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Interference Channels with Source Cooperation

Vinod M. Prabhakaran, Pramod Viswanath

arXiv:0905.3109v2cs.ITmath.IT

TL;DR

The paper asks how source cooperation can manage interference in a two-user Gaussian interference channel, where ordinary orthogonalization or treating interference as noise can be far from optimal. It develops a cooperative coding scheme based on Han and Kobayashi superposition coding, characterizes sum-rate within 20 bits, and applies the scheme to a symmetric channel with noiseless feedback.

  • Problem

    Orthogonalization and treating interference as noise can be far from optimal, while prior cooperation gains were established only coarsely in the asymptotic many-radio regime.

    Source: §I · p. 1

  • Method

    A block-Markov scheme builds on Han and Kobayashi superposition coding and adds cooperative-public and cooperative-private messages.

    Source: §I · p. 2, §V · p. 10, §IV · p. 10

  • Results

    The Gaussian source-cooperation sum-rate is characterized within 20 bits, and the same strategy obtains the symmetric noiseless-feedback sum-capacity within 19 bits.

    Source: §I · pp. 1–2

  • Takeaways & Limitations

    The paper shows that full-duplex source cooperation can provide interference-management gains in a two-user Gaussian channel.

    Source: §I · p. 1

Abstract

from arXiv · show

The role of cooperation in managing interference - a fundamental feature of the wireless channel - is investigated by studying the two-user Gaussian interference channel where the source nodes can both transmit and receive in full-duplex. The sum-capacity of this channel is obtained within a gap of a constant number of bits. The coding scheme used builds up on the superposition scheme of Han and Kobayashi (1981) for the two-user interference channel without cooperation. New upperbounds on the sum-capacity are also derived. The same coding scheme is shown to obtain the sum-capacity of the symmetric two-user Gaussian interference channel with noiseless feedback within a constant gap.

I. INTRODUCTION

The paper studies whether full-duplex cooperation between sources can improve interference management in the two-user Gaussian interference channel. It characterizes the sum-rate within 20 bits and extends the coding strategy to symmetric Gaussian channels with noiseless feedback.

I. INTRODUCTION

  • Prior cooperation results showed near co-located MIMO performance only in the asymptotic regime of very many radios.That result concerns scaling of symmetric capacity as the number of radios grows.Source: §I · p. 1
  • The Gaussian source-cooperation channel’s sum-rate is characterized within 20 bits, with an example tightening the gap to 6 bits.The paper also provides recipes to improve the lower and upper bounds.Source: §I · p. 1
  • Orthogonalizing cooperative links into noisefree bit-pipes is not generally an efficient solution for this problem.The capacity region for source conferencing over orthogonal links was characterized within a constant gap in.Source: §I · p. 2
  • The coding scheme extends Han and Kobayashi superposition coding with cooperative-public and cooperative-private messages.Cooperative-public messages are decoded by both destinations and the other source; cooperative-private messages are decoded by the intended destination and the other source.Source: §I · p. 2
  • The same coding strategy characterizes the symmetric Gaussian interference channel with noiseless output feedback within 19 bits, while [21] obtains a better bound.The paper identifies this feedback problem as closely related to source cooperation.Source: §I · p. 2

II. PROBLEM STATEMENT

The problem statement defines a two-user channel in which sources transmit to their destinations while also receiving signals from one another. Source transmissions are causal, depending on each source’s message and what it has heard through the previous time instance.

II. PROBLEM STATEMENT

  • In the Gaussian model, destinations receive mixtures of both sources’ signals plus independent, unit-variance Gaussian noise.The sources also receive one another’s signals, and the source links are reciprocal; the model uses unit power constraints when both sources have the same power constraint.Source: §II · p. 3
  • Each source’s transmission can depend only on its message and everything it has heard up to the previous time instance.This imposes causality on the encoding functions.Source: §II · p. 3
  • The model also considers a linear deterministic channel with finite-field input vectors and shift-matrix channel maps.The source-to-source links are set equal to model reciprocity.Source: §II · p. 4

A. Sum-rate Characterization

The Gaussian channel’s sum-capacity is characterized within 20 bits of the minimum of five upper-bound quantities. For the linear deterministic channel, the sum-capacity is characterized exactly as a minimum of quantities.

