PaperlayerView exact revision on arXiv

Source-linked AI summary

Massive MIMO has Unlimited Capacity

Emil Björnson, Jakob Hoydis, Luca Sanguinetti

arXiv:1705.00538v4cs.IT

TL;DR

Pilot contamination was believed to impose a finite capacity limit as the antenna count grows, based on simplistic channel models and suboptimal processing. This paper analyzes multicell MMSE precoding and combining under more general covariance conditions. It proves that capacity increases without bound when contaminating users' covariance matrices are asymptotically linearly independent, and that covariance diagonals can suffice when they are linearly independent.

  • Problem

    Pilot contamination was believed to create a finite asymptotic spectral-efficiency and capacity limit, especially under i.i.d. Rayleigh fading and conventional processing.

  • Method

    The paper analyzes MMSE combining and precoding using Rayleigh channel models with user covariance matrices, including multicell extensions and diagonal-only covariance information.

  • Results

    Capacity increases without bound as M →∞ under pilot contamination when contaminating users have asymptotically linearly independent covariance matrices.

  • Takeaways & Limitations

    The finite capacity limit is not general: small spatial-correlation or large-scale-fading variations can make covariance matrices linearly independent and enable unbounded asymptotic capacity.

  • Takeaways & Limitations

    Full covariance knowledge is assumed for most results, although linearly independent covariance diagonals are sufficient under the stated theorems.

Abstract

from arXiv · show

The capacity of cellular networks can be improved by the unprecedented array gain and spatial multiplexing offered by Massive MIMO. Since its inception, the coherent interference caused by pilot contamination has been believed to create a finite capacity limit, as the number of antennas goes to infinity. In this paper, we prove that this is incorrect and an artifact from using simplistic channel models and suboptimal precoding/combining schemes. We show that with multicell MMSE precoding/combining and a tiny amount of spatial channel correlation or large-scale fading variations over the array, the capacity increases without bound as the number of antennas increases, even under pilot contamination. More precisely, the result holds when the channel covariance matrices of the contaminating users are asymptotically linearly independent, which is generally the case. If also the diagonals of the covariance matrices are linearly independent, it is sufficient to know these diagonals (and not the full covariance matrices) to achieve an unlimited asymptotic capacity.

I. INTRODUCTION

Massive MIMO can raise cellular spectral efficiency through array gain and spatial multiplexing, but pilot reuse creates coherent interference under conventional assumptions. The paper argues that this finite-limit view is generally incorrect: MMSE processing achieves unbounded spectral efficiency when contaminating users have asymptotically linearly independent covariance matrices.

  • Massive MIMO reinforces desired signal power by M through coherent combining or precoding, while noise and independent interference do not increase.
  • Pilot reuse is necessary because pilot resources are limited by channel coherence time, and it correlates desired and interfering channel estimates.Under MR processing and i.i.d. Rayleigh fading, coherent interference also scales with M, producing a finite SE limit.
  • Prior MMSE asymptotic analyses also obtained finite SE limits under i.i.d. Rayleigh fading, where spatial correlation is absent.M-MMSE uses channel estimates from all cells, whereas S-MMSE uses only estimates from the serving cell.
  • Practical covariance matrices generally lack orthogonal support, so pilot decontamination methods requiring orthogonal covariance subspaces apply only to unlikely special cases.Random covariance matrices are almost surely not orthogonal in support, and measurements report irregular angular support.
  • The analysis models Rayleigh channels through covariance matrices that capture macroscopic effects and spatial correlation, with pilot observations corrupted by receiver noise.The two-user setup illustrates how MMSE processing rejects coherent interference between pilot-sharing users.
  • The paper proves that M-MMSE combining yields unbounded instantaneous SINR when the channel covariance matrices are asymptotically linearly independent.The condition requires the covariance-matrix differences to retain energy proportional to M; both users can then achieve SE growing as log2(M).

C. Downlink Data Transmission

The downlink uses MMSE precoding to transmit user-specific signals while treating random channel terms as noise in an achievable spectral-efficiency bound. Under the same covariance conditions as the uplink, both users' downlink spectral efficiencies and capacity increase without bound as M grows.

  • The transmitted signal combines normalized information-bearing signals with user-specific precoding vectors under a normalized downlink power allocation.
  • The achievable downlink bound treats random received-signal terms as noise while retaining the deterministic average precoded channel as the desired signal.
  • The downlink SE requires each user to know the average precoded channel and total variance, not an instantaneous downlink channel estimate.
  • MMSE precoding produces an effective downlink SINR that increases unboundedly as M →∞.
  • The downlink SE, and therefore capacity, increases without bound under the same assumptions as the uplink.Its asymptotic growth is proportional to log2(M), and the second user can simultaneously achieve unbounded SE.

