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

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

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

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

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ν-1Ν not taken into account ! If not, i.e. when Discrete data in a finite interval (removal of linear trend)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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)

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)

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 :

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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”

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

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”

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”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

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:

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:

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

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:

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:

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:

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

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:

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:

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:

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

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)

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)

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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