Gaussian Interference Networks: Sum Capacity in the Low Interference Regime and New Outer Bounds on the Capacity Region

V. Sreekanth Annapureddy; Venugopal V. Veeravalli

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Gaussian Interference Networks: Sum Capacity in the Low Interference Regime and New Outer Bounds on the Capacity Region

V. Sreekanth Annapureddy, Venugopal V. Veeravalli

arXiv:0802.3495v2Feb 24, 2008cs.IT

TL;DR

The paper addresses the open problem of characterizing Gaussian interference-network capacity regions. It develops improved genie-aided outer bounds and related information-theoretic tools, then shows that treating interference as noise achieves sum capacity in low-interference regimes, including extensions beyond two users.

Problem
Characterizing the capacity region of Gaussian interference networks remains an open information-theoretic problem outside established strong-interference settings.
Method
The paper develops improved genie-aided outer bounds using wider classes of genie signals, entropy inequalities, and generalized constructions for larger networks.
Results
Treating interference as noise achieves the sum capacity in low-interference regimes for two-user and larger Gaussian interference networks.
Takeaways & Limitations
The total interference threshold below which treating interference as noise is sum-capacity optimal can be higher for networks with more users than for two-user channels.

Abstract

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Establishing the capacity region of a Gaussian interference network is an open problem in information theory. Recent progress on this problem has led to the characterization of the capacity region of a general two user Gaussian interference channel within one bit. In this paper, we develop new, improved outer bounds on the capacity region. Using these bounds, we show that treating interference as noise achieves the sum capacity of the two user Gaussian interference channel in a low interference regime, where the interference parameters are below certain thresholds. We then generalize our techniques and results to Gaussian interference networks with more than two users. In particular, we demonstrate that the total interference threshold, below which treating interference as noise achieves the sum capacity, increases with the number of users.

I. INTRODUCTION

The paper develops improved outer bounds for Gaussian interference networks and shows that treating interference as noise achieves sum capacity when interference is sufficiently low. The results extend to networks with more than two users, where the total interference threshold can increase.

I. INTRODUCTION: The capacity region outside very strong and strong interference regimes remains an open problem, despite the Han-Kobayashi scheme providing the best known achievable region.The Han-Kobayashi scheme splits messages into private and common parts and requires coordinated multi-user encoding and decoding.
I. INTRODUCTION: At sufficiently low interference levels, treating interference as noise with single-user encoders and decoders achieves the sum capacity without loss.The paper establishes this by matching a genie-aided outer bound to the sum rate achieved by treating interference as noise.
I. INTRODUCTION: A wider class of genie signals, combined with the entropy power inequality, yields tighter outer bounds on the entire two-user capacity region.The bounding technique extends prior genie-aided methods and produces bounds tighter than existing ones.
I. INTRODUCTION: The techniques extend to many-to-one, one-to-many, and arbitrary Gaussian interference networks with more than two users.The paper introduces a genie construction that provides multiple genie signals to each receiver.
I. INTRODUCTION: For a three-user symmetric channel, treating interference as noise can remain optimal even when total interference-to-noise ratio exceeds the two-user threshold.The paper tightens the bound for this setting and demonstrates channels with this property.

III. MATHEMATICAL PRELIMINARIES

This section develops information inequalities used to establish the paper's outer bounds. It extends worst-case noise results to multiple independent inputs and establishes auxiliary Gaussian and Markov-chain lemmas.

III. MATHEMATICAL PRELIMINARIES: The section introduces a generalized maximum-entropy result for noisy observations of random vectors under covariance constraints.This result is used as a mathematical preliminary for the new outer bounds.
III. MATHEMATICAL PRELIMINARIES: The entropy power inequality provides the scalar worst-case noise result, with equality achieved by i.i.d. Gaussian inputs.The corresponding scalar statement concerns random sequences under an average power constraint.
III. MATHEMATICAL PRELIMINARIES: The vector worst-case noise result identifies i.i.d. Gaussian vectors as achieving equality under an average covariance constraint.The section notes that this vector result does not follow from the entropy power inequality unless the covariance is a scaled identity.
III. MATHEMATICAL PRELIMINARIES: A multi-input extension shows that independent i.i.d. Gaussian sequences achieve equality in the relevant worst-case noise inequality.The result is intended for outer bounds on interference networks with more than two users.
III. MATHEMATICAL PRELIMINARIES: Additional lemmas characterize conditional entropy with correlated Gaussian side information and conditional mutual information through Markov chains.For Gaussian variables, conditional independence from multiple side-information components is equivalent to conditional independence from their combined side information.

