1. Signals and Spectral Methods
in Geoinformatics
Lecture 2:
Fourier Series

2. Development of a function
defined in an interval
into Fourier Series
Jean Baptiste Joseph Fourier

3. REPRESENTING A FUNCTION BY NUMBERS
f (t) t Τ
coefficients α1, α2, ...
of the function
f = a1φ1+ a2 φ
known base functions
φ1, φ2, ...
function f

4. The base functions of Fourier series+1 –1 Τ +1 +1 +1 +1 –1 –1 –1 –1 Τ Τ Τ Τ +1 +1 +1 +1 –1 –1 –1 –1 Τ Τ Τ Τ

5. Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:

6. Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:
Every base function has:
period:

7. Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:
Every base function has:
period:
frequency:

8. Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:
Every base function has:
period:
frequency:
angular frequency:

9. Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:
Every base function has:
period:
frequency:
angular frequency:
fundamental period
fundamental frequency
fundamental angular frequency

10. term angular frequencies
Development of a real function f(t) defined in the interval [0,T ] into Fourier series
3 alternative forms:
Every base function has:
period:
frequency:
angular frequency:
fundamental period
fundamental frequency
fundamental angular frequency
term periods
term frequencies
term angular frequencies

11. Development of a real function f(t) defined in the interval [0,T ] into Fourier series
simplest form:

12. Development of a real function f(t) defined in the interval [0,T ] into Fourier series
simplest form:
Fourier basis (base functions):

13. Development of a real function f(t) defined in the interval [0,T ] into Fourier series
simplest form:
Fourier basis (base functions):

14. An example for the development of a function+1 –1
f (x)
An example for the development of a function
in Fourier series
Separate analysis of each term for k = 0, 1, 2, 3, 4, …

15. +1 –1
f (x)
k = 0
base function +1 –1

16. +1 –1
f (x)
k = 0
contribution of term +1 –1

17. +1 –1
f (x)
k = 1
base functions +1 –1 +1 –1

18. +1 –1
f (x)
k = 1
contributions of term +1 –1 +1 –1

19. +1 –1
f (x)
k = 2
base functions +1 –1 +1 –1

20. +1 –1
f (x)
k = 2
contributions of term +1 –1 +1 –1

21. +1 –1
f (x)
k = 3
base functions +1 –1 +1 –1

22. +1 –1
f (x)
k = 3
contributions of term +1 –1 +1 –1

23. +1 –1
f (x)
k = 4
base functions +1 –1 +1 –1

24. +1 –1
f (x)
k = 4
contributions of term +1 –1 +1 –1

25. +1 –1
f (t)

26. Exploiting the idea of function othogonality
vector:
orthogonal vector basis
inner product:

27. Exploiting the idea of function othogonality
vector:
orthogonal vector basis
inner product:
Computation of vector components:

28. Exploiting the idea of function othogonality
vector:
orthogonal vector basis
inner product:
Computation of vector components:

29. Exploiting the idea of function othogonality
vector:
orthogonal vector basis
inner product:
Computation of vector components:

30. Exploiting the idea of function othogonality
vector:
orthogonal vector basis
inner product:
Computation of vector components:

31. Exploiting the idea of function othogonality
vector:
orthogonal vector basis
inner product:
Computation of vector components:

32. Orthogonality of Fourier base functions
Inner product of two functions:
Fourier basis:
Orthogonality relations (km):
Norm (length) of a function:

33. Computation of Fourier series coefficients
Ortjhogonality relations (km):

34. Computation of Fourier series coefficients
Ortjhogonality relations (km):

35. Computation of Fourier series coefficients
Ortjhogonality relations (km):
0 for km
0 for km

36. Computation of Fourier series coefficients
Ortjhogonality relations (km):
0 for km
0 for km

37. Computation of Fourier series coefficients
Ortjhogonality relations (km):
0 for km
0 for km

38. Computation of Fourier series coefficients
Ortjhogonality relations (km):
0 for km
0 for km

39. Computation of Fourier series coefficients
of a known function:

40. Computation of Fourier series coefficients

41. Computation of Fourier series coefficients
change of
notation

42. Computation of Fourier series coefficients
change of
notation

43. Computation of Fourier series coefficients

44. Computation of Fourier series coefficients

45. Alternative forms of Fourier series (polar forms)
Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk !
ρk = «length»
θk = «azimuth»
φk = «direction angle»
φk + θk = 90

46. Alternative forms of Fourier series (polar forms)
Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk !
ρk = «length»
θk = «azimuth»
φk = «direction angle»
φk + θk = 90

47. Alternative forms of Fourier series (polar forms)
Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk !
ρk = «length»
θk = «azimuth»
φk = «direction angle»
φk + θk = 90

48. Alternative forms of Fourier series (polar forms)

49. Alternative forms of Fourier series (polar forms)

50. Alternative forms of Fourier series (polar forms)
θk = phase (sin)

51. Alternative forms of Fourier series (polar forms)
θk = phase (sin)

52. Alternative forms of Fourier series (polar forms)
θk = phase (sin)
φk = phase (cosine)

53. Fourier series for a complex function
Fourier series of real functions:
«real» part
«imaginary» part

54. Fourier series for a complex function
Fourier series of real functions:
«real» part
«imaginary» part
setting

55. Fourier series for a complex function
Fourier series of real functions:
«real» part
«imaginary» part
setting

56. Fourier series for a complex function
Implementation of complex symbolism:

57. Fourier series for a complex function
Implementation of complex symbolism:

58. Fourier series for a complex function
Implementation of complex symbolism:

59. Fourier series for a complex function
Implementation of complex symbolism:

60. Fourier series for a complex function
Implementation of complex symbolism:

61. Complex form of Fourier series
Development of a complex function
into a Fourier series
with complex base functions
and complex coefficients
Computation of complex coefficients
for a known complex function

