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Massive MIMO for Maximal Spectral Efficiency: How Many Users and Pilots Should Be Allocated?

Emil Björnson, Erik G. Larsson, Mérouane Debbah

arXiv:1412.7102v3cs.ITcs.NI

TL;DR

The paper studies how many UEs should be scheduled to maximize massive-MIMO spectral efficiency for a fixed antenna count, despite position-dependent conventional expressions. It derives tractable position-independent expressions and finds that large systems should devote about half their frame to pilots, while processing and interference conditions affect practical choices.

  • Problem

    The paper asks how many UEs should be scheduled to maximize spectral efficiency for a fixed number of antennas, since conventional expressions depend strongly on UE positions.

  • Method

    The paper derives uplink and downlink spectral-efficiency expressions using power control, random UE locations, arbitrary pilot allocation, and MR, ZF, and P-ZF processing.

  • Results

    The SE-optimal K* approaches S/2β as M grows, implying that half the frame should be used for pilots; simulations observed 5% to 40% pilot allocation for M ≤1000.

  • Takeaways & Limitations

    ZF is often best for per-cell spectral efficiency, while P-ZF is mainly useful under strong inter-cell interference and MR schedules the most UEs.

Abstract

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Massive MIMO is a promising technique to increase the spectral efficiency (SE) of cellular networks, by deploying antenna arrays with hundreds or thousands of active elements at the base stations and performing coherent transceiver processing. A common rule-of-thumb is that these systems should have an order of magnitude more antennas, $M$, than scheduled users, $K$, because the users' channels are likely to be near-orthogonal when $M/K > 10$. However, it has not been proved that this rule-of-thumb actually maximizes the SE. In this paper, we analyze how the optimal number of scheduled users, $K^\star$, depends on $M$ and other system parameters. To this end, new SE expressions are derived to enable efficient system-level analysis with power control, arbitrary pilot reuse, and random user locations. The value of $K^\star$ in the large-$M$ regime is derived in closed form, while simulations are used to show what happens at finite $M$, in different interference scenarios, with different pilot reuse factors, and for different processing schemes. Up to half the coherence block should be dedicated to pilots and the optimal $M/K$ is less than 10 in many cases of practical relevance. Interestingly, $K^\star$ depends strongly on the processing scheme and hence it is unfair to compare different schemes using the same $K$.

I. INTRODUCTION

The paper asks how many UEs should be scheduled per cell to maximize massive MIMO spectral efficiency, and develops tractable system-level analysis for this problem.

  • The paper studies the previously unanswered multi-cell question of how many UEs should be scheduled per cell to maximize spectral efficiency.
  • It derives SE expressions for uplink and downlink transmission with random UE locations and power control, supporting joint network optimization.The expressions are independent of instantaneous UE positions and can provide network-wide performance for symmetric topologies.
  • The analysis considers MR, ZF, and distributed full-pilot zero-forcing processing that suppresses parts of inter-cell interference without BS signaling.P-ZF uses all estimated pilot directions and trades array gain for additional inter-cell interference mitigation.
  • The system model includes random UE positions, a design parameter K for active users, and frames constrained by users’ channel coherence.Each BS has M antennas and communicates with K single-antenna UEs selected from Kmax users.
  • Statistics-aware power control inverts average channel attenuation, equalizing average effective channel gain across UEs.This policy is intended to provide uniform user experience, save UE energy, and avoid near-far blockage.

B. Downlink

The downlink reuses uplink channel measurements through TDD reciprocity, while pilot-based estimation must address inter-cell interference and pilot contamination.

  • B. Downlink: Downlink transmission uses channel reciprocity in calibrated TDD systems, so BSs need no downlink pilots or CSI feedback.The BS uses uplink channel measurements for downlink processing and can select downlink power control through estimated CSI.
  • B. Downlink: Perfect network-wide synchronization is assumed, although distant asynchronous interference is treated as practically insuppressible and expected to be weak.The analyzed processing can suppress strong interference from nearby neighboring-cell tiers.
  • A. Pilot-Based Channel Estimation: Pilot contamination limits CSI quality because reused pilots are affected by inter-cell interference, making corresponding interference difficult to reject.Users sharing a pilot produce parallel channel estimates at the BS.
  • A. Pilot-Based Channel Estimation: The paper derives channel-estimation results for arbitrary pilot reuse, allowing cells to use subsets of a pilot book spanning B symbols.The pilot book consists of B orthogonal signals with unit-magnitude entries.
  • A. Pilot-Based Channel Estimation: The MMSE estimator targets effective power-controlled uplink channels and supports arbitrary pilot allocation, while its error is summarized by an estimation-error covariance matrix.The mean-squared error is the trace of that covariance matrix.

