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Pointwise Comparison of PCM and Σ Δ Quantization

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Abstract

A quantitative comparison of Pulse Code Modulation (PCM) and Sigma–Delta (Σ Δ) quantization methods is made in the setting of finite frames. Frames allow for redundant, overcomplete signal decompositions. PCM and Σ Δ are two industry-standard quantization methods, and the setting of finite frames is appropriate for a host of modern applications. Previous results for this comparison are known for upper error bounds, where Σ Δ performs better in the setting of frames, as opposed to orthonormal bases, where PCM is optimal. We answer the following question: For which signals x is the PCM error, that is, the norm of the difference between x and its PCM approximant, less than the Σ Δ error? We prove that, typically, in the setting of frames, Σ Δ outperforms PCM, but not always.

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Authors and Affiliations

  1. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    John J. Benedetto

  2. School of Engineering and Science, Jacobs University, Bremen, Germany

    Onur Oktay

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  1. John J. Benedetto
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  2. Onur Oktay
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Correspondence to John J. Benedetto.

Additional information

Communicated by Ronald A. DeVore.

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Benedetto, J.J., Oktay, O. Pointwise Comparison of PCM and Σ Δ Quantization. Constr Approx 32, 131–158 (2010). https://doi.org/10.1007/s00365-010-9083-x

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  • DOI: https://doi.org/10.1007/s00365-010-9083-x

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