  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.
