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Sigma Delta Quantization with Harmonic Frames and Partial Fourier Ensembles

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Abstract

Sigma Delta (\(\Sigma \Delta \)) quantization, a quantization method first surfaced in the 1960s, has now been widely adopted in various digital products such as cameras, cell phones, radars, etc. The method features a great robustness with respect to quantization noises through sampling an input signal at a Super-Nyquist rate. Compressed sensing (CS) is a frugal acquisition method that utilizes the sparsity structure of an objective signal to reduce the number of samples required for a lossless acquisition. By deeming this reduced number as an effective dimensionality of the set of sparse signals, one can define a relative oversampling/subsampling rate as the ratio between the actual sampling rate and the effective dimensionality. When recording these “compressed” analog measurements via Sigma Delta quantization, a natural question arises: will the signal reconstruction error previously shown to decay polynomially as the increase of the vanilla oversampling rate for the case of band-limited functions, now be decaying polynomially as that of the relative oversampling rate? Answering this question is one of the main goals in this direction. The study of quantization in CS has so far been limited to proving error convergence results for Gaussian and sub-Gaussian sensing matrices, as the number of bits and/or the number of samples grow to infinity. In this paper, we provide a first result for the more realistic Fourier sensing matrices. The main idea is to randomly permute the Fourier samples before feeding them into the quantizer. We show that the random permutation can effectively increase the low frequency power of the measurements, thus enhance the quality of \(\Sigma \Delta \) quantization.

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Acknowledgements

The author would like to thank Rayan Saab, Özgur Yılmaz and Joe-Mei Feng for valuable discussion about the topic. R.  Wang was funded in part by an NSERC Collaborative Research and Development Grant DNOISE II (22R07504).

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  1. Michigan State University, East Lansing, USA

    Rongrong Wang

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  1. Rongrong Wang
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Correspondence to Rongrong Wang.

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Communicated by Roman Vershynin.

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Wang, R. Sigma Delta Quantization with Harmonic Frames and Partial Fourier Ensembles. J Fourier Anal Appl 24, 1460–1490 (2018). https://doi.org/10.1007/s00041-017-9582-2

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  • DOI: https://doi.org/10.1007/s00041-017-9582-2

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