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Two-user real AWGN MAC with perfect causal output feedback

Definition

Use the channel, independent messages, noise law, average joint error, powers $P_{1}, P_{2}$ and nats/use convention of Two-user real AWGN multiple-access channel and no-feedback code. A length- $n$ perfect-feedback code has a Borel-measurable joint decoder $g : R^{n} \to [M_{1}] \times [M_{2}]$ . Its user- $j$ encoder is a deterministic Borel-measurable family $f_{j i} : [M_{j}] \times R^{i - 1} \to R$ , $1 \leq i \leq n$ , with $X_{j i} = f_{j i} (M_{j}, Y^{i - 1})$ ; at $i = 1$ the history is the singleton empty tuple. Both senders receive the same noiseless past output, and there is no message sharing or preshared common randomness. The same expected block-power constraint is imposed.

Facts & Assumptions

Given: A channel and code convention as in Two-user real AWGN multiple-access channel and no-feedback code.

[F1]

Two-user real AWGN multiple-access channel and no-feedback code: messages are independent and uniform, the channel is $Y_{i} = X_{1 i} + X_{2 i} + Z_{i}$ with fresh independent $N (0, N)$ noise, the no-feedback decoder is Borel-measurable, and costs are expected block powers.

Proof

1.1givenF1algebra

For each $i$ , the domain $[M_{j}] \times R^{i - 1}$ records exactly user $j$ 's message and the common output history strictly before time $i$ , so the maps are causal and use no future output; Borel measurability makes each input a measurable function of that history.

2.1F1step 1.1algebra∎

Given a message pair and a past-output history, the Borel encoder maps determine the next inputs and the Gaussian channel supplies a measurable probability kernel. Iterating these kernels for the finite blocklength defines a normalized joint law; the Borel decoder makes the error event measurable, so its average error and the expected powers are well-defined as in the no-feedback model.

Depends on

Two-user real AWGN multiple-access channel and no-feedback code

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 18, AWGN MAC with feedback (standard reference, not scraped)
Amos Lapidoth and Michèle Wigger, On the AWGN MAC with Imperfect Feedback, §III-A (standard reference, not scraped)