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Cell-Free Massive MIMO versus Small Cells

Hien Quoc Ngo, Alexei Ashikhmin, Hong Yang, Erik G. Larsson, Thomas L. Marzetta

arXiv:1602.08232v2cs.IT

TL;DR

The paper addresses how much distributed Cell-Free Massive MIMO can improve over noncooperating small cells under realistic channel estimation, pilot, and power-control conditions. It derives throughput bounds and max-min power-control methods, finding substantially higher 95%-likely per-user throughput and greater robustness to correlated shadow fading, while requiring more backhaul.

  • Problem

    A comprehensive comparison of distributed Massive MIMO and small-cell systems accounting for imperfect CSI, pilot assignment, and power control was not available.

  • Method

    The paper derives closed-form throughput bounds for Cell-Free Massive MIMO and develops pilot-assignment and max-min power-control algorithms for downlink and uplink operation.

  • Results

    Cell-Free Massive MIMO significantly outperforms small-cell systems in 95%-likely per-user throughput and is more robust to shadow fading correlation.

  • Takeaways & Limitations

    With shadowing correlation, Cell-Free 95%-likely per-user throughputs are an order of magnitude higher than those of small-cell systems, but small cells require much less backhaul.

Abstract

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A Cell-Free Massive MIMO (multiple-input multiple-output) system comprises a very large number of distributed access points (APs)which simultaneously serve a much smaller number of users over the same time/frequency resources based on directly measured channel characteristics. The APs and users have only one antenna each. The APs acquire channel state information through time-division duplex operation and the reception of uplink pilot signals transmitted by the users. The APs perform multiplexing/de-multiplexing through conjugate beamforming on the downlink and matched filtering on the uplink. Closed-form expressions for individual user uplink and downlink throughputs lead to max-min power control algorithms. Max-min power control ensures uniformly good service throughout the area of coverage. A pilot assignment algorithm helps to mitigate the effects of pilot contamination, but power control is far more important in that regard. Cell-Free Massive MIMO has considerably improved performance with respect to a conventional small-cell scheme, whereby each user is served by a dedicated AP, in terms of both 95%-likely per-user throughput and immunity to shadow fading spatial correlation. Under uncorrelated shadow fading conditions, the cell-free scheme provides nearly 5-fold improvement in 95%-likely per-user throughput over the small-cell scheme, and 10-fold improvement when shadow fading is correlated.

I. INTRODUCTION

The paper introduces Cell-Free Massive MIMO, in which distributed single-antenna APs jointly serve users without cells or cell boundaries. It develops analytical and optimization tools and compares the approach quantitatively with small-cell systems.

  • System concept: Cell-Free Massive MIMO uses many distributed APs to serve fewer users cooperatively over the same time-frequency resources.APs and users each have one antenna, and the APs cooperate phase-coherently through a backhaul network.
  • System concept: The system limits CPU–AP information exchange to payload data and slowly varying power-control coefficients, without sharing instantaneous CSI.Channels are estimated locally from uplink pilots and used for downlink precoding and uplink detection.
  • Scope and trade-offs: The paper notes that alternative linear processing could improve performance but would require more backhaul than maximum-ratio processing.The complexity–performance tradeoff is left for future study.
  • Analysis and algorithms: The paper derives finite-system capacity lower bounds that account for channel-estimation errors, power control, and non-orthogonal pilots.It uses conjugate beamforming on the downlink and matched filtering on the uplink.
  • Analysis and algorithms: Max-min power-control algorithms maximize the smallest user rate, with globally optimal downlink and uplink solutions obtained through SOCP and linear-program sequences.This design emphasizes uniformly good per-user service rather than sum throughput.
  • Evaluation: The study compares Cell-Free Massive MIMO and small-cell systems under uncorrelated and correlated shadow fading, addressing an insufficiently quantified comparison in prior work.Earlier comparisons often considered collocated Massive MIMO or assumed perfect CSI.

