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Physically degraded two-receiver discrete memoryless broadcast channel

Definition

A finite physically degraded two-receiver DM broadcast channel has finite nonempty input/output alphabets $X, Y_{1}, Y_{2}$ and kernels $W_{1} (y_{1} ∣ x)$ and $W_{2} (y_{2} ∣ y_{1})$ such that $W (y_{1}, y_{2} ∣ x) = W_{1} (y_{1} ∣ x) W_{2} (y_{2} ∣ y_{1}) .$ Receiver 1 is called the stronger receiver. This factorization fixes a joint coupling of the outputs and gives the Markov chain $X \to Y_{1} \to Y_{2}$ for every input law. No feedback is part of this definition.

Facts & Assumptions

Given: Finite nonempty alphabets and finite stochastic kernels $W_{1}, W_{2}$ .

[F1]

Finite stochastic kernels: each row of each kernel is a probability vector.

Proof

1.1givenF1algebra

For every $x$ , summing $W_{1} (y_{1} ∣ x) W_{2} (y_{2} ∣ y_{1})$ over $y_{1}, y_{2}$ gives $\sum_{y_{1}} W_{1} (y_{1} ∣ x) = 1$ by [F1], so the displayed joint law is a stochastic kernel.

2.1givenF1algebra∎

The factorization makes the conditional law of $Y_{2}$ given $(X, Y_{1})$ equal to $W_{2} (\cdot ∣ Y_{1})$ ; hence $X$ and $Y_{2}$ are conditionally independent given $Y_{1}$ , which is exactly $X \to Y_{1} \to Y_{2}$ .

Depends on

Finite stochastic kernels

Used by

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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)