Multi-user two-way deterministic modulo 2 adder channels — When adaptation is useless

Abstract

In two-way channels nodes are both sources and destinations of messages, allowing them to "adapt" or "interact" in the sense that their next channel input may be a function of their past received signals. This "adaptation" and how to best exploit it lies at the heart of two-way communication problems, rendering them particularly complex and challenging. It would be useful to know when adaptation is not beneficial from a capacity perspective. Certain examples exist: it is known that for the pointto-point two-way modulo 2 adder channel, and the point-to-point Gaussian two-way channel, adaptation does not increase capacity. In this work we show that the same is true for certain classes of deterministic multi-user two-way channels. In particular, we consider a class of multi-user two-way modulo 2 adder channels, which include the two-way modulo 2 adder MAC/BC channel, the two-way modulo 2 adder interference channel, and the twoway modulo 2 adder Z channel. For all three channel models we obtain the capacity region, which may be achieved using simple time-sharing.

On the capacity of multi-user two-way linear deterministic channels

“…This path is "blocked" by the partial adaptation assumption, as node 3 could not adapt to carry M 12 . However, it should be pointed out that this is not necessary in general: adaptation in the two-way modulo 2 adder IC is useless [14], but the path is not blocked.…”

“…In all these channel models only two messages are present and the backwards feedback links, whether perfect, noisy, or interfering still serve only to further rates in the forward direction. The tradeoff between sending new information versus feedback on each of the links is not addressed; the only other example of such a 4-message two-way interference channel besides our prior work [14], [15] is in Section VI of [13], where an example of a linear deterministic scheme in a specific regime is provided.…”

“…The two-way versions of the binary modulo 2 MAC/BC, Z and IC were first considered in [14], where adaptation was shown to be useless -time-sharing achieved capacity. Here we extend the limited results on multi-user two-way channels to the more general class of linear deterministic channel models.…”

On constant gaps for the two-way Gaussian interference channel

“…where g jk , for j, k ∈ {1, 2, 3, 4} are the complex channel gains. We assume the power constraints E[|X j | 2 ] ≤ P j = 1, j ∈ {1, 2, 3, 4}, and independent, identically distributed complex Gaussian noise Z j ∼ CN (0, 1) at all nodes j ∈ (1,2,3,4), which may be done…”

“…In particular, in [3] we obtained the capacity regions of the deterministic, binary modulo 2 adder models for all three channels, where it was shown that adaptation at the nodes does not increase the capacity regions beyond non-adaptive schemes. In follow-up work in [4] we considered a slightly more general class of deterministic channels: the linear deterministic channels in the spirit of [5].…”