Convolution Idempotents With a Given Zero-Set

Computing the Discrete Fourier Transform of signals with spectral frequency support

“…Lemma 1: when N is a power of 2, the submatrix F −1 (I, J ) is unitary iff for some set of pivots L 1) J corresponds to a conforming digit table with pivots L, 2) I corresponds to a conforming digit table with pivots N/2L. Lemma 1 is a direct consequence of earlier works [11], [12]; we discuss more about this in Section V.…”

“…Thus, the zero set contains all the indices in Z N whose gcd with N is in D J . The proof is elementary: we direct the interested reader to the references ( [24, Theorem 2.1], [9], [12], [25]) for the proof and details. We refer to the set D J often as zero-set divisors of h J .…”

“…When J is spectral and N is a power of 2, we provide a (deterministic) algorithm to compute the DFT of signals in B J that has complexity O(k log k). Our algorithm uses recent results on the structure of spectral sets [10]- [12] in terms of the binary expansion of the indices in J . Our algorithm reads k entries of the vector x at specific locations chosen according to the structure of the spectral set J .…”

Some results on convolution idempotents

“…We then present some results when N is a product of two primes. This is a followup to our work in [2] where we characterized all possible idempotents with given zero sets, in the case when N is a prime power.…”

“…In [6], we also gave a complete characterization of all solutions to i(D) when N = p M is a prime power, using base−p expansions of elements of J . In this work, we build a case for solving i(D) when N has more than one prime factor, and generalize some of the results of [2]. The main contribution here is to characterize all solutions to i(D) when N = pq and D = {1}, {p, q}, {1, p} or {1, q} (Theorem 1, Corollary 1, 2).…”