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Fourier Transformation

Fourier Transformation. Fourier Transformasjon. f(x). F( u ). Continuous Fourier Transform Def. The Fourier transform of a one-dimentional function f(x). The Inverse Fourier Transform. Continuous Fourier Transform Def - Notation. The Fourier transform

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Fourier Transformation

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  1. Fourier Transformation Fourier Transformasjon f(x) F(u)

  2. Continuous Fourier TransformDef The Fourier transform of a one-dimentional function f(x) The Inverse Fourier Transform

  3. Continuous Fourier TransformDef - Notation The Fourier transform of a one-dimentional function f(x) The inverse Fourier Transform of F(u)

  4. Continuous Fourier TransformAlternative Def

  5. Continuous Fourier TransformExample - cos(2ft)

  6. Continuous Fourier TransformExample - cos(t)

  7. Continuous Fourier TransformExample - sin(t)

  8. Continuous Fourier TransformExample - Delta-function

  9. Continuous Fourier TransformExample - Gauss function

  10. Signals and Fourier TransformFrequency Information FT FT FT

  11. Stationary / Non-stationary signals Stationary FT Non stationary FT The stationary and the non-stationary signal both have the same FT. FT is not suitable to take care of non-stationary signals to give information about time.

  12. Transient SignalFrequency Information Constant function in [-3,3]. Dominating frequency  = 0 and some freequency because of edges. Transient signal resulting in extra frequencies > 0. Narrower transient signal resulting in extra higher frequencies pushed away from origin.

  13. Transient SignalNo Information about Position Moving the transient part of the signal to a new position does not result in any change in the transformed signal. Conclusion: The Fourier transformation contains information of a transient part of a signal, but only the frequency not the position.

  14. Inverse Fourier Transform [1/3] Theorem: Proof:

  15. Inverse Fourier Transform [2/3] Theorem: Proof:

  16. Inverse Fourier Transform [3/3]

  17. Properties

  18. Fourier Transforms of Harmonic and Constant Function

  19. Fourier Transforms of Some Common Functions

  20. Even and Odd Functions [1/3] Def Every function can be split in an even and an odd part Every function can be split in an even and an odd part and each of this can in turn be split in a real and an imaginary part

  21. Even and Odd Functions [2/3] 1. Even component in f produces an even component in F 2. Odd component in f produces an odd component in F 3. Odd component in f produces an coefficient -j

  22. Even and Odd Functions [3/3]

  23. The Shift Theorem

  24. The Similarity Theorem

  25. The Convolution Theorem

  26. ConvolutionEdge detection

  27. The Adjoint of the Fourier Transform Theorem: Suppose f and g er are square integrable. Then: Proof:

  28. Plancherel Formel - The Parselval’s Theorem Theorem: Suppose f and g are square integrable. Then: Proof:

  29. The Rayleigh’s TheoremConservation of Energy The energy of a signal in the time domain is the same as the energy in the frequency domain

  30. The Fourier Series Expansionu a discrete variable - Forward transform Suppose f(t) is a transient function that is zero outside the interval [-T/2,T/2] or is considered to be one cycle of a periodic function. We can obtain a sequence of coefficients by making a discrete variable and integrating only over the interval.

  31. The Fourier Series Expansionu a discrete variable - Inverse transform The inverse transform becomes:

  32. The Fourier Series Expansioncn coefficients

  33. The Fourier Series Expansionzn, an, bn coefficients

  34. The Fourier Series Expansionan,bn coefficients

  35. Fourier SeriesPulse train Pulse train approximated by Fourier Serie N = 1 N = 2 N = 5 N = 10

  36. Fourier SeriesPulse train – Java program

  37. Pulse Train approximated by Fourier Serie f(x) square wave (T=2) N=1 N=2 N=10

  38. Fourier SeriesZig tag Zig tag approximated by Fourier Serie N = 1 N = 2 N = 5 N = 10

  39. Fourier SeriesNegative sinus function Negative sinus function approximated by Fourier Serie N = 1 N = 2 N = 5 N = 10

  40. Fourier SeriesTruncated sinus function Truncated sinus function approximated by Fourier Serie N = 1 N = 2 N = 5 N = 10

  41. Fourier SeriesLine Line approximated by Fourier Serie N = 1 N = 2 N = 5 N = 10 N = 50

  42. Fourier SeriesJava program for approximating Fourier coefficients Approximate functions by adjusting Fourier coefficients (Java program)

  43. The Discrete Fourier Transform - DFTDiscrete Fourier Transform - Discretize both time and frequency Continuous Fourier transform Discrete frequency Fourier Serie Discrete frequency and time Discrete Fourier Transform

  44. The Discrete Fourier Transform - DFTDiscrete Fourier Transform - Discretize both time and frequency { fi } sequence of length N, taking samples of a continuous function at equal intervals

  45. Continuous Fourier Transform in two DimensionsDef The Fourier transform of a two-dimentional function f(x,y) The Inverse Fourier Transform

  46. The Two-Dimensional DFT and Its Inverse

  47. Fourier Transform in Two DimensionsExample 1

  48. Fourier Transform in Two DimensionsExample 2

  49. End

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