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Energy and Spectral Efficiency of Very Large Multiuser MIMO Systems

Hien Quoc Ngo, Erik G. Larsson, Thomas L. Marzetta

arXiv:1112.3810v2cs.IT

TL;DR

The paper asks how very large MU-MIMO arrays can reduce uplink power while operating with imperfect pilot-based CSI and practical receivers. It derives finite-antenna capacity bounds and quantifies spectral- and energy-efficiency tradeoffs. The results show that large arrays can improve both efficiencies substantially, with simple linear processing such as MRC or ZF.

  • Problem

    The paper examines uplink power savings and efficiency tradeoffs in very large MU-MIMO systems using imperfect CSI and practical receiver structures.

  • Method

    The paper derives finite-antenna uplink capacity bounds for MRC, ZF, and MMSE and analyzes spectral and energy efficiency with pilot-based channel estimates.

  • Results

    Very large MIMO can increase spectral efficiency by one or two orders of magnitude and energy efficiency by three orders of magnitude using simple linear processing and uplink-pilot channel estimates.

  • Takeaways & Limitations

    Large antenna arrays can substantially improve spectral and energy efficiency while supporting simple BS processing and distributed per-antenna MRC implementation.

Abstract

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A multiplicity of autonomous terminals simultaneously transmits data streams to a compact array of antennas. The array uses imperfect channel-state information derived from transmitted pilots to extract the individual data streams. The power radiated by the terminals can be made inversely proportional to the square-root of the number of base station antennas with no reduction in performance. In contrast if perfect channel-state information were available the power could be made inversely proportional to the number of antennas. Lower capacity bounds for maximum-ratio combining (MRC), zero-forcing (ZF) and minimum mean-square error (MMSE) detection are derived. A MRC receiver normally performs worse than ZF and MMSE. However as power levels are reduced, the cross-talk introduced by the inferior maximum-ratio receiver eventually falls below the noise level and this simple receiver becomes a viable option. The tradeoff between the energy efficiency (as measured in bits/J) and spectral efficiency (as measured in bits/channel use/terminal) is quantified. It is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single-antenna system.

I. INTRODUCTION

This paper analyzes uplink power savings in very large MU-MIMO systems, deriving finite-antenna capacity bounds for imperfect and perfect CSI with MRC, ZF, and MMSE receivers. It also studies spectral-efficiency and energy-efficiency tradeoffs, including simple MRC processing and pilot-based CSI acquisition.

  • Scope and contributions: The paper analyzes uplink power savings in very large MU-MIMO systems and derives capacity bounds for finite numbers of base-station antennas.The analysis considers both single-cell and multicell systems, focusing on single-cell performance because it is comprehensible and bounds multicell performance.
  • Capacity bounds: Finite-M lower bounds are derived for achievable uplink rates with MRC, ZF, and MMSE receivers under perfect and imperfect CSI.The bounds explicitly address realistic receiver structures for very large MIMO systems.
  • Power scaling: 1/M with perfect CSI and 1/√M with pilot-based CSI are the respective transmit-power scalings as M grows without bound.These scalings hold even with simple linear receivers.
  • Efficiency tradeoff: In the low-transmit-power regime with imperfect CSI, spectral efficiency and energy efficiency can increase simultaneously.The paper identifies this as a regime where the usual tradeoff can be overcome.
  • Efficiency tradeoff: Very high spectral efficiency can coexist with orders-of-magnitude transmit-power reductions using simple MRC, even after accounting for pilot-based CSI acquisition.MRC can also be implemented in a distributed per-antenna manner.

B. Review of Some Results on Very Long Random Vectors

The section reviews random-vector limits underlying favorable propagation, where large-array channel vectors become nearly orthogonal and support predictable rate behavior. It also relates channel geometry to lower and upper bounds on achievable rate.

  • Random-vector limits: Independent zero-mean random vectors exhibit law-of-large-numbers convergence and central-limit-theorem behavior as their dimension grows.The notation distinguishes almost-sure convergence from convergence in distribution.
  • Favorable propagation: Large-array channel measurements approximate favorable propagation, supporting the assumption that channel vectors become pairwise orthogonal.The paper notes experimental justification for this assumption.
  • Favorable propagation: Favorable propagation is desirable because large MIMO arrays can make small-scale fading effects average out and reduce intracell interference.The review connects channel-vector geometry with the benefits of large arrays.
  • Rate bounds: The rate bounds depend on the channel singular values, with the lower bound attained by a rank-one line-of-sight channel.The upper bound occurs when channel columns are mutually orthogonal and have equal norms.

III. ACHIEVABLE RATE AND ASYMPTOTIC (M →∞) POWER EFFICIENCY

The section derives achievable uplink-rate bounds for linear detectors and analyzes how transmit power scales with the number of base-station antennas. It focuses on the large-array regime, including perfect CSI and the associated rate and efficiency behavior.