B. Achievable UL Spectral Efficiencies

The paper derives achievable uplink spectral-efficiency bounds for arbitrary pilot reuse, random user locations, and linear combining schemes. Closed-form expressions expose how pilot contamination, inter-user interference, and interference suppression shape performance.

  • Pilot allocation: Pilot reuse factor β partitions cells into subsets sharing the same K pilot sequences, while different subsets use different pilots.All UEs within a cell use distinct pilots, and the SE interference terms sum over cells sharing pilots.
  • Achievable-SE bound: The ergodic achievable SE is a lower bound on ergodic capacity obtained by averaging over channel realizations and random interfering-user locations.The bound treats the effective channel mean as desired signal, uncorrelated signal and interference as noise, and noise as worst-case Gaussian.
  • Combining schemes: MR relies on large-M channel quasi-orthogonality for passive interference rejection, whereas ZF actively orthogonalizes the K intra-cell channels.ZF cancels some interference but reduces the array gain from M to M − K.
  • Closed-form results: Theorem 1 provides closed-form per-cell uplink SE expressions for MR and ZF, with interference depending on the combining scheme and pilot-sharing cells.The expressions are conservative lower bounds and depend on pilot allocation and propagation parameters.
  • Combining schemes: P-ZF orthogonalizes all B estimated pilot directions to mitigate parts of inter-cell interference using only locally estimated CSI.Compared with conventional ZF, P-ZF loses array gain B instead of K and requires no inter-base-station signaling.

C. Achievable DL Spectral Efficiencies

The paper extends the achievable-SE analysis to downlink precoding using channel reciprocity and position-dependent statistical power control. Uplink-downlink duality allows the downlink to achieve the same SINRs as the uplink with appropriately selected power coefficients.

  • Downlink model: Downlink precoding uses the M channel inputs to coherently enhance desired signals and suppress interference for each UE.The precoding vectors are based on acquired CSI and are normalized by their average squared norm for analytic tractability.
  • Downlink model: The average transmit power q_jk depends on UE positions but not instantaneous channel realizations.This statistical power-control assumption supports tractable downlink SE expressions.
  • Achievable-SE bound: Lemma 3 gives an ergodic achievable downlink SE with an effective SINR based on the expectations available to each UE.The UE is assumed not to know the instantaneous channels.
  • Uplink-downlink duality: Uplink-downlink duality provides downlink power control coefficients that achieve the same SINRs as in the uplink with the same total transmit power.The power is allocated differently across UEs, enabling joint uplink-downlink analysis.
  • Precoding schemes: MR, ZF, and P-ZF are used for downlink precoding, with P-ZF fully distributed because each base station uses only locally estimated CSI.P-ZF actively rejects both intra-cell and inter-cell interference, unlike conventional ZF, which rejects only intra-cell interference.

D. Finite and Asymptotic Analysis

The analysis characterizes spectral efficiency in the large-antenna regime and derives the asymptotically optimal number of scheduled UEs. It shows that pilot allocation and pilot reuse strongly shape the optimum, while finite-antenna systems require separate investigation.

