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Capacity Bounds for the Gaussian Interference Channel

Abolfazl S. Motahari, Amir K. Khandani

arXiv:0801.1306v1cs.IT

TL;DR

Characterizing the Gaussian interference channel remains difficult because the full Han–Kobayashi region involves unknown optimal distributions and many time-sharing degrees of freedom. This paper develops admissible-channel outer bounds and Gaussian Han–Kobayashi analyses, deriving sum-capacity results for weak and mixed channels and simplifying achievable-region characterization.

  • Problem

    The full Han–Kobayashi achievable region is difficult to characterize because optimal distributions are unknown and Gaussian restrictions still involve many time-sharing degrees of freedom.

  • Method

    The paper introduces admissible interference channels and uses an extremal-inequality-based outer-bounding technique alongside Gaussian Han–Kobayashi analyses.

  • Results

    The paper derives sum capacity for weak channels in a parameter range and mixed channels across the full parameter range, while obtaining tighter outer bounds and Gaussian Han–Kobayashi region characterizations.

  • Takeaways & Limitations

    For the stated channel regimes, Gaussian codebooks, interference-as-noise decoding, and three-band Han–Kobayashi schemes achieve the reported capacity or region results.

Abstract

from arXiv · show

The capacity region of the two-user Gaussian Interference Channel (IC) is studied. Three classes of channels are considered: weak, one-sided, and mixed Gaussian IC. For the weak Gaussian IC, a new outer bound on the capacity region is obtained that outperforms previously known outer bounds. The sum capacity for a certain range of channel parameters is derived. For this range, it is proved that using Gaussian codebooks and treating interference as noise is optimal. It is shown that when Gaussian codebooks are used, the full Han-Kobayashi achievable rate region can be obtained by using the naive Han-Kobayashi achievable scheme over three frequency bands (equivalently, three subspaces). For the one-sided Gaussian IC, an alternative proof for the Sato's outer bound is presented. We derive the full Han-Kobayashi achievable rate region when Gaussian codebooks are utilized. For the mixed Gaussian IC, a new outer bound is obtained that outperforms previously known outer bounds. For this case, the sum capacity for the entire range of channel parameters is derived. It is proved that the full Han-Kobayashi achievable rate region using Gaussian codebooks is equivalent to that of the one-sided Gaussian IC for a particular range of channel parameters.

I. INTRODUCTION … C. Han-Kobayashi Achievable Region

The paper studies capacity bounds for two-user Gaussian interference channels, developing tighter outer bounds and characterizing Gaussian Han–Kobayashi regions through convex-analytic and frequency-sharing constructions. It establishes capacity and optimality results for weak, one-sided, and mixed regimes while formalizing the channel model and achievable-region framework.

  • I. INTRODUCTION: The paper introduces admissible interference channels to obtain tighter outer bounds for weak and mixed Gaussian ICs, and derives weak-channel sum capacity over a specified parameter range.The approach relies on an extremal inequality and also identifies cases where Gaussian codebooks with interference treated as noise are optimal.
  • I. INTRODUCTION: For weak Gaussian ICs, the paper derives sum capacity when users treat interference as noise and transmit at their highest rates, then obtains an improved outer bound.It also shows that time-sharing or concavification of the basic Han–Kobayashi region yields the same enlarged region.
  • A. The Two-user Interference Channel: The two-user Gaussian IC is modeled with interference gains a and b, standard Gaussian noises, and transmitter power constraints P1 and P2; channel classes are determined by these gains.The strong-interference capacity region is characterized by individual bounds R1 ≤ γ(P1), R2 ≤ γ(P2), and a sum-rate bound.
  • B. Support Functions: The preliminary framework uses support functions to represent closed convex achievable regions and compare set inclusions through their support-function inequalities.Boundary points of compact convex regions correspond to maximizers of the support function.
  • C. Han-Kobayashi Achievable Region: The full Han–Kobayashi region CHK is the strongest inner bound, but its optimal input distributions remain unknown; G restricts codebook generation to Gaussian distributions.The basic Gaussian region G0 depends on power allocations P1, P2 and common-message fractions α, β, while αP1 and βP2 allocate common-message power.
  • C. Han-Kobayashi Achievable Region: The Gaussian HK construction enlarges G0 through time-sharing and alternative convexification procedures, including regions G1 and G2 based on different combinations of parameterized polytopes.G2 corresponds to dividing the frequency band into sub-bands, assigning each band its own powers and α, β values.
  • C. Han-Kobayashi Achievable Region: The Gaussian-region constructions satisfy the inclusion chain G0 ⊆ G1 ⊆ G2 ⊆ G ⊆ CHK ⊆ C, establishing their relationship to the full HK region and capacity region.G is closed, bounded, and convex, and its interior boundary can be characterized using its support function.