IV. TWO USER INTERFERENCE CHANNEL: EXISTING BOUNDS

The paper reviews Gaussian interference networks and focuses on the two-user channel, whose capacity region is known in very strong and strong interference settings but remains unresolved in weak interference.

IV. TWO USER INTERFERENCE CHANNEL: EXISTING BOUNDS: A Gaussian interference network consists of transmitter-receiver pairs in which each receiver seeks only its corresponding transmitter's information.The two-user channel is parameterized by transmit powers and cross-channel gains.
IV. TWO USER INTERFERENCE CHANNEL: EXISTING BOUNDS: The two-user capacity region is known in very strong and strong interference settings, where both receivers can decode all transmitted messages.In these regimes, the capacity region equals that of a compound multiple access channel.

A. Inner bounds

The section reviews simple and sophisticated achievable strategies and genie-aided outer bounds for the two-user Gaussian interference channel. Treating interference as noise supplies a simple inner bound, while genie constructions support tighter converse bounds.

A. Inner bounds: With no interference, single-user Gaussian codebooks achieve capacity, while time or frequency orthogonalization provides an alternative simple strategy.The paper uses treating interference as noise when interference is sufficiently low.
A. Inner bounds: The treating-interference-as-noise strategy gives a lower bound on the sum capacity based on each user's signal-to-interference-plus-noise ratio.The displayed expression contains the terms 1 + P1 over 1 + h2_12P2 and 1 + P2 over 1 + h2_21P1.
A. Inner bounds: Sophisticated schemes exploit interference structure, but the Han-Kobayashi region remains formidable despite later simplifications.These schemes split messages and jointly decode selected information from the interfering user.
A. Inner bounds: The broadcast channel outer bound is a tightened version of the Z-channel sum-rate outer bound and is used to tighten the ETW outer bound.The paper gives a direct proof of this connection before developing later bounds.
A. Inner bounds: A genie-aided channel is an outer bound because receivers may ignore the supplied side information.The paper restricts attention to linear side information with additive Gaussian noise that is i.i.d. in time.

D. Etkin, Tse and Wang (ETW) Outer Bound [9]

The ETW outer bound uses genie signals to upper-bound the capacity region of the two-user Gaussian interference channel. It relies on worst-case noise relations and bounds involving individual and weighted sums of user rates.

The genie signals are selected so worst-case noise relations and related identities support the outer-bound derivations.
The bounds use Gaussian inputs to maximize the relevant entropy expressions under the stated power constraints.
The ETW outer bound applies to a two-user Gaussian interference channel with h12 ≤1 and h21 ≤1.
R1 + R2, 2R1 + R2, and R1 + 2R2 are bounded using mutual-information expressions involving genie signals.
The resulting outer bound has a form similar to a simplified Han–Kobayashi region and is within one bit in a special case.
For one-sided interference, the broadcast-channel outer bound is identified as a tightened version of the Z-channel sum-rate outer bound.

F. Tightening the Outer Bounds

The paper tightens existing outer bounds using generalized genie signals and EPI-based arguments. These bounds establish that treating interference as noise achieves sum capacity below explicit interference thresholds.

Generalized genie signals extend the earlier genie class by using worst-case noise relations rather than canceling terms.
Treating interference as noise achieves sum capacity in a low but nonzero interference regime.
For the symmetric channel, the threshold is characterized by the condition |h + h^3P| ≤0.5.
The new outer bound matches the inner bound obtained by treating interference as noise when interference is below a threshold.
A useful genie makes Gaussian inputs optimal for the genie-aided channel, while a smart genie does not increase the Gaussian-input sum rate.
A genie that is both useful and smart yields an outer bound equal to the sum rate achieved by treating interference as noise.
In the high-SNR asymptotic regime, the INR threshold in dB equals one third of the SNR.

B. Asymmetric Interference Channel

For the asymmetric Gaussian interference channel, the paper constructs an asymmetric genie and gives conditions under which treating interference as noise achieves sum capacity.

The asymmetric channel uses interference parameters h12 and h21 and an asymmetric genie with correlations ρ1 and ρ2.
The resulting sum capacity is expressed through the two users’ signal-to-noise terms, including 1 + P1 over 1 + h2_12P2 and 1 + P2 over 1 + h2_21P1.
Treating interference as noise achieves sum capacity when the asymmetric genie is simultaneously useful and smart under the stated conditions.

