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Cooperative Algorithms for MIMO Interference Channels

Steven W. Peters, Robert W. Heath

arXiv:1002.0424v2cs.IT

TL;DR

The paper addresses interference-alignment designs that neglect uncoordinated interference and can deliver suboptimal finite-SNR throughput. It proposes three colored-noise-aware precoding algorithms with different performance and complexity tradeoffs, finding that accounting for co-channel interference can help, although ignoring it is advantageous in some cases.

  • Problem

    Prior interference-alignment methods neglect uncoordinated interference and colored noise, while alignment can have suboptimal finite-SNR throughput.

  • Method

    The paper develops minimum interference-plus-noise leakage, joint-MMSE, and convergent maximum-SINR precoding designs for MIMO interference channels.

  • Results

    The algorithms exhibit different performance–complexity tradeoffs, and simulations show that co-channel interference can benefit from precoders designed to account for it, although ignoring it is sometimes advantageous.

  • Takeaways & Limitations

    The proposed designs can outperform one another in different operating regimes, so precoder choice depends on the relevant performance and complexity tradeoff.

Abstract

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Interference alignment is a transmission technique for exploiting all available degrees of freedom in the interference channel with an arbitrary number of users. Most prior work on interference alignment, however, neglects interference from other nodes in the network not participating in the alignment operation. This paper proposes three generalizations of interference alignment for the multiple-antenna interference channel with multiple users that account for colored noise, which models uncoordinated interference. First, a minimum interference-plus-noise leakage algorithm is presented, and shown to be equivalent to previous subspace methods when noise is spatially white or negligible. A joint minimum mean squared error design is then proposed that jointly optimizes the transmit precoders and receive spatial filters, whereas previous designs neglect the receive spatial filter. This algorithm is shown to be a generalization of previous joint MMSE designs for other system configurations such as the broadcast channel. Finally, a maximum signal-to-interference-plus-noise ratio algorithm is developed that is proven to converge, unlike previous maximum SINR algorithms. The latter two designs are shown to have increased complexity due to non-orthogonal precoders, more required iterations, or more channel state knowledge than the min INL or subspace methods. The sum throughput performance of these algorithms is simulated in the context of a network with uncoordinated co-channel interferers not participating in the alignment protocol. It is found that a network with cochannel interference can benefit from employing precoders designed to consider that interference, but in some cases, ignoring the co-channel interference is advantageous.

I. INTRODUCTION

The paper addresses finite-SNR limitations of interference alignment and neglected colored noise by proposing alternative precoding designs with explicit performance–complexity tradeoffs.

  • Motivation: Interference alignment achieves maximum capacity scaling but can have suboptimal finite-power sum rate because transmitters prioritize interference cancellation over their own links.This motivates relaxing perfect alignment to improve nonasymptotic throughput.
  • Motivation: Existing subspace algorithms target alignment and neglect colored noise, which can arise from co-channel interference.Their asymptotic objective is therefore incomplete in finite-SNR and externally interfered regimes.
  • Proposed designs: The paper proposes a minimum interference-plus-noise leakage design that incorporates colored noise while retaining orthogonal precoders and reducing to subspace methods with white or negligible noise.The design is also characterized as a special type of MMSE design and is suitable for quantized CSI.
  • Proposed designs: A joint-MMSE design jointly optimizes transmit precoders and receive filters, improving sum rate over subspace methods at the cost of greater computational complexity, more iterations, and nonorthogonal precoders.The design generalizes prior joint-MMSE approaches for point-to-point and multiuser settings.
  • Proposed designs: A convergent maximum-SINR algorithm directly targets sum-rate-related performance and can often increase throughput, but requires more channel state information and produces nonorthogonal precoders.Its result can differ from the earlier approximate maximum-SINR algorithm for a particular channel realization despite similar average performance.
  • Evaluation: The three algorithms span performance–complexity regimes, and simulations with uncoordinated interferers show that different methods can outperform one another depending on operating conditions.The simulations include colored noise caused by an uncoordinated interferer and enumerate regimes where each algorithm performs best.

II. SYSTEM MODEL

The system is a synchronized, narrowband, block-fading K-user MIMO interference channel with linear precoding, coordinated interference, and arbitrary receiver noise covariance.

