Demonstrate How FFT Works In Matrix Form

The discrete Fourier transform of a vector v is equivalent to fft_matrix_1.gif, where F is the matrix defined as follows:

fft_matrix_2.gif

fft_matrix_3.gif

fft_matrix_4.gif

fft_matrix_5.gif

F can be decomposed into log n+1 sparse matrics. 
That’s why FFT can be done in O(n log n) computational time.

fft_matrix_6.gif

fft_matrix_7.gif

Spikey Created with Wolfram Mathematica 8.0