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A New Outer Bound and the Noisy-Interference Sum-Rate Capacity for Gaussian Interference Channels

Xiaohu Shang, Gerhard Kramer, Biao Chen

arXiv:0712.1987v2cs.IT

TL;DR

The paper addresses the unresolved capacity region of Gaussian interference channels with weak or moderate interference. It develops an improved genie-aided outer bound and shows that, under specified channel and power conditions, treating interference as noise achieves sum-rate capacity, with additional corner-point results for certain mixed-interference channels.

  • Problem

    The Gaussian interference-channel capacity region remains unknown for weak or moderate interference.

  • Method

    The paper develops a genie-aided outer bound that improves existing bounds and does not require either receiver to decode the other transmitter’s messages.

  • Results

    Under channel-coefficient and power-constraint conditions, treating interference as noise achieves sum-rate capacity; another condition yields a mixed-interference corner point.

  • Takeaways & Limitations

    For the defined noisy-interference channels, simple single-user transmission and detection is sum-rate optimal.

Abstract

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A new outer bound on the capacity region of Gaussian interference channels is developed. The bound combines and improves existing genie-aided methods and is shown to give the sum-rate capacity for noisy interference as defined in this paper. Specifically, it is shown that if the channel coefficients and power constraints satisfy a simple condition then single-user detection at each receiver is sum-rate optimal, i.e., treating the interference as noise incurs no loss in performance. This is the first concrete (finite signal-to-noise ratio) capacity result for the Gaussian interference channel with weak to moderate interference. Furthermore, for certain mixed (weak and strong) interference scenarios, the new outer bounds give a corner point of the capacity region.

I. INTRODUCTION

The Gaussian interference channel capacity region remains unknown for weak or moderate interference, despite several existing inner and outer bounds. This paper develops a genie-aided outer bound that improves prior bounds and yields sum-rate capacity results under channel and power conditions.

  • I. INTRODUCTION: The Gaussian interference-channel capacity region is known only for three special cases, while the weak-to-moderate interference regime remains unresolved.Existing known cases include one absent cross-link with the other gain at least one, or the reverse configuration.
  • I. INTRODUCTION: Existing approaches include superposition coding with joint decoding, simplified Han-Kobayashi inner bounds, and several genie-aided outer bounds.Prior outer bounds use receiver cooperation, reduced noise, message-decoding side information, degraded broadcast channels, or other genie-aided constructions.
  • I. INTRODUCTION: None of the previously described outer bounds is known to be tight for the general Gaussian interference channel.Some prior bounds approximate capacity within one bit or a factor of two, while their relative performance varies with SNR.
  • I. INTRODUCTION: The paper introduces a new genie-aided outer bound that improves the bounds of and using a recently proposed extremal inequality.Unlike the method used in [10, Theorem 1], the new construction does not require either receiver to decode the other transmitter’s messages.
  • I. INTRODUCTION: Under channel-coefficient and power-constraint conditions, the paper obtains sum-rate capacity by treating weak interference as noise.For noisy interference, single-user transmission and detection is sum-rate optimal.
  • I. INTRODUCTION: For a > 1 and 0 < b < 1 under another condition, user 1 first recovers user 2’s messages, while user 2 decodes only its own messages.This establishes a capacity-region corner point for a mixed weak-and-strong interference configuration.

II. MAIN RESULTS

The paper introduces a new genie-aided outer bound for Gaussian interference channels, combining techniques that can be further improved by using different genie signals at the receivers. The bound yields tighter sum-rate bounds in several settings and supports results for mixed interference.

  • Theorem 1 provides a new outer bound on the capacity region of Gaussian interference channels.
  • The bounds are obtained by providing different genie-aided signals to the receivers, with the active bound depending on channel conditions and the rate pair.
  • For Z-ICs, bounds (2) and (3) are outer bounds because a Z-IC is equivalent to a degraded interference channel.
  • The new bounds are equivalent to prior bounds obtained with transmitter power sharing, although the derivation implicitly assumes such sharing.
  • Bounds (2) and (3) are tighter than the first and second sum-rate bounds of [11, Theorem 3], respectively.
  • The new outer bound is not always tighter than the bound in for every rate point.

