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Achieving "Massive MIMO" Spectral Efficiency with a Not-so-Large Number of Antennas

Hoon Huh, Giuseppe Caire, Haralabos C. Papadopoulos, Sean A. Ramprashad

arXiv:1107.3862v2cs.IT

TL;DR

The paper addresses a dimensionality bottleneck associated with channel coherence block length and studies network-MIMO performance using large-system asymptotic analysis. The analysis closely approximates finite-dimensional performance, while the proposed architecture achieves comparable spectral efficiency with roughly one-tenth as many base-station antennas.

  • Problem

    High-SNR capacity is limited by a dimensionality bottleneck associated with the fading coherence block length.

  • Method

    The paper uses large-system asymptotic analysis to study network-MIMO performance and analyzes pilot contamination through theorems on MMSE channel estimation.

  • Results

    The asymptotic analysis provides a very accurate approximation of finite-dimensional system performance, and the proposed architecture achieves comparable spectral efficiencies with a 10-fold reduction in base-station antennas.

  • Takeaways & Limitations

    The proposed architecture achieves spectral efficiencies comparable to Massive MIMO with roughly 500 to 50 base-station antennas.

  • Takeaways & Limitations

    The numerical comparison assumes the same system model as and a 20 MHz bandwidth.

Abstract

from arXiv · show

The main focus and contribution of this paper is a novel network-MIMO TDD architecture that achieves spectral efficiencies comparable with "Massive MIMO", with one order of magnitude fewer antennas per active user per cell. The proposed architecture is based on a family of network-MIMO schemes defined by small clusters of cooperating base stations, zero-forcing multiuser MIMO precoding with suitable inter-cluster interference constraints, uplink pilot signals reuse across cells, and frequency reuse. The key idea consists of partitioning the users population into geographically determined "bins", such that all users in the same bin are statistically equivalent, and use the optimal network-MIMO architecture in the family for each bin. A scheduler takes care of serving the different bins on the time-frequency slots, in order to maximize a desired network utility function that captures some desired notion of fairness. This results in a mixed-mode network-MIMO architecture, where different schemes, each of which is optimized for the served user bin, are multiplexed in time-frequency. In order to carry out the performance analysis and the optimization of the proposed architecture in a clean and computationally efficient way, we consider the large-system regime where the number of users, the number of antennas, and the channel coherence block length go to infinity with fixed ratios. The performance predicted by the large-system asymptotic analysis matches very well the finite-dimensional simulations. Overall, the system spectral efficiency obtained by the proposed architecture is similar to that achieved by "Massive MIMO", with a 10-fold reduction in the number of antennas at the base stations (roughly, from 500 to 50 antennas).

I. INTRODUCTION

The paper addresses antenna and training-overhead limits in MU-MIMO by proposing a TDD network-MIMO architecture that approaches Massive MIMO spectral efficiency with fewer antennas. It combines bin-specific scheme optimization, large-system analysis, and fairness-oriented scheduling.

  • Motivation: FDD training overhead grows linearly with cooperating transmit antennas, restricting MU-MIMO gains from large antenna arrays.In TDD, uplink pilot overhead instead scales with active users per cell and is independent of cooperating BS antennas.
  • Motivation: TDD performance can improve significantly by increasing BS antennas when the number of scheduled users is fixed.
  • Analysis and results: One order of magnitude fewer antennas per active user achieves spectral efficiencies comparable with Massive MIMO.
  • Architecture: The proposed architecture uses small cooperating-BS clusters, LZFBF, inter-cluster interference constraints, pilot reuse, and frequency reuse.
  • Architecture: Users are partitioned into geographically determined bins, each assigned an optimized network-MIMO scheme and scheduled across time-frequency slots for fairness.
  • Analysis and results: The resulting mixed-mode architecture multiplexes bin-optimized schemes, while large-system analysis provides closed-form rates for rapid system optimization.Its asymptotic predictions match finite-dimensional simulations very well.

II. SYSTEM MODEL

The system model defines a configurable family of clustered network-MIMO schemes and maps geographically equivalent user bins to optimized schemes. Scheduling selects bins over time-frequency resources according to a fairness-oriented network utility.

