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Signals and Spectral Methods in Geoinformatics: Discrete Fourier Transforms

This lecture discusses the application of discrete Fourier transforms in geoinformatics, highlighting the challenges in real-world data analysis and the impact of discrete data. It covers the relationship between functions and discrete values, Fourier series expansion, the Discrete Fourier Transform (DFT), the inverse DFT, aliasing, the Discrete Time Fourier Transform (DTFT), and the discrete convolution theorem.

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Signals and Spectral Methods in Geoinformatics: Discrete Fourier Transforms

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  1. SignalsandSpectral Methods in Geoinformatics Lecture 4: Discrete Fourier Transforms

  2. Spectral methods for discrete data or from mathematical “convenience” to the difficulties of real applications

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

  4. Discrete data in a finite interval

  5. Discrete data in a finite interval 0 1 2 3 4 5 Ν-1 Ν not taken into account ! If not, i.e. when (removal of linear trend)

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

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

  8. Discrete data in a finite interval

  9. Discrete data in a finite interval Set: where:

  10. Discrete data in a finite interval

  11. Discrete data in a finite interval

  12. Discrete data in a finite interval

  13. Discrete data in a finite interval

  14. Discrete data in a finite interval

  15. Discrete data in a finite interval

  16. Discrete data in a finite interval

  17. Discrete data in a finite interval

  18. Discrete data in a finite interval

  19. The DiscreteFourier Transform (DFT) = sum of terms with frequencies The coefficientFm corresponds tocmaffected by the coefficientscm+jΝ of all corresponding higher frequencies aliasing

  20. The DiscreteFourier Transform (DFT)

  21. The DiscreteFourier Transform (DFT) System of Ν equations with Ν unknowns and unique solution:

  22. The DiscreteFourier Transform (DFT)

  23. The DiscreteFourier Transform (DFT) DFT = Discrete Fourier Transform invDFT = Inverse Discrete Fourier Transform

  24. Proof of Forij : Fori = j :

  25. The DiscreteFourier Transform (DFT) DFT numbers frequencies inv-DFT

  26. Discrete data on an infinite interval

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

  28. Discrete data on an infinite interval

  29. Discrete data on an infinite interval

  30. Discrete data on an infinite interval

  31. Discrete data on an infinite interval

  32. Discrete data on an infinite interval

  33. Discrete data on an infinite interval

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

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

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

  37. Discrete data on an infinite interval Proof of theDTFT:

  38. Discrete data on an infinite interval Proof that theDTFT satisfies theinvDTFT (already defined) Proof of theDTFT:

  39. Discrete data on an infinite interval Usual simplification:

  40. Discrete data on an infinite interval Usual simplification: Notation:

  41. Discrete convolution definition: notation: property:

  42. Discrete convolution Mathematical mapping: The valuegn of the discrete functiong for any particular n follows by multiplying eachvaluefk of the discrete functionf with a factor (weight) hn-k which depends on the “distance”n-k between the particularn and the varyingk(-∞<k<+∞). Thus each valuegn of the functiong is a “weighted mean”of the values fk with weightshn-k defined by the functionhn.

  43. The discrete convolution theorem

  44. The discrete convolution theorem PROOF

  45. Discrete convolution

  46. Discrete convolution

  47. Discrete convolution

  48. Discrete convolution sum of “columns”

  49. Discrete convolution sum of “columns”

  50. Discrete convolution - Example

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