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On the Optimality of Gaussian Code-books for Signaling over a Two-Users Weak Gaussian Interference Channel

Amir K. Khandani

TL;DR

The paper proves that the capacity region of a two-user Gaussian interference channel in the weak regime can be achieved with single-letter Gaussian code-books. By applying calculus of variations to incrementally reallocate power between public and private messages, it shows Gaussian densities maximize the weighted sum-rate and that at most two time-sharing phases suffice to realize the boundary, with the Han-Kobayashi region achieving the optimum boundary. The results hinge on zero-mean Gaussian densities for core/compound random variables, a Pareto-minimal power reallocation ensuring a unique boundary point, and a reduction to single-letter signaling without requiring vector encoding. Consequently, Gaussian single-letter signaling is optimal for realizing the entire capacity boundary (and HK with Gaussian inputs suffices), with applicability to the general interference channel beyond the weak regime.

Abstract

This article shows that the capacity region of a two users weak Gaussian interference channel can be achieved using single letter Gaussian code-books. The approach relies on traversing the boundary in incremental steps. Starting from a corner point with Gaussian code-books, and relying on calculus of variation, it is shown that the end point in each step is achieved using Gaussian code-books. Optimality of Gaussian code-books is first established by limiting the random coding to independent and identically distributed scalar (single-letter) samples. Then, it is shown that the value of any optimum solution for vector inputs does not exceed that of the single-letter case. It is also shown that the maximum number of phases needed to realize the optimum time-sharing is two. It is established that the solution to the Han-Kobayashi achievable rate region, with single letter Gaussian code-books, achieves the optimum boundary. Even though the article focuses on weak interference, the results are applicable to the general case.

On the Optimality of Gaussian Code-books for Signaling over a Two-Users Weak Gaussian Interference Channel

TL;DR

The paper proves that the capacity region of a two-user Gaussian interference channel in the weak regime can be achieved with single-letter Gaussian code-books. By applying calculus of variations to incrementally reallocate power between public and private messages, it shows Gaussian densities maximize the weighted sum-rate and that at most two time-sharing phases suffice to realize the boundary, with the Han-Kobayashi region achieving the optimum boundary. The results hinge on zero-mean Gaussian densities for core/compound random variables, a Pareto-minimal power reallocation ensuring a unique boundary point, and a reduction to single-letter signaling without requiring vector encoding. Consequently, Gaussian single-letter signaling is optimal for realizing the entire capacity boundary (and HK with Gaussian inputs suffices), with applicability to the general interference channel beyond the weak regime.

Abstract

This article shows that the capacity region of a two users weak Gaussian interference channel can be achieved using single letter Gaussian code-books. The approach relies on traversing the boundary in incremental steps. Starting from a corner point with Gaussian code-books, and relying on calculus of variation, it is shown that the end point in each step is achieved using Gaussian code-books. Optimality of Gaussian code-books is first established by limiting the random coding to independent and identically distributed scalar (single-letter) samples. Then, it is shown that the value of any optimum solution for vector inputs does not exceed that of the single-letter case. It is also shown that the maximum number of phases needed to realize the optimum time-sharing is two. It is established that the solution to the Han-Kobayashi achievable rate region, with single letter Gaussian code-books, achieves the optimum boundary. Even though the article focuses on weak interference, the results are applicable to the general case.
Paper Structure (17 sections, 14 theorems, 123 equations, 7 figures)

This paper contains 17 sections, 14 theorems, 123 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Literature Survey
  3. Summary of Main Results
  4. Analysis Limited to Single Letter Code-books
  5. Formulation for Single Letter Inputs
  6. Boundary for Single Letter Code-books
  7. Optimality of Single Letter Code-books
  8. Converse Results
  9. Optimality of the HK Region with Gaussian Code-books
  10. Constrained Maximization of Entropy Functions
  11. Entropy Term Involving a Single Density Function
  12. Effect of Scaling of Random Variables
  13. First Order Variations
  14. Second Order Variations
  15. Functional of Compound Random Variables
  16. ...and 2 more sections

Key Result

Theorem 1

For $\mu<1$, consider a set of consecutive steps, in counterclockwise direction, along the boudnary of the single letter capacity region for the component GIC based on Eq2. Corresponding values for $\mathbf{\Upsilon}$ in Eq3 will be monotonically decreasing, while $\mathbf{\Gamma}$ in Eq4 will be mo

Figures (7)

  • Figure 1: Two users Gaussian Interference Channel (GIC) with $a<1$ and $b<1$.
  • Figure 2: An example for power reallocation and its corresponding step along the boundary.
  • Figure 3: Channel models depicting decoding methods discussed in Theorem \ref{['V4Th3']} where \ref{['F4V4L']}(a) corresponds to successive decoding of $U_2$, $U_1$ and $V_2$ at $Y_2$, and \ref{['F4V4L']}(b) corresponds to joint decoding of $(U_1,U_2)$ followed by decoding of $V_1$ at $Y_1$.
  • Figure 4: Example of a channel where the stationary solution for mutual information may result in a maximum or a minimum, according to the statistical mean of $\tilde{Z}$.
  • Figure 5: $\mathbf{\Upsilon}$ and $\mathbf{\Gamma}$ as a function of $\omega$ (related to Theorem \ref{['Th10']}).
  • ...and 2 more figures

Theorems & Definitions (28)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 18 more