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Quantization for Spectral Super-Resolution

Abstract

We show that the method of distributed noise-shaping beta-quantization offers superior performance for the problem of spectral super-resolution with quantization whenever there is redundancy in the number of measurements. More precisely, we define the over-sampling ratio \(\lambda \) as the largest integer such that \(\lfloor M/\lambda \rfloor - 1\ge 4/\Delta \), where M denotes the number of Fourier measurements and \(\Delta \) is the minimum separation distance associated with the atomic measure to be resolved. We prove that for any number \(K\ge 2\) of quantization levels available for the real and imaginary parts of the measurements, our quantization method combined with either TV-min/BLASSO or ESPRIT guarantees reconstruction accuracy of order \(O(M^{1/4}\lambda ^{5/4} K^{- \lambda /2})\) and \(O(M^{3/2} \lambda ^{1/2} K^{- \lambda })\), respectively, where the implicit constants are independent of M, K and \(\lambda \). In contrast, naive rounding or memoryless scalar quantization for the same alphabet offers a guarantee of order \(O(M^{-1}K^{-1})\) only, regardless of the reconstruction algorithm.

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Notes

  1. Some super-resolution papers assume that the Fourier samples are indexed by \(\{-N,\dots ,N\}\) for some integer N, whereas we assume the frequencies lie in \(\{0,\dots ,M-1\}\). Both settings are equivalent, but we need to be careful with the transcription process: Our number of measurements is M, which corresponds to \(2N+1\) in some other papers.

  2. The numerical constant 0.3298 that appears in (3.1) is double the one found in [19], again due to our convention that the Fourier samples are from \(\{0,1,\dots ,M-1\}\).

  3. When the noise energy is sufficiently small, there is a unique optimal permutation. In the subsequent results, our assumptions guarantee that this is the case.

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Acknowledgements

Weilin Li gratefully acknowledges support from the AMS-Simons Travel Grant.

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Correspondence to Weilin Li.

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Communicated by Ronald DeVore.

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Güntürk, C.S., Li, W. Quantization for Spectral Super-Resolution. Constr Approx (2022). https://doi.org/10.1007/s00365-022-09574-5

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  • DOI: https://doi.org/10.1007/s00365-022-09574-5

Keywords

  • Quantization
  • Super-resolution
  • Spectral estimation
  • Total variation
  • ESPRIT

Mathematics Subject Classification

  • 94A12
  • 94A20