DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Definition

Fix powers $P_{1}, P_{2} \geq 0$ and noise variance $N > 0$ . At use $i$ the real channel is $Y_{i} = X_{1 i} + X_{2 i} + Z_{i},$ where the noises are independent of the messages and of one another, each with Lebesgue density $ϕ_{N} (z) = (2 π N)^{- 1 / 2} e^{- z^{2} / (2 N)}$ . The normalization is proved in step 1.1 below. A length- $n$ no-feedback code has independent uniform messages $M_{j} \in [M_{j}]$ , separate deterministic encoders $f_{j} : [M_{j}] \to R^{n}$ , and a Borel-measurable joint decoder $g : R^{n} \to [M_{1}] \times [M_{2}]$ , where the finite message sets have their discrete sigma-algebras. The encoders are measurable because their domains are finite. Write $X_{j}^{n} = f_{j} (M_{j})$ . Its average joint error is $P_{e} = \Pr [g (Y^{n}) \neq (M_{1}, M_{2})]$ and its expected block-power constraints are $n^{- 1} \sum_{i} E X_{j i}^{2} \leq P_{j}$ , separately for $j = 1, 2$ . The rate is $R_{j} = n^{- 1} \ln M_{j}$ nats/use. Achievability means a sequence with error tending to zero and limiting lower rates at least the specified pair; the capacity region is the closure of all achievable pairs.

Facts & Assumptions

Given: $P_{1}, P_{2} \geq 0$ , $N > 0$ , independent finite uniform messages and fresh independent Gaussian noises.

[F1]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ : $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ .

[F2]

Substitution: if $φ$ is differentiable on $[c, d]$ with $φ^{'}$ integrable and $f$ is continuous on an interval containing $φ ([c, d])$ , then $\int_{φ (c)}^{φ (d)} f = \int_{c}^{d} (f \circ φ) φ^{'}$ : for a differentiable substitution with integrable derivative and continuous integrand, the oriented integral changes by the derivative factor.

[F3]

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ : each nonnegative real has a unique nonnegative square root.

[F4]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: a nonnegative measurable function on a sigma-finite product has equal iterated and product integrals.

Proof

1.1givenF1F2F3algebra

For each finite $A > 0$ , apply [F2] on the bounded interval $[- A / \sqrt{2 N}, A / \sqrt{2 N}]$ to the linear substitution $x = \sqrt{2 N} t$ . It gives $\int_{- A}^{A} (2 π N)^{- 1 / 2} e^{- x^{2} / (2 N)} d x = \frac{1}{\sqrt{π}} \int_{- A / \sqrt{2 N}}^{A / \sqrt{2 N}} e^{- t^{2}} d t .$ As $A \to \infty$ , the right side tends to $\sqrt{π} / \sqrt{π} = 1$ by the improper-integral statement [F1]; hence the Gaussian density has total mass one.

2.1F4step 1.1algebra

Tonelli's theorem [F4] gives the product density $\prod_{i = 1}^{n} (2 π N)^{- 1 / 2} e^{- z_{i}^{2} / (2 N)}$ total mass one; its coordinates are independent and have the required Gaussian law.

3.1givenstep 2.1algebra∎

The finite message sets, deterministic encoders, Borel decoder and channel law define a joint distribution and a measurable decoder error event; the displayed expected costs and error probability are therefore well-defined, and the closure convention specifies the operational region.

Depends on

Used by

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Sources

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 15, §15.3 (standard reference, not scraped)
Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 4, Gaussian and discrete MACs (standard reference, not scraped)