Procedures for computing the discrete Fourier transform on staggered blocks

In this work the problem of evaluating successively the discrete Fourier transform (DFT) on ordered sets of
N
elements staggered of
M
is considered. Three procedures for solving such a problem are given, of which two are recursive and one nonrecursive. The complexity of each procedure, in number of complex multiplications, is about
(N/2) \log_{2} 4M
.