TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pending

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.

Exact strong-interference region for the two-user Gaussian interference channel

Statement

For the real or complex Gaussian interference channel of Real and complex two-user Gaussian interference channels, assume the strong regime $∣ h_{12} ∣ \geq ∣ h_{11} ∣, ∣ h_{21} ∣ \geq ∣ h_{22} ∣ .$ Then the capacity region is the set of nonnegative rate pairs satisfying $R_{1} \leq κ \log_{2} (1 + {S N R}_{1}), R_{2} \leq κ \log_{2} (1 + {S N R}_{2}),$ $R_{1} + R_{2} \leq κ \min {\log_{2} (1 + {S N R}_{1} + {I N R}_{1}), \log_{2} (1 + {S N R}_{2} + {I N R}_{2})},$ where $κ = 1 / 2$ for real and $1$ for complex channels.

Facts & Assumptions

Given: A length- $n$ interference code with independent messages, the channel and power constraints of [F1], and the two gain inequalities in the statement.

[A1]

The Axiom of Choice: AC is inherited through the Gaussian random-coding and scalar-channel information suppliers.

[F1]

Real and complex two-user Gaussian interference channels: noises are independent unit-variance real or circular complex Gaussian; the input power constraints and SNR/INR conventions are as displayed there.

[F2]

Random coding achieves the finite-dimensional Gaussian multiple-access region: independent Gaussian codebooks achieve both receivers' multiple-access regions.

[F3]

Fano bound for finite messages and density observations: a finite message estimated from a density observation with error $P_{e}$ has conditional entropy at most $\ln 2 + P_{e} \ln ∣ M ∣$ .

[F4]

Mutual information tensorizes over a memoryless vector channel: for a memoryless channel and any dependent input vector, vector mutual information is at most the sum of its coordinate mutual informations.

[F5]

Conditional mutual-information chain rule and data processing for standard-Borel variables: conditional mutual information obeys the chain rule and the data-processing inequality for standard-Borel variables, including finite message side information.

[F6]

Power bound for real and complex scalar Gaussian channels: for either scalar Gaussian model, $I (X; a X + Z) \leq κ \ln (1 + ∣ a ∣^{2} E ∣ X - E X ∣^{2})$ nats, with equality for the corresponding Gaussian inputs.

Proof

1.1A1F1F3F4F5F6givenalgebra

For the individual bound on $R_{1}$ , give $M_{2}$ to receiver 1. Fano gives $n R_{1} \ln 2 \leq I (M_{1}; Y_{1}^{n} ∣ M_{2}) + o (n)$ in nats. For each fixed $M_{2}$ , $X_{2}^{n}$ is fixed; translating $Y_{1}^{n}$ by the known vector $- h_{21} X_{2}^{n}$ is invertible, so [F5] in both directions preserves this conditional mutual information. The shifted channel from $X_{1}^{n}$ is memoryless. The codeword map is deterministic and the channel law depends on $M_{1}$ only through $X_{1}^{n}$ , so [F5] in both Markov directions gives $I (M_{1}; Y_{1}^{n} ∣ M_{2}) = I (X_{1}^{n}; h_{11} X_{1}^{n} + Z_{1}^{n} ∣ M_{2})$ . Apply [F4] for each $M_{2}$ and average to bound this by $\sum_{i} I (X_{1, i}; h_{11} X_{1, i} + Z_{1, i})$ . By [F6], each term is at most $κ \ln (1 + ∣ h_{11} ∣^{2} p_{1, i})$ nats, where $p_{1, i} = E ∣ X_{1, i} ∣^{2}$ . Jensen's inequality and the power constraint give $R_{1} \leq κ \log_{2} (1 + {S N R}_{1})$ . Exchanging users gives the stated $R_{2}$ bound.