  • III. ACHIEVABLE RATE AND ASYMPTOTIC (M →∞) POWER EFFICIENCY: The paper restricts attention to linear MRC, ZF, and MMSE detectors in the regime 1 ≪ K ≪ M, treating perfect and estimated CSI separately.Maximum-likelihood detection has exponentially growing complexity in K.
  • Achievable-rate bounds: The detector output is modeled with a noise-plus-interference term, yielding an achievable-rate lower bound after linear separation and Jensen’s inequality.The rate analysis assumes ergodic channels and coding over many channel realizations, potentially across frequency using coded OFDM.
  • Perfect CSI: With perfect CSI, transmit power per user can scale as 1/M while preserving the performance of a SISO link with power E_u.The resulting link has no intracell interference or fast fading, while serving K users increases spectral efficiency K times.

2) Zero-Forcing Receiver:

This section derives rate bounds for zero-forcing and MMSE detection using SINR distributions and analytic approximations. It establishes MMSE optimality for the stated achievable-rate expression.

  • 2) Zero-Forcing Receiver:: For ZF, the achievable uplink rate is lower bounded under Rayleigh fading when M ≥ K + 1.The bound is obtained from the ZF orthogonality relation and Jensen’s inequality.
  • 3) Minimum Mean-Squared Error Receiver:: With perfect CSI, Rayleigh fading, and MMSE, the kth user’s achievable-rate lower bound is approximated using a Gamma approximation for the SINR distribution.The approximation is combined with a Gamma-function identity to obtain the stated closed form.
  • 3) Minimum Mean-Squared Error Receiver:: For MMSE detection, the detector maximizes the achievable rate in the paper’s rate expression.The result follows from Cauchy–Schwarz equality for the MMSE detector.

B. Imperfect Channel State Information

The section analyzes uplink transmission when the base station estimates channels from pilots rather than having perfect CSI. It shows that estimation changes the power-scaling law because data and pilot signals both lose power.

  • B. Imperfect Channel State Information: Channel estimates are obtained from orthogonal uplink pilot sequences of length τ, with MMSE estimation performed at the base station.The pilots satisfy Φ^HΦ = I_K and are transmitted simultaneously by the users.
  • B. Imperfect Channel State Information: Pilot signals cannot exploit the full number of receive antennas because channel estimation is performed separately for each receive antenna.The analysis explicitly incorporates this per-antenna estimation constraint.
  • B. Imperfect Channel State Information: With imperfect CSI and large M, user transmit power can scale inversely with the square root of M while maintaining a fixed rate.Proposition 5 relates the resulting performance to an interference-free SISO link without fast fading.
  • B. Imperfect Channel State Information: In the rank-one propagation case, the power-scaling laws still hold, but multiplexing gains do not materialize because intracell interference cannot be canceled as M grows.This is identified as the most unfavorable propagation case.

1) Maximum-Ratio Combining:

This section derives achievable-rate lower bounds for MRC, ZF, and MMSE with imperfect CSI and Rayleigh fading. It also compares their large-array behavior and identifies MMSE as rate-optimal for the stated expression.

  • 1) Maximum-Ratio Combining:: With imperfect CSI, Rayleigh fading, and MRC, the kth user’s achievable uplink rate has a lower bound for M ≥ 2.As M →∞, this bound equals the exact limit from Proposition 5.
  • 2) Zero-Forcing Receiver:: With imperfect CSI and ZF processing, the kth user’s achievable uplink rate is bounded when M ≥ K + 1.The large-M rate and lower bound converge to the same value as MRC and Proposition 5.
  • 3) MMSE Receiver:: The MMSE receiver is optimal because it maximizes the achievable rate expression for the estimated-channel model.The section then uses an approximate SINR distribution to obtain a closed-form lower bound.
  • 3) MMSE Receiver:: An approximate lower bound is derived for the MMSE achievable rate under imperfect CSI and Rayleigh fading.The resulting expression uses parameters obtained from equations involving the channel-estimation setting.
  • B. Imperfect Channel State Information: The single-cell analysis motivates extending the power-scaling question to multicell MU-MIMO, where reduced user power also reduces other-cell interference.The cited passage presents this as an intuitive argument for retaining the single-cell scaling law.

C. Power-Scaling Law for Multicell MU-MIMO Systems

In multicell MU-MIMO, increasing the antenna count suppresses ordinary intercell interference and preserves the power-scaling law, but pilot contamination leaves residual interference when CSI is imperfect.

  • As M grows large, interference from other cells disappears under the analyzed multicell model.
  • The uplink SINR converges to a constant value when M grows large.
  • The single-cell M power-scaling law remains valid in multicell systems.
  • Pilot reuse contaminates channel estimates with intercell interference from other cells.
  • Pilot-contamination-induced intercell interference grows with M at the same rate as the desired signal, creating residual interference.

IV. ENERGY-EFFICIENCY VERSUS SPECTRAL-EFFICIENCY TRADEOFF

The paper analyzes how energy efficiency and spectral efficiency interact in uplink MU-MIMO, especially with imperfect CSI. Their relationship can reverse at low transmit power, enabling joint improvements in both efficiencies within a limited regime.