A. Sum-rate Characterization

  • The linear deterministic channel’s sum-capacity equals the minimum of a set of quantities, including expressions in the channel parameters.The supplied expressions include a maximum-based term and a term with a condition on differences between channel parameters.Source: A. Sum-rate Characterization · p. 4
  • Within 20 bits, a sum-rate is achievable relative to the minimum of five upper-bound quantities for the Gaussian channel with source cooperation.The theorem states the sum-capacity is at most that minimum.Source: A. Sum-rate Characterization · pp. 4–5
  • The Gaussian channel’s sum-capacity is upper-bounded by the minimum of five quantities.Source: A. Sum-rate Characterization · p. 4

B. Gains from cooperation

For the symmetric channel with hI = √hD, the sum-capacity is characterized within 6 bits for sufficiently large hD. As cooperation-link strength grows, the gains occur in three distinct regimes, with plateaus between them.

B. Gains from cooperation

  • Within 6 bits, C characterizes the sum-capacity for sufficiently large hD in the specified symmetric channel.The result holds for any ϵ > 0, with sum-capacity in (C − 6 − ϵ, C + ϵ).Source: B. Gains from cooperation · p. 5
  • For log |hC|2 / log |hD|2 ≤ 1/2, every 3 dB increase in cooperation-link strength raises sum-capacity by 2 bits.Without cooperation, time-sharing essentially achieves the sum-capacity.Source: B. Gains from cooperation · p. 5
  • For 1/2 < log |hC|2 / log |hD|2 ≤ 1, the earlier linear gain saturates and does not resume until the cooperation link matches the direct link.The plotted quantity is normalized sum-capacity as cooperation-link strength varies in the hD → ∞ limit.Source: B. Gains from cooperation · p. 5, Figure 2 · p. 6
  • For 1 < log |hC|2 / log |hD|2 ≤ 3/2, capacity rises by 2 bits per 6 dB increase, then becomes flat after approaching cooperation capacity.The plot shows normalized sum-capacity against the cooperation-link strength ratio in the stated limit.Source: B. Gains from cooperation · p. 5, Figure 2 · p. 6

IV. CODING SCHEME: ILLUSTRATIVE EXAMPLES

The examples illustrate how cooperative-public and cooperative-private messages combine with Han–Kobayashi messages to exploit source cooperation. They also motivate a coding scheme that generalizes the signal arrangements beyond these hand-picked cases.

Illustrative examples

  • The examples distinguish cooperative-public messages, decoded by both sources and destinations, from cooperative-private messages decoded only by the intended destination.Uncoded examples illustrate key scheme features, but are not sufficient for all deterministic channels or Gaussian channels.Source: §IV · pp. 6, 10

Example 1

  • Example 1 achieves sum-rate 6 by having sources exchange cooperative-public signals and destinations decode backward, while private signals avoid interfering destinations.The setup has nD = 4, nI = 2, nC = 1; without cooperation, time-sharing achieves sum-capacity nD.Source: §IV · pp. 6–7

Example 2

  • Example 2 achieves sum-rate 7 and adds ordinary public messages, which both destinations decode, alongside cooperative-public and private messages.The transmitted signals can be viewed as a superposition of three codewords with separate power allocations, motivating a random linear-code generalization.Source: §IV · pp. 6, 9, 10, Figure 3 · p. 8

Example 3

  • In Example 3, a strong cooperation link enables sources to share cooperative-private signals in advance and use precoding to avoid interference at the unintended destination.With nD = 4, nI = 3, and nC = 5, the scheme achieves sum-rate 6 versus sum-capacity 5 without cooperation; this uncoded construction relies on the field characteristic not being 2.Source: §IV · pp. 9–10

Generalization

  • The examples motivate replacing uncoded signal arrangements with linear coding over a sufficiently large extension field, preserving the message-variable rates while enabling decoding at sources and destinations.The paper then develops coding schemes for discrete-memoryless interference channels with source cooperation.Source: §IV · p. 10

V. CODING SCHEMES

The coding schemes extend Han–Kobayashi superposition coding through block-Markov cooperation and backward decoding. Different schemes trade off cooperative-public and cooperative-private message handling, with their combination left unexplored for the paper’s sum-rate focus.

Coding theorem

  • The coding theorem uses block-Markov coding built on Han and Kobayashi superposition coding, with elements of decode-and-forward and backward decoding.The schemes apply to memoryless interference channels with source cooperation.Source: §V · pp. 10, 12

Scheme (a)

  • Scheme (a) has sources decode and retransmit cooperative-public interference, alongside unaided public and private messages.The scheme is closely related to Example 1 and decodes only the cooperative-public part at the source, treating other messages as noise.Source: §V · pp. 13–14

Scheme (b)

  • Scheme (b) additionally shares cooperative-private messages in advance, using joint source decoding; it can outperform scheme (a) at strong cooperation-link strengths.At low cooperation-link strengths, scheme (a) can perform better than scheme (b) without cooperative-private messages.Source: §V · pp. 13–14

Design choices and scope

  • Cooperative-private messages are used when the cooperation link exceeds the direct link; simple precoding is used, while dirty-paper coding may improve the gap.Scheme (c) combines the approaches asymmetrically, but the paper does not explore a fully combined scheme because its focus is sum-rate results.Source: §V · p. 14

VI. INTERFERENCE CHANNEL WITH FEEDBACK

The section relates the symmetric Gaussian interference channel with noiseless output feedback to the source-cooperation model. It applies the cooperation result to establish a constant-gap sum-capacity characterization for feedback.