D. Interpretation and Generality

Asymptotic linear independence of pilot-contaminated channel estimates enables interference rejection while preserving array gain. This condition is generally supported by realistic covariance variations and extends beyond the two-user setting.

  • Interpretation: Linearly independent channel estimates permit combining vectors that null pilot-contaminated interference while retaining desired-signal gain.The same principle applies to precoding, eliminating coherent downlink interference under the stated construction.
  • Interpretation: M-MMSE combining and precoding retain array gain while rejecting coherent interference, and therefore outperform the illustrative ZF construction in SINR.ZF provides the intuitive interference-rejection argument; MMSE has at least as high SINR.
  • Examples: N = αM with 0 < α < 1 yields a non-zero asymptotic covariance-separation limit and an O(M)-rank subspace where diagonal covariances differ.When N is constant, the corresponding expression instead converges to zero.
  • Examples: Random antenna-wise fading perturbations make covariance matrices asymptotically linearly independent, so the required condition is generally satisfied in irregular propagation environments.The argument treats covariance matrices equal up to scaling as non-robust to random perturbations.
  • Examples: For distributed arrays, b11b22 ≠ b12b21 makes the covariance matrices asymptotically linearly independent, yielding unbounded uplink and downlink SE with MMSE or ZF.This condition also determines whether the associated matrix inverse exists.
  • Practical implications: The MMSE results require knowledge of deterministic channel statistics, including the covariance matrices and their relevant sums.Covariance matrices may be estimated from channel realizations over multiple resource blocks.

E. Achievable SE with Partial Knowledge of Covariance Matrices

Element-wise MMSE uses only covariance diagonals for channel estimation and combining. Under asymptotic linear independence of those diagonals, uplink and downlink spectral efficiencies grow without bound.

  • Estimator: Element-wise MMSE estimates each channel entry separately using only the main diagonals of the covariance matrices.These diagonals can be estimated efficiently with a sample count that need not grow with M.
  • Combining: The approximate MMSE combining vector is diagonal-matrix based and coincides with ordinary MMSE combining when the covariance matrices are diagonal.The analysis uses a use-and-then-forget spectral-efficiency bound that applies to arbitrary estimation and combining schemes.
  • Condition: Diagonal covariance profiles must be asymptotically linearly independent for the partial-knowledge result.This is formalized as Assumption 3 on the diagonal matrices D1 and D2.
  • Result: Theorem 3 states that the uplink SINR increases unboundedly as M →∞ under the stated assumptions.The corresponding uplink SEs for both users, and similarly the downlink SE, increase without bound even when only covariance diagonals are known.

III. ASYMPTOTIC SPECTRAL EFFICIENCY IN MULTICELL MASSIVE MIMO

In multicell Massive MIMO, pilot reuse correlates channel estimates across cells and creates pilot contamination. Multicell MMSE uses all relevant channel estimates and achieves unbounded asymptotic spectral efficiency under covariance-independence conditions.

  • System model: Pilot reuse across cells contaminates channel estimates because users sharing pilots produce correlated estimates at the base station.The multicell model uses one shared pilot for the kth user across each of L cells.
  • Combining schemes: M-MMSE combining maximizes the instantaneous effective SINR using channel estimates from users in all cells.S-MMSE uses only own-cell estimates and replaces inter-cell estimate products by their averages.
  • Assumptions: Assumptions 4 and 5 require nonvanishing covariance traces, bounded spectral norms, and asymptotic linear independence among pilot-sharing covariance matrices.The covariance-independence condition implies the corresponding estimated channels are asymptotically linearly independent.
  • Main result: M →∞ makes the uplink SINR γul_jk increase unboundedly under M-MMSE combining and the stated assumptions.Theorem 4 therefore establishes unbounded uplink SE despite pilot contamination.
  • Interpretation: Multicell ZF can reject interference from pilot-contaminating users while retaining growing array gain, and M-MMSE inherits unbounded SE because it is optimal.The construction nulls the subspace spanned by the relevant interfering channel estimates.

B. Downlink Data Transmission

Downlink transmission assigns each user a normalized precoding vector and can use M-MMSE precoding derived from multicell combining vectors. Under the stated assumptions, downlink SINR and spectral efficiency grow without bound as the antenna count increases, including with approximate schemes using covariance diagonals.