  • Scheme family: The scheme family varies cooperating-BS cluster size and shape, pilot reuse, frequency reuse, and downlink linear precoding.
  • User bins: Users are partitioned into bins according to cellular position, with statistically equivalent users grouped together.Symmetric location sets provide users with the same distances from all BSs in a cluster.
  • User bins: Each user bin is associated with the optimal network-MIMO scheme from the family.
  • Scheduling: A scheduler allocates user bins in time-frequency to maximize a suitable concave, componentwise non-decreasing utility of ergodic user rates.The utility can represent proportional fairness.
  • Cellular layout: The model considers single-cell processing or small clusters of 2 BSs in one dimension and 3 BSs in two dimensions.Larger clusters do not improve performance because of training overhead and higher complexity.
  • Scheduling: Round-robin scheduling can select equal-sized active-user subsets within each statistically equivalent group, sharing group spectral efficiency evenly.

B. Channel statistics and received signal model

The channel model uses block-fading Rayleigh channels with distance-dependent covariance and clustered transmission. Each active user receives equal power, while intra-cluster and inter-cluster interference are represented explicitly.

  • Channel model: The channel covariance depends on user and BS locations through the distance-dependent gain function g(x, b).
  • Channel model: Channel coefficients undergo independent small-scale Rayleigh fading across BS antennas, subbands, and users.
  • Transmission: Active users receive equal transmit power 1/S, giving each cluster total transmit power N.
  • Block fading: Channels remain constant over coherence blocks of length LN signal dimensions, with each block corresponding to a scheduling slot.
  • Interference: Intra-cluster interference comes from other columns of the local codeword matrix, while signals from other active clusters form inter-cluster interference.
  • Frequency reuse: Frequency reuse F>1 restricts cluster transmission to a fraction 1/F of system bandwidth and is modeled through adjusted noise variance.

III. UPLINK TRAINING AND CHANNEL ESTIMATION

Uplink pilots provide TDD channel-state information through orthogonal training within scheduled groups and controlled reuse across groups. Pilot reuse creates contamination that affects channel estimates and motivates the analysis of alternative precoders.

  • Pilot design: Each scheduled active user transmits an uplink pilot, requiring SN orthogonal pilot signals for channel estimation at serving clusters.
  • Pilot design: Pilot length is LP=QS, where Q is an integer reuse factor optimized for each cluster and user-bin configuration.
  • Pilot reuse: Q training codebooks are assigned periodically, so groups separated by Q reuse the same pilot codebook.Pilot reuse and frequency reuse may use different factors.
  • Channel estimation: The training observation superposes channels from users sharing a pilot, producing pilot contamination.The resulting MMSE estimate is correlated with channels associated with the same reused pilot.
  • Pilot contamination: With LSUBF, pilot contamination sends nonvanishing interfering power toward unintended users and yields an interference-limited system.
  • Pilot contamination: For the proposed LZFBF schemes, pilot contamination is quantified by Theorems 2 and 3.

IV. MU-MIMO PRECODERS AND ACHIEVABLE RATES

The paper constructs beamformers from estimated channel matrices, using either no zero forcing, serving-cluster zero forcing, or additional inter-cluster constraints to mitigate interference. Pilot reuse determines which out-of-cluster channels can be estimated, with examples using masking and zeroed channel subvectors.

  • Beamforming framework: Beamforming matrices are calculated from estimated channel matrices, with schemes differing by their employed beamforming design.The considered family includes LZFBF and related single-user beamforming schemes.
  • Beamforming cases: J = 0 imposes no zero-forcing constraints, whereas J = 1 applies classical zero forcing within the serving cluster.For J = 1, each beam is orthogonal to the estimated channels of the other active users in that cluster.
  • Inter-cluster interference: J > 1 adds zero-forcing constraints at J −1 neighboring clusters to mitigate inter-cluster interference.This provides an alternative to frequency reuse and can also be used jointly with it.
  • Pilot reuse: Inter-cluster constraints require controllers to estimate out-of-cluster users’ channels, enabled when those users use distinct pilot codebook indices.The construction is restricted by the pilot reuse factor: J > 1 can be used only if Q is larger than 1.