VI. INTERFERENCE CHANNEL WITH FEEDBACK

  • The feedback model differs from source cooperation because each source receives noiseless feedback from its destination.The sources’ transmissions are deterministic functions of their messages and past observations.Source: §VI · p. 14
  • The source-cooperation result applies to the symmetric feedback channel: its achievability proof depends on noise marginals, while the converse uses a feedback upper bound.The source-cooperation converse assumes reciprocal cooperation-link magnitudes, a condition covered by the symmetric channel.Source: §VI · p. 15
  • The symmetric Gaussian interference channel with noiseless output feedback has sum-capacity within 19 bits of the expression in Proposition 5.The proposition states this result for the sum-capacity of the symmetric, Gaussian channel with output feedback.Source: §VI · p. 15

A. Reversibility

The destination-cooperation channel is formed by swapping source and destination roles and reversing link directions. Its sum-capacity matches the source-cooperation channel in the deterministic model and is within 50 bits in the Gaussian model.

A. Reversibility

  • Destination cooperation adds causal transmissions from destinations, which are functions of what each has received up to the previous time instant.For this comparison, nodes 3 and 4 are the sources and nodes 1 and 2 are the destinations.Source: A. Reversibility · p. 16
  • In the linear deterministic model, source and destination cooperation have the same sum-capacity expression.The paper identifies this as a reversibility property of its achievable strategy and notes that broader settings merit investigation.Source: A. Reversibility · p. 16
  • The Gaussian source- and destination-cooperation channels have sum-capacities within 50 bits, irrespective of channel coefficients.The comparison is between the two channels in Figures 5a and 5b.Source: A. Reversibility · p. 16

B. Dependence on channel state information

The paper states when its schemes need full channel state information and outlines the block-Markov random-coding proof of Theorem 4, including backwards decoding at destinations.

B. Dependence on channel state information

  • Theorem 4(a) needs only channel strengths to both destinations, and only for a cooperative link weaker than all destination links; Theorems 4(b) and 4(c) require full channel state information.The paper’s general assumption is full channel state information at sources and destinations.Source: B. Dependence on channel state… · p. 17

APPENDIX A PROOF OF THEOREM 4

  • Theorem 4’s proof uses block-Markov coding; destinations decode backwards, and the random-coding argument makes error probability vanish as blocklength grows when its rate conditions hold.The proof also has sources decode cooperative messages from the preceding block.Source: Appendix A · pp. 17–19
  • Scheme (c) combines the earlier schemes, with only destination 3 receiving a private message sent cooperatively by both sources.Accordingly, it uses an S1 codebook but no S2 codebook.Source: Appendix A · p. 21

APPENDIX B PROOF OF ACHIEVABILITY OF THEOREM 1

The achievability proof divides the deterministic-channel sum-rate into four regimes of cooperation-link strength and uses different schemes across them. The proof establishes achievability in each regime through the corresponding rate conditions.

Regime (ii)

  • In regime (ii), the sum-rate expression is constant, so achievability follows from regime (i).This regime is nmin < nC ≤ min(n1,3, n2,4).Source: Appendix B · p. 22

Regime (iv)

  • In regime (iv), applying Theorem 4(b) yields a sum-rate equal to the minimum of u2(nC), u3(nC), u4(nC), and u5.This regime has nC greater than max(n1,3, n2,4).Source: Appendix B · pp. 22, 26

Regime (i)

  • In regime (i), the selected auxiliary variables and rates achieve sum-rate u1(nC) under the stated restricted regime.When condition (13) does not hold, the proof instead establishes achievability at nC = 0.Source: Appendix B · pp. 22–23

Regime (iii)

  • Fourier-Motzkin elimination shows that sum-rate min(u1(0), u2(0), u3(0), u4(0), u5) is achievable; in regime (iii), assuming n1,3 ≤ nC ≤ n2,4, Theorem 4(c) is applied for n2,4 ≥ n1,4 and n2,4 < n1,4.Source: Appendix B · p. 23

APPENDIX C PROOF OF ACHIEVABILITY OF THEOREM 2

The appendices establish achievability within constant gaps and derive sum-rate upperbounds for source cooperation. They also apply these results to the feedback bound and a symmetric Gaussian example, with regime-dependent gaps and an upperbound of C + ϵ for sufficiently large hD.