  • Downlink Data Transmission: Downlink transmission assigns each UE a data signal and a unit-normalized precoding vector with normalized transmit power.The precoding vector satisfies E{∥w_li∥^2}=1, so the allocated signal power is ρ_dl.
  • Downlink Data Transmission: M-MMSE precoding selects downlink precoding vectors from the multicell M-MMSE combining vectors.This choice is motivated by uplink-downlink duality because joint multicell precoding optimization is difficult.
  • Downlink Data Transmission: Under Assumptions 4 and 5, Theorem 5 states that M-MMSE precoding yields the stated downlink SINR result.The proof uses arguments from Theorem 2 and results from Appendix F for Theorem 4.
  • Downlink Data Transmission: All network UEs achieve asymptotically unbounded downlink spectral efficiency despite equal power allocation and no instantaneous precoded-channel estimation.The key requirement is asymptotic linear independence between desired-user and pilot-contaminating users' channel estimates.
  • Approximate M-MMSE Combining and Precoding: Approximate M-MMSE combining uses only covariance diagonals, whose asymptotic linear independence is sufficient for unbounded uplink spectral efficiency and unlimited capacity.The result is generally satisfied because small random covariance-element variations can provide asymptotic linear independence.
  • Approximate M-MMSE Combining and Precoding: Under Assumptions 4 and 6, Theorem 6 states that approximate M-MMSE combining produces SINR that increases unboundedly as M →∞.The proof is omitted and follows along the lines of Theorem 3.

IV. NUMERICAL RESULTS

The numerical results show that modest spatial correlation or large-scale fading variations make covariance matrices linearly independent, enabling M-MMSE to achieve unbounded spectral efficiency despite pilot contamination. Conventional schemes saturate or perform poorly, while diagonal-only covariance information closely approaches full-MMSE performance in the downlink.

  • Covariance models: The three covariance models produce eigenvalue variations; one-ring is rank-deficient, whereas exponential correlation and large-scale fading variation are full-rank.The simulations emphasize that linear independence, rather than rank deficiency, is sufficient for the main results.
  • Simulation setup: In the challenging four-cell setup, cell-edge users have similar but non-identical angles and distances, creating severe pilot contamination.The setup uses L = 4 cells, K = 2 UEs per cell, pilot length τp = K, and coherence blocks of τc = 200 channel uses.
  • A. Uplink: M-MMSE uplink spectral efficiency grows without bound with M, while S-MMSE and MR converge to finite limits under exponential correlation with r = 0.5.M-ZF performs poorly because suppressing interference also removes much of the desired signal; M-MMSE balances interference suppression and coherent combining.
  • A. Uplink: Avoiding pilot contamination by time splitting yields unbounded growth more slowly because four-cell scheduling imposes a pre-log factor of 1/4.The results therefore identify time splitting as inefficient even for the small L = 4 system studied.
  • A. Uplink: At M = 200, small large-scale fading variations over the array make covariance matrices linearly independent and give M-MMSE substantial gains over S-MMSE and MR.With σ = 0, the covariance matrices are scaled identities and M-MMSE provides no benefit; measured variations around 4 dB correspond to σ ≈ 4.

APPENDIX A – USEFUL RESULTS

The appendix supplies matrix and quadratic-form tools used to establish asymptotic SINR growth. Under the stated assumptions, these tools imply unbounded uplink spectral efficiency.

  • Useful lemmas: Quadratic forms with independent vectors converge to normalized traces, while cross terms vanish asymptotically.The deviations have moment order O(M^-p/2).
  • Useful lemmas: The matrix inversion lemma provides rank-one and low-rank inverse updates for Hermitian matrices.These identities are repeatedly used to manipulate MMSE expressions.
  • Asymptotic coefficients: lim infM β22 > 0 ensures lim infM δ1 > 0 after minimizing over λ2 and substituting the resulting bound.The infimum is attained at λ2 = β12/β22.
  • Asymptotic coefficients: Assumption 2 yields the corollary needed to control the trace-based coefficients appearing in the proof.The argument uses λ = [λ1, λ2]^T and evaluates the condition for i = 1, 2.
  • Conclusion: γul_1 grows almost surely without bound, so log2(1 + γul_1) and the expected uplink spectral efficiency also diverge.The logarithm is strictly increasing, and almost-sure divergence of non-negative random variables implies divergence of their expected value.

APPENDIX C – PROOF OF COROLLARY 1 IN APPENDIX B

This appendix proves the corollary by expanding the uplink SINR and bounding its constituent terms. The resulting bounds establish unbounded uplink and downlink SINR growth under the relevant assumptions.