VI. TWO-USER INTERFERENCE CHANNEL: OUTER BOUNDS TO THE CAPACITY REGION

The paper develops EPI-based tightenings of the two-user Gaussian interference-channel outer bounds. The resulting theorem combines tightened bounds on the sum rate and weighted rate sums.

The EPI-based approach applies the entropy power inequality to a generalized class of genie signals instead of relying only on the earlier worst-case noise argument.
Theorem 3 outer-bounds the capacity region using the tightened bounds in Lemmas 13 and 14 together with Lemma 10.
Lemma 13 tightens the outer bound on R1 + R2 for channels with h12 ≤1 and h21 ≤1.
The derivation introduces slack variables, applies the EPI, and eliminates them to obtain the tightened bounds.
Lemma 14 tightens the outer bounds on 2R1 + R2 and R1 + 2R2.

A. Numerical Results

The numerical comparisons show that the EPI-based ETW outer bound is tighter than earlier outer bounds. In low-interference cases, the inner and outer bounds meet, establishing sum capacity.

The comparisons use the EPI-based ETW bound alongside the original ETW, broadcast-channel, and Han–Kobayashi-related bounds.
The EPI-based ETW outer bound contains the original ETW and broadcast-channel outer bounds as special cases, making it tighter.
In Figure 5’s low-interference setting, the inner and outer bounds meet at one point, yielding the sum capacity.
Figure 6 illustrates a higher-interference case where the relevant low-interference condition fails and the inner and outer bounds do not meet.

A. Many-to-one and One-to-many interference channels

The paper analyzes many-to-one and one-to-many Gaussian interference channels as special network cases. Under stated channel conditions, treating interference as noise achieves their sum capacity.

The many-to-one channel has only one receiver experiencing interference, while the one-to-many channel has only one interfering transmitter.
Many-to-one: For the many-to-one channel, treating interference as noise achieves the sum capacity under the conditions of Theorem 4.
One-to-many: For the one-to-many channel, treating interference as noise achieves the sum capacity under the conditions of Theorem 5.

B. Vector genie

The paper constructs a vector genie that supplies multiple side-information signals to each receiver and uses it to derive outer bounds for arbitrary Gaussian interference networks. For the symmetric three-user channel, these bounds establish sum-capacity optimality of treating interference as noise under numerically testable conditions.

Construction: The vector genie gives each receiver multiple side-information signals and generalizes the two-user ETW genie.
Outer bound: The genie construction includes an interference-free signal at each receiver, a property used in deriving the outer bound.
Outer bound: For any ordering function, the vector genie is useful and yields an outer bound on the network sum capacity.
Three-user channel: For the symmetric three-user channel, treating interference as noise achieves sum capacity when the covariance and genie parameters satisfy Theorem 7’s conditions.
Three-user channel: The paper does not provide an explicit three-user threshold equation as a function of P; admissible h values are found numerically.
Three-user channel: The vector genie raises the three-user INRtotal threshold by more than 1 dB relative to the scalar genie.
Three-user channel: The three-user INRtotal threshold obtained with the vector genie exceeds the two-user INR threshold.
Implication: The reported thresholds are lower bounds on the optimal threshold, although the authors believe the threshold increases with the number of users.

VIII. CONCLUSIONS

The paper develops improved genie-aided outer bounds and extends treating-interference-as-noise optimality to networks with more than two users. It derives closed-form low-interference conditions, tightens the symmetric three-user bound, and shows that total interference thresholds can exceed the two-user threshold, while optimal scaling remains open.

New genie-aided outer bounds improve the capacity-region analysis for the two-user Gaussian interference channel.
Treating interference as noise achieves sum capacity in a low-interference regime for Gaussian interference networks with more than two users.
Closed-form expressions characterize the low-interference regime for many-to-one and one-to-many interference channels.
The vector genie generalizes the ETW genie to arbitrary Gaussian interference networks by providing multiple side-information signals to each receiver.
Correlating vector-genie noise terms tightens the outer bound and establishes sum-capacity results for a three-user symmetric interference channel.For computational-complexity reasons, the analysis considers only the three-user symmetric case.
The total interference threshold can be higher than in the two-user case, but the optimal threshold as a function of the number of interferers remains unanswered.