D. Concavification Versus Time-Sharing

This section establishes when time-sharing and concavification produce identical achievable regions, and bounds the numbers of operating points needed for each construction. For Gaussian interference channels treating interference as noise, both constructions coincide with cardinality q = q′ = M + 1.

  • D. Concavification Versus Time-Sharing: If D0 has the unique minimizer property, then time-sharing and concavification coincide: D = D2.A polymatroid D0 satisfies this property, yielding D = D2.
  • D. Concavification Versus Time-Sharing: For M-user systems, time-sharing needs q < M + K + 1, or q ≤ M + K when Ψ(P) is continuous.Here M is the dimension of P and K is the dimension of Ψ(P).
  • D. Concavification Versus Time-Sharing: For Gaussian codebooks with interference treated as noise, the time-sharing cardinality is less than 2M.In this setting, continuity of Ψ(P) and Theorem 2 yield the bound.
  • D. Concavification Versus Time-Sharing: Concavification requires at most q′ ≤ M + 1 operating points to characterize boundary points, independently of the number of rate-region inequalities.The bound follows by viewing the support function as the concavification of g(c, P).
  • D. Concavification Versus Time-Sharing: For the same Gaussian interference channel class, D2 = D and the two cardinalities are equal: q = q′ = M + 1.Thus, Gaussian codebooks with interference treated as noise make the two constructions equivalent.

E. Extremal Inequality

This section establishes Gaussian optimality for a scalar-noise, trace-constrained extremal inequality and characterizes the optimizer across noise-ordering and multiplier regimes. The resulting optimization is then used repeatedly in the remainder of the paper.

  • Optimization method: The analysis specializes the general covariance-constrained problem to isotropic noises N1I and N2I with a trace constraint, then solves it using covariance decomposition and KKT conditions.For N1 ≤ N2, the KKT conditions imply equal eigenvalues, reducing the optimizer to an isotropic covariance.
  • N1 ≤ N2: For N1 ≤ N2, the optimal input is iid Gaussian for every µ ≥ 0, with covariance and power determined by the stated µ-dependent regimes.The solution uses full permissible power in one regime and less than the permissible power when µ exceeds the corresponding threshold.
  • Reusable optimization: The paper defines a scaled extremal optimization fh(P, N1, N2, a, µ) and applies Lemma 1 to evaluate it under the condition N1 ≤ N2/a.The scaling follows from h(AX) = log(|A|) + h(X), producing the transformed objective used later.

III. ADMISSIBLE CHANNELS · A. Classes of Admissible Channels · 1) Class A1:

The paper constructs admissible interference channels whose capacity regions contain that of the Gaussian IC and whose tractable capacity bounds yield outer bounds. It then defines Class A1, a two-receiver channel family designed to bound σC′(µ, 1), and proves an upper-bound lemma under additional parameter constraints.

