[Paper Review] Relaxing the Gaussian AVC

Anand D. Sarwate, Michael Gastpar|arXiv (Cornell University)|Sep 13, 2012

Wireless Communication Security Techniques34 references3 citations

TL;DR

This paper revisits the Gaussian Arbitrarily Varying Channel (GAVC) by relaxing its worst-case interference model, showing that small common randomness (O(log n) bits) enables capacity-achieving rates, known interference at the transmitter can increase capacity via dirty paper coding-like strategies, and rank-constrained jamming in MIMO systems allows higher rates than full-rank interference, with exact capacity derived for 2×2×1 MIMO setups under specific conditions.

ABSTRACT

The arbitrarily varying channel (AVC) is a conservative way of modeling an unknown interference, and the corresponding capacity results are pessimistic. We reconsider the Gaussian AVC by relaxing the classical model and thereby weakening the adversarial nature of the interference. We examine three different relaxations. First, we show how a very small amount of common randomness between transmitter and receiver is sufficient to achieve the rates of fully randomized codes. Second, akin to the dirty paper coding problem, we study the impact of an additional interference known to the transmitter. We provide partial capacity results that differ significantly from the standard AVC. Third, we revisit a Gaussian MIMO AVC in which the interference is arbitrary but of limited dimension.

Motivation & Objective

• To understand the impact of worst-case interference in point-to-point communication and how relaxing adversarial assumptions improves achievable rates.
• To investigate whether limited common randomness between transmitter and receiver can recover the capacity of randomized codes in the Gaussian AVC.
• To explore how known interference at the transmitter can be exploited to counteract adversarial jamming and increase reliable communication rates.
• To analyze the capacity of MIMO Gaussian AVCs when the jammer's interference is constrained to low rank, particularly in the 2×2×1 case.
• To determine whether geometric power allocation strategies in rank-constrained MIMO AVCs achieve optimal or near-optimal rates.

Proposed method

• Introduces a relaxation of the Gaussian AVC by allowing the encoder and decoder to share O(log n) bits of common randomness, enabling them to simulate randomized codes without full randomness.
• Analyzes a variant where the transmitter knows an additional interference signal, using a dirty paper coding-inspired strategy to pre-cancel the interference and improve reliability.
• Considers a MIMO Gaussian AVC with a jammer constrained to a single-antenna (rank-1) signal, modeling the channel as a set of parallel subchannels with different noise powers.
• Derives achievable rates by optimizing power allocation across subchannels under the constraint that the jammer can only affect one subchannel at a time.
• Uses waterfilling-like power allocation strategies to maximize mutual information under power and rank constraints on the jammer.
• Establishes capacity bounds and exact rates in limiting cases (e.g., Λ→∞, Γ,Λ→∞ with fixed ρ=Γ/Λ), using geometric and information-theoretic arguments.

Experimental results

Research questions

• RQ1 Can a sub-exponential amount of common randomness (O(log n)) suffice to achieve the randomized coding capacity of the Gaussian AVC?
• RQ2 Does the presence of a known interference signal at the transmitter increase the achievable rate in an adversarial jamming environment?
• RQ3 What is the capacity of a MIMO Gaussian AVC when the jammer is restricted to a low-rank interference signal, particularly in the 2×2×1 case?
• RQ4 Is the optimal power allocation strategy for the transmitter and jammer in the rank-constrained MIMO AVC a saddle-point solution, or does the adversarial nature prevent such a characterization?
• RQ5 How do the asymptotic behaviors of the achievable rate scale as interference power Λ and signal power Γ grow with fixed ratio ρ=Γ/Λ?

Key findings

• O(log n) bits of common randomness are sufficient for the encoder and decoder to achieve the randomized coding capacity of the Gaussian AVC, effectively neutralizing worst-case jamming.
• In the presence of a known interference signal, the transmitter can use a dirty paper coding-like strategy to achieve a higher rate than the standard deterministic GAVC, with capacity gains observed in special cases.
• For the (2,2,1) MIMO Gaussian AVC, the exact capacity is derived under full randomized coding, with the optimal jammer strategy being to jam the noisier subchannel.
• As Λ→∞, the achievable rate converges to R = (1/2)log(1 + Γ/(σ₁² + σ₂²)), which is strictly less than the upper bound R_ub = (1/2)log(1 + Γ/σ₂²), indicating a fundamental loss due to adversarial jamming.
• When both Γ and Λ grow with fixed ratio ρ=Γ/Λ, the achievable rate scales as R(ρ,Γ) = O(log Γ) + (1/2)log(1 + ρ/2), showing logarithmic growth in signal power.
• The paper conjectures that the optimal jamming strategy always targets the strongest subchannel (smallest σ²), and that the dirty paper coding approach is optimal for known interference.