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Linear Precoding Based on Polynomial Expansion: Large-Scale Multi-Cell MIMO Systems

Abla Kammoun, Axel Müller, Emil Björnson, Mérouane Debbah

arXiv:1310.1799v2cs.IT

TL;DR

Large-scale multi-cell MIMO needs precoding that preserves RZF-like performance without its costly matrix inversion, while multi-cell RZF parameter optimization remains difficult. The paper uses TPE precoding, random-matrix deterministic equivalents, and offline coefficient optimization, achieving higher throughput than suboptimal RZF in simulations with TPE order 3.

  • Problem

    RZF requires large matrix inversion, and optimizing its regularization parameters in realistic multi-cell systems is difficult.

  • Method

    The paper approximates RZF matrix inversion with TPE precoding, derives deterministic user-rate equivalents, and optimizes polynomial coefficients using weighted max-min fairness.

  • Results

    TPE precoding achieves higher throughput than RZF in certain scenarios, using a TPE order of only 3.

  • Takeaways & Limitations

    Optimized polynomial coefficients can provide both lower complexity and better throughput than the evaluated RZF scheme in multi-cell settings.

Abstract

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Large-scale MIMO systems can yield a substantial improvement in spectral efficiency for future communication systems. Due to the finer spatial resolution achieved by a huge number of antennas at the base stations, these systems have shown to be robust to inter-user interference and the use of linear precoding is asymptotically optimal. However, most precoding schemes exhibit high computational complexity as the system dimensions increase. For example, the near-optimal RZF requires the inversion of a large matrix. This motivated our companion paper, where we proposed to solve the issue in single-cell multi-user systems by approximating the matrix inverse by a truncated polynomial expansion (TPE), where the polynomial coefficients are optimized to maximize the system performance. We have shown that the proposed TPE precoding with a small number of coefficients reaches almost the performance of RZF but never exceeds it. In a realistic multi-cell scenario involving large-scale multi-user MIMO systems, the optimization of RZF precoding has thus far not been feasible. This is mainly attributed to the high complexity of the scenario and the non-linear impact of the necessary regularizing parameters. On the other hand, the scalar weights in TPE precoding give hope for possible throughput optimization. Following the same methodology as in the companion paper, we exploit random matrix theory to derive a deterministic expression for the asymptotic SINR for each user. We also provide an optimization algorithm to approximate the weights that maximize the network-wide weighted max-min fairness. The optimization weights can be used to mimic the user throughput distribution of RZF precoding. Using simulations, we compare the network throughput of the TPE precoding with that of the suboptimal RZF scheme and show that our scheme can achieve higher throughput using a TPE order of only 3.

I. INTRODUCTION

Large-scale multi-cell MIMO improves spatial interference mitigation but makes conventional precoding computationally expensive. The paper applies TPE precoding, deterministic rate analysis, and offline coefficient optimization to address this trade-off.

  • I. INTRODUCTION: Large antenna arrays improve spatial resolution, reducing vulnerability to inter-user interference and enabling predictable throughput in large-system regimes.Random matrix theory provides deterministic approximations of otherwise stochastic performance quantities.
  • I. INTRODUCTION: RZF and other advanced linear precoders require matrix inversion with cubic complexity in min(M, K), making implementation prohibitive at large dimensions.MRT has lower complexity but does not actively suppress interference and may require an order of magnitude more antennas to approach RZF performance.
  • I. INTRODUCTION: TPE precoding approximates the RZF matrix inverse with a (J −1)-degree polynomial, creating a low-complexity multistage implementation.Changing J provides a transition from MRT at J = 1 toward RZF at J = min(M, K).
  • I. INTRODUCTION: The multi-cell model incorporates user-specific covariance matrices, imperfect CSI, pilot contamination, and cell-specific power constraints.The TPE order Jj may differ across cells to reflect cell size, performance requirements, and hardware resources.
  • I. INTRODUCTION: Deterministic equivalents of achievable user rates depend on channel statistics rather than instantaneous channel realizations, enabling offline coefficient optimization.The joint coefficient optimization is mathematically related to multicast beamforming optimization.