  • III. ADMISSIBLE CHANNELS: Admissible channels are introduced to contain the Gaussian IC capacity region while retaining a tractable capacity expression or outer bound.The tightest outer bound is obtained by intersecting the capacity regions of all admissible channels, motivating tractable subclasses.
  • III. ADMISSIBLE CHANNELS: An interference channel is admissible when deterministic functions of its outputs satisfy the required mutual-information inequalities, and genie-aided channels form a subclass.For genie-aided channels, choosing each function to recover the original output makes the admissibility inequalities hold trivially.
  • III. ADMISSIBLE CHANNELS: Capacity-region boundary points are characterized through support-function optimization, with non-axis points represented by nonnegative weights c1 and c2 satisfying c1 + c2 = 1.The paper uses optimization problems whose solutions correspond to boundary points and then seeks upper bounds on σC′(µ, 1) and σC′(1, µ).
  • A. Classes of Admissible Channels: Class A1 is designed to upper-bound σC′(µ, 1) using one transmit and receive antenna for User 1 and one transmit plus two receive antennas for User 2.The channel is represented by User 1’s output and User 2’s two outputs, with Gaussian noise variances N21 and N22 and power constraints P1 and P2.
  • 1) Class A1:: Class A1 uses deterministic functions f1 and f2 to impose admissibility conditions on its parameters, and includes the one-sided Gaussian IC when g2 = 0, N21 →∞, and N22 = 1.The one-sided channel arises by removing the link between Transmitter 1 and Receiver 2.
  • 1) Class A1:: Additional constraints narrow the admissible Class A1 channels but are required to obtain a closed-form upper bound on σC′(µ, 1).Lemma 3 applies to the channels modeled by (73) and satisfying (79).

2) Class A2:

Class A2 defines admissible channels through two linear functions and parameter constraints, including a one-sided Gaussian IC as a limiting case. Under additional constraints, Lemma 4 establishes the required upper bound for the modeled channels.

  • 2) Class A2:: Class A2 channels are characterized using two linear functions f1 and f2.
  • 2) Class A2:: When g1 = 0, Class A2 reduces to the one-sided Gaussian IC by taking N12 →∞ and N11 = 1.
  • 2) Class A2:: The channel modeled by (87) is admissible when its corresponding parameters satisfy the stated conditions and additional Class A2 constraints.
  • 2) Class A2:: Lemma 4 provides the required upper bound for Class A2 channels modeled by (87) and satisfying (93).

3) Class B:

Class B constructs admissible Gaussian channels that upper-bound both σC(µ, 1) and σC(1, µ), then proves the resulting sum-capacity bounds are tight. In the relevant parameter range, Gaussian codebooks with treating interference as noise achieve the Class B sum capacity, a property previously observed by Etkin et al..

  • Class B: The constructed channel is designed to upper-bound both σC(µ, 1) and σC(1, µ) under the transmitters’ power constraints.The two transmitters satisfy power constraints P1 and P2, respectively.
  • Class B: Admissibility requires nonnegative parameters g1 and g2 satisfying the specified equalities, with additional constraints enabling computable outer bounds.Adding constraints reduces the set of admissible channels but permits outer bounds on σC′(µ, 1) and σC′(1, µ).
  • Class B: Lemma 5 establishes the key outer bound for channels modeled by (95) and satisfying the imposed constraints.The proof uses Fano’s inequality, Gaussian extremality, Jensen’s inequality, and Lemma 1 to bound the relevant entropy terms.
  • Class B: For the stated range of µ, the derived outer bounds become tight because treating interference as noise achieves the corresponding rates.The paper identifies this tightness as a property first observed by Etkin et al..
  • Class B: The Class B sum capacity is attained with Gaussian codebooks when receivers treat interference as noise.The theorem states that this strategy achieves the sum capacity in Class B.

4) Class C:

For Class C channels, the paper uses the model in (73) and modifies the constraints so receiver 2 obtains a less noisy version of user 1’s signal. This enables receiver 2 to decode both users’ messages and yields Lemma 6’s bound.

  • Class C: Class C uses the model in (73), with admissibility determined by the corresponding channel parameters.
  • Class C: Changing the constraints makes receiver 2’s observation of user 1’s signal less noisy after decoding its own signal.This less-noisy relationship is the basis for receiver 2 decoding user 1’s signal in addition to its own.
  • Class C: The Class C outer-bound argument establishes Lemma 6 by exploiting receiver 2’s ability to decode both users’ messages.The proof concludes by bounding the component terms and summing the resulting inequalities.
  • Class C: The proof shows that one resulting constraint is redundant because receiver 2 can jointly use both decoded observations when conditioning on user 2’s input.

