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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 4: Discrete Fourier Transforms 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 Spectral methods for discrete data or from mathematical “convenience” to the difficulties of real applications

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 CONTINUOUS DATA FOR ALL VALUES DISCRETE DATA IN A FINITE INTERVAL Fourier transform Fourier series The only realistic case for real applications

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 Discrete data in a finite interval

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 012345Ν-1Ν not taken into account ! If not, i.e. when Discrete data in a finite interval (removal of linear trend)

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 Relation between function and discrete values Fourier series expansion in the interval [0, Τ]: Discrete data in a finite interval

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 The known discrete values impose restrictions on the possible values of the Fourier series coefficients, since they must satisfy the following N conditions Relation between function and discrete values Fourier series expansion in the interval [0, Τ]: Discrete data in a finite interval

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 Discrete data in a finite interval

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 Set: where: Discrete data in a finite interval

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 Discrete data in a finite interval

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 Discrete data in a finite interval

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 Discrete data in a finite interval

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 Discrete data in a finite interval

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 Discrete data in a finite interval

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 Discrete data in a finite interval

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 Discrete data in a finite interval

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 Discrete data in a finite interval

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 Discrete data in a finite interval

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 The Discrete Fourier Transform (DFT) = sum of terms with frequencies The coefficient F m corresponds to c m affected by the coefficients c m+jΝ of all corresponding higher frequencies aliasing

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 The Discrete Fourier Transform (DFT)

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 System of Ν equations with Ν unknowns and unique solution: The Discrete Fourier Transform (DFT)

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 Discrete Fourier Transform (DFT)

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 DFT = Discrete Fourier Transform invDFT = Inverse Discrete Fourier Transform The Discrete Fourier Transform (DFT)

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 Proof of For i = j : For i  j :

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 DFT inv-DFT numbers frequencies The Discrete Fourier Transform (DFT)

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 Discrete data on an infinite interval

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 Discrete data on an infinite interval Unknown function Known discrete values 1 23 11 22 33 0

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 Discrete data on an infinite interval

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 Discrete data on an infinite interval

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 Discrete data on an infinite interval

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 Discrete data on an infinite interval

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 Discrete data on an infinite interval

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 Discrete data on an infinite interval

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 aliasing For frequencies  ω Ν /2 < ω < ω Ν /2, smaller in absolute value than the Nyquist value ω Ν the discrete spectrum F Δt (ω) differs from the corresponding continuous spectrum F(ω), due to the superimposition of the spectra of all “higher” frequencies F(ω  kωΝ) outside the interval  ω Ν /2 < ω < ω Ν /2 (aliasing) Discrete data on an infinite interval

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 (relation with the continuous Fourier transform) Inverse Discrete-Time Fourier transform (invDTFT) Discrete data on an infinite interval

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 Discrete Time Fourier Transform (DTFT ) (relation with the continuous Fourier transform) Inverse Discrete-Time Fourier transform (invDTFT) Discrete data on an infinite interval

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 Proof of the DTFT: Discrete data on an infinite interval

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 Proof that the DTFT satisfies the invDTFT (already defined) Proof of the DTFT: Discrete data on an infinite interval

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 Usual simplification: Discrete data on an infinite interval

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 Notation: Usual simplification: Discrete data on an infinite interval

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 Discrete convolution property: definition: notation:

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 Mathematical mapping: The value g n of the discrete function g for any particular n follows by multiplying each value f k of the discrete function f with a factor (weight) h n-k which depends on the “distance” n-k between the particular n and the varying k (-∞<k<+∞ ). Thus each value g n of the function g is a “weighted mean” of the values f k with weights h n-k defined by the function h n. Discrete convolution

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 The discrete convolution theorem

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 PROOF The discrete convolution theorem

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 Discrete convolution

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 Discrete convolution

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 Discrete convolution

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 sum of “columns” Discrete convolution

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 Discrete convolution sum of “columns”

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 Discrete convolution - Example

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 Discrete convolution - Example sum of “columns”

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 Discrete convolution - Example sum of “columns”

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 Discrete convolution - Example

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 Discrete convolution - Example

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 Discrete convolution - Example

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 Discrete convolution - Example

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 Discrete convolution - Example

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 Discrete convolution - Example

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 Discrete convolution - Example

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 Discrete convolution - Example

