Fast Fourier Transform

A bunch of smart people

Outline (hopefully)

• Finite discrete signals

• Linear shift-invariant filters

• Impulse responses

• Circular convolutions

• Discrete Fourier Transform

• The Fast Fourier Transform

Continuous signals

Finite signals

• Usually come from regular discretization

Finite signals

• Usually come from regular discretization

• Might as well think of them as vectors!

Discrete Fourier Transform

Complexity O(N2)

FFT is just an O(N log N) algorithm

Transformations on vectors

Linear (i.e., matrices)

Linear (i.e., matrices)

Shift invariant

(in a circular way)

Transformations on vectors

Closer look at shift invariance

Closer look at shift invariance

L is shift invariant iff

The canonic basis of RN

is the canonic basis for

plays the role of Dirac's delta or impulse

How does matrix L look like?

h is the impulse response of L

It captures all information about L!

How does matrix L look like?

How do we multiply L by v?

This is why we care about convolutions!

The circular convolution

Complexity O(N2)

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)

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!

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)])

Better convolution strategy

F-1v F-1LF F-1vO(N)

product

LvO(N2)

vconvolution

FFT IFFT

But first...

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!

The eigenvalues of P

Indeed, P has a full set!

The eigenvectors wk of P

The eigenvectors wk of P

Better convolution strategy

F-1v F-1LF F-1vO(N)

product

LvO(N2)

vconvolution

FFT IFFT

Not only does F exist. It does not depend on L!

How do F and F-1 look like?

We can also verify that...

How to compute F-1v?

Finally!!!

The eigenvalues of L

Finally!!!

Discrete Fourier Transform

FFT is just an O(N log N) algorithm

Better convolution strategy

F-1v F-1LF F-1vO(N)

product

LvO(N2)

vconvolution

FFT IFFT

Fast Fourier Transform

Fast Fourier Transform

• Coley and Tukey, 1965

• Gauss, 1805

Better convolution strategy

F-1v F-1LF F-1vO(N)

product

LvO(N2)

vconvolution

FFT IFFT

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)])