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The capacity region of classes of product broadcast channels

Yanlin Geng, Amin Gohari, Chandra Nair, Yuanming Yu

Abstract

We establish a new outer bound for the capacity region of product broadcast channels. This outer bound matches Marton's inner bound for a variety of classes of product broadcast channels whose capacity regions were previously unknown. These classes include product of reversely semi-deterministic and product of reversely more-capable channels. A significant consequence of this new outer bound is that it establishes, via an example, that the previously best known outer-bound is strictly suboptimal for the general broadcast channel. Our example is comprised of a product broadcast channel with two semi-deterministic components in reverse orientation.

The capacity region of classes of product broadcast channels

Abstract

We establish a new outer bound for the capacity region of product broadcast channels. This outer bound matches Marton's inner bound for a variety of classes of product broadcast channels whose capacity regions were previously unknown. These classes include product of reversely semi-deterministic and product of reversely more-capable channels. A significant consequence of this new outer bound is that it establishes, via an example, that the previously best known outer-bound is strictly suboptimal for the general broadcast channel. Our example is comprised of a product broadcast channel with two semi-deterministic components in reverse orientation.

Paper Structure

This paper contains 24 sections, 16 theorems, 124 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions of some classes of broadcast channels
  4. Organization and summary of results
  5. The UV outer bound is not tight
  6. Definitions and preliminary results
  7. Optimal sum rate for product of reversely semi-deterministic channels
  8. The UV outer bound is strictly suboptimal
  9. Capacity regions for classes of product broadcast channels
  10. An outer bound for product channels
  11. An achievable region for a product broadcast channel
  12. Capacity regions for new classes of product broadcast channels
  13. On Marton's inner bound and $\lambda$-sum-rate
  14. Two letter Marton's inner bound
  15. Properties of $\lambda$-sum-rate
  16. ...and 9 more sections

Key Result

Lemma 1

The following min-max theorem holds: This implies that the sum rate of Marton's inner bound can be calculated using any of the three above expressions.

Figures (2)

  • Figure 1: A broadcast channel
  • Figure 2: A reversely semi-deterministic product broadcast channel. Remark: The published version of this paper contains an incorrect channel diagram. The calculations are correct and correspond to the channel shown here. The authors apologize for this oversight.

Theorems & Definitions (55)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Definition 4
  • Claim 1
  • proof
  • Theorem 1
  • proof
  • ...and 45 more