωkt]
have a more detailed contribution to fN(t) a k increases
 As Ν increases more details are added to the Fourier series expansion
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
Characteristics of the Fourier series expansion
 The coefficients ak, bk become generally smaller as k increases
 The base functions cosωkt, sinωkt have larger frequency ωk = kωT
and smaller period Τk = T/k (i.e. more detail) as k increases
 The terms [ak cosωkt + bk sinωkt]
have a more detailed contribution to fN(t) a k increases
 As Ν increases more details are added to the Fourier series expansion
 For a sufficient large Ν (which?) fN(t) ia a satisfactory approximation
to f(t) within a particular application
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The finite sum of the Fourier series expansion
as the best approximation of a function within an interval
Question : In an expansion with finite number of terms Ν, of the form
N
f N (t )  A0   [ Ak cos
k 1
2 kt
2 kt
 Bk sin
]
T
T
which are the values of the coefficients Α0, Ak, Bk for which
the sum fN(t) best approximates f(t), in the sense that
T
||  f (t ) ||  || f (t )  f N (t ) ||   [ f (t )  f N (t )]2 dt  min
2
2
0
A. Dermanis
A0 ,{ Ak , Bk }
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The finite sum of the Fourier series expansion
as the best approximation of a function within an interval
Question : In an expansion with finite number of terms Ν, of the form
N
f N (t )  A0   [ Ak cos
k 1
2 kt
2 kt
 Bk sin
]
T
T
which are the values of the coefficients Α0, Ak, Bk for which
the sum fN(t) best approximates f(t), in the sense that
T
||  f (t ) ||  || f (t )  f N (t ) ||   [ f (t )  f N (t )]2 dt  min
2
2
0
Answer :
A. Dermanis
A0 ,{ Ak , Bk }
The Fourier coefficients a0, ak, bk
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The finite sum of the Fourier series expansion
as the best approximation of a function within an interval

Question : What is the meaning of the symbol

in the Fourier series expansion?
k 1
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The finite sum of the Fourier series expansion
as the best approximation of a function within an interval

Question : What is the meaning of the symbol

in the Fourier series expansion?
k 1

ANSWER : The symbol

means that we can choose
k 1
a sufficiently large Ν, so that we can make satisfactorily small
the error
δf(t) = f(t) – fN(t)
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
The finite sum of the Fourier series expansion
as the best approximation of a function within an interval

Question : What is the meaning of the symbol

in the Fourier series expansion?
k 1

ANSWER : The symbol

means that we can choose
k 1
a sufficiently large Ν, so that we can make satisfactorily small
the error
δf(t) = f(t) – fN(t)
Specifically:
For every small ε there exists a corresponding integer Ν = Ν(ε) such that
||  f (t ) ||  || f (t )  f N (t ) || 
T
2
[
f
(
t
)

f
(
t
)]
dt  
N

0
small mean square error !
A. Dermanis
Signals and Spectral Methods in Geoinformatics
Aristotle University of Thessaloniki – Department of Geodesy and Surveying
END
A. Dermanis
Signals and Spectral Methods in Geoinformatics