B. Sum-rate capacity for noisy interference

The paper characterizes sum-rate capacity for Gaussian interference channels satisfying the paper’s noisy-interference condition. Under this condition, treating interference as noise achieves the sum-rate capacity, while very strong interference represents an opposite decoding extreme.

  • Theorem 2 gives the sum-rate capacity for Gaussian interference channels satisfying condition (16).
  • For a Z-IC with a = 0 and 0 < b < 1, condition (16) holds and the sum capacity is given by (17).
  • Theorem 2 follows from Theorem 1 with µ = 1, and the genie-aided bound is tight under condition (16).
  • Treating interference as noise achieves the sum-rate capacity for channels satisfying condition (16), which depends on both channel gains and powers.
  • The noisy-interference and very-strong-interference regimes are extremes: one decodes while treating interference as noise, whereas the other decodes interference before or together with intended messages.
  • Noisy interference is characterized as weaker than the paper’s cited definition of weak interference.

C. Capacity region corner point

For mixed Gaussian interference channels with a > 1 and 0 < b < 1, under the stated condition, the sum-rate capacity is achieved at a capacity-region corner point. The result includes degraded and additional channel cases and uses a simple decoding scheme.

  • For IC(a, b, P1, P2) with a > 1 and 0 < b < 1, the sum-rate capacity is characterized under the theorem’s condition.
  • A symmetric result follows by swapping a and b, and P1 and P2.
  • The capacity-achieving scheme has user 1 transmit at its maximum rate while user 2 uses a rate decodable at both receivers with single-user detection.
  • The resulting rate pair is a corner point of the capacity region and achieves the sum-rate capacity when condition (24) holds.
  • The degraded IC with ab = 1 and 0 < b < 1 is a special case of Theorem 3.
  • Theorem 3 also applies when ab > 1 with arbitrary positive P1, and when ab < 1 subject to P1 ≤ a−1.

D. State of the Art

The paper derives Theorems 2 and 3 from a genie-based outer bound and summarizes their coverage across Gaussian interference-channel regimes. The resulting regime classification links channel-gain regions and power constraints to sum-rate capacity expressions.

  • Theorems 2 and 3 are direct consequences of Theorem 1, which is derived by providing receivers with additional genie information.
  • Four curves, including ab = 1 and b ≤ 1, divide the channel-gain plane into seven regimes.
  • Table I lists the sum-rate capacity for each regime under specified power constraints.

III. PROOFS OF THE MAIN RESULTS

The proof section establishes Gaussian extremal-inequality tools and applies them to covariance-constrained optimization problems. These results support the outer-bound and sum-rate-capacity theorems.

  • The proofs use extremal inequalities introduced in prior work and present them as supporting lemmas.
  • Lemmas 1 and 2 state that Gaussian X is optimal for related optimization problems under positive-semidefinite matrix constraints.
  • The optimization variables are Gaussian vectors with specified positive-definite covariance matrices, and X is independent of the auxiliary Gaussian vectors.
  • A general optimization problem is reduced to Lemma 1 when μ ≥ 1 and to Lemma 2 when μ < 1, yielding Gaussian optimality in both cases.
  • Corollaries 1 and 2 extend the optimization results from matrix constraints to trace constraints for the corresponding parameter regimes.
  • The proofs use eigenvalue decomposition of the constraint matrix and establish the associated Gaussian covariance characterization.

1. If the discrete or continuous random

This result shows that a conditional Gaussian variable can be replaced by an equivalent Gaussian variable with the same variance. The replacement is used within the proof’s entropy and independence arguments.

  • The lemma assumes Gaussian variables, specified covariance conditions, and independence of X from the relevant auxiliary variables.
  • The proof introduces an independently distributed Gaussian W′ and uses its matching distributional relationship with (U, V).
  • Lemma 3 shows that U|V can be replaced by an equivalent Gaussian random variable with the same variance.

