[Paper Review] Capacity Region of the Finite-State Multiple Access Channel with and without Feedback

Haim H. Permuter, Tsachy Weissman|ArXiv.org|Aug 2, 2007

Wireless Communication Security Techniques29 references4 citations

TL;DR

This paper establishes the capacity region of the finite-state multiple access channel (FS-MAC) with arbitrary time-invariant feedback using directed information. It proves that for FS-MACs with input-independent states, feedback does not enlarge the capacity region, generalizing Alajaji’s point-to-point result to multi-user settings and showing that source-channel separation holds for a class of decomposable MACs.

ABSTRACT

The capacity region of the Finite-State Multiple Access Channel (FS-MAC) with feedback that may be an arbitrary time-invariant function of the channel output samples is considered. We characterize both an inner and an outer bound for this region, using Masseys's directed information. These bounds are shown to coincide, and hence yield the capacity region, of FS-MACs where the state process is stationary and ergodic and not affected by the inputs. Though `multi-letter' in general, our results yield explicit conclusions when applied to specific scenarios of interest. E.g., our results allow us to: - Identify a large class of FS-MACs, that includes the additive mod-2 noise MAC where the noise may have memory, for which feedback does not enlarge the capacity region. - Deduce that, for a general FS-MAC with states that are not affected by the input, if the capacity (region) without feedback is zero, then so is the capacity (region) with feedback. - Deduce that the capacity region of a MAC that can be decomposed into a `multiplexer' concatenated by a point-to-point channel (with, without, or with partial feedback), the capacity region is given by $\sum_{m} R_m \leq C$, where C is the capacity of the point to point channel and m indexes the encoders. Moreover, we show that for this family of channels source-channel coding separation holds.

Motivation & Objective

• To characterize the capacity region of the finite-state multiple access channel (FS-MAC) when feedback is available as an arbitrary time-invariant function of channel outputs.
• To establish both inner and outer bounds on the feedback capacity region using directed information, and show they coincide under specific conditions.
• To determine whether feedback increases the capacity region for FS-MACs where the state process is stationary, ergodic, and independent of inputs.
• To extend known results on feedback capacity from point-to-point channels to the multiple access channel setting, particularly for memory-impairing noise models.
• To investigate the validity of source-channel coding separation in a class of MACs composed of a multiplexer followed by a point-to-point channel with or without feedback.

Proposed method

• Uses Masseys’s directed information to derive inner and outer bounds on the feedback capacity region of the FS-MAC.
• Applies the information-spectrum method and code-tree constructions to model causal encoding and feedback dependence in multi-terminal systems.
• Employs concatenation of code-trees (an extension of Kramer’s method for discrete memoryless MACs) to handle the feedback-dependent encoding process.
• Leverages Gallager’s finite-state channel framework and Lapidoth-Telatar’s compound channel techniques to analyze the asymptotic behavior of directed information terms.
• Uses a modified union bound with exponentiation and optimization over auxiliary parameters to bound error probabilities in the decoding process.
• Applies Lemma 21 on the equality of max and min directed information terms under indecomposability to show convergence of bounds to zero.

Experimental results

Research questions

• RQ1 Does feedback increase the capacity region of a finite-state multiple access channel when the state process is stationary, ergodic, and independent of the inputs?
• RQ2 What is the precise characterization of the feedback capacity region for a general FS-MAC with arbitrary feedback functions?
• RQ3 Can the capacity region of a MAC that decomposes into a multiplexer followed by a point-to-point channel be expressed as a sum-rate constraint involving the point-to-point channel capacity?
• RQ4 Under what conditions does source-channel coding separation hold for feedback-capacity-achieving schemes in MACs?
• RQ5 Does feedback provide any rate gain in the additive mod-q MAC with memory in the noise, even when feedback is available?

Key findings

• For FS-MACs with stationary, ergodic, and input-independent states, the inner and outer bounds on the feedback capacity region coincide, yielding a complete characterization.
• Feedback does not increase the capacity region of the additive mod-q MAC with memory in the noise, regardless of the memory structure of the noise.
• If the capacity region of an FS-MAC without feedback is zero, then the feedback capacity region is also zero, under the same state conditions.
• For MACs that can be decomposed as a multiplexer followed by a point-to-point channel (with or without feedback), the capacity region is given by the sum-rate constraint $\sum_m R_m \leq C$, where $C$ is the point-to-point channel capacity.
• Source-channel coding separation holds for the class of decomposable MACs where the channel is a multiplexer followed by a point-to-point channel with feedback.
• The directed information-based bounds converge to zero under the stated conditions, proving tightness of the characterization and validating the use of code-trees in feedback-aided multi-user systems.