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.

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.

We consider the problem of recovering N length vectors h that vanish on a given set of indices and satisfy h*h = h. We give some results on the structure of such h when N is a product of two primes, and investigate some bounds and their connections to certain graphs defined on ZN.