  • Network model: Each transmitter–receiver pair communicates over a channel where every transmitter reaches every receiver, but each transmitter’s data is intended for only its paired receiver.The model contains K transmit–receive pairs and non-negligible links.
  • Network model: Transmitter k uses M_k antennas to send S_k ≤ M_k spatial streams, while receiver k has N_k ≥ S_k antennas.In the symmetric case, all users use M transmit antennas, N receive antennas, and S streams.
  • Channel assumptions: The narrowband block-fading model keeps every link static during one transmission but allows channels to change between successive transmissions.Users are synchronized, with no frequency offsets, and linear precoding is performed independently for each channel realization.
  • Signal model: The received signal contains the desired transmission, coordinated interference from other transmitters, and Gaussian noise with spatial covariance R_k.No assumptions are made on the noise power or covariance at any receiver.
  • Performance metric: The instantaneous sum-rate metric captures total network throughput, but the paper assumes ideal nonlinear decoding for fair comparison across precoder designs.High-performance linear receiver design is left for future work except in the MMSE design.
  • Interference alignment: Interference alignment confines the sum of interfering streams to an N_k − S_k dimensional receive subspace, allowing receiver k to resolve its S_k desired streams with linear processing.This is the core subspace-based interference-suppression mechanism.

III. ITERATIVE ALGORITHMS VIA ALTERNATING MINIMIZATION

This section introduces three iterative metrics that approximate sum-rate maximization and uses alternating minimization to optimize precoders and receive processing matrices.

  • The section proposes three metrics intended to achieve better finite-SNR rates than previous work.
  • Each algorithm optimizes a global objective over K precoders and K receive processing matrices.The objective depends on the precoders, all channel matrices, and design-specific receiver processing matrices.
  • Because no closed-form global solution is known, the designs use alternating minimization over the 2K variables.Variables are optimized one at a time while the remaining variables are held fixed.
  • Except for the proposed maximum SINR design, precoders can generally be derived in parallel during each iteration.

A. Subspace Optimization

The subspace method minimizes coordinated interference leakage by designing orthogonal precoders and receive subspaces, while leaving final desired-stream separation to a subsequent spatial filter.

  • Subspace Optimization: The subspace approach precodes signals so coordinated interference is nearly orthogonal to each receiver’s intended signal subspace.The objective minimizes residual coordinated interference after projection by the receive-subspace basis.
  • Subspace Optimization: The precoders have orthogonal columns under per-stream power constraints, supporting channel-state feedback.
  • Subspace Optimization: The receiver processing matrices in these designs mainly simplify precoder optimization rather than directly serving as high-performance spatial equalizers.High-performance linear receiver design is left to future work except in the joint-MMSE algorithm.
  • Subspace Optimization: The method uses alternating minimization because each precoder and interference-free receive subspace depend on one another.
  • Subspace Optimization: After coordinated interference cancellation, a separate linear spatial filter and MIMO equalizer process the desired streams.The receiver forms G_k = Φ_kW_k, with W_k designed to equalize the desired signal.

B. Minimum Interference Plus Noise Leakage (INL)

Minimum interference-plus-noise leakage extends subspace alignment to colored noise by minimizing coordinated interference and noise leakage, while reducing to ordinary subspace alignment for white or negligible noise.

  • Minimum Interference Plus Noise Leakage (INL): Colored noise from uncoordinated interference can bias ordinary IA subspaces toward aligning with noise as well as coordinated interference.
  • Minimum Interference Plus Noise Leakage (INL): The INL objective minimizes total post-processing coordinated interference power together with noise leakage.
  • Minimum Interference Plus Noise Leakage (INL): For fixed receive subspaces, INL yields the same precoder optimization as ordinary interference leakage and aligns coordinated interference with dominant noise directions.Rank-one spatially correlated noise can lead the algorithm to align interference with the interferer’s dominant direction.
  • Minimum Interference Plus Noise Leakage (INL): When noise is spatially white at every receiver, minimizing INL is equivalent to minimizing ordinary interference leakage.
  • Minimum Interference Plus Noise Leakage (INL): As ρ_k →∞ or ||R_k||_F →0 for all k, INL converges to ordinary IA and has the same SNR scaling.
  • Minimum Interference Plus Noise Leakage (INL): The INL objective will generally have a positive global minimum unless the number of streams is reduced below the network’s degrees of freedom.Zero minimum leakage requires singular noise covariances and receive subspaces lying in their null spaces; adapting stream counts is outside the paper’s scope.