  • Finite and Asymptotic Analysis: The uplink and downlink sum spectral efficiency can be optimized jointly and divided arbitrarily between the two directions using positive payload fractions.The same result applies because the uplink and downlink expressions differ only through their allocation fractions.
  • Finite and Asymptotic Analysis: As M →∞, MR, ZF, and P-ZF achieve the same effective-SINR limit for fixed K and B.The asymptotic limit depends on the cells sharing pilots with the reference cell.
  • Finite and Asymptotic Analysis: Pilot contamination asymptotically depends only on cells that interfered during pilot transmission, so strongly interfering cells should use different pilot subsets.This motivates selecting pilot reuse patterns that separate cells with large interference contributions.
  • Finite and Asymptotic Analysis: K⋆ is one of the closest integers to S/(2β) when M is large, so the optimal number of scheduled UEs is proportional to the coherence-block length.For β = 1, K⋆ = S/2; for β = 3, K⋆ = S/6.
  • Finite and Asymptotic Analysis: The asymptotically optimal spectral efficiency increases linearly with S, while the optimal pilot reuse factor balances SINR improvement against pre-log loss.Larger β reduces the number of pilot-sharing interferers but also reduces the pre-log factor.
  • Finite and Asymptotic Analysis: B = S/2 for every β at the asymptotic optimum, allocating half the coherence frame to pilot transmission.The result follows because the gain from adding a UE outweighs the pre-log loss when at least half the frame remains for data.

IV. OPTIMIZING NUMBER OF UES IN HEXAGONAL NETWORKS

The finite-system analysis uses an infinitely large symmetric hexagonal network with non-universal pilot reuse and a pathloss-based channel model. Power control makes the resulting spectral efficiencies independent of user positions under the stated assumptions.

  • IV. OPTIMIZING NUMBER OF UES IN HEXAGONAL NETWORKS: The network model uses symmetric hexagonal cells with universal payload reuse and an infinitely large grid to avoid edge effects.Each cell has six first-tier interfering neighbors, twelve second-tier neighbors, and so on.
  • IV. OPTIMIZING NUMBER OF UES IN HEXAGONAL NETWORKS: The pilot book has size B = βK, allowing non-universal pilot reuse to mitigate pilot contamination from neighboring cells.The hexagonal geometry permits symmetric reuse factors such as β ∈ {1, 3, 4, 7, 9, 12, 13, ...}.
  • IV. OPTIMIZING NUMBER OF UES IN HEXAGONAL NETWORKS: The simulations use pathloss exponent κ ≥ 2 and average-SNR parameter ρ/σ², with power control making SEs independent of UE positions.The cell radius and pathloss reference value cancel under the stated distribution assumptions.

A. Optimizing SE for Different Interference Levels

The study optimizes scheduled users and pilot reuse for average, best-case, and worst-case inter-cell interference. Interference conditions substantially affect achievable SE and the preferred processing and allocation choices.

  • Simulation setup: The simulations optimize SE over integer K and pilot reuse factor β for each antenna count M.The coherence block is set to S = 400, with SNR ρ/σ2 = 5 dB and pathloss exponent κ = 3.7.
  • Interference scenarios: The average, best-case, and worst-case scenarios represent increasing severity of inter-cell interference based on interfering-UE locations.The average case uses uniformly distributed UE locations; the extreme cases place interfering UEs at cell edges nearest to or farthest from the victim BS.
  • Interference scenarios: The average case is probably most applicable in practice, whereas the best and worst cases provide optimistic and pessimistic boundaries.The best case gives an upper bound for coordinated scheduling, while the worst case cannot occur simultaneously with respect to all neighboring cells.
  • Results: The optimized SE differs strongly across interference scenarios, while average-case SEs are rather similar for MR, ZF, and P-ZF when 10 ≤ M ≤ 200.ZF excels under best-case interference, P-ZF under worst-case interference, and at least M = 105 antennas is needed to approach the asymptotic limit.
  • Results: As M increases, the optimal K⋆ generally increases and β decreases; asymptotically, K⋆ approaches 67, 200, and 50 for average, best, and worst cases.These values correspond to β = 3, β = 1, and β = 4, respectively.

B. Impact of System Parameters

The parameter study examines formula accuracy, pilot reuse, user-level SE, antenna-to-user ratios, SNR, and coherence-block length. These parameters change both achievable SE and the operating point that maximizes it.

