Fast Fourier Transform: the most important numerical algorithm of our lifetime

Transcription

1 VU Signal and Image Processing Fast Fourier Transform: the most important numerical algorithm of our lifetime Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/17s/ Raphael 1 Sahann

2 Overview More 2D Fourier Transform Intuition Fast Fourier Transform motivation how it works trivia Fourier Transform in Biology? 2

3 Frequency Spectrum 3

4 Frequency Spectrum 4

5 Frequency Spectrum angle: 16.7 degrees 5

6 Overview More 2D Fourier Transform Intuition Fast Fourier Transform motivation how it works trivia Fourier Transform in Biology? 6

7 Usage FFT is used in almost every electronic device Telecommunication is only possible because of FFT All media applications A huge number of scientific applications Physics: partial differential equations Meteorology: weather simulations 7

8 Discrete Fourier Transform - DFT DFT takes one period of DTFS Analysis Synthesis Warning: periodicity is not eliminated. Evaluating outside the intervals will not give zero. 26 Hrvoje Bogunovic, Raphael Sahann 8

9 DFT vs FFT DFT: more general: 9

10 DFT vs FFT general form of the DFT: with entries: 10

11 DFT vs FFT Consider 1s of audio signal, sampled at 44.1 khz Matrix F has dimensions NxN, which is * = billion floating point operations order of O(n 2 ) FFT can do the same calculation in O(n log(n)) * = /4 million floating point operations 11

12 DFT vs FFT is identical to evens odds 12

13 DFT vs FFT F4 is close to F2 * 13

14 DFT vs FFT in general: 14

15 DFT vs FFT in general: 15

16 DFT vs FFT For this example FFT is 200 times faster than regular DFT 16

17 Fast Fourier Transform The speedup for powers of 2 using the FFT is so significant that the fastest way for signals that do not fit into a power of 2 is to pad them with zeroes until their size is at the next power of 2 17

18 Fast Fourier Transform The first version dates back to 1805 when J.C.F. Gauss used a similar algorithm to interpolate the orbits of Pallas and Juno from sample observations Cooley and Tukey are attributed with publishing the general FFT in 1965 The idea was originally used by Tukey (Princeton) to analyse sensor data from sensors placed along the borders of the USSR to detect nuclear tests 18

19 Fast Fourier Transform Soon after it was discovered that the algorithm had a wide range of applications and Cooley (IBM) implemented it Since Tukey did not work at IBM the patentability of the algorithm was doubted and it went into public domain 19

20 Fast Fourier Transform FFT was included in the Top 10 algorithms of the 20th century by IEEE journal Computing in Science and Engineering Gilbert Strang called it the most important numerical algorithm of our lifetime 20

21 Overview More 2D Fourier Transform Intuition Fast Fourier Transform motivation how it works trivia Fourier Transform in Biology? 21

22 An Historic Fourier Transform Taken by Rosalind Franklin, this image sparked Watson and Crick s insight into the double helix. Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): (c) Prof. Dennis Freeman, MIT,

23 An Historic Fourier Transform This is an x-ray crystallographic image of DNA, and it shows the Fourier transform of the structure of DNA. Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): (c) Prof. Dennis Freeman, MIT,

24 An Historic Fourier Transform High-frequency bands indicate repeating structure of base pairs. b 1/b Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): (c) Prof. Dennis Freeman, MIT,

25 An Historic Fourier Transform Low-frequency bands indicate a lower frequency repeating structure. h 1/h Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): (c) Prof. Dennis Freeman, MIT,

26 An Historic Fourier Transform Tilt of low-frequency bands indicates tilt of low-frequency repeating structure: the double helix! Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): (c) Prof. Dennis Freeman, MIT,

27 Simulation Easy to calculate relation between structure and Fourier transform. 42 Prof. Dennis Freeman, MIT,