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Achieving Gaussian Vector Broadcast Channel Capacity with Scalar Lattices

M. Yusuf Şener, Gerhard Kramer, Shlomo Shamai, Ronald Böhnke, Wen Xu

TL;DR

This work demonstrates that a scalar-lattice dirty paper coding scheme, augmented with noise whitening and SVD to yield parallel scalar channels, can achieve any rate tuple in the capacity region of a $K$-receiver Gaussian MIMO broadcast channel. By employing $M$-ary ASK, a modulo operation with interval $A$, and truncated Gaussian shaping, the scheme can approximate capacity as $M$ and $A$ grow, even when the receiver noises yield a mixture of Gaussian and shaped components. The analysis develops finite-$M$ extensions of prior lemmas and theorems, bounding input power and entropy and showing that the resultant output densities approach uniformity on the modulo interval, thereby preserving capacity-achieving behavior. The results offer a practical lattice-based approach to MIMO broadcast channel capacity with reduced complexity compared to full Marton/Costa-type schemes, with future work comparing performance against channel inversion and exploring concrete coding implementations.

Abstract

A coding scheme with scalar lattices is applied to K-receiver, Gaussian, vector broadcast channels with K independent messages, one for each receiver. The method decomposes each receiver channel into parallel scalar channels with known interference and applies dirty paper coding with a modulo interval, amplitude shift keying (ASK), and probabilistic shaping to each scalar channel. The achievable rate tuples include all points inside the capacity region by choosing truncated Gaussian shaping, large ASK alphabets, and large modulo intervals.

Achieving Gaussian Vector Broadcast Channel Capacity with Scalar Lattices

TL;DR

This work demonstrates that a scalar-lattice dirty paper coding scheme, augmented with noise whitening and SVD to yield parallel scalar channels, can achieve any rate tuple in the capacity region of a -receiver Gaussian MIMO broadcast channel. By employing -ary ASK, a modulo operation with interval , and truncated Gaussian shaping, the scheme can approximate capacity as and grow, even when the receiver noises yield a mixture of Gaussian and shaped components. The analysis develops finite- extensions of prior lemmas and theorems, bounding input power and entropy and showing that the resultant output densities approach uniformity on the modulo interval, thereby preserving capacity-achieving behavior. The results offer a practical lattice-based approach to MIMO broadcast channel capacity with reduced complexity compared to full Marton/Costa-type schemes, with future work comparing performance against channel inversion and exploring concrete coding implementations.

Abstract

A coding scheme with scalar lattices is applied to K-receiver, Gaussian, vector broadcast channels with K independent messages, one for each receiver. The method decomposes each receiver channel into parallel scalar channels with known interference and applies dirty paper coding with a modulo interval, amplitude shift keying (ASK), and probabilistic shaping to each scalar channel. The achievable rate tuples include all points inside the capacity region by choosing truncated Gaussian shaping, large ASK alphabets, and large modulo intervals.
Paper Structure (14 sections, 3 theorems, 33 equations, 5 figures)

This paper contains 14 sections, 3 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Symmetric and Unimodal Functions
  5. Gaussian Vector Broadcast Channels
  6. Model and Coding Schemes
  7. Capacity-Achieving Scheme
  8. Parallel DPC with Scalar Lattices
  9. Proof of Lemma 2 in csener2024-dpc-capacity
  10. Extension of Theorem 1 in csener2024-dpc-capacity
  11. Power and Differential Entropy of ${\tilde{X}}_{k,i}$
  12. Density and Differential Entropy of $Y'$
  13. Conclusions
  14. Acknowledgements

Key Result

Lemma 1

The convolution $f*g(.)$ of two symmetric functions $f(.)$ and $g(.)$ is symmetric.

Figures (5)

  • Figure 1: Curves of \ref{['eq:p-bounds']} for $A=6$, $M=4$, and truncated Gaussian shaping.
  • Figure 2: Curves of \ref{['eq:pz-bounds']} for $A=6$, $M=4$, and truncated Gaussian shaping.
  • Figure 3: $f(x)$ for $A=M=6$. The sampling points are shifted by $x=0.2$ and are located at the top left corner of each of the six bars.
  • Figure 4: $f(x)$ for $A=M=6$. The area of the red bars is less than the area of $f(x)$, which is here 1.
  • Figure 5: $f(x)$ for $A=M=6$. The area of the red, green, and blue bars is greater than the area of $f(x)$, which is here 1.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma 3