[Paper Review] Multiple Access Channel with Partial and Controlled Cribbing Encoders

Haim H. Permuter, Himanshu Asnani|arXiv (Cornell University)|Mar 21, 2011

Wireless Communication Security Techniques11 references3 citations

TL;DR

This paper introduces a novel model of the multiple access channel (MAC) with partial cribbing encoders, where each encoder observes a deterministic function of the other's output with or without delay. It derives single-letter capacity regions for causal and strictly-causal partial cribbing using block-Markov coding, rate splitting, and backward decoding, and further extends the model to action-dependent cribbing, showing that time-sharing is suboptimal in 'to crib or not to crib' scenarios.

ABSTRACT

In this paper we consider a multiple access channel (MAC) with partial cribbing encoders. This means that each of two encoders obtains a deterministic function of the other encoder output with or without delay. The partial cribbing scheme is especially motivated by the additive noise Gaussian MAC since perfect cribbing results in the degenerated case of full cooperation between the encoders and requires an infinite entropy link. We derive a single letter characterization of the capacity of the MAC with partial cribbing for the cases of causal and strictly causal partial cribbing. Several numerical examples, such as quantized cribbing, are presented. We further consider and derive the capacity region where the cribbing depends on actions that are functions of the previous cribbed observations. In particular, we consider a scenario where the action is "to crib or not to crib" and show that a naive time-sharing strategy is not optimal.

Motivation & Objective

• To address the limitations of perfect cribbing in Gaussian MACs, where infinite-rate feedback is required, by introducing partial cribbing via deterministic functions of encoder outputs.
• To characterize the capacity region of the MAC with partial cribbing under causal and strictly-causal information causality models.
• To extend the framework to action-dependent cribbing, where the decision to crib is a function of past observations and carries a cost.
• To demonstrate that naive time-sharing strategies are suboptimal in action-dependent cribbing scenarios.

Proposed method

• Uses block-Markov coding and backward decoding techniques adapted from perfect cribbing models to achieve the capacity region.
• Introduces rate splitting to handle the asymmetric decoding capability when one encoder can only decode part of the other's message.
• Models action-dependent cribbing where the action (e.g., 'to crib or not to crib') is a function of past cribbed signals and influences the current cribbing function.
• Employs superposition coding and Shannon’s strategies to manage interference and feedback in the partial cribbing setting.
• Derives single-letter capacity expressions by optimizing over auxiliary random variables and action distributions.
• Uses convex optimization and grid search to numerically evaluate capacity under action constraints.

Experimental results

Research questions

• RQ1 What is the capacity region of a MAC with strictly-causal partial cribbing, where each encoder observes a deterministic function of the other’s output with delay?
• RQ2 How does the capacity region change when one encoder has causal and the other has strictly-causal partial cribbing?
• RQ3 Can the capacity of the MAC be improved by making the cribbing decision itself a function of past observations and messages?
• RQ4 Is a time-sharing strategy optimal for action-dependent cribbing, or can adaptive action selection yield higher rates?
• RQ5 How does partial cribbing in the Gaussian MAC differ from perfect cribbing, and what is the capacity under quantized feedback?

Key findings

• The capacity region for strictly-causal partial cribbing is characterized using a single-letter expression involving auxiliary random variables and rate splitting.
• For causal and strictly-causal mixed cribbing (Case B), the capacity region is derived and shown to be strictly larger than the sum of individual MAC capacities.
• In the action-dependent cribbing model, the capacity is given by a maximin expression over action probabilities and coding parameters, with explicit expression for the binary-symmetric Z-channel example.
• The capacity for the Z-channel case with Γ=0 is analytically derived as H_b(1/5) - 2/5 ≈ 0.154 bits per channel use.
• The capacity for Γ=1 is found numerically as max_α min(α, H_b(α)), yielding approximately 0.188 bits per channel use.
• The paper shows that a time-sharing strategy is suboptimal in the 'to crib or not to crib' scenario, as adaptive action selection can yield higher rates.