A. Proof of Theorem 1

The proof derives a genie-aided outer bound using Gaussian auxiliary variables and entropy inequalities, then specializes it under power and correlation constraints to obtain rate constraints.

  • Genie construction: The proof defines correlated Gaussian noise variables whose correlations and variances parameterize the genie construction.The auxiliary variables are i.i.d. across channel uses and satisfy feasibility conditions on variances and correlations.
  • Genie-aided bound: Starting from Fano’s inequality, the proof upper-bounds reliable rates with mutual-information expressions involving genie-provided Gaussian observations.The argument decomposes the resulting terms using auxiliary variables and conditional entropies.
  • Gaussian extremality: Gaussian input distributions are shown to be optimal for the relevant mutual-information expressions under the block power constraints.Concavity and Jensen’s inequality extend the single-letter bounds to block power allocations.
  • Resulting constraints: The entropy calculations yield explicit logarithmic rate constraints, including a bound of 2 log(1 + bP1 + P2).The displayed constraints are obtained from the Gaussian entropy evaluations and the preceding conditional-information inequalities.
  • Constraint simplification: The derived constraints are then simplified using redundancy relations to obtain the stated outer-bound conditions.The proof identifies redundant cases and concludes that the remaining expressions establish the corresponding theorem constraint.

B. Proof of Theorem 2

The proof establishes the noisy-interference sum-rate result by showing that equality in the outer bound is achieved when both receivers treat interference as noise.

  • Achievability: Treating interference as noise at both receivers achieves equality in the relevant sum-rate outer-bound expression.This identifies single-user detection as sufficient for the stated sum-rate result.
  • Feasibility conditions: Feasibility requires the auxiliary variances and correlation parameters to satisfy nonnegativity and bounded-correlation conditions.The proof introduces at least one feasible parameter pair before applying the derived inequalities.
  • Case reduction: The proof reduces the alternatives generated by the inequalities to the condition stated in Theorem 2.Conditions (53) and (54) exclude two alternatives, while the remaining case matches condition (16).

C. Proof of Theorem 3

The proof extends the outer-bound argument to a mixed-interference regime and shows that the resulting sum-rate bound is achievable when one receiver decodes the other user first.

  • Outer-bound extension: The outer-bound inequality remains valid for a broader range because its proof requires only 0 < b < 1, including cases with a > 1.The argument then specializes the bound to obtain the stated sum-rate upper bound.
  • Achievability: The sum-rate upper bound is achievable when the condition in (25) holds.The construction assigns user 2 a specified rate and uses successive decoding at receiver 1.
  • Decoding strategy: Receiver 1 decodes user 2’s message before decoding its own messages, yielding the claimed rate expressions.This decoding order produces the rate pair associated with the theorem’s corner-point result.

IV. NUMERICAL EXAMPLES

The numerical examples compare the proposed outer bound with existing bounds and inner bounds across channel parameters and power levels, showing tightness in noisy-interference cases and a nonmonotone sum-rate pattern.

  • Capacity-region comparison: In the example of Fig. 4, the proposed outer bound coincides with the inner bound at the sum-rate point because the channel has noisy interference.The figure compares inner and outer bounds, including ETW, Kramer, and Han–Kobayashi-based bounds.
  • Symmetric-channel bounds: For symmetric channels across the displayed power levels, the upper bounds remain tight up to point A.Figures 5–8 cover different values of P1 = P2.
  • Comparison with prior bounds: At high power, the ETW bound approaches the proposed bound but retains a gap.This comparison is reported for the large-power numerical examples.
  • Dependence on interference gain: The sum capacity is not monotone in the interference gain a, as indicated by the reported bump in the lower bound.The figures address whether the sum-rate capacity decreases with a or exhibits a bump between 0 and 1.
  • Conclusions and extensions: The paper derives the outer bound by a genie-aided method and obtains sum-rate capacities for channels satisfying condition (16) or (24).The discussion also identifies extensions to parallel and MIMO Gaussian interference channels.
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