1.2F1givenconstructalgebra

First suppose both cross gains are nonzero. Receiver 1 first forms its estimate ${\hat{M}}_{1}$ . On the event ${\hat{M}}_{1} = M_{1}$ , it reconstructs $X_{1}^{n}$ and subtracts $h_{11} X_{1}^{n}$ , obtaining $h_{21} X_{2}^{n} + Z_{1}^{n}$ . Since $∣ h_{21} ∣ \geq ∣ h_{22} ∣$ , it scales this residual by $h_{22} / h_{21}$ and adds independent Gaussian noise of variance $1 - ∣ h_{22} / h_{21} ∣^{2}$ ; for complex channels use circular Gaussian noise. Adding back $h_{12} X_{1}^{n}$ gives a simulated ${\tilde{Y}}_{2}^{n}$ . For the error analysis, construct the same vector using the true $M_{1}$ : it then has exactly the law of receiver 2's $Y_{2}^{n}$ , and on ${\hat{M}}_{1} = M_{1}$ it equals the receiver's simulation. Hence the joint error is at most $P_{e, 1} + P_{e, 2}$ . Since the message pair is finite, the randomized decoder has a deterministic MAP decoder with no larger average error. Exchanging users and using $∣ h_{12} ∣ \geq ∣ h_{11} ∣$ gives the symmetric joint decoder at receiver 2.

2.1F1F3F4F5F6step 1.2algebra

In the nonzero-cross-gain case of step 1.2, apply the density-observation Fano bound [F3] to the joint decoder at receiver 1. Let $M = (M_{1}, M_{2})$ and $X = (X_{1}^{n}, X_{2}^{n})$ . The deterministic codeword map and channel law give both Markov directions between $M$ and $X$ with observation $Y_{1}^{n}$ , so [F5] yields $I (M; Y_{1}^{n}) = I (X; Y_{1}^{n})$ . The two input vectors are independent because their messages are independent. By [F4], $n (R_{1} + R_{2}) \ln 2 \leq I (X_{1}^{n}, X_{2}^{n}; Y_{1}^{n}) + o (n) \leq \sum_{i} I (X_{1, i}, X_{2, i}; Y_{1, i}) + o (n)$ . The output variance at time $i$ is at most $1 + ∣ h_{11} ∣^{2} E ∣ X_{1, i} ∣^{2} + ∣ h_{21} ∣^{2} E ∣ X_{2, i} ∣^{2}$ , because the independent inputs have zero cross covariance after centering. These single-coordinate mutual informations are finite by the power bound and [F6].

3.1F1F5F6step 1.1step 1.2step 2.1algebracases

For each coordinate put $S_{i} = h_{11} X_{1, i} + h_{21} X_{2, i}$ . The channel output depends on the input pair only through $S_{i}$ , so [F5] in both directions gives $I (X_{1, i}, X_{2, i}; Y_{1, i}) = I (S_{i}; S_{i} + Z_{1, i})$ . Independence gives $Var (S_{i}) \leq ∣ h_{11} ∣^{2} p_{1, i} + ∣ h_{21} ∣^{2} p_{2, i}$ ; [F6] therefore bounds this term by $κ \ln (1 + ∣ h_{11} ∣^{2} p_{1, i} + ∣ h_{21} ∣^{2} p_{2, i})$ nats. Jensen's inequality and the average power constraints yield $R_{1} + R_{2} \leq κ \log_{2} (1 + {S N R}_{1} + {I N R}_{1})$ . The joint decoder at receiver 2 analogously gives $R_{1} + R_{2} \leq κ \log_{2} (1 + {S N R}_{2} + {I N R}_{2})$ . If $h_{21} = 0$ , then $h_{22} = 0$ , so $R_{2} = 0$ by step 1.1; the first sum bound is the $R_{1}$ individual bound and the second is no tighter because ${I N R}_{2} \geq {S N R}_{1}$ . If $h_{12} = 0$ , then $h_{11} = 0$ , so $R_{1} = 0$ and the second sum bound is the $R_{2}$ individual bound while the first is no tighter because ${I N R}_{1} \geq {S N R}_{2}$ . Thus both sum bounds hold also when a cross gain vanishes.

4.1F1F2step 1.1algebra∎

Use the same independent Gaussian codebook ensemble at the two receivers. The finite-receiver construction in the proof of [F2] sums the two ensemble-average errors and jointly selects codebooks meeting both deterministic power constraints, so one codebook pair achieves the intersection of the two Gaussian MAC regions. The strong gain inequalities make each unintended-receiver individual constraint at least as large as the corresponding intended-receiver constraint; the two sum constraints are exactly those displayed. Therefore every rate pair strictly inside the claimed region is achievable, and closure removes strictness.

Depends on

Used by

Dependency tree · two levels

23 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, Lecture Notes on Network Information Theory, arXiv:1001.3404v4 (standard reference, not scraped)
M. H. M. Costa and Abbas A. El Gamal, The Capacity Region of the Discrete Memoryless Interference Channel with Strong Interference, IEEE Transactions on Information Theory 33(5), 1987 (standard reference, not scraped)