C. Mean Squared Error Minimization

The joint-MMSE design jointly optimizes transmit precoders and receive filters by minimizing estimation error, extending MMSE formulations beyond point-to-point and broadcast settings.

  • Mean Squared Error Minimization: The interference-channel formulation jointly designs multiple transmitters’ precoders and multiple receivers’ filters, unlike point-to-point or single-sided multiuser designs.
  • Mean Squared Error Minimization: Joint MMSE designs the receive spatial filters directly and minimizes the expected sum of estimation errors between each ŝ_k and s_k.
  • Mean Squared Error Minimization: The orthogonality constraint is relaxed to a total power inequality so the precoders can satisfy KKT conditions.
  • Mean Squared Error Minimization: With no inter-user interference, the design is equivalent to point-to-point MMSE, and at high SNR its precoders and receivers diagonalize their information links.
  • Mean Squared Error Minimization: The MMSE receiver directly estimates s_k, whereas other designs require a conventional MIMO receiver after spatial projection.
  • Mean Squared Error Minimization: The joint-MMSE design is harder to implement because its precoder optimization lacks a closed form, produces nonorthogonal precoders, and may require more computation.

D. Signal-to-Interference-Plus-Noise-Ratio Maximization

The maximum-SINR section defines a global objective that maximizes network sum signal power relative to interference, including inter-stream interference, and optimizes it by alternating updates. The resulting design can outperform simpler methods but may require more information and implementation effort.

  • Objective: The proposed maximum-SINR formulation divides total network signal power by total interference power, including inter-user and inter-stream interference after processing.
  • Algorithm: Unlike the previous maximum-SINR approach, the proposed algorithm alternates optimization of a global performance function, providing a convergence guarantee.
  • Optimization: The design can reduce post-processing interference or increase signal power because its objective jointly rewards a smaller denominator and larger numerator.
  • Optimization: With fixed remaining variables, generalized eigenvectors solve the column updates that maximize the global SINR function.
  • Comparison: For any channel realization and initialization, the proposed and earlier algorithms may produce identical results or either may achieve the larger sum rate, despite similar average performance.
  • Implementation trade-offs: The proposed design requires more network knowledge, while running both algorithms and selecting the better realization-specific design can exceed either algorithm individually.

E. Convergence and Initialization

The section examines convergence and initialization because iterative objectives may converge without yielding globally optimal throughput or stable precoder designs. Multiple initializations can therefore matter substantially for the achieved rate.

  • Initialization: Different initializations can produce drastically different throughputs even when an algorithm reaches the same final objective or cost function.
  • Initialization: For the tested (2,2,3) MIMO interference channel, MMSE varied most across ten random initializations at 40 dB, but this behavior was not general.
  • Initialization: Random initializations performed as well as the tested intelligent initialization, and multiple runs are recommended when throughput is the priority.
  • Convergence: All algorithms are guaranteed to converge because their bounded objectives improve monotonically, but global-optimum convergence is not guaranteed.
  • Convergence: Convergence of the cost function does not automatically establish convergence of the precoder designs.

IV. SIMULATIONS

The simulations compare iterative precoding algorithms under scaled or fixed uncoordinated interference, showing that accounting for interference can improve throughput but that the best design depends on the operating regime. The min-INL algorithm offers a complexity-performance tradeoff, while MMSE and max-SINR can perform better in some scenarios.

