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The Fast Fourier Transform (FFT)

Module by: Justin Romberg. E-mail the author

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Summary: This module describes the fast Fourier Transform (FFT).

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Introduction

The Fast Fourier Transform (FFT) is an efficient O(NlogN) algorithm for calculating DFTs

  • originally discovered by Gauss in the early 1800's
  • rediscovered by Cooley and Tukey at IBM in the 1960's
  • C.S. Burrus, Rice University's very own Dean of Engineering, literally "wrote the book" on fast DFT algorithms.
The FFT exploits symmetries in the WW matrix to take a "divide and conquer" approach. We won't talk about the actual FFT algorithm here, see these notes if you are interested in reading a little more on the idea behind FFT.

Speed Comparison

How much better is O(NlogN) than O( N2 N 2 )?

Figure 1: This figure shows how much slower the computation time of an O(NlogN) process grows.
Figure 1 (fft_f1.png)
Table 1
NN 1010 100100 10001000 106106 109109
N2N2 100 104104 106106 10121012 10181018
NlogNNN 11 200200 30003000 6×1066106 9×1099109

Say you have a 1 MFLOP machine (a million "floating point" operations per second). Let N=1million=106 N 1 million 10 6 .

An O( N2 N 2 ) algorithm takes 10121012 flors → 106 10 6 seconds ≃ 11.5 days.

An O( NlogN N N ) algorithm takes 6×106 6 10 6 Flors → 6 seconds.

Note:

N=1million N 1 million is not unreasonable.

Example 1

3 megapixel digital camera spits out 3×106 3 10 6 numbers for each picture. So for two NN point sequences fnfn and hnhn. If computing fn⊛hn ⊛ f n h n directly: O( N2 N 2 ) operations.

taking FFTs -- O(NlogN)

multiplying FFTs -- O(N)

inverse FFTs -- O(NlogN).

the total complexity is O(NlogN).

Note:

FFT + digital computer were almost completely responsible for the "explosion" of DSP in the 60's.

Note:

Rice was (and still is) one of the places to do research in DSP.

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