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 Discrete convolution - Example

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 CONTINUOUS DATA INFINITE DATA DISCRETE DATA FINITE DATA Fourier Transform FT Fourier Series FS Discrete Fourier Transform DFT ( also DΤFS ) Discrete-Time Fourier Transform DTFT Discrete-Time Fourier Series

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 CONTINUOUS DATA INFINITE DATA DISCRETE DATA FINITE DATA Fourier Transform FT Fourier Series FS Discrete Fourier Transform DFT Discrete-Time Fourier Transform DTFT REALISTIC CASE

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 CONTINUOUS DATA INFINITE DATA DISCRETE DATA FINITE DATA Fourier Transform FT Fourier Series FS Discrete Fourier Transform DFT Discrete-Time Fourier Transform DTFT REALISTIC CASE CONVOLUTION THEOREM

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 Implementation of the convolution theorem to the realistic case (discrete data in a finite interval) Ν1Ν1 012345

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 DFT: invDFT: Discrete data in a finite interval

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 DFT: invDFT: 1 Discrete data in a finite interval Computation of values outside the interval 0  k  N  1

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 DFT: invDFT: Discrete data in a finite interval Computation of values outside the interval 0  k  N  1 Periodic reproduction of the values within the interval 0  k  N  1

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 DFT: invDFT: Computation of values outside the interval 0  k  N  1 Periodic reproduction of the values within the interval 0  k  N  1 Discrete data in a finite interval

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 DFT: invDFT: Discrete data in a finite interval Computation of values outside the interval 0  k  N  1 Periodic reproduction of the values within the interval 0  k  N  1

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 DFT: invDFT: DΤFT: invDΤFT: periodic extension known values Discrete data in a finite interval

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 DFT: DΤFT: periodic extension known values Discrete data in a finite interval

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 DFT: DΤFT: periodic extension known values Discrete data in a finite interval

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 Implementation of the discrete convolution theorem periodic extension discrete convolution theorem

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 discrete convolution theorem DFTinvDFT Implementation of the discrete convolution theorem

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 DFTinvDFT periodic extension ATTENTION: Convolution is applied on the periodic extensions and not on the original values MEANING ? Implementation of the discrete convolution theorem discrete convolution theorem

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 Direct definition of - Special case: Computation for 0  n  N-1: Συνήθως Implementation of the discrete convolution theorem

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 3 cases: Implementation of the discrete convolution theorem Computation for 0  n  N  1:

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 values needed outside the interval [ 0, Ν  1 ] Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1:

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 Replacement with same values for n  k+N Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1:

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 DISADVANTAGE: g n is influenced by “distant” values of f n Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1: Replacement with same values for n  k+N

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 values needed only within the interval [ 0, Ν  1 ] Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1:

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 values needed outside the interval [ 0, Ν  1 ] Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1:

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 Replacement with same values for n  k  N Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1:

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 Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1: Replacement with same values for n  k  N DISADVANTAGE: g n is influenced by “distant” values of f n

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 DISADVANTAGE: g n is influenced by “distant” values of f n Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1:

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 NO PROBLEM Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1:

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 DISADVANTAGE: g n is influenced by “distant” values of f n Implementation of the discrete convolution theorem 3 cases: Computation for 0  n  N  1:

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 Zero padding (A way to avoid the influence of distant values) Assignment of Κ zeros before and after Νέο 0 και Ν  1 The distant values of f n which affect g n are now zero ! ZERO CONTRIBUTION

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 Zero padding (A way to avoid the influence of distant values)

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 The “distant” values of f n are now zero ZERO CONTRIBUTION Zero padding (A way to avoid the influence of distant values)

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 The “distant” values of f n are now zero ZERO CONTRIBUTION Zero padding (A way to avoid the influence of distant values)

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 ZERO PADDING BUT... DFT discrete convolution theorem Zero padding (A way to avoid the influence of distant values)

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 ZERO PADDING BUT... DFT discrete convolution theorem Zero padding (A way to avoid the influence of distant values) DIFFERENT COEFFICIENTS

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 ZERO PADDING BUT... DFT discrete convolution theorem Zero padding (A way to avoid the influence of distant values) ADDITIONAL COEFFICIENTS

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 The realistic approach Computation of the valuesonly for

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 Computation of the valuesonly for The realistic approach

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 END

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