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Private-message code for a physically degraded DM broadcast channel

Definition

For the channel of Physically degraded two-receiver discrete memoryless broadcast channel, a length- $n$ private-message code has independent uniform messages $M_{1} \in [M_{1}]$ and $M_{2} \in [M_{2}]$ , a single deterministic encoder $f : [M_{1}] \times [M_{2}] \to X^{n}$ , and decoders $g_{j} : Y_{j}^{n} \to [M_{j}]$ . Receiver $j$ estimates only $M_{j}$ . Average joint error is $\Pr [{\hat{M}}_{1} \neq M_{1} or {\hat{M}}_{2} \neq M_{2}]$ . Rates are $R_{j} = n^{- 1} \ln M_{j}$ nats/use; capacity means the closure of rate pairs achievable with vanishing joint error. No feedback is present.

Facts & Assumptions

Given: A physically degraded finite broadcast channel and independent finite private messages.

[F1]

Physically degraded two-receiver discrete memoryless broadcast channel: the block law is memoryless and has $X \to Y_{1} \to Y_{2}$ at each use.

Proof

1.1givenF1

The single encoder fixes the input word for each message pair; the finite product law in [F1] then defines both receiver output sequences jointly.

2.1givenstep 1.1algebra∎

Both decoder error events are measurable subsets of finite output alphabets, and their union is the stated joint error; the rate and closure conventions are therefore well-defined.

Depends on

Physically degraded two-receiver discrete memoryless broadcast channel

Used by

Physically degraded two-user DM broadcast capacity region Theorem

Dependency tree · two levels

2 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

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 15, §§15.6.1–15.6.3 (standard reference, not scraped)
Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 5, Degraded Broadcast Channels (standard reference, not scraped)