Dynamic SFT with Structured Measurements: Fast Queries, Fast Updates, Provable Guarantees

Yang Cao, Zhao Song

Keywords: sparse Fourier transform

Abstract: The sparse Fourier transform typically proceeds in two stages: frequency estimation and signal estimation. The first recovers the set of frequencies from noisy time-domain samples; the second constructs their corresponding magnitudes. In most methods, signal estimation is only approximate and depends on the frequencies identified in the first stage. In this paper, we study a complementary question: given access to an oracle that returns the exact magnitude for any queried frequency, what is the minimum number of oracle calls needed to perform a sparse Fourier transform? For an n-point discrete Fourier transform, the naive approach queries all n frequencies. We design the first algorithm that requires only $o(n)$ oracle invocations. We further complement this upper bound with a lower bound, derived using tools from computational complexity.

Submission Number: 80