Permutation and Circuland Matrices and the Fast Fourier Transform

Abstract

This paper provides a description of the Fast Fourier Transform and its connection with the circulant and permutation matrices. It is written for the case where the number of discrete time samples is equal to the number of discrete frequency samples but is otherwise not restricted. The paper demonstrates that since the modal matrix of a permutation matrix contains only one bit of information, the evaluation of the discrete Fourier Transform involves considerably fewer than N² multiplications where N is the number of samples involved and is also the order of the matrices involved.