  • Simulation setup: The simulations use i.i.d. Gaussian channel coefficients, perfect CSI, flat channels, and Monte Carlo evaluation of expected sum rate.Five random initializations are tested per channel realization, retaining the initialization maximizing the objective.
  • Scaled uncoordinated interference: With scaled rank-one interference at all receivers, MMSE can let two transmitters turn off and retain one degree of freedom, unlike the other algorithms.The MMSE power inequality constraint permits this behavior, so its sum capacity scales linearly with ρ.
  • Complexity-performance tradeoffs: At high ρ, incorporating colored noise yields a roughly 20% sum-rate increase, while min-INL retains orthogonal precoders and lower complexity than MMSE and max-SINR.MMSE and max-SINR are the two best-performing algorithms in the cited scenario but use nonorthogonal precoders.
  • Fixed uncoordinated interference: With fixed uncoordinated interference, all algorithms except random beamforming perform similarly and exploit the network’s maximum degrees of freedom.At fixed iteration counts, MMSE may fail to scale at high ρ because it appears to require more iterations to converge.
  • Interference at one receiver: When interference reaches only one receiver, max-SINR and min-INL suffer at high ρ relative to IA and MMSE because they optimize average network performance.MMSE can reduce the affected transmitter’s power, while maximizing sum rate may favor ignoring external interference or using a power inequality constraint.
  • Interfering-link path loss: With no external interferer, IA throughput remains constant as interfering-link path loss decreases, whereas other iterative algorithms exploit the reduced interference and converge to IA at high interference power.The data-link SNR is fixed at 40 dB in this comparison.

V. CONCLUSIONS AND FUTURE WORK

The paper concludes that direct interference alignment is often undesirable in practical environments with unequal link strengths, whereas max-SINR and MMSE adapt more flexibly. The min-INL algorithm is presented as a lower-complexity compromise, but implementation overhead remains a limitation.

  • Conclusions: Direct interference alignment is suboptimal when one or more links have relatively little energy, making max-SINR or MMSE preferable in most environments.These metrics adapt between interference alignment at high interference power and SVD precoding when interference is absent.
  • Conclusions: The min-INL algorithm trades some performance for simpler solutions and orthogonal precoders while improving over IA with uncoordinated interference or colored noise.The paper identifies it as a practical compromise among the three proposed algorithms.
  • Future work and limitations: Nonorthogonality, absent closed-form solutions, power-constraint optimization, and additional channel-state knowledge increase implementation complexity.MMSE requires power-constraint optimization, while max-SINR requires more channel-state knowledge per iteration.
  • Future work: Future work targets reducing solution overhead and evaluating whether interference-alignment gains justify the required overhead and feedback in cellular networks.The paper states that overhead and feedback analyses remain necessary.

APPENDIX A DERIVATION OF MEAN SQUARED ERROR MINIMIZATION

The MMSE optimization is solved by alternating minimization, using KKT conditions to update receive combiners and transmit precoders while enforcing power constraints.

  • Optimization procedure: KKT conditions solve each alternating-minimization step with all but one variable fixed.The receive combiners are unconstrained, so setting the MMSE gradient to zero automatically satisfies their KKT conditions.
  • Receive-combiner update: For fixed precoders and Lagrange multipliers, each receive combiner is obtained by setting the MMSE gradient to zero.
  • Transmit-precoder update: Transmit-precoder updates must satisfy stationarity, feasibility, complementary slackness, and dual feasibility conditions.The derivation explicitly requires all conditions in (43)–(46) to hold.
  • Power constraint: If the multiplier is zero, the unconstrained solution is optimal; otherwise, the precoder is scaled to meet its Frobenius-norm power constraint.
  • Power constraint: Bisection can find a positive multiplier because the precoder norm decreases monotonically with that multiplier.No closed-form solution for the multiplier is known in the active-constraint case.

APPENDIX B DERIVATION OF SINR MAXIMIZATION

The SINR maximization derivation updates precoder and receive-equalizer columns through generalized Rayleigh-quotient optimization, holding the remaining variables fixed.

  • Coordinate updates: The SINR optimization operates on individual columns of each precoder and spatial equalizer.
  • Precoder update: When updating one precoder column, all other precoders, columns, and receive-combining matrices remain fixed.
  • Precoder update: The column objective becomes a generalized Rayleigh quotient whose solution is a generalized eigenvector of numerator and denominator matrices.
  • Rayleigh-quotient formulation: The numerator and denominator terms independent of the updated column are grouped separately during the quotient reformulation.
  • Receive-combiner update: Receive-combiner columns follow the same derivation, using unit-norm constraints that remove the corresponding multiplier terms.
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