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Optimal Beamforming in Interference Networks with Perfect Local Channel Information

Rami Mochaourab, Eduard Jorswieck

arXiv:1004.4492v2cs.ITmath.IT

TL;DR

Efficient Pareto resource allocation is difficult in interference networks where transmitters affect intended and unintended receivers. The paper analyzes each transmitter’s power gain-region to characterize efficient beamforming strategies, finding that rank-1 beamforming and a low-dimensional parameterization suffice, with power control required in some cases.

  • Problem

    Designing Pareto-optimal resource allocation is difficult in interference networks, where transmitters can degrade unintended receivers’ performance.

    Source: §I · p. 2

  • Method

    The paper analyzes each transmitter’s power gain-region and parameterizes Pareto-relevant strategies using single-stream beamforming and real-valued parameters.

    Source: §I · p. 4, §III · p. 9

  • Results

    The gain-region boundary is convex and achieved with single-stream beamforming, and efficient strategies require an additional power parameter in some antenna-to-receiver regimes.

    Source: §I · p. 4, A. The number of transmit antennas… · p. 13, B. The number of transmit antennas… · p. 13

  • Takeaways & Limitations

    The parameterization reduces resource-allocation complexity and can support low-complexity coordination between transmitters.

    Source: §I · p. 2

Abstract

from arXiv · show

We consider settings in which T multi-antenna transmitters and K single-antenna receivers concurrently utilize the available communication resources. Each transmitter sends useful information only to its intended receivers and can degrade the performance of unintended systems. Here, we assume the performance measures associated with each receiver are monotonic with the received power gains. In general, the systems' joint operation is desired to be Pareto optimal. However, designing Pareto optimal resource allocation schemes is known to be difficult. In order to reduce the complexity of achieving efficient operating points, we show that it is sufficient to consider rank-1 transmit covariance matrices and propose a framework for determining the efficient beamforming vectors. These beamforming vectors are thereby also parameterized by T(K-1) real-valued parameters each between zero and one. The framework is based on analyzing each transmitter's power gain-region which is composed of all jointly achievable power gains at the receivers. The efficient beamforming vectors are on a specific boundary section of the power gain-region, and in certain scenarios it is shown that it is necessary to perform additional power allocation on the beamforming vectors. Two examples which include broadcast and multicast data as well as a cognitive radio application scenario illustrate the results.

I. INTRODUCTION

The paper develops a general framework for characterizing Pareto-relevant transmission strategies in multiuser interference networks, addressing the difficulty of designing efficient resource allocation. It uses each transmitter’s power gain-region to represent efficient strategies with low-dimensional parameters.

I. INTRODUCTION

  • Interference limits wireless performance, while finding efficient Pareto operating points is difficult; even maximum sum-rate and proportional-fair optimization in the MISO interference channel are strongly NP-hard.Source: §I · p. 2
  • Prior work characterizes efficient beamforming in the two-user MISO interference channel, while broader multiuser settings are not straightforward and existing ITC- and Lagrangian-based characterizations do not generalize to this framework.Source: §I · pp. 2–3
  • The framework characterizes Pareto-relevant transmission strategies in general multiuser settings and parameterizes them with real values to reduce resource-allocation complexity and support low-complexity coordination.Source: §I · pp. 2, 4
  • For a single transmitter, the power gain-region contains jointly achievable receiver power gains; its relevant boundary strategies are characterized using beamforming parameters, with power control needed in some cases.When transmit antennas are at least as numerous as receivers, full power achieves all boundary points; when antennas are fewer, an additional parameter varies transmit power.Source: §I · p. 4

II. SYSTEM AND CHANNEL MODEL

  • The model considers T transmitters with N_k antennas and K single-antenna receivers sharing a spectral band, with each transmitter assumed to know its channels to all receivers perfectly.Source: §II · pp. 5–6

A. Assumptions on Performance Measure

The section assumes receiver utility rises with power from intended transmitters and falls with power from unintended transmitters, then illustrates that model in a network with broadcast, multiple-access, multicast, and interference links.

A. Assumptions on Performance Measure

B. Example Setting

III. POWER GAIN REGION

The power gain-region collects jointly achievable receiver gains for one transmitter and is compact and convex. Its boundary identifies extreme gain combinations relevant to efficient operation, with power reduction needed in some antenna-limited cases.

III. POWER GAIN REGION

  • The gain-region is compact and convex, enabling boundary characterization through programming problems based on hyperplane separation.Source: §III · p. 9
  • Boundary points represent extreme achievable receiver gains; improving gain in one direction there requires decreasing gain in another direction.Source: §III · p. 9
  • For one intended and one unintended receiver, direction [1, −1] selects the boundary from maximum intended gain to minimum unintended gain, with MRT and ZF at the respective extremes.Source: §III · p. 10, Figure 2 · p. 10

A. The number of transmit antennas satisfies N ≥K

The gain-region’s boundary identifies extreme jointly achievable receiver power gains, and its points can be characterized with rank-1 beamforming. The required power allocation depends on the relationship between antenna and receiver counts.

