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A numerical asymmetric Gaussian channel in the strong regime

Example

Use the complex Gaussian channel with $P_{1} = P_{2} = 1$ , unit noise variance, $h_{11} = 1$ , $h_{22} = 3 / 2$ , $h_{21} = 2$ , and $h_{12} = 5 / 2$ in the convention of Real and complex two-user Gaussian interference channels. Then ${S N R}_{1} = 1, {S N R}_{2} = \frac{9}{4}, {I N R}_{1} = 4, {I N R}_{2} = \frac{25}{4},$ and the exact capacity region is $R_{1} \geq 0, R_{2} \geq 0, R_{1} \leq 1, R_{2} \leq \log_{2} \frac{13}{4}, R_{1} + R_{2} \leq \log_{2} 6$ bits per use.

Facts & Assumptions

Given: The complex Gaussian interference channel and parameters in the example.

[A1]

Assume AC (The Axiom of Choice), as required by the Gaussian information and coding suppliers used in [F2].

[F1]

Real and complex two-user Gaussian interference channels: ${I N R}_{1} = ∣ h_{21} ∣^{2} P_{2}$ and ${I N R}_{2} = ∣ h_{12} ∣^{2} P_{1}$ ; complex scalar rates use $κ = 1$ .

[F2]

Exact strong-interference region for the two-user Gaussian interference channel: if $∣ h_{12} ∣ \geq ∣ h_{11} ∣$ and $∣ h_{21} ∣ \geq ∣ h_{22} ∣$ , capacity is the stated intersection of the two Gaussian MAC regions.

Proof

1.1A1F1F2givenalgebra

Assume [A1] and use the complex channel model [F1] with the stated unit power budgets and unit-variance noises. The gains satisfy $∣ h_{12} ∣ = 2.5 \geq 1 = ∣ h_{11} ∣$ and $∣ h_{21} ∣ = 2 \geq 1.5 = ∣ h_{22} ∣$ , so the strong-regime hypothesis of [F2] holds. The displayed SNR and INR values follow by squaring the gains and multiplying by unit powers.

2.1F1F2step 1.1algebra∎

The individual constraints in [F2] become $R_{1} \leq \log_{2} (2) = 1$ and $R_{2} \leq \log_{2} (1 + 9 / 4) = \log_{2} (13 / 4)$ . The two sum constraints are $\log_{2} (1 + 1 + 4) = \log_{2} 6$ and $\log_{2} (1 + 9 / 4 + 25 / 4) = \log_{2} (19 / 2)$ ; since $6 < 19 / 2$ , the first is active. This gives exactly the stated region.

Depends on

Used by

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Sources

Abbas El Gamal and Young-Han Kim, Lecture Notes on Network Information Theory, arXiv:1001.3404v4 (standard reference, not scraped)