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Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals.

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Presentation on theme: "Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals."— Presentation transcript:

1 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Lecture 3: Fourier Transform Signals and Spectral Methods in Geoinformatics

2 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform and inverse Fourier transform direct inverse

3 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics from the number domain to the frequency domain Fourier transform and inverse Fourier transform direct inverse

4 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform and inverse Fourier transform from the frequency domain to the number domain direct inverse

5 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series in the interval [ 0, Τ ] Fourier transform in the interval (- ,+  ) inverse direct

6 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics From Fourier series to Fourier transform

7 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics From Fourier series to Fourier transform Change of notation

8 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics From Fourier series to Fourier transform Change of notation

9 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics From Fourier series to Fourier transform Change of notation

10 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics From Fourier series to Fourier transform

11 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics From Fourier series to Fourier transform

12 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics From Fourier series to Fourier transform

13 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics From Fourier series to Fourier transform

14 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics inverse direct From Fourier series to Fourier transform

15 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series in a continuously increasing interval Τ  ∞

16 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series in a continuously increasing interval Τ  ∞

17 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series in a continuously increasing interval Τ  ∞

18 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series in a continuously increasing interval Τ  ∞

19 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series in a continuously increasing interval Τ  ∞

20 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 3 Fourier series in a continuously increasing interval Τ  ∞

21 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier series in a continuously increasing interval Τ  ∞

22 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The Fourier series expansion of a function in a continuously larger interval Τ, provides coefficients for continuously denser frequencies ω k. As the length of the interval Τ tends to infinity, the frequencies ω k 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 discrete frequencies ω k wit step Δω = 2π / Τ continuous frequencies - all possible values (      ) Fourier series in a continuously increasing interval Τ  ∞

23 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform of a complex function

24 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform of a complex function

25 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform of a complex function

26 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform of a complex function

27 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform of a complex function

28 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics complex form real form Fourier transform of a complex function

29 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Notation Usual (mathematically incorect) notation directinverse Fourier transform of a complex function

30 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform of a real function Complex function:

31 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Real function: Fourier transform of a real function Complex function:

32 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Real function: Fourier transform of a real function Complex function:

33 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics cosine transform sine transform Real function: Fourier transform of a real function Complex function:

34 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics cosine transform sine transform Real function: Fourier transform of a real function Complex function:

35 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics even function odd function Fourier transform of a real function

36 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Fourier transform in polar form |F(ω)||F(ω)|

37 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics amplitude spectrum phase spectrum polar form: Fourier transform in polar form

38 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics even function odd function evenodd amplitude spectrum phase spectrum Fourier transform in polar form

39 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linearity Properties of the Fourier transform

40 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linearity Symmetry Properties of the Fourier transform

41 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linearity Symmetry Time translation Properties of the Fourier transform

42 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linearity Symmetry Time translation Properties of the Fourier transform

43 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linearity Symmetry Time translation Properties of the Fourier transform

44 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Properties of the Fourier transform Phase translation

45 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Phase translation Modulation theorem Properties of the Fourier transform

46 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem Proof: Properties of the Fourier transform

47 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem Proof: Properties of the Fourier transform

48 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem Proof: Properties of the Fourier transform

49 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem Proof: Properties of the Fourier transform

50 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem Proof: Properties of the Fourier transform

51 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem Proof: Properties of the Fourier transform

52 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem Proof: Properties of the Fourier transform

53 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem signal carrier frequency Properties of the Fourier transform

54 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Modulation theorem signal modulated signal carrier frequency Properties of the Fourier transform

55 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (amplitude) spectrum of signal (amplitude) spectrum of modulated signal Properties of the Fourier transform Modulation theorem

56 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (amplitude) spectrum of signal (amplitude) spectrum of modulated signal Properties of the Fourier transform Modulation theorem

57 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (amplitude) spectrum of signal (amplitude) spectrum of modulated signal Properties of the Fourier transform Modulation theorem

58 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (amplitude) spectrum of signal (amplitude) spectrum of modulated signal Properties of the Fourier transform Modulation theorem

59 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Change of time scale: Properties of the Fourier transform

60 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Change of time scale: Differentiation theorem with respect to time: Properties of the Fourier transform

61 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Change of time scale: Differentiation theorem with respect to time: Properties of the Fourier transform

62 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Change of time scale: Differentiation theorem with respect to time: Differentiation theorem with respect to frequency: Properties of the Fourier transform

63 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The Dirac delta function δ(t)

64 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics area =1 The Dirac delta function δ(t)

65 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics area =1 The Dirac delta function δ(t)

66 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics area =1 The Dirac delta function δ(t)

67 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics area =1 The Dirac delta function δ(t)

68 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The Dirac delta function δ(t)

69 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics = average value of φ in the interval The Dirac delta function δ(t)

70 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics = average value of φ in the interval The Dirac delta function δ(t)

71 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The Dirac delta function δ(t)

72 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 1 The Dirac delta function δ(t)

73 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 1 The Dirac delta function δ(t)

74 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Heaviside step functionDirac delta function 1 1 The Dirac delta function δ(t)

75 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Properties of the Fourier transform involving the Dirac delta function δ(t)

76 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Properties of the Fourier transform involving the Dirac delta function δ(t)

77 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution definition: notation:

78 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution definition: notation:

79 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution property: definition: notation:

80 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Mathematical mapping: Convolution

81 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Mathematical mapping: 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 (-∞

82 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Mathematical mapping: 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 (-∞

83 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution

84 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution

85 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution

86 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution

87 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution

88 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics area Convolution

89 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution area

90 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example area

91 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

92 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

93 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

94 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

95 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

96 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

97 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

98 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

99 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution - Example

100 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution properties

101 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Convolution properties

102 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Convolution is replaced by a simple multiplication in the frequency domain !

103 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Proof:

104 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Proof: The convolution theorem

105 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Proof: The convolution theorem

106 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Proof: The convolution theorem

107 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Proof: The convolution theorem

108 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Proof: The convolution theorem

109 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics CONVOLUTION THEOREM for frequencies

110 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics CONVOLUTION THEOREM for frequencies

111 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics PARSEVAL THEOREM

112 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Similar relation for Fourier series PARSEVAL THEOREM

113 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Comparison with Similar relation for Fourier series PARSEVAL THEOREM

114 Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics END


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