B. Model of Imperfect Channel State Information at BSs

The system uses uplink pilots in synchronized TDD cells to estimate downlink channels, with pilot reuse causing contamination from neighboring cells. These estimates support the precoding schemes.

  • B. Model of Imperfect Channel State Information at BSs: Each cell uses mutually orthogonal uplink pilots internally, while pilot reuse across cells causes neighboring-cell pilot contamination.The MMSE channel estimate is formed from the contaminated pilot observations.
  • B. Model of Imperfect Channel State Information at BSs: The effective training SNR ρtr enters the MMSE channel estimation model used by the base stations.The estimated channel is characterized through its channel covariance and estimation covariance.
  • B. Model of Imperfect Channel State Information at BSs: The estimated channels for all users in a cell are collected into a matrix and used by the considered precoding schemes.The estimated channel vectors remain complex Gaussian under Rayleigh fading with MMSE estimation.

III. REVIEW ON REGULARIZED ZERO-FORCING PRECODING

RZF is a widely used heuristic whose large-system SINRs admit deterministic equivalents, but optimizing its multi-cell regularization parameters remains difficult. The paper reviews this analysis as context for TPE precoding.

  • III. REVIEW ON REGULARIZED ZERO-FORCING PRECODING: RZF provides a closed-form heuristic for multi-cell precoding, while optimal linear precoding under imperfect CSI remains unknown.RZF is also known by several equivalent beamforming interpretations.
  • III. REVIEW ON REGULARIZED ZERO-FORCING PRECODING: The regularization parameters ϕj and Zj balance intended-channel gain against interference suppression and can depend on SNR, CSI quality, and system dimensions.Their optimization is difficult in general multi-cell scenarios, where prior work used intuitive parameter choices.
  • III. REVIEW ON REGULARIZED ZERO-FORCING PRECODING: Random matrix theory supplies deterministic equivalents for quantities involving one or two resolvent matrices, supporting asymptotic SINR analysis.Theorem 1 handles one resolvent occurrence, while subsequent results address products involving two resolvents.
  • III. REVIEW ON REGULARIZED ZERO-FORCING PRECODING: RZF user SINRs converge to deterministic quantities in the large-(M, K) regime that depend only on channel statistics.These quantities are used to characterize RZF performance asymptotically.

A. Complexity Issues of RZF Precoding

Although RZF SINRs have deterministic large-system limits, repeatedly recomputing its large matrix inverse remains computationally intractable. TPE precoding distributes computation through simpler matrix-vector operations.

  • A. Complexity Issues of RZF Precoding: RZF requires recomputing a large-dimensional matrix inverse hundreds of times per second under coherence times of a few milliseconds.Matrix inversion scales cubically in the matrix rank.
  • A. Complexity Issues of RZF Precoding: TPE precoding avoids per-coherence-interval precomputation and divides the computation into simple matrix-vector multiplications.These operations can be distributed uniformly over time and implemented with lower complexity.

IV. TRUNCATED POLYNOMIAL EXPANSION PRECODING

TPE precoding approximates the RZF matrix inverse with a truncated polynomial, providing a tunable low-complexity family from MRT to RZF. In the multi-cell setting, its scalar coefficients and extensions support performance-oriented design under realistic channel and interference conditions.

  • TPE construction: TPE precoding approximates the RZF matrix inverse with a truncated matrix polynomial implemented through low-complexity multistage hardware.The polynomial has degree J−1, while changing J controls the transition between simpler and more accurate precoding.
  • TPE construction: The TPE order J_j determines hardware complexity and spans MRT at J_j=1 to RZF at J_j=min(M,K).For intermediate orders, the polynomial coefficients become design parameters selected for a system-performance objective.
  • Approximation properties: TPE approximation error per user terminal depends on J_j rather than system dimensions, so the order need not scale with M and K.This follows from approximating each eigenvalue inversion with a J_j-term Taylor expansion.
  • Multi-cell extension: The proposed multi-cell framework can accommodate arbitrary deterministic interference-suppression matrices Z_j by applying TPE to rotated channels.The extension may require different power constraints in the SINR optimization.
  • Multi-cell extension: Asymptotic SINR analysis is provided for the multi-cell TPE scheme, enabling coefficient design through deterministic performance expressions.The analysis is introduced after extending the framework to the multi-cell setting.

