On the Best Lattice Quantizers

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Authors

  • Erik Agrell
  • Bruce Allen

Research Organisations

External Research Organisations

  • Chalmers University of Technology
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Details

Original languageEnglish
Pages (from-to)7650-7658
Number of pages9
JournalIEEE Transactions on Information Theory
Volume69
Issue number12
Publication statusPublished - 30 Jun 2023

Abstract

A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization error: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any lattice whose mean square error cannot be decreased by a small perturbation of the generator matrix, and (ii) for an optimal product of lattices that are themselves locally optimal in the sense of (i). We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the best known upper bound.

Keywords

    Dither autocorrelation, laminated lattice, lattice theory, mean square error, moment of inertia, normalized second moment, product lattice, quantization constant, quantization error, vector quantization, Voronoi region, white noise

ASJC Scopus subject areas

Cite this

On the Best Lattice Quantizers. / Agrell, Erik; Allen, Bruce.
In: IEEE Transactions on Information Theory, Vol. 69, No. 12, 30.06.2023, p. 7650-7658.

Research output: Contribution to journal › Article › Research › peer review

Agrell E, Allen B. On the Best Lattice Quantizers. IEEE Transactions on Information Theory. 2023 Jun 30;69(12):7650-7658. doi: 10.1109/TIT.2023.3291313
Agrell, Erik ; Allen, Bruce. / On the Best Lattice Quantizers. In: IEEE Transactions on Information Theory. 2023 ; Vol. 69, No. 12. pp. 7650-7658.
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