  • Formula validation: For K = 10, the closed-form MR and ZF expressions closely match Monte Carlo simulations, while P-ZF has a few percent deviation.The P-ZF formula is based on a lower bound, so its actual performance is slightly better than reported.
  • Pilot reuse: Pilot reuse factors β = 1 and β = 3 provide the highest SEs for M ≤ 1000, with broad regions where their SEs are nearly equal.This robustness simplifies cell planning and scheduling based on user load.
  • User-level performance: The optimized SE per UE is around 1 bit/s/Hz for MR, 1–2.5 bit/s/Hz for ZF, and 1–3 bit/s/Hz for P-ZF.MR has the lowest SE per scheduled UE, while P-ZF has the highest.
  • Antenna-to-user ratio: Practical optimized operating points generally target 2–8 BS antennas per scheduled UE, rather than the order-of-magnitude rule-of-thumb.Some MR operating points even have M/K⋆ < 1.
  • SNR: The SE saturates already at an SNR of 5 dB, while ZF and P-ZF are more sensitive to SNR than MR.Their active interference suppression requires higher CSI estimation quality.
  • Coherence block: For M = 100, increasing S above 500 gives relatively small gains, whereas for M = 500 it enables more scheduled UEs and major SE improvements.At higher user loads, imperfect-CSI-limited intra-cell interference becomes dominant and the benefit of P-ZF diminishes.

V. SPECTRAL EFFICIENCIES WITH HARDWARE IMPAIRMENTS

The paper extends its SE analysis to non-ideal transceiver hardware by modeling signal attenuation and Gaussian distortion noise. Simulations compare ideal hardware with hardware impairments across antenna counts.

  • Impairment model: Hardware impairments are modeled by reducing the original signals by 1 − ϵ2 and replacing the removed power with Gaussian distortion noise.The impairment parameter ϵ is interpreted as error vector magnitude, with typical LTE values 0 ≤ ϵ ≤ 0.17.
  • Analysis: Theorem 4 provides a jointly achievable UL/DL SE expression under MR, ZF, or P-ZF processing with hardware impairments.The expression incorporates the impairment-dependent interference terms into the effective SINR.
  • Analysis: As M → ∞ with finite K and B, the effective SINRs approach an upper limit under the impairment model.The asymptotic result follows from the earlier processing-specific SINR expressions with additional impairment factors.
  • Simulation results: For ϵ = 0.1, hardware impairments cause only a tiny SE difference for M < 5000 but a substantial difference at larger M.The divergence at high M reflects different asymptotic SE limits for ideal and impaired hardware.

VI. CONCLUSION

The paper derives position-independent UL/DL SE expressions and analyzes the SE-optimal number of scheduled UEs in massive MIMO. Its asymptotic and simulation results revise common allocation expectations for pilots and antennas.

  • VI. CONCLUSION: The new SE expressions remove dependence on instantaneous UE positions through power control and averaging over random UE locations, enabling joint UL/DL optimization.The expressions support fixed-M optimization of the scheduled-user count K.
  • VI. CONCLUSION: As M → ∞, the SE-optimal user count approaches K∗ = S/(2β), irrespective of processing scheme.Consequently, the pilot-symbol count B = βK∗ approaches S/2.
  • VI. CONCLUSION: Half the coherence frame should be devoted to pilots asymptotically, while simulations with M ≤ 1000 allocate 5% to 40% of the frame to pilots.The asymptotic SE limit itself is not reached at practical antenna counts.
  • VI. CONCLUSION: High per-cell SE is obtained by scheduling many simultaneous UEs, even though SE per UE may be only 1–4 bit/s/Hz.P-ZF gives the highest per-UE performance, whereas MR schedules the largest number of UEs.

APPENDIX: COLLECTION OF PROOFS

The appendix derives achievable spectral-efficiency and SINR expressions for MR, ZF, and P-ZF processing, using channel expectations, Jensen’s inequality, and matrix-based uplink–downlink power control.

  • Theorem 1 computes MR expectations over channel realizations and substitutes them into the SINR expressions to obtain the MR spectral-efficiency formula.
  • Jensen’s inequality moves user-location expectations into the SINR denominator, producing an achievable lower bound used in the MR, ZF, and P-ZF results.
  • ZF derivations use properties of Wishart matrices and the ZF principle to evaluate channel expectations and account for pilot-sharing interference.
  • For P-ZF, the appendix follows the ZF procedure and derives the corresponding expression under jointly Gaussian channels and colored noise.
  • The matrix equation Dσ2 = q − DΨq expresses downlink SINR conditions, and solving it yields transmit powers that achieve the target SINRs.
  • Uplink–downlink duality achieves the same SINRs and total transmit power in both links, while hardware distortion changes the SINR through (1 − ϵ2) factors.
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