IV. WEAK GAUSSIAN INTERFERENCE CHANNEL · A. Sum Capacity

For the weak Gaussian interference channel, the paper derives the sum capacity over a parameter range where treating interference as noise is optimal. It also establishes a tighter outer bound and shows that time-sharing and concavification coincide for Gaussian codebooks.

  • IV. WEAK GAUSSIAN INTERFERENCE CHANNEL: The weak-channel section also reports a tighter outer bound than previously known and equality between time-sharing and concavification for Gaussian codebooks.These are stated as additional contributions alongside the sum-capacity result.
  • A. Sum Capacity: The sum-capacity derivation reduces the Class B-channel optimization by first minimizing over g1 and g2, leaving constraints 0 < S1 < 1 and 0 < S2 < 1.The objective is the Class B sum capacity, while the constraints ensure channel admissibility and validate the bound.
  • A. Sum Capacity: Within this parameter range, treating interference as noise achieves the sum capacity.The optimization objective becomes independent of S1 and S2 under the stated condition, and the resulting value is achievable by treating interference as noise.
  • A. Sum Capacity: Theorem 5 establishes the sum capacity of the two-user weak Gaussian IC for a specified range of channel parameters.The result follows by characterizing feasible auxiliary parameters and proving the corresponding upper and lower bounds coincide.
  • A. Sum Capacity: The paper proves the feasible parameter sets D and D′ are equal, completing the characterization needed for the sum-capacity result.The proof establishes both inclusions, D′ ⊆ D and D ⊆ D′.
  • A. Sum Capacity: The sum-capacity result for the weak Gaussian IC was established independently in and, with related work noted in.The paper explicitly records these independent derivations.
  • A. Sum Capacity: For the symmetric Gaussian IC, Figure 7 identifies the admissible parameter region where treating interference as noise is optimal.The plotted region is the set of parameters for which this strategy obtains the sum capacity.
  • A. Sum Capacity: For fixed P, Figure 8 shows that the upper bound and the treating-interference-as-noise lower bound coincide up to a certain value of a.This demonstrates tightness of the bounds over that portion of the symmetric-channel parameter range.

B. New Outer Bound · C. Han-Kobayashi Achievable region

For the weak Gaussian interference channel, the paper derives a new family of weighted-rate outer bounds and characterizes the Gaussian Han–Kobayashi region using at most three dimensions. The new outer bound is tighter than previously known bounds, while time-sharing and concavification yield the same achievable region.

  • B. New Outer Bound: The new outer-bound construction upper-bounds σC(μ, 1) and σC(1, μ) using channels from Classes A1, A2, and B.The method combines bounds from different channel classes to control both weighted-rate directions.
  • B. New Outer Bound: Theorem 6 bounds every achievable weak-Gaussian-IC rate pair through μ1R1 + R2 ≤ W(μ1) and R1 + μ2R2 ≤ W̃(μ2) for all μ1, μ2 ≥ 1.The bounds use minima of two auxiliary support-function bounds for each weighted-rate direction.
  • C. Han-Kobayashi Achievable region: The region G0 has four extreme points in the interior of the first quadrant, and none of its defining inequalities is redundant.Figure 9 displays the possible extreme points of G0.
  • C. Han-Kobayashi Achievable region: G0 has the unique minimizer property: the dual-program minimizers are independent of P1, P2, α, and β.This independence supports the reduction used to characterize the achievable region.
  • C. Han-Kobayashi Achievable region: For weighted direction (μ, 1), σD0 equals (μ − 2)ψ1 + ψ4 when μ > 2 and (2 − μ)ψ3 + (μ − 1)ψ4 when 1 ≤ μ ≤ 2.These formulas identify the relevant extreme-point support values in the two μ ranges.
  • C. Han-Kobayashi Achievable region: For weighted direction (1, μ), σD0 equals (μ − 2)ψ2 + ψ5 when μ > 2 and (2 − μ)ψ3 + (μ − 1)ψ5 when 1 ≤ μ ≤ 2.The corresponding dual solutions are likewise specified separately for the two μ ranges.
  • C. Han-Kobayashi Achievable region: Theorem 7 shows that time-sharing and concavification produce the same region, with the Gaussian Han–Kobayashi region characterized by power allocation over at most three dimensions.The proof attributes the three-dimension limit to the unique minimizer property and the frequency-band result.
  • C. Han-Kobayashi Achievable region: The new outer bound is tighter than previously known bounds in the plotted symmetric weak Gaussian IC comparisons.The comparison is reported for Figures 10 and 11, though only Figure 10’s caption is supplied here.

