Capacity Region of the Finite-State Multiple Access Channel with and without Feedback

TL;DR

This paper characterizes the capacity region of finite-state multiple access channels with feedback, providing explicit bounds and conditions under which feedback influences capacity, with implications for specific channel classes.

Contribution

It derives exact capacity regions for FS-MACs with feedback using directed information, including cases where feedback does not enlarge capacity and for channels decomposable into multiplexers.

Findings

Feedback does not enlarge capacity for certain FS-MACs including additive mod-2 noise MAC.

If capacity without feedback is zero, then capacity with feedback is also zero.

For channels decomposable into multiplexers and point-to-point links, the capacity region is characterized by a sum-rate bound.

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.