Mikrokontroler dan Antar Muka (14)

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Mikrokontroler dan Antar Muka (14)

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

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