A. The number of transmit antennas satisfies N ≥K

  • With N ≥ K and independent channel vectors, full-power transmission reaches all upper boundary points, including points where gains at other receivers are nulled.When N > K, part of the boundary also lies between an axis point and the origin.Source: A. The number of transmit antennas… · p. 11
  • Rank-1 transmit covariance matrices achieve the gain-region boundary in any direction, so its boundary strategies can be represented by beamforming vectors.The gain-region is compact and convex.Source: A. The number of transmit antennas… · p. 12
  • K − 1 real parameters characterize upper boundary points in a direction, with parameters between zero and one.The boundary plotted for three antennas is convex and touches each axis at one point.Source: A. The number of transmit antennas… · pp. 12–13

B. The number of transmit antennas satisfies N < K

  • When N < K, full-power transmission cannot null every other receiver, so reaching all gain-region boundary points requires power control.The optimal allocation is full power for a positive largest eigenvalue, zero for a negative one, and any allocation from zero to one when it is zero.Source: B. The number of transmit antennas… · pp. 13–14
  • With two antennas, the illustrated gain-region has a cone-like shape, with a full-power outer boundary and flat surfaces produced by varying transmission power.The variable-power surfaces connect the origin to boundary points.Source: B. The number of transmit antennas… · pp. 14–15, Figure 4 · p. 15

IV. PARETO BOUNDARY CHARACTERIZATION

The framework selects each transmitter’s relevant gain-region boundary according to its intended and unintended receivers. Theorem 2 characterizes beamformers that achieve Pareto-optimal utility points.

IV. PARETO BOUNDARY CHARACTERIZATION

  • All Pareto-optimal points in the utility region can be achieved by the beamforming vectors characterized in Theorem 2.The framework combines maximizing gains at intended receivers with minimizing gains at unintended receivers.Source: §IV · pp. 15–16
  • The characterization uses T(K − 1) real parameters when N_k > |K(k)|, and TK parameters when N_k ≤ |K(k)|; each lies between zero and one.The direction for each transmitter’s relevant gain-region boundary is determined by its intended and unintended receiver sets.Source: §IV · p. 16

A. Example Revisited

The example applies the framework to a broadcast and multicast setting and to the K-user MISO interference channel. Its parametrization identifies Pareto-optimal operating points in both settings.

A. Example Revisited

  • The broadcast and multicast example uses seven parameters in [0, 1] to generate points on the Pareto boundary, but not weakly Pareto-optimal points.The plotted parameters are varied on a grid with step length 0.05.Source: A. Example Revisited · pp. 16–17

A. Multiple-input Single-output Interference Channel

  • In the K-user MISO IC, Theorem 2 reaches all Pareto-boundary rate-region points with a real-valued parametrization using the same number of parameters as.The paper describes this parametrization as lower-dimensional than the complex-valued one in.Source: A. Multiple-input Single-output… · p. 18
  • Fig. 6 plots the three-user MISO IC rate-region boundary at SNRs of −10 dB and 30 dB using beamformers from Theorem 2.The plotted parameter grid uses step length 0.05; points are filtered to retain visible, nondominated points.Source: A. Multiple-input Single-output… · p. 18, Figure 6 · p. 18

1) Maximum ratio transmission:

The section relates maximum-ratio and zero-forcing transmission to the beamforming parameterization, then applies null-shaping constraints to underlay cognitive radio. In that setting, the constraints characterize Pareto-efficient strategies with parameter complexity similar to Theorem 2.

1) Maximum ratio transmission:

  • Maximum-ratio transmission is the unique Nash-equilibrium strategy of each transmitter and corresponds to λk,ℓ=1 for k=ℓ and 0 otherwise.This is the noncooperative-game outcome.Source: 1) Maximum ratio transmission · p. 19

2) Zero-forcing transmission:

  • Zero-forcing transmission corresponds to λk,ℓ=0 for k=ℓ and λk,ℓ>0 for k≠ℓ.The passage identifies this parameter setting as the zero-forcing strategy.Source: 2) Zero-forcing transmission · p. 19

B. Noncooperative Underlay Cognitive Radio

  • Null-shaping constraints address underlay operation where secondary users share resources with primary users subject to limits on primary-user QoS degradation.When no primary-user interference is allowed, the constraints are called null-shaping constraints; the transmitter must have at least as many antennas as primary receivers.Source: B. Noncooperative Underlay Cognitive… · pp. 19–20
  • All Pareto-boundary points of the utility region are reachable with the null-shaping-based strategies, whose parameter count matches Theorem 2.The design uses K−1 null-shaping constraints per transmitter; the strategy is MRT satisfying those constraints, and noncooperative transmitters can achieve efficient points.Source: B. Noncooperative Underlay Cognitive… · p. 20

VI. EXTENSION TO MULTIPLE ANTENNAS AT THE RECEIVERS

The extension to multiple-antenna receivers applies when receive beamforming does not depend on multiple transmit beamformers, but not to the stated MMSE case. The conclusion summarizes the gain-region framework and its parameterization, including when power control is required.

