A two-stage representation of DFT and its applications

A two-stage representation in terms of preprocessing and postprocessing of DFT is developed by vector transformation of sines and cosines into new basis functions using Möbius inversion of number theory. The preprocessing matrix, with elements 1, -1, and 0, is obtained by replacing
\cos 2\pin / N
and
\sin 2\pin / N
by
\mu(n / N + 1 / 4)
and
\mu(n/N)
, respectively, where
\mu(\cdot)
is the bipolar rectangular wave function. The postprocessing matrix is block diagonal where each block is a circular correlation and consists of the new basis functions. The two-stage representation has been found very useful in applications such as parallel implementation of DFT and signal/image recognition.