  • Proof setup: The proof begins by lower-bounding the argument associated with UE 1 and applying the matrix inversion lemma.The same reasoning is later used for the downlink.
  • Main bound: The denominator in the key bound is uniformly bounded above and independent of λ2, making Assumption 2 sufficient for the required inequality.The result for i = 2 follows by interchanging the UE indices.
  • Coefficient bounds: The coefficients β′11, β′22, and β′12 are non-negative because they are traces of products of positive-semidefinite matrices.This supports the subsequent bounds on the SINR terms.
  • Coefficient bounds: Assumption 1 and Lemma 3 provide boundedness and positive lower limits for the quantities entering the SINR bound.The proof applies these properties to both desired-signal and interference-related terms.
  • Conclusion: γul_1 and γdl_1 grow unboundedly as M →∞, yielding the corresponding unbounded spectral-efficiency behavior.The downlink conclusion follows by the same arguments used for the uplink.

APPENDIX E – PROOF OF THEOREM 3

The appendix analyzes the EW-MMSE estimator, whose estimate and error are correlated. Under Assumption 3, the derived bounds show that uplink SINR and spectral efficiency grow without bound.

  • Estimator model: The EW-MMSE estimate and estimation error are Gaussian with covariance expressions determined by Dk, Λ, Q, and Rk.Unlike MMSE estimation, the estimate and error are correlated.
  • SINR expansion: The correlation between the estimate, error, and combining vector is explicitly accounted for when expanding the SINR terms.The proof uses E{ĥk ĥH_l} and related covariance identities.
  • Asymptotic bound: Assumption 3 implies a positive lower limit for υ1, while the remaining components of υ′1 remain uniformly bounded.This combination supplies the key asymptotic SINR bound.
  • Conclusion: γul_1 grows unboundedly as M →∞, and SEul_1 therefore also grows without bound.The conclusion follows directly from the derived SINR behavior.

APPENDIX F – PROOF OF THEOREM 4

The appendix develops the proof using matrices of estimated channels from pilot-contaminating users and repeated matrix-inversion steps. The supplied passages mainly describe this proof setup.

  • Proof setup: The proof collects estimated channels from pilot-contaminating users into matrices and applies the matrix inversion lemma.The matrices exclude either the indexed cell or user as required by the derivation.
  • Asymptotic handling: Independence between the indexed estimated channel and the remaining channel matrix permits quadratic-form asymptotics through Lemma 3.The normalized cross term converges to zero under the stated independence.
  • Asymptotic handling: Assumption 4 ensures that the covariance matrices of the estimated channels have uniformly bounded spectral norm.This boundedness is obtained using Lemma 4.

1 M ˆHH

Under Assumption 5, the constructed matrix Cjk becomes invertible as M grows, enabling a positive asymptotic bound and unbounded growth of the relevant performance quantities.

  • Cjk is defined through vectors and becomes invertible as M →∞ under Assumption 5.Corollary 3 states the asymptotic invertibility condition for Cjk.
  • The required infimum exists for sufficiently large M because Cjk is asymptotically invertible.The infimum is then characterized using bjk and Cjk.
  • lim infM δjk > 0, which implies that γuljk grows almost surely without bound and that SEul increases without bound.The positive lower bound on δjk is obtained by substituting the asymptotic infimum into the preceding condition.
  • The auxiliary quantity jk also grows unboundedly as M →∞ under the preceding boundedness conditions.This follows from the stated limits on δ1 and δ2 and the corresponding asymptotic argument.

APPENDIX G – PROOF OF COROLLARY 3 IN APPENDIX F

The proof establishes Cjk's invertibility by viewing it as a Gramian and showing that its generating vectors are asymptotically linearly independent under the stated condition.

  • Cjk is a Gramian formed from inner products of the vectors {ujlk: ∀l ≠ j}.Thus, asymptotic invertibility is equivalent to asymptotic linear independence of these vectors.
  • Condition (89) implies that the full vector set {ujlk: ∀l} is asymptotically linearly independent.The argument rewrites condition (89) in compact form before applying the independence result.
  • Because every subset of a finite linearly independent set is linearly independent, the vectors excluding l = j are also asymptotically linearly independent.This establishes the condition needed for Cjk's asymptotic invertibility.
  • Therefore, under Assumption 5, Cjk is invertible as M →∞.This completes the proof of Corollary 3.
Loading 1705.00538v4…