"Unlimited Sensing, Gaussian Integers and Prony’s Method"Pavlícek, VáclavParametric sampling of exponential sums is a classic and important problem widely studied in harmonic analysis with applications across different disciplines of science and engineering. The problem was first studied by Prony who proposed a method for estimating K exponentials from 2K samples, provided the sampling step is sufficiently small. Prony’s method works on the assumption that signals with arbitrary amplitude range can be captured. However, this may turn out to be a fragile assumption in practice often leading to information loss due to saturation or clipping. To overcome this gap between theory and practice, we pose the problem of frequency estimation from modulo samples. This strategy is at the heart of the Unlimited Sensing Framework. On the one hand, modulo samples generated via novel hardware prevent signal clipping. On the other hand, it necessitates the development of an alternative recovery strategy that mimics Prony’s method but can tackle folded samples. To this end, we derive a sampling theorem that utilizes a multi-channel architecture that enables estimating K frequencies from 6K folded samples without any sampling rate requirement. We present a constructive proof that leads to a novel recovery algorithm combining the Chinese Remainder Theorem with the Gaussian integers. Furthermore, we demonstrate that the proposed method is theoretically guaranteed against the bounded noise, which we validate via numerical experiments. This is a joint work with Ruiming Guo and Ayush Bhandari at Imperial College London. |
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