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The Fast Fourier Transform (FFT)

Module by: Justin Romberg

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

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)
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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