B. Achievable group spectral efficiency

The section derives large-system achievable group spectral efficiencies for LSUBF and LZFBF under single-cell and multicell processing. The results express performance through geometry-, frequency-, and pilot-reuse-dependent coefficients, while multicell processing requires an interference-power upper bound.

  • Definitions and regime: User spectral efficiency is defined in bit/s/Hz for a user at location x + c, and the derived results apply in the limit N →∞.Theorems 1–3 cover LSUBF and LZFBF cases under the stated system parameters.
  • LSUBF: Theorem 1 gives an achievable group spectral efficiency for LSUBF as N →∞ for specified sets and system parameters.Its coefficients depend on system geometry, frequency reuse, and pilot reuse, but not on S or M.
  • Single-cell LZFBF: Theorem 2 gives an achievable group spectral efficiency for single-cell LZFBF, with the relevant neighboring clusters selected by proximity.The construction defines E(x) as the J −1 closest non-serving cluster centers, or the empty set when J = 1.
  • Asymptotic comparison: As M →∞, the single-cell LZFBF expression coincides with the corresponding LSUBF limit.In this Massive MIMO regime, LZFBF yields no advantage over LSUBF.
  • Multicell LZFBF: For C > 1, pilot contamination prevents an exact asymptotic inter-cluster-interference power expression because beamforming and channel vectors have complicated statistical dependence.Theorem 3 instead provides an achievable rate based on an upper bound on inter-cluster interference power.

V. SCHEDULING AND FAIRNESS

The architecture partitions users into geographically defined bins, optimizes the network-MIMO scheme separately for each bin, and schedules bins over time-frequency resources using a concave fairness utility. The asymptotic closed forms make this optimization computationally efficient.

  • User bins: Users are partitioned into K bins defined by sets of symmetric locations that uniformly discretize the cellular coverage region.Users within each bin are handled as a group for spectral-efficiency optimization and scheduling.
  • Per-bin optimization: Each bin’s spectral efficiency is obtained by maximizing over cluster size, frequency reuse F, loading S, pilot reuse Q, and beamforming parameter J.The objective combines pilot-dimensionality overhead with the data-phase spectral efficiency from Theorems 1–3.
  • Per-bin optimization: The optimization is subject to JS ≤ CM, a constraint that becomes relevant for LZFBF precoding.The search spans the discrete family of network-MIMO schemes.
  • Scheduling and fairness: A scheduler allocates bins across time-frequency slots to maximize a componentwise non-decreasing concave network utility of bin spectral efficiencies.With randomized or round-robin user selection, users in a bin share an equal average fraction of ρkR⋆(Xk).
  • Scheduling and fairness: Proportional fairness yields equal bin allocations, ρk = 1/K, while minimum-rate objectives can impose max-min fairness.The framework also includes α-fairness utilities as a family containing these scheduling rules as special cases.

VI. NUMERICAL RESULTS AND DISCUSSION

The numerical results show that bin-specific network-MIMO schemes adapt performance to user location, while large-system analysis accurately predicts finite-dimensional behavior. The bin-optimized architecture matches the reference throughput with substantially fewer base-station antennas.

  • Bin optimization: The proposed architecture partitions users into homogeneous bins and serves each bin with a tailored network-MIMO scheme.The scheduler can therefore select schemes according to users’ geographic locations.
  • Location-dependent schemes: The (1, 1, 1) scheme with Q = 1 performs best near cell centers, whereas C = 2, J = 2, or F = 2 performs better at cell edges.The location-dependent optimum supports multiplexing different schemes across user bins.
  • Large-system approximation: Large-system analysis closely matches Monte Carlo results, even for very small system dimensions.The reported agreement motivates presenting subsequent results using closed-form large-system expressions.
  • Comparison with the reference: The proposed architecture’s gain over the (1, 1, 0), Q = 1 reference ranges from about 40% to 580%, depending on user location.This comparison uses the two-dimensional layout with M = 50.
  • Throughput versus antenna factor: The bin-optimized architecture improves throughput for every value of M, while (3, 3, 1) is strongest for M < 20 and (1, 1, 1) becomes best as M increases.The comparison is made under proportional-fair scheduling and the stated LTE-TDD coherence-block assumptions.
  • Throughput versus antenna count: For finite antenna counts, the proposed architecture achieves the reference throughput with a 10-fold reduction in base-station antennas, roughly from 500 to 50.The reference is the scheme associated with the prior comparison and its infinite-antenna limit can require more than 10000 antennas per BS.