62. Ortjhogonality of the complex basis
Conjugateς z* of a complex number z :
inner product:

63. Ortjhogonality of the complex basis
Conjugateς z* of a complex number z :
inner product:

64. Ortjhogonality of the complex basis
Conjugateς z* of a complex number z :
inner product:

65. Ortjhogonality of the complex basis
Conjugateς z* of a complex number z :
inner product:

66. Ortjhogonality of the complex basis
Conjugateς z* of a complex number z :
inner product:

67. Fourier series of a real function using complex notation
Implementation of complex symbolism:

68. Fourier series of a real function using complex notation
Implementation of complex symbolism:

69. Fourier series of a real function using complex notation
Implementation of complex symbolism:

70. Fourier series of a real function using complex notation
Implementation of complex symbolism:

71. Fourier series of a real function using complex notation
Implementation of complex symbolism:

72. Fourier series of a real function using complex notation

73. Fourier series of a real function using complex notation
f (t) = real function

74. Fourier series of a real function using complex notation
f (t) = real function

75. Fourier series of a real function using complex notation
f (t) = real function

76. Fourier series of a real function using complex notation
f (t) = real function

77. Fourier series of a real function using complex notation
f (t) = real function
The imaginary part disappears !

78. Fourier series of a real function: Real and complex form

79. Fourier series of a real function: Real and complex form

80. Fourier series of a real function: Real and complex form

81. Fourier series of a real function: Real and complex form

82. Fourier series of a real function: Real and complex form

83. Extension of the function f (t) outside the interval [0, T ]2T 3T –T
–2T
The extension is a periodic function, with period Τ
for every integer n
CAUSE OF USUAL MISCONCEPTION: “Fourier series expansion
deals with periodic functions»

84. Fourier series on the circle
(naturally periodic domain) θ
(angle)

85. Fourier series on the circle
(naturally periodic domain) θ
(angle)

86. Fourier series on the circle
(naturally periodic domain) θ
(angle)

87. Fourier series on the plane
Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)
Base functions: Ty Tx

88. Fourier series on the plane
Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)
Base functions:
(angular frequencies along x and y ) Ty Tx

89. Fourier series on the plane
Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)
Base functions:
(angular frequencies along x and y ) Ty Tx

90. Fourier series on the plane
Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)
Base functions:
(angular frequencies along x and y ) Ty Tx

91. Fourier series on the plane
Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)

92. Fourier series on the plane
Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)
Equivalent to double Fourier series: First along x and then along y (or vice-versa)

93. Fourier series on the plane
Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)
Equivalent to double Fourier series: First along x and then along y (or vice-versa)

94. Fourier series on the plane
Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)
Equivalent to double Fourier series: First along x and then along y (or vice-versa)

95. Fourier series on the plane
Inner product:
for every Α = a,b,c,d
and B = a,b,c,d
Orthogonal Fourier basis ! ή

96. Fourier series on the plane
Inner product:
for every Α = a,b,c,d
and B = a,b,c,d
Orthogonal Fourier basis ! ή
Computation of coefficients:

97. Fourier series on the plane
Complex form:

98. Fourier series on the plane
Complex form:
Fourier series in n dimensions

99. Fourier series in n dimensions

100. Fourier series in n dimensions
In matrix notation:
domain of definition:
(orthogonal hyper-parallelepiped)
(parallelepiped volume)

101. Fourier series in n dimensions
In matrix notation:
domain of definition:
(orthogonal hyper-parallelepiped)
(parallelepiped volume)

102. Fourier series on any interval [Α, Β]

103. Approximating a function by a finite Fourier series expansion
Question : What is the meaning of the symbol in the Fourier series expansion?

104. Approximating a function by a finite Fourier series expansion
Question : What is the meaning of the symbol in the Fourier series expansion?
Certainly not that infinite terms must be summed! This is impossible!

105. Approximating a function by a finite Fourier series expansion
Question : What is the meaning of the symbol in the Fourier series expansion?
Certainly not that infinite terms must be summed! This is impossible!
In practice we can use only
a finite sum
with a «sufficiently large» integer Ν

106. 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

107. 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 !!!

108. 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 in the interval [0,Τ]

109. Characteristics of the Fourier series expansion
 The coefficients ak, bk become generally smaller as k increases

110. 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

111. 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

112. 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

113. 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

114. as the best approximation of a function within an interval
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
which are the values of the coefficients Α0, Ak, Bk for which
the sum fN(t) best approximates f(t), in the sense that

115. as the best approximation of a function within an interval
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
which are the values of the coefficients Α0, Ak, Bk for which
the sum fN(t) best approximates f(t), in the sense that
Answer : The Fourier coefficients a0, ak, bk

116. as the best approximation of a function within an interval
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?

117. as the best approximation of a function within an interval
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?
ANSWER : The symbol means that we can choose
a sufficiently large Ν, so that we can make satisfactorily small
the error
δf(t) = f(t) – fN(t)

118. as the best approximation of a function within an interval
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?
ANSWER : The symbol means that we can choose
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
small mean square error !

119. END