A. Uplink Training

Uplink training occurs within each coherence block, allowing every AP to estimate all user channels locally from simultaneously transmitted pilot sequences. Non-orthogonal pilots create pilot contamination when the training duration is insufficient for pairwise orthogonality.

  • Training procedure: Each coherence interval includes an uplink training phase of duration τ_cf, with τ_cf < τ_c, during which all users transmit pilots simultaneously.The APs use these pilots to estimate channels before payload transmission.
  • Channel estimation: Each AP receives a pilot vector and projects it onto a user’s pilot sequence to obtain a channel-estimation statistic.For arbitrary pilots this statistic can be suboptimal, but it is sufficient when pilots are identical or orthogonal.
  • Pilot contamination: When τ_cf ≥ K, pairwise orthogonal pilots eliminate the interfering term in the channel estimate.In general, limited coherence intervals require τ_cf < K, so non-orthogonal pilots are used.
  • Pilot contamination: Non-orthogonal pilot signals from other users degrade channel estimates, producing pilot contamination.The APs perform channel estimation independently and do not exchange channel estimates.
  • Data transmission: The APs use local channel estimates for conjugate beamforming on the downlink and matched filtering for uplink detection.Downlink signals are precoded locally, while matched-filtered uplink observations are forwarded to the CPU.

III. PERFORMANCE ANALYSIS

As the number of APs grows, channel vectors become orthogonal under the analyzed conditions, removing non-coherent interference, small-scale fading, and noise. Residual interference is tied to pilot non-orthogonality and disappears with orthogonal pilots.

  • Asymptotic behavior: As M → ∞, the channels between users and APs become orthogonal under fixed large-scale fading coefficients.The convergence analysis is conditioned on the deterministic coefficients {β_mk}.
  • Asymptotic behavior: Conjugate beamforming and matched filtering make non-coherent interference, small-scale fading, and noise disappear as M grows without bound.This mirrors the corresponding asymptotic behavior in collocated Massive MIMO.
  • Residual interference: In the large-M limit, the downlink received signal contains the desired signal plus interference originating from non-orthogonal pilot sequences.The remaining interference is therefore associated with pilot non-orthogonality.
  • Residual interference: Orthogonal pilot sequences eliminate the remaining interference and noise in the received signal.Similar asymptotic results hold for the uplink.

B. Achievable Rate for Finite M

The paper derives finite-dimensional closed-form achievable downlink and uplink rates for Cell-Free Massive MIMO using statistical channel knowledge and effective-noise decompositions. The resulting expressions account for desired signal, beamforming uncertainty, interference, channel estimation, and per-AP power constraints.

  • B. Achievable Rate for Finite M: The rate decomposition distinguishes desired signal, beamforming gain uncertainty, and interference from other users.These terms are identified in the downlink signal model and used to form an achievable-rate bound.
  • B. Achievable Rate for Finite M: Closed-form achievable downlink and uplink rate expressions are derived for finite numbers of APs and users.The downlink uses conjugate beamforming, while the uplink uses statistical channel knowledge for detection.
  • B. Achievable Rate for Finite M: The downlink rate expression is valid for any finite M and K under conjugate beamforming.It is obtained by treating the remaining terms as effective noise after separating the desired signal.
  • B. Achievable Rate for Finite M: Cell-Free and collocated Massive MIMO differ through spatially varying large-scale fading and individual AP power constraints.The collocated special case uses equal large-scale fading across APs and a total base-station power constraint.
  • B. Achievable Rate for Finite M: Under statistical channel knowledge, the achievable rate is close to the genie-aided rate based on instantaneous channel knowledge.The reported small gap is attributed to channel hardening, so downlink training is not necessary.

2) Achievable Uplink Rate:

The section presents an achievable uplink-rate result for finite Cell-Free Massive MIMO systems with matched-filter detection, alongside pilot-assignment procedures used to manage pilot contamination. A greedy method iteratively improves the pilot of the lowest-downlink-rate user.