V. ONE-SIDED GAUSSIAN INTERFERENCE CHANNELS · A. Sum Capacity · B. Outer Bound

The one-sided Gaussian IC is specialized to the weak-interference case a < 1, whose capacity region remains incompletely characterized. The section presents an alternative proof of Sato’s outer bound, states the Gaussian-codebook Han–Kobayashi region, and confirms the sum-capacity point obtained by treating interference as noise.

  • V. ONE-SIDED GAUSSIAN INTERFERENCE CHANNELS: For one-sided Gaussian ICs, b = 0 eliminates interference at Receiver 2; the strong subclass a ≥ 1 is fully characterized, while the weak subclass a < 1 remains open.The analysis therefore assumes a < 1 throughout.
  • V. ONE-SIDED GAUSSIAN INTERFERENCE CHANNELS: Costa’s result equates the weak one-sided IC capacity region with that of a degraded IC after an appropriate parameter change, enabling Sato’s degraded-IC outer bound to apply.The section also provides an alternative proof of this outer bound and characterizes the full Gaussian-codebook Han–Kobayashi achievable region.
  • A. Sum Capacity: The one-sided Gaussian IC sum capacity is attained at Sason’s stated extreme-point rate pair.This result is stated as Theorem 8 and identifies the sum-capacity point on the capacity-region boundary.
  • B. Outer Bound: Sato’s outer bound requires every rate pair in the weak one-sided IC capacity region to satisfy the theorem’s bound for all β ∈ [0, 1], with P = P1/a + P2.The bound is transferred from the degraded Gaussian IC through Costa’s equivalence.
  • B. Outer Bound: Setting µ = 1 in the supporting-function argument shows that the same achievable point is the one-sided Gaussian IC sum-capacity point, yielding an alternative proof of Sason’s result.The proof characterizes other boundary points through weighted sums after noting that User 2 transmits at its maximum rate at sum capacity.
  • B. Outer Bound: Treating interference as noise at Receiver 1 achieves equality in the outer bound for 1 ≤ µ ≤ P2 + 1/a, placing the resulting point in the capacity region.This establishes achievability of the relevant boundary point under the stated range of µ.
  • B. Outer Bound: The outer-bound proof completes the equivalence between the theorem’s inequality description and its dual convex-region representation.It uses the closedness and convexity of the auxiliary region E2.

C. Han-Kobayashi Achievable Region

The Han–Kobayashi achievable regions for the weak one-sided Gaussian interference channel are characterized through G0, G1, G2, and G. The analysis shows G2 = G and establishes a constructive boundary characterization for G1.

  • C. Han-Kobayashi Achievable Region: For the one-sided channel, User 1 contributes only a private message, corresponding to α = 1, enabling an explicit characterization of G0.The absence of a link from Transmitter 1 to Receiver 2 removes User 1’s common-message component.
  • C. Han-Kobayashi Achievable Region: G2 equals G because G0 has the unique minimizer property.G0 is described as a pentagon with two extreme points in the first quadrant, and the unique minimizer property is verified.
  • C. Han-Kobayashi Achievable Region: Figure 12 compares different bounds for the one-sided Gaussian interference channel.The comparison uses P1 = 1, P2 = 7, and a = 0.4.
  • C. Han-Kobayashi Achievable Region: G1 is represented by rate pairs satisfying the lemma’s constraints for all β′ ∈ [0, 1], and G1 is convex.The characterization is completed by showing the relevant extreme points of G0 lie in the stated set for every β ∈ [0, 1].
  • C. Han-Kobayashi Achievable Region: Every boundary point of G1 can be achieved using superposition coding and successive decoding.The proof establishes achievability after showing both inclusions between the lemma’s set and G1.

