Computing the Discrete Fourier Transform of signals with spectral frequency support

We consider the problem of finding the Discrete Fourier Transform (DFT) of
$N$
-length signals with known frequency support of size
$k$
. When
$N$
is a power of 2 and the frequency support is a spectral set, we provide an
$O(k\log k)$
algorithm to compute the DFT. Our algorithm uses some recent characterizations of spectral sets and is a generalization of the standard radix-2 algorithm.