VI. EXTENSION TO MULTIPLE ANTENNAS AT THE RECEIVERS

  • The framework can apply when receive beamforming vectors do not depend on the transmit beamforming vector, including vectors fixed before transmission.The passage gives SVD-based receive beamforming as an example.Source: §VI · p. 21
  • The framework does not apply when receive beamforming vectors depend on several transmit beamforming vectors, as with the linear MMSE receiver.The stated limitation concerns coupling through the receive filters.Source: §VI · p. 21

VII. CONCLUSIONS

  • The paper characterizes Pareto-efficient beamforming by studying each transmitter’s power-gain region and shows that its boundary is convex and achievable with single-stream beamforming.For at least as many transmit antennas as receivers, full-power transmission achieves all boundary points using K−1 parameters; with fewer antennas, an additional parameter controls transmit power.Source: §VII · p. 22

APPENDIX A

The appendices establish compactness and convexity of the power-gain region, characterize its boundary, and show that rank-1 covariance matrices achieve that boundary.

APPENDIX A

  • The power-gain region is closed, bounded, and convex, hence compact.The appendix uses continuity and the compact feasible covariance set for closedness, and convex combinations of feasible covariance matrices for convexity.Source: Appendix A · pp. 22–23

APPENDIX B

  • When K≤N, every boundary point in a specified direction can be achieved with a transmit covariance matrix using the full power constraint.The construction adds power in a null-space direction while preserving gains at the other receivers.Source: Appendix B · pp. 23–24

APPENDIX C

  • Rank-1 transmit covariance matrices achieve the boundary of the power-gain region.The optimization solution assigns the full eigenvalue weight to the largest-eigenvalue component.Source: Appendix C · p. 24

APPENDIX D

The appendices identify the largest-eigenvalue solution for a directional boundary problem and use a contradiction argument to connect transmitter gain-region boundaries to Pareto-boundary utility points.

APPENDIX D

  • For the directional boundary optimization, the solution is the eigenvector corresponding to the largest eigenvalue of the Hermitian matrix Z.Z need not be positive semidefinite because the direction can contain negative components.Source: Appendix D · p. 25

APPENDIX E

  • All Pareto-boundary utility points are achieved when each transmitter chooses a beamforming vector on its gain-region boundary in the specified direction.The proof argues that an interior gain tuple could be replaced by a boundary tuple that improves at least one receiver without changing the others, contradicting Pareto optimality.Source: Appendix E · p. 26

APPENDIX F

Appendix F shows that, when the transmitter has at least as many antennas as receivers and the channel vectors are linearly independent, the stated multiplier conditions yield beamforming vectors satisfying the zero-forcing condition. It also notes that the result is not guaranteed if a cross-channel multiplier is zero.

APPENDIX F

  • The zero-forcing condition is imposed on transmitter k’s beamforming vector, denoted w_ZF.Source: Appendix F · p. 26
  • With N_k ≥ K, linearly independent channel vectors, and zero same-index but positive cross-channel multipliers, the largest-eigenvalue beamforming vector satisfies the zero-forcing condition.The proof uses the matrix whose largest-eigenvalue eigenvector determines the beamforming vector.Source: Appendix F · pp. 26–27
  • If any cross-channel multiplier λ_ℓ is zero, the largest eigenvector does not necessarily produce zero gain at that receiver.Source: Appendix F · p. 27

APPENDIX G

Appendix G states that beamforming vectors from (29) achieve the same gains as those from (21). Its proof uses properties of the associated Hermitian matrices and their eigenvectors.

APPENDIX G

  • Beamforming vectors in (29) achieve the same gains as beamforming vectors in (21).Source: Appendix G · p. 27
  • M_k is the sum of Hermitian matrices A_k and B_k, with A_k positive semidefinite and B_k negative semidefinite; the proof applies Weyl’s eigenvalue inequality.Source: Appendix G · p. 27
  • The ordered eigenvalues of M_k include zero eigenvalues, and the proof states that N_k ≥ |K| yields zero eigenvalues when the space dimension exceeds the receiver count.Source: Appendix G · pp. 27–28
  • The eigenvectors of M_k form an orthonormal set; G_k(λ) collects the first N_k−1 eigenvectors, and Z_k(λ) excludes those satisfying (52).Source: Appendix G · p. 28
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