  • 2) Achievable Uplink Rate:: An achievable uplink rate for user k is given for any finite numbers of APs and users with matched-filter detection.The result is stated as a closed-form expression associated with equation (27).
  • 2) Achievable Uplink Rate:: The achievable-rate figure plots rate per user against the number of APs for different K values.The caption identifies achievable rate as the plotted quantity and AP count as the horizontal variable.
  • 2) Achievable Uplink Rate:: When τcf < K, users must share non-orthogonal pilots, creating pilot contamination; when τcf ≥ K, orthogonal pilots are assigned.Random assignment can place nearby users on the same pilot, causing strong contamination.
  • 2) Achievable Uplink Rate:: Greedy pilot assignment repeatedly changes the pilot of the lowest-downlink-rate user to reduce its contamination effect.The algorithm begins with random assignment and iterates for a predetermined number of iterations.
  • 2) Achievable Uplink Rate:: The CPU can recompute greedy pilot assignments on the large-scale fading timescale and communicate only pilot indices to users.This reduces the required central-unit signaling for pilot updates.

B. Power Control

The paper uses max-min power control to provide uniformly good service by maximizing the smallest user rate. The downlink and uplink problems admit efficient globally optimal solution procedures based on convex or linear feasibility subproblems.

  • B. Power Control: Max-min power control maximizes the smallest user rate and is performed at the CPU on the large-scale fading timescale.The stated purpose is to provide uniformly good service regardless of geographical location.
  • B. Power Control: In the downlink, the power-control coefficients maximize the minimum user rate subject to per-AP power constraints.At the optimum, all users receive the same rate.
  • B. Power Control: The downlink optimization is quasi-concave and can be solved by bisection with convex feasibility problems.Slack-variable reformulation establishes the equivalent optimization structure.
  • B. Power Control: The uplink max-min problem is quasi-linear and can be solved efficiently by bisection with linear feasibility problems.For a fixed target rate, the constraints are linear.

V. SMALL-CELL SYSTEM

The small-cell baseline assigns each user to one AP and models separate downlink and uplink channel estimation because the effective channel does not harden. Its power control is formulated as a max-min rate problem, with handovers excluded for analytical tractability.

  • V. SMALL-CELL SYSTEM: The paper compares small-cell and Cell-Free Massive MIMO systems using achievable downlink and uplink rates.The small-cell model and power control are introduced for the later performance comparison.
  • V. SMALL-CELL SYSTEM: Small-cell uplink max-min power control is formulated as an optimization problem and solved with a bisection algorithm.The algorithm repeatedly tests convex feasibility over a target objective range.
  • V. SMALL-CELL SYSTEM: Each small-cell user is served by one AP selected from the available APs using the largest average received useful signal power.AP selection proceeds user by user in random order, and already selected APs become unavailable.
  • V. SMALL-CELL SYSTEM: The analysis assumes a short enough interval that handovers between APs do not occur.This enables rigorous analysis, but the authors state that the resulting small-cell performance figures may be overoptimistic.
  • V. SMALL-CELL SYSTEM: Because the small-cell effective channel is a single Rayleigh scalar, channel hardening does not occur.Both users and APs therefore estimate their effective channel gains for demodulation, requiring uplink and downlink training.
  • V. SMALL-CELL SYSTEM: Small-cell downlink transmission sends each user's symbol from its single selected AP without conjugate beamforming.The user estimates the channel from downlink pilots before detecting the desired signal.

1) Achievable Downlink Rate:

The downlink and uplink power-control problems maximize the minimum user throughput subject to bounded power-control coefficients. Both formulations are quasi-linear programs solvable by bisection.

  • Max-min power control is formulated for the downlink with each coefficient constrained between 0 and 1.
  • The downlink reformulation remains equivalent because the relevant rate expression is monotonically increasing in the channel-dependent quantity.
  • Bisection solves the resulting downlink quasi-linear program.
  • For the uplink, APs estimate channels from user pilots and use the estimates to detect desired signals and derive achievable rates.
  • The uplink max-min power-control problem is likewise a quasi-linear program with coefficients constrained between 0 and 1, solvable by bisection.

