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Lossless transmission of two correlated IID sources over a DM-MAC

Definition

Let $(S_{1 t}, S_{2 t})_{t \geq 1}$ be the $k = 2$ case of the finite IID multisource in Finite-alphabet IID multisource, with finite nonempty component alphabets and common joint law $P_{S_{1} S_{2}}$ . A length- $n$ lossless source-channel code over a DM-MAC as in Two-user discrete memoryless multiple-access channel has separate deterministic encoders $f_{j} : S_{j}^{n} \to X_{j}^{n}$ and one decoder $g : Y^{n} \to S_{1}^{n} \times S_{2}^{n}$ . Its error is $\Pr [g (Y^{n}) \neq (S_{1}^{n}, S_{2}^{n})]$ . Reliable transmission means a sequence of such codes with error tending to zero. All source entropies are in nats and use Finite conditional entropy.

Facts & Assumptions

Given: A finite IID source pair and the finite DM-MAC.

[F1]

Two-user discrete memoryless multiple-access channel: the channel block law is $\prod_{i} W (y_{i} ∣ x_{1 i}, x_{2 i})$ .

[F2]

Finite conditional entropy: finite conditional entropy omits zero-mass conditioning values and is a finite weighted sum.

[F3]

Finite-alphabet IID multisource: the source pair has block law $P_{S_{1} S_{2}}^{\otimes n}$ and the component sequences are the separate terminal observations.

Proof

1.1givenF1F3

The finite IID source law [F3] and deterministic encoders assign a finite input pair to each source pair; [F1] then gives a normalized conditional distribution of $Y^{n}$ .

2.1givenF2step 1.1algebra∎

The decoder error event is a subset of a finite source-output space, so its probability and the vanishing-error reliability convention are well-defined; the entropy convention in [F2] is finite.

Depends on

Used by

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Sources

Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 15, Joint Source–Channel Coding (standard reference, not scraped)
Abbas El Gamal and Thomas M. Cover, Multiple User Information Theory, Proceedings of the IEEE 68 (1980), §VII (standard reference, not scraped)