[Paper Review] On the Capacity of Gaussian Weak Interference Channels with Degraded Message sets

Wei Wu, Sriram Vishwanath+1 more|ArXiv.org|May 17, 2006

Wireless Communication Security Techniques2 references4 citations

TL;DR

This paper characterizes the capacity region of a two-user Gaussian interference channel with weak interference and degraded message sets, where one transmitter knows both messages. By leveraging dirty paper coding and cooperative transmission, the authors derive a closed-form outer bound that serves as a capacity region for this cooperative scenario and provides a tighter outer bound for the non-cooperative case, particularly when interference is weak (|b| ≤ 1).

ABSTRACT

This paper is motivated by a sensor network on a correlated field where nearby sensors share information, and can thus assist rather than interfere with one another. We consider a special class of two-user Gaussian interference channels (IFCs) where one of the two transmitters knows both the messages to be conveyed to the two receivers. Both achievability and converse arguments are provided for a channel with Gaussian inputs and Gaussian noise when the interference is weaker than the direct link (a so called weak IFC). In general, this region serves as an outer bound on the capacity of weak IFCs with no shared knowledge between transmitters.

Motivation & Objective

• To study the capacity region of a two-user Gaussian interference channel where one transmitter knows both messages, modeling scenarios like sensor networks with correlated data.
• To provide a capacity-achieving scheme for weak interference channels (|b| ≤ 1) under partial transmitter cooperation.
• To establish that this cooperative capacity region serves as an outer bound for the non-cooperative Gaussian IFC, improving upon prior bounds in certain regimes.
• To explore the potential of using lossy functions of messages instead of full message knowledge to potentially tighten outer bounds for non-cooperative IFCs.
• To generalize insights from this two-user model to multi-user interference channels in future work.

Proposed method

• Formulates the Gaussian IFC with inputs X₁, X₂ and outputs Y₁, Y₂, where Y₁ = X₁ + aX₂ + Z₁ and Y₂ = bX₁ + X₂ + Z₂, with |b| ≤ 1 for weak interference.
• Introduces a structured auxiliary random variable U to model the common message knowledge at T₁, enabling joint encoding and precoding.
• Applies dirty paper coding at T₁ to pre-cancel interference seen at R₂ by treating X₂ as known interference, leveraging channel state knowledge.
• Derives the capacity region using mutual information expressions: R₁ ≤ I(X₁;Y₁|X₂,U) and R₂ ≤ I(U,X₂;Y₂), with Gaussian input distributions.
• Uses a Lagrangian optimization framework to maximize R₁ + μR₂ over input distributions, proving optimality of i.i.d. Gaussian inputs via comparison to MIMO broadcast channel outer bounds.
• Establishes the capacity region as the convex hull of the closure of all rate pairs satisfying the derived mutual information bounds.

Experimental results

Research questions

• RQ1 What is the capacity region of a two-user Gaussian interference channel with weak interference and one transmitter possessing both messages?
• RQ2 Can dirty paper coding and cooperative transmission at the transmitter with message knowledge improve the achievable rate region in weak interference settings?
• RQ3 How does this cooperative capacity region compare to known outer bounds for the non-cooperative Gaussian IFC?
• RQ4 Can the use of partial message knowledge (e.g., lossy functions) at transmitters yield tighter outer bounds for the non-cooperative IFC?
• RQ5 Is the capacity region of this cooperative IFC a valid outer bound for the non-cooperative IFC, and how tight is it compared to existing bounds?

Key findings

• The capacity region of the Gaussian IFC with T₁ knowing both messages and |b| ≤ 1 is characterized as the convex hull of the closure of rate pairs satisfying R₁ ≤ ½ log(1 + αP₁) and R₂ ≤ ½ log(1 + (hΣhᵀ)/(1 + b²αP₁)), with h = [b√(1−α)P₁, √P₂].
• When α = 1, the rate pair (½ log(1 + P₁), ½ log((1 + P₁b² + P₂)/(1 + b²P₁))) matches the extreme point of Kramer’s outer bound, validating tightness at this point.
• When α = 0, the region reduces to the MISO channel capacity for transmitting W₂, confirming the full cooperation limit.
• The derived region provides a tighter outer bound than Carleial’s bound when the power allocation parameter α is close to 1, particularly in weak interference regimes.
• The outer bound based on transmitter genie-aiding (this work) is less tight than Kramer’s receiver-genie-aided bound overall, but offers a complementary perspective.
• The capacity region for the case where T₂ knows both messages (|a| ≤ 1) is derived via symmetry, yielding R₁ ≤ ½ log((1 + P₁ + 2|a|√((1−β)P₁P₂) + a²P₂)/(1 + a²βP₂)) and R₂ ≤ ½ log(1 + βP₂), 0 ≤ β ≤ 1.