VI. NUMERICAL RESULTS AND DISCUSSIONS

The evaluation models spatially correlated shadow fading and compares Cell-Free Massive MIMO with small-cell systems under matched simulation conditions. It examines throughput distributions, pilot assignment, and max-min power control for downlink and uplink transmission.

  • Simulation setup: The study randomly places M APs and K users in a wrapped square area and evaluates Cell-Free Massive MIMO against small-cell systems.Wrapping supplies eight neighboring copies to reduce boundary effects.
  • Shadowing correlation model: Shadow fading combines AP-specific and user-specific components controlled by δ, with δ = 0 and δ = 1 representing opposite correlation extremes.The components are modeled by independent Gaussian variables associated with nearby obstructing objects.
  • Shadowing correlation model: 20–200 m is the typical decorrelation-distance range, and shorter distances correspond to less stationary environments.The correlation model has been validated theoretically and experimentally.
  • Throughput and evaluation procedure: Net throughput includes channel-estimation overhead through training fractions within each coherence interval.Cell-Free and small-cell systems use separate training durations, τcf and τsc, respectively.

C. Results and Discussions

Cell-Free Massive MIMO consistently outperforms small-cell systems in per-user throughput, especially under correlated shadow fading and with power control. Performance depends on user/AP density, training duration, pilot assignment, and distributed power allocation.

  • Throughput comparison: 7 times higher 95%-likely downlink net throughput is achieved by Cell-Free Massive MIMO without shadow fading correlation.The Cell-Free downlink is about 14 Mbits/s versus about 2.1 Mbits/s for small cells.
  • Throughput comparison: 10 times higher 95%-likely downlink net throughput is achieved by Cell-Free Massive MIMO with shadow fading correlation.Small-cell systems are more affected by shadow fading correlation because correlated shadowing reduces the benefit of selecting the best APs.
  • Pilot assignment: Greedy pilot assignment improves 95%-likely net throughput by about 20% compared with random pilot assignment.The comparison uses otherwise corresponding settings in Figures 3–4 and Figures 7–8.
  • Distributed serving: About 10–20 of 100 APs effectively serve each user when at least 95% of allocated power is considered.Larger τcf reduces pilot contamination, improves channel estimates, and allows more APs to usefully serve each user.
  • System scaling: Reducing K or τcf decreases performance, while increasing M improves both Cell-Free and small-cell performance.Cell-Free systems benefit from favorable propagation and array gain, whereas small cells benefit from diversity gain; Cell-Free remains significantly better for all M.

VII. CONCLUSION

The paper analyzes Cell-Free Massive MIMO with imperfect channel estimation, non-orthogonal pilots, and power control, and compares it with small-cell systems under two shadow-fading models. Cell-Free systems deliver substantially higher throughput and greater robustness to shadow-fading correlation, while requiring more backhaul.

  • The analysis incorporates channel estimation, pilot non-orthogonality, and power control when comparing Cell-Free and small-cell systems.
  • Cell-Free Massive MIMO significantly outperforms small-cell systems in throughput and is more robust to shadow fading correlation.With correlated shadowing, its 95%-likely per-user throughput is an order of magnitude higher.
  • Small-cell systems require much less backhaul than Cell-Free Massive MIMO.

APPENDIX

The appendix derives closed-form rate expressions using channel-estimation properties and establishes quasi-concavity of the power-control optimization problem. The derivation relies on independence, zero-mean errors, and convex upper-level sets.

  • Channel-estimation error and the MMSE channel estimate are independent, enabling expectation and variance simplifications.
  • Substituting intermediate terms into the achievable-rate expression yields the stated closed-form rate result.
  • The power-control objective is quasi-concave because its upper-level sets and constraint set are convex.The upper-level set is represented as a second-order cone.
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