Fast%20Fourier%20Transform

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Fast Fourier Transform

• A bunch of smart people

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Outline (hopefully)

• Finite discrete signals
• Linear shift-invariant filters
• Impulse responses
• Circular convolutions
• Discrete Fourier Transform
• The Fast Fourier Transform

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Continuous signals
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Finite signals

• Usually come from regular discretization

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Finite signals

• Usually come from regular discretization
• Might as well think of them as vectors!

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Discrete Fourier Transform
Complexity O(N2)
FFT is just an O(N log N) algorithm
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Transformations on vectors
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Transformations on vectors
Linear (i.e., matrices)
Shift invariant
(in a circular way)
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Closer look at shift invariance
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Closer look at shift invariance
L is shift invariant iff
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The canonic basis of RN
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How does matrix L look like?
h is the impulse response of L
It captures all information about L!
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How does matrix L look like?
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How do we multiply L by v?
This is why we care about convolutions!
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The circular convolution
Complexity O(N2)
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What we have so far

• We discretized a continuous function turning it
  into a vector in RN
• We defined a class of transformations from RN to
  RN that we care about
• Each linear shift-invariant transformations L can
  be written as a circular convolution
• The convolution is with the impulse response h of
  the transformation L
• We can compute it in O(N2)

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This is too abstract!

• Want to make a recording of your voice sound as
  if you were inside a bathroom?
• Your bathroom transform sound in a linear, shift
  invariant way.
• Record the sound of a clap in the bathroom.
• Record your voice outside.
• Convolve the two signals!

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Easy to do in Matlab

• a wavread('bathroom-ip.wav')
• b wavread('laugh.wav')
• a a / max(abs(a))
• b b / max(abs(b))
• c conv(a, b)
• c c / max(abs(c))
• p audioplayer(c, 44000)
• play(p, 1 length(c))

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Better convolution strategy
O(N2)
Lv
v
convolution
FFT
IFFT
F-1LF F-1v
O(N)
F-1v
product
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Diagonalizing L

• Need a basis of eigenvectors for L. Trick is to
  look at P first!
• Assume for now that P has distinct eigenvalues.
• If wi is an eigenvector of P, then it is also an
  eigenvector of L!

• Lwi is an eigenvector of P, with the same
  eigenvalue
• We must have

• wi is also eigenvector of L (for some other
  eigenvalue)
• Since P has a full set of wi, L is diagonalized
  by the same wi!

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The eigenvalues of P
Indeed, P has a full set!
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The eigenvectors wk of P
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The eigenvectors wk of P
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Better convolution strategy
O(N2)
Lv
v
convolution
FFT
IFFT
F-1LF F-1v
O(N)
F-1v
product
Not only does F exist. It does not depend on L!
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How do F and F-1 look like?
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How to compute F-1v?
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The eigenvalues of L
Finally!!!
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Discrete Fourier Transform
FFT is just an O(N log N) algorithm
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Better convolution strategy
O(N2)
Lv
v
convolution
FFT
IFFT
F-1LF F-1v
O(N)
F-1v
product
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Fast Fourier Transform
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Fast Fourier Transform

• Coley and Tukey, 1965

• Gauss, 1805

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Better convolution strategy
O(N2)
Lv
v
convolution
FFT
IFFT
F-1LF F-1v
O(N)
F-1v
product
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Easy to do in Matlab

• a wavread('bathroom-ip.wav')
• b wavread('laugh.wav')
• a a / max(abs(a))
• b b / max(abs(b))
• c fftfilt(a, b)
• c c / max(abs(c))
• p audioplayer(c, 44000)
• play(p, 1 length(c))