[Paper Review] The State-Dependent Multiple-Access Channel with States Available at a Cribbing Encoder

Shraga I. Bross, Amos Lapidoth|arXiv (Cornell University)|Jul 22, 2010

Wireless Communication Security Techniques8 references4 citations

TL;DR

This paper studies a state-dependent multiple-access channel where one encoder (Encoder 2) causally cribbs the channel inputs of the other encoder (Encoder 1), while also having non-causal knowledge of the state sequence. It derives the capacity region for both strictly causal and causal cribbing scenarios, generalizing prior models by removing the assumption that messages are pre-shared, and establishes a complete characterization using a novel Markov chain and strong typicality-based coding scheme.

ABSTRACT

The two-user discrete memoryless state-dependent multiple-access channel (MAC) models a scenario in which two encoders transmit independent messages to a single receiver via a MAC whose channel law is governed by the pair of encoders' inputs and by an i.i.d. state random variable. In the cooperative state-dependent MAC model it is further assumed that Message 1 is shared by both encoders whereas Message 2 is known only to Encoder 2 - the cognitive transmitter. The capacity of the cooperative state-dependent MAC where the realization of the state sequence is known non-causally to the cognitive encoder has been derived by Somekh-Baruch et. al. In this work we dispense of the assumption that Message 1 is shared a-priori by both encoders. Instead, we study the case in which Encoder 2 cribs causally from Encoder 1. We determine the capacity region for both, the case where Encoder 2 cribs strictly causal and the case where Encoder 2 cribs causally from Encoder 1.

Motivation & Objective

• To model a more realistic cooperative state-dependent multiple-access channel where Encoder 2 acquires Encoder 1's inputs causally via cribbing, rather than assuming pre-shared messages.
• To determine the capacity region of the state-dependent MAC when Encoder 2 has access to both the state sequence and past or current inputs of Encoder 1.
• To generalize existing results on cooperative MACs by removing the assumption of pre-shared messages and instead modeling causal information exchange via cribbing.
• To provide a complete characterization of the capacity region for both strictly causal and causal cribbing scenarios.
• To establish a coding scheme that leverages cribbed information and state knowledge to achieve optimal rates using Markov chains and strong typicality arguments.

Proposed method

• Formalizes the state-dependent MAC with a cribbing encoder by defining two variants: strictly causal (Encoder 2 uses past inputs of Encoder 1) and causal (Encoder 2 uses current and past inputs).
• Introduces auxiliary random variables $ U_k $ and $ V_k $ to model the causal dependency of Encoder 2’s input on past inputs and state, establishing a Markov chain $ U_k ightarrow S_k V_k ightarrow X_{1,k} $.
• Uses strong typicality and joint typicality arguments to analyze error probability and derive outer bounds, relying on standard information-theoretic tools such as the AEP and typical set enumeration.
• Applies Lemma 2 (conditional independence in typical sets) to bound the probability of decoding error when using a time-sharing strategy based on the auxiliary variables.
• Derives the capacity region by characterizing the set of all achievable rate pairs $ (R_1, R_2) $ satisfying a set of mutual information constraints involving the auxiliary variables and channel inputs.
• Establishes that the capacity region is the closure of the set of all rate tuples satisfying $ R_1 eq 0 $, $ R_2 eq 0 $, and $ R_1 + R_2 eq 0 $, with constraints involving $ I(U_k; Y_k | V_k, S_k) $, $ I(X_{2,k}; Y_k | X_{1,k}, S_k, V_k) $, and $ I(U_k; S_k) $.

Experimental results

Research questions

• RQ1 What is the capacity region of a state-dependent multiple-access channel when one encoder causally observes the other encoder’s inputs and has non-causal state knowledge?
• RQ2 How does the capacity region change when the cribbing encoder uses strictly causal versus causal knowledge of the other encoder’s inputs?
• RQ3 Can the capacity region be characterized without assuming pre-shared messages between encoders, under a more practical cribbing model?
• RQ4 What role do auxiliary random variables play in achieving the capacity region under causal cribbing and state knowledge?
• RQ5 How does the use of strong typicality and Markov chain constraints improve the characterization of achievable rates in this model?

Key findings

• The capacity region for the strictly causal cribbing case is characterized by a set of mutual information constraints involving auxiliary variables $ U_k $, $ V_k $, and the channel inputs and outputs.
• For the causal cribbing case, the capacity region is similarly characterized but with the encoding function of Encoder 2 depending on current and past inputs of Encoder 1, leading to a different Markov structure.
• The capacity region is shown to be the closure of the set of all rate tuples $ (R_1, R_2) $ satisfying $ R_1 eq 0 $, $ R_2 eq 0 $, and $ R_1 + R_2 eq 0 $, with constraints involving $ I(U_k; Y_k | V_k, S_k) $, $ I(X_{2,k}; Y_k | X_{1,k}, S_k, V_k) $, and $ I(U_k; S_k) $.
• The use of auxiliary random variables $ U_k $ and $ V_k $ enables the characterization of the capacity region by capturing the causal dependency of Encoder 2’s input on Encoder 1’s past inputs and the state.
• The outer bound is tight and matches the inner bound, proving that the derived region is indeed the capacity region for both strictly causal and causal cribbing models.
• The results generalize prior work on cooperative MACs by removing the assumption of pre-shared messages and instead modeling practical information exchange via causal cribbing.