VII. CONCLUSIONS

The paper proposes a mixed-mode network-MIMO architecture that selects bin-optimized schemes across time-frequency resources and achieves strong spectral efficiency with asymptotic analysis validated by simulations.

  • Architecture: The architecture partitions users into geographically determined bins and serves each bin with an independently optimized MU-MIMO transmission.The bins are scheduled as separate transmissions whose parameters are tailored to their user locations.
  • Architecture: The scheme family ranges from conventional single-user beamforming to zero-forcing beamforming with additional constraints on neighboring cells.The optimization can tailor pilot reuse, frequency reuse, user loading, cooperative cluster size, and beamforming type to each bin.
  • Analysis: The analysis uses a large-system limit in which all system dimensions grow with fixed ratios and derives closed-form spectral-efficiency expressions for each scheme and bin.The expressions support computationally efficient system optimization.
  • Results: The asymptotic performance predictions match very well with finite-dimensional simulations.This agreement supports using the large-system analysis for the proposed finite-dimensional design evaluation.
  • Results: The different schemes achieve their best spectral efficiency at different user locations, motivating adaptive scheme selection across the coverage region.The resulting mixed-mode architecture serves each user bin with the scheme optimized for that bin.
  • Architecture: Different schemes are multiplexed in the time-frequency plane to adapt service across the coverage region.The scheduler allocates slots to bins and selects among schemes ranging from single-cell processing to joint network-MIMO processing.

APPENDIX A

Appendix A derives normalized group spectral efficiency by decomposing the received signal into useful signal, intra-cluster interference, inter-cluster interference, and noise terms.

  • Signal decomposition: The derivation focuses on a reference cluster and expresses the received data-phase signal for users in a served location group.The resulting normalized group spectral efficiency is evaluated using symmetry across clusters and subbands.
  • Signal decomposition: The useful signal and interference powers are evaluated using MMSE channel estimates, beamforming representations, and large-system limits.The SINR numerator and denominator converge to deterministic limits as N →∞.
  • Interference analysis: Interference is separated into self-interference, intra-cluster interference, and inter-cluster interference, with pilot-sharing contributions treated separately.The inter-cluster term distinguishes clusters whose users share the reference pilot from those that do not.

APPENDIX B

Appendix B analyzes zero-forcing beamforming in the large-system regime, using block-structured channel matrices and symmetry properties to derive useful-signal and spectral-efficiency expressions.

  • Beamforming model: The analysis covers linear zero-forcing beamforming with or without inter-cluster interference constraints.Each cluster imposes beamforming constraints across active users and, depending on the scheme, neighboring clusters.
  • Beamforming model: The beamforming matrix is organized into independent blocks whose entries have block-dependent variances or are identically zero.The blocks correspond to base-station antennas and users at locations where zero-forcing constraints are imposed.
  • Large-system behavior: 116 ZF dimensionality limit: as the constraint-to-antenna ratio JS/(CM) tends to 1, the effective useful signal term vanishes.The analysis assumes JS < CM before approaching this limit.
  • Large-system behavior: Under symmetric user and base-station placements, a constant partial-trace property gives a large-system limit of 1/C independent of the base-station index.This property is used in evaluating terms involving the beamforming matrix.
  • Spectral-efficiency results: The appendix derives achievable normalized group spectral-efficiency expressions for both C = 1 and C > 1 cases.For C > 1, the result uses bounds because certain beamforming and estimated-channel terms are statistically dependent and lack simple closed forms.
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