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One-bit Sigma-Delta modulation on the circle

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Abstract

Manifold models in data analysis and signal processing have become more prominent in recent years. In this paper, we will look at one of the main tasks of modern signal processing, namely, at analog-to-digital (A/D) conversion in connection with a simple manifold model — the circle. We will focus on Sigma-Delta modulation which is a popular method for A/D conversion of bandlimited signals that employs coarse quantization coupled with oversampling. Classical Sigma-Delta schemes provide mismatches and large errors at the initialization point if the signal to be converted is defined on the circle. In this paper, our goal is to get around these problems for Sigma-Delta schemes. Our results show how to design an update for the first and the second order schemes based on the reconstruction error analysis such that for the updated scheme the reconstruction error is improved.

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Funding

This work was supported by the DFG Collaborative Research Center TRR109, Discretization in Geometry and Dynamics.

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Authors and Affiliations

  1. Department of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85748, Garching, Germany

    Olga Graf & Felix Krahmer

  2. Helmholtz Imaging, Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607, Hamburg, Germany

    Sara Krause-Solberg

Authors
  1. Olga Graf
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  2. Felix Krahmer
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  3. Sara Krause-Solberg
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Correspondence to Olga Graf.

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Communicated by: Yang Wang

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Graf, O., Krahmer, F. & Krause-Solberg, S. One-bit Sigma-Delta modulation on the circle. Adv Comput Math 49, 32 (2023). https://doi.org/10.1007/s10444-023-10032-4

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  • DOI: https://doi.org/10.1007/s10444-023-10032-4

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