APPENDIX C PROOF OF ACHIEVABILITY OF THEOREM 2

  • In regime (i), when condition (13) does not hold, Etkin-Tse-Wang yields achievability within 9 bits; overall, this regime achieves the upperbound within 20 bits.When condition (13) holds, the proof uses Gaussian auxiliaries and obtains a sum-rate within at most 13 bits in the restricted range of nC; regime (ii) follows from regime (i).Source: Appendix C · pp. 27–28
  • In regime (iii), the upperbound is achievable within at most 16 bits, with the proof covering both orderings of |h2,4| and |h1,4|.For each ordering, Fourier–Motzkin elimination gives a sum-rate within 9 bits of the stated minimum of u′ terms.Source: Appendix C · pp. 30–31
  • Theorem 2’s upperbound is achievable within a constant gap of 20 bits across all four regimes.In regime (iv), the minimum of u′_2(nC), u′_3(nC), u′_4(nC), and u′_5 is achievable within 7 bits.Source: Appendix C · p. 33

APPENDIX D SOURCE COOPERATION: UPPERBOUNDS

  • Appendix D introduces four sum-rate upperbounds that together imply the upperbounds in Theorems 1 and 2.The first upperbound begins from Fano’s inequality.Source: Appendix D · p. 34
  • The Appendix D proof derives linear-deterministic and Gaussian sum-rate upperbounds, including cut-set bounds, and notes several ways to tighten the constant gap.The authors mention accounting for destination-noise and input correlations and optimizing the input covariance matrix, but do not pursue these refinements.Source: Appendix D · pp. 35, 40, 41

APPENDIX E UPPERBOUNDS FOR THE GAUSSIAN INTERFERENCE CHANNEL WITH FEEDBACK

  • 19 bits is the overall gap to upperbound (12) achieved by Appendix C’s scheme for the symmetric channel with noiseless feedback.The proof’s achieved rate is within 13 bits of u′2, which is within 5 bits of (12); the achievability argument carries over because it depends only on noise marginals.Source: Appendix E · p. 41

APPENDIX F A GAUSSIAN EXAMPLE

  • For the symmetric Gaussian channel with hI = √hD, the appendix gives several sum-rate upperbounds and notes that u1 is slightly stronger than the Theorem 2 bound.The u1 strengthening follows directly by specializing the Appendix D proof.Source: A GAUSSIAN EXAMPLE · p. 42
  • For sufficiently large hD, the upperbounds imply that sum-capacity is at most C + ϵ for any ϵ > 0.The passage notes that one of the upperbounds is not important for this channel.Source: A GAUSSIAN EXAMPLE · p. 42
  • Within 6 bits, Theorem 4(a) achieves the upperbound for hC ≤ √hD when hD is sufficiently large.The construction chooses independent, zero-mean Gaussian auxiliary random variables.Source: A GAUSSIAN EXAMPLE · pp. 42–43
  • For √hD ≤ hC ≤ hD, u2 = log 2(1 + hD) dominates the other bounds and is independent of hC, so achievability follows from the hC ≤ √hD regime.This holds for sufficiently large hD.Source: A GAUSSIAN EXAMPLE · p. 43
  • Within 5 bits, Theorem 4(b) achieves the upperbound for hC > hD when hD is sufficiently large.The construction uses independent, zero-mean Gaussian auxiliary random variables and constant U1, U2.Source: A GAUSSIAN EXAMPLE · p. 43

APPENDIX G PROOF OF THEOREM 6

The appendix compares constant-gap bounds for Gaussian interference channels with source and destination cooperation. It concludes that both sum-capacities are within 50 bits.

APPENDIX G PROOF OF THEOREM 6

  • The destination-cooperation theorem bounds sum-capacity by the minimum of five quantities.The passage introduces Theorem 7 as the destination-cooperation analog of Theorem 2.Source: Appendix G · p. 43
  • The primed and unprimed quantities differ by at most a constant number of bits, while their respective minima are equal.Source: Appendix G · p. 44
  • Both Gaussian interference channels—with source cooperation and destination cooperation, respectively—have sum-capacities within at most 50 bits.The comparison follows from Theorems 2 and 7.Source: Appendix G · p. 44
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