A. Large-Scale Approximations of the SINRs

The paper derives large-scale deterministic approximations for the SINRs of multi-cell TPE precoding. These results reduce random SINR quantities to recursively computable expressions based on channel statistics and polynomial-expansion derivatives.

  • Asymptotic SINR analysis: In the large-(M,K) regime, each user terminal’s SINR can be approximated by a deterministic term depending only on channel statistics.The analysis is stated under the paper’s asymptotic assumptions.
  • Recursive computation: The derivation uses derivatives of T_ℓ(t) and δ_ℓ,k(t), computed recursively up to order 2J_ℓ−1.These derivatives enter the asymptotic SINR expressions.
  • Asymptotic results: Theorem 4 establishes asymptotic behavior for the auxiliary quantities X_j,m(t) and Z_ℓ,j,m(t), including variance convergence for fixed derivative order.The theorem applies under Assumptions A-1 and A-5 in the regime defined by Assumption A-4.
  • Asymptotic results: Corollary 5 converts the auxiliary-function limits into an asymptotic expression for the SINR of each user.The result follows in the asymptotic regime of Theorem 4.
  • Deterministic equivalents: Asymptotic equivalents of a_j,m and B_ℓ,j,m replace the random quantities appearing in the SINR expression for all terminals and cells.Their finite dimensions make it sufficient to approximate the expected value of each element through resolvent-matrix analysis.

B. Optimization of the System Performance

The paper optimizes TPE coefficients for weighted max-min fairness using deterministic SINR equivalents, then solves an approximate quasi-convex formulation through semidefinite relaxation and bisection.

  • The TPE coefficients are optimized using asymptotic SINR equivalents while preserving per-cell transmit-power constraints.
  • Weighted max-min fairness balances system throughput, user fairness, and computational complexity by maximizing the minimum weighted user rate.
  • The resulting coefficient problem is NP-hard, so the paper focuses on finding a sensible approximate solution rather than the global optimum.
  • Semidefinite relaxation replaces rank-one coefficient matrices with positive semidefinite matrices, producing a tractable relaxed optimization problem for fixed fairness level.
  • Bisection searches for the largest feasible minimum weighted rate because the relaxed problem is convex for fixed ξ and becomes stricter as ξ increases.
  • When relaxed matrices have rank greater than one, principal eigenvectors are scaled to satisfy the power constraints and recover rank-one approximations.
  • The optimization complexity is polynomial in user count and TPE orders, independent of antenna count, and can therefore be performed offline.
  • User weights can be selected as RZF-achieved rates to make TPE approximate the RZF user-throughput distribution.

V. SIMULATION EXAMPLE

The simulations evaluate TPE precoding against RZF in a realistic three-cell deployment, varying antennas, polynomial order, regularization, training SNR, and finite-size accuracy. TPE can outperform suboptimal RZF in the tested multi-cell setting, while deterministic equivalents remain accurate for finite dimensions.

  • Deployment and setup: The study models three cells with grouped users, imperfect CSI from uplink pilots, normalized downlink power, and cell-specific channel statistics.The deployment uses a three-sector site with L = 3 cells and G = 2 user groups per cell.
  • TPE order and antennas: Increasing the antenna count and TPE order raises user rates, but gains beyond order Jj = 4 are negligible in the tested scenario.The simulations use K = 40 users per cell and M ∈ {80, 160, 240, 320, 400}.
  • Comparison with RZF: TPE precoding achieves higher user rates than suboptimal RZF for all Jj ≥ 5 in the tested comparison.The reported advantage is attributed to optimized polynomial coefficients that enable some inter-cell coordination.
  • Regularization sensitivity: RZF provides the highest performance when its regularization coefficient is chosen very carefully, whereas TPE remains competitive in user performance and implementation complexity.This comparison uses K = 100, M = 250, and J = 5 while varying the common regularization coefficient.
  • Training quality: Both precoding schemes achieve higher performance as the effective training SNR ρtr increases.The experiment evaluates K = 100, M = 250, J ∈ {3, 5}, and ϕ = 0.01.
  • Asymptotic accuracy: Deterministic equivalents yield good accuracy even for finite system dimensions when empirical and theoretical user rates are compared.The comparison uses TPE with Jj = 5 and RZF while varying M.

