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Mikrokontroler dan Antar Muka (14)
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Mikrokontroler dan Antar Muka (14)

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  • 1. Pendahuluan
    BambangHeruIswanto, Dr.rer.nat
    Sesion #01
    JurusanFisika
    FakultasMatematikadanIlmuPengetahuanAlam
  • 2. Outline
    Transforms
    Fourier Transform
    Time Domain and Frequency Domain
    Discrete Fourier Transform
    Fast Fourier Transform
    Applications
    Applications: Frequency Domain In Images
    Other Applications of the DFT
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
    2
    04/01/2011
  • 3. 04/01/2011
    3
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 4. Transforms
    Transform:
    In mathematics, a function that results when a given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits. (Britannica)
    Can be thought of as a substitution
    04/01/2011
    4
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 5. Transforms
    Example of a substitution:
    Original equation: x + 4x² – 8 = 0
    Familiar form: ax² + bx + c = 0
    Let: y = x²
    Solve for y
    x = ±√y
    4
    04/01/2011
    5
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 6. Fourier Transform
    Property of transforms:
    They convert a function from one domain to another with no loss of information
    Fourier Transform:
    converts a function from the time (or spatial) domain to the frequency domain
    04/01/2011
    6
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 7. Time Domain and Frequency Domain
    Time Domain:
    Tells us how properties (air pressure in a sound function, for example) change over time:
    Amplitude = 100
    Frequency = number of cycles in one second = 200 Hz
    04/01/2011
    7
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 8. Time Domain and Frequency Domain
    Frequency domain:
    Tells us how properties (amplitudes) change over frequencies:
    04/01/2011
    8
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 9. Time Domain and Frequency Domain
    Example:
    Human ears do not hear wave-like oscilations, but constant tone
    Often it is easier to work in the frequency domain
    04/01/2011
    9
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 10. Time Domain and Frequency Domain
    In 1807, Jean Baptiste Joseph Fourier showed that any periodic signal could be represented by a series of sinusoidal functions
    In picture: the composition of the first two functions gives the bottom one
    04/01/2011
    10
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 11. Time Domain and Frequency Domain
    04/01/2011
    11
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 12. Fourier Transform
    Because of the
    property:
    Fourier Transform takes us to the frequency domain:
    04/01/2011
    12
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 13. Discrete Fourier Transform
    In practice, we often deal with discrete functions (digital signals, for example)
    Discrete version of the Fourier Transform is much more useful in computer science:
    O(n²) time complexity
    04/01/2011
    13
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 14. Fast Fourier Transform
    Many techniques introduced that reduce computing time to O(n log n)
    Most popular one: radix-2 decimation-in-time (DIT) FFT Cooley-Tukey algorithm:
    (Divide and conquer)
    04/01/2011
    14
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 15. Applications
    In image processing:
    Instead of time domain: spatial domain (normal image space)
    frequency domain: space in which each image value at image position F represents the amount that the intensity values in image I vary over a specific distance related to F
    04/01/2011
    15
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 16. Applications: Frequency Domain In Images
    If there is value 20 at the point that represents the frequency 0.1 (or 1 period every 10 pixels). This means that in the corresponding spatial domain image I the intensity values vary from dark to light and back to dark over a distance of 10 pixels, and that the contrast between the lightest and darkest is 40 gray levels
    04/01/2011
    16
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 17. Applications: Frequency Domain In Images
    Spatial frequency of an image refers to the rate at which the pixel intensities change
    In picture on right:
    High frequences:
    Near center
    Low frequences:
    Corners
    04/01/2011
    17
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 18. Other Applications of the DFT
    Signal analysis
    Sound filtering
    Data compression
    Partial differential equations
    Multiplication of large integers
    04/01/2011
    18
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  • 19. Percobaan DFT mengunakan Labview
    04/01/2011
    19
    © 2010 Universitas Negeri Jakarta | www.unj.ac.id |