VI. MIXED GAUSSIAN INTERFERENCE CHANNELS … C. Han-Kobayashi Achievable Region

For mixed Gaussian interference channels with a < 1 and b ≥ 1, the paper characterizes sum capacity across the full parameter range, develops a new outer bound, and analyzes the Han-Kobayashi achievable region. The achievable region is equivalent to that of a corresponding one-sided channel when 1 ≤ ab, while certain capacity facets require both common and private signaling by User 2.

  • A. Sum Capacity: The mixed Gaussian IC sum capacity is characterized for the entire range of channel parameters, extending independent results limited to a certain parameter range [25].The converse uses upper bounds from the two underlying one-sided Gaussian ICs, while achievability lets Receiver 2 decode and remove Transmitter 1’s common message.
  • A. Sum Capacity: Depending on whether 1 + P2 ≤ b + abP2, the sum capacity equals that of the one-sided weak or strong Gaussian IC, respectively.The sum-capacity point has User 2 transmitting at its maximum rate R2 = γ(P2), enabling characterization of other boundary points through weighted-rate optimization.
  • B. New Outer Bound: Any achievable rate pair lies in the intersection E1 ∩ E2 of the capacity regions of the two underlying one-sided Gaussian ICs, with an additional inequality holding for all 1 ≤ µ.E1 comes from removing the link from Transmitter 1 to Receiver 2, and E2 from removing the link from Transmitter 2 to Receiver 1.
  • C. Han-Kobayashi Achievable Region: For the Han-Kobayashi scheme, User 1 assigns all power to a common message, while User 2 splits its power into common and private components using β and 1 − β.The region is analyzed through three parameter cases determined by 1 + P2 ≤ b + abP2 and the comparison between 1 − a and abP1.
  • C. Han-Kobayashi Achievable Region: When 1 ≤ ab, the Gaussian Han-Kobayashi achievable region equals that of the one-sided Gaussian IC obtained by removing the interfering link from Transmitter 1 to Receiver 2.Under 1 ≤ ab, the condition 1 + P2 ≤ b + abP2 holds for all P1 and P2, making the enlarged region equivalent to the one-sided region.
  • C. Han-Kobayashi Achievable Region: In Cases II and III, Region E3 is a capacity-region facet obtainable when Transmitter 2 uses both common and private messages.This facet is highlighted as a surprising consequence of splitting User 2’s transmission across both message types.
  • C. Han-Kobayashi Achievable Region: The paper compares different bounds for the mixed Gaussian IC across Cases I and II using parameter settings shown in Figures 14 and 15.Figure 14 uses P1 = 7, P2 = 7, a = 0.6, b = 2; Figure 15 uses P1 = 7, P2 = 7, a = 0.4, b = 1.5.

VII. CONCLUSION

The paper characterizes capacity-region bounds for weak, one-sided, and mixed Gaussian interference channels using admissible channels as the main outer-bound tool. It establishes sum-capacity results, Gaussian-codebook Han–Kobayashi characterizations, and new or improved outer bounds across these channel classes.

  • Overall scope: Across the three channel classes, the study considers sum capacities, inner bounds, and outer bounds, with admissible channels serving as the main tool for deriving outer bounds.The channel classes are weak, one-sided, and mixed Gaussian interference channels.
  • Weak Gaussian IC: For the weak Gaussian IC, the sum capacity is attained by Gaussian codebooks with interference treated as noise over a certain parameter range, and a new outer bound is tighter than Kramer’s and ETW’s bounds.The work also reduces the computational complexity of the Han–Kobayashi achievable region.
  • One-sided Gaussian IC: For the one-sided Gaussian IC, the paper gives an alternative proof of Sato’s outer bound and derives the full Gaussian-codebook Han–Kobayashi achievable region.
  • Mixed Gaussian IC: For the mixed Gaussian IC, the sum capacity is derived for the entire parameter range, alongside an outer bound that outperforms ETW’s bound.The Gaussian-codebook full Han–Kobayashi region is also shown equivalent to the one-sided IC region for a particular range of channel gains.
  • Capacity-region structure: A capacity-region facet is derived for a certain parameter range and can be achieved when one transmitter uses both common and private messages.This provides a specific structural insight into how message types can jointly attain part of the capacity region.
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