VI. CONCLUSION

The paper extends TPE precoding to large-scale multi-cell MIMO and uses deterministic SINR expressions to optimize polynomial coefficients offline. Numerical results show that TPE can provide lower complexity and higher throughput than RZF in certain scenarios.

  • Contribution: The paper generalizes TPE precoding from single-cell to large-scale multi-cell MIMO systems.TPE approximates RZF’s regularized channel inversion with a truncated polynomial expansion.
  • Validation: The deterministic approximations support optimization of polynomial coefficients and are validated by comparing empirical and theoretical user rates.Figure 5 is used to assess the asymptotic accuracy of the approximations.
  • Model and optimization: The model incorporates user-specific channel statistics, pilot contamination, different TPE orders across cells, and cell-specific power constraints.The resulting asymptotic SINR expressions depend only on channel statistics and support offline coefficient optimization.
  • Main result: In certain scenarios, TPE achieves both lower complexity and better throughput than RZF because its polynomial coefficients are optimized while RZF regularization is not.The conclusion specifically contrasts optimized TPE coefficients with the unavailable corresponding multi-cell RZF optimization.

APPENDIX A SOME USEFUL RESULTS

This appendix collects matrix identities, convergence inequalities, rank-one perturbation tools, and intermediate deterministic-equivalent calculations used in the analysis. These results establish asymptotic control of the random quantities underlying the user-rate expressions.

  • Resolvent tools: Resolvent identities and rank-one perturbation results provide algebraic tools for manipulating matrices after removing individual channel columns.The appendix introduces common inverses of resolvents and a rank-one perturbation lemma.
  • Convergence bounds: Quadratic-form convergence lemmas control Gaussian vector expressions involving independent random matrices with bounded spectral norm.The appendix states moment bounds and a simplified inequality for these quadratic forms.
  • Target quantities: The analysis targets deterministic equivalents for E[Xj,m(t)] and E[Zj,m(t)] using resolvent matrices with selected user contributions removed.The notation and sequential decomposition are introduced before proving the asymptotic limits.
  • Asymptotic result: The derived limit shows E[Xj,m(t)] converges to δj,m(t)/(1 + tδj,m(t)) as M and K grow jointly.The convergence statement is obtained after applying the appendix lemmas and dominated convergence.
  • Interference terms: The remaining interference-related terms are decomposed into auxiliary quantities and controlled with quadratic-form convergence before substitution yields the desired result.The proof treats Uℓ,j,m(t) and Vℓ,j,m(t) separately and then combines their deterministic equivalents.

APPENDIX C PROOF OF COROLLARY 5

The proof extends convergence of deterministic equivalents from the functions themselves to their derivatives. It uses analyticity, subsequence convergence, and a decomposition controlling behavior near t = 0.

  • Starting point: The proof begins from convergence of Xj,m(t) to its deterministic equivalent and treats the corresponding Zℓ,j,m(t) argument analogously.The appendix then studies convergence of derivatives through analytic-function arguments.
  • Analytic argument: Analyticity on C\R− and boundedness on compact subsets allow Montel’s theorem to produce convergent subsequences.Because the limiting function vanishes on R+, analyticity is used to extend the conclusion.
  • Control near zero: The extension to t = 0 decomposes the derivative difference into terms whose magnitudes are separately bounded by epsilon-dependent controls.The proof selects M sufficiently large and η sufficiently small to combine the bounds.

APPENDIX D ALGORITHM FOR COMPUTING TℓAND δℓ,m.

The appendix presents an iterative procedure for computing the first D derivatives of deterministic equivalents at t = 0. It initializes per-user quantities and recursively updates matrix and scalar terms for each derivative order.

  • The algorithm computes the first D derivatives of deterministic equivalents at t = 0.
  • It initializes δ(0) ℓ,k from the trace of Φℓ,ℓ,k, while setting g(0) ℓ,k and Q(0) to zero.
  • For each derivative order i = 1,...,D, the procedure recursively updates Q(i) and T(i) ℓ using previously computed terms.
  • After each matrix update, it recursively computes f (i) ℓ,k, g(i) ℓ,k, and δ(i) ℓ,k for every user k.
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