Fast DFT Computation for Signals With Structured Support

Suppose an
$N-$
length signal has known frequency support of size
$k$
. Given access to samples of this signal, how fast can we compute the DFT? The answer to this question depends on the structure of the frequency support. We first identify some frequency supports for which (an ideal)
$O(k \log k)$
complexity can be achieved, which we refer to as homogeneous sets. We give a generalization of the radix-2 FFT that enables
$O(k\log k)$
computation of signals with homogeneous frequency support. We use homogeneous sets as building blocks to construct more complex support structures for which the complexity of
$O(k\log k)$
is achievable. Applying these ideas, we present an
$O(k\log ^{2}k)$
algorithm for computing the DFT of signals whose frequency support is additively structured. We also present partial converses.