  • Energy efficiency is spectral efficiency divided by transmit power, and increasing spectral efficiency typically reduces energy efficiency.
  • In one operating regime, energy efficiency and spectral efficiency can increase jointly, so no tradeoff occurs.
  • This joint-improvement regime is probably of less practical interest.
  • With imperfect CSI, energy efficiency does not always decrease as spectral efficiency increases.
  • At high p_u, energy efficiency decreases as p_u increases, whereas at low p_u it increases with spectral efficiency.
  • Maximum-Ratio Combining: At low p_u, doubling spectral efficiency or doubling M increases energy efficiency by 1.5 dB.

2) Zero-Forcing Receiver:

The paper derives and evaluates energy- and spectral-efficiency behavior for multicell systems using MRC and ZF under imperfect CSI. Larger arrays reduce required power, while receiver differences depend on operating power and target spectral efficiency.

  • The multicell analysis derives spectral- and energy-efficiency expressions for imperfect CSI with MRC and ZF receivers.
  • Spectral efficiency decreases as the intercell interference factor β or the number of cells L increases.
  • Power-scaling law: At p_u = E_u/M, imperfect-CSI spectral efficiency decreases to 0 as M increases, unlike perfect CSI.
  • Receiver comparison: At p_u proportional to 1/M, MRC performs almost as well as ZF for large M, while MMSE remains better and close to ZF.
  • Doubling M cuts required power by approximately 3 dB with perfect CSI and 1.5 dB with imperfect CSI.
  • When M/K ⪆ 6, MRC is within 1 dB of ZF or MMSE for perfect CSI and within 3 dB for imperfect CSI.

2) Energy Efficiency versus Spectral Efficiency Tradeoff :

The paper evaluates energy–spectral-efficiency tradeoffs by optimizing transmit power, users, and training under fixed spectral efficiency. Moderately large multiuser arrays substantially outperform a single-antenna reference, though receiver choice matters at high spectral efficiency.

  • Energy efficiency is normalized against a single-antenna reference mode, while curves optimize transmit power and pilot length for fixed spectral efficiency.
  • Reducing transmit power from 10 dB to −10 dB improves energy efficiency 100-fold without reducing spectral efficiency.
  • At the same comparison point, energy efficiency rises by factors of 55 and 75 for M = 50 and M = 100, respectively.
  • Receiver comparison: At high spectral efficiency, ZF outperforms MRC because MRC is limited by intracell interference, causing its spectral efficiency to approach a constant.
  • Training optimization: At low transmit power, more coherence-interval time is allocated to training than payload transmission.

B. Multicell MU-MIMO Systems

The multicell analysis examines energy and spectral efficiency under pilot contamination and shows that large-scale MIMO can improve both efficiencies substantially, while receiver rankings depend on contamination and power regime.

  • Multicell evaluation: The multicell study evaluates energy and spectral efficiency for a system with L = 7 cells and jointly optimizes τ, K, and p_u for each spectral-efficiency target.The reference mode is the single-cell configuration used in Fig. 6.
  • Pilot contamination: β increases from 0.11 to 0.32, reducing spectral efficiency by a factor of 3 and energy efficiency by a factor of 2.7 at p_u = 10 dB.The increase in β corresponds to increased pilot contamination.
  • Pilot contamination: Below 10 bits/s/Hz spectral efficiency, pilot contamination has little effect on system performance.This conclusion is stated for the multicell setting examined in the paper.
  • Receiver comparison: Under high pilot contamination in a multicell environment, MRC achieves better performance than ZF.In general, ZF benefits from intracell-interference cancellation, but that advantage diminishes with strong pilot contamination.
  • Overall conclusions: Very large MIMO can increase cell sum-rate by one or two orders of magnitude and energy efficiency by three orders of magnitude versus a single-antenna system.The reported operating regime includes about 100 antennas serving about 50 terminals simultaneously.
  • Overall conclusions: These gains remain possible with MRC or ZF, channel estimates from uplink pilots, and training that uses half of the channel coherence interval.MRC additionally supports distributed per-antenna detector implementation.

APPENDIX

The appendix supplies distributional and algebraic steps underlying the paper’s finite-antenna uplink bounds, alongside simulation figures for spectral efficiency, power scaling, and energy-efficiency tradeoffs.

  • Analytical derivations: The appendix uses Gaussian channel properties and central complex Wishart matrices in deriving finite-antenna uplink results.The derivation substitutes intermediate expressions into earlier equations to obtain the stated results.
  • Simulation comparisons: The simulations compare MRC, ZF, and MMSE under perfect and imperfect CSI across different numbers of BS antennas.The figures report spectral-efficiency bounds and numerically evaluated values.
  • Power scaling: Figures 4 and 5 evaluate transmit power required for 1 and 2 bits/channel use per user with K = 10 users.The comparisons include perfect and imperfect CSI as functions of the BS antenna count M.
  • Energy-efficiency tradeoff: Figure 6 compares normalized energy efficiency versus spectral efficiency for MRC and ZF with imperfect CSI, using M = 50 and M = 100 antenna configurations.The number of users, transmit power, and training fraction are optimized for the large-array curves.
  • Training optimization: Figure 7 reports the optimal number of users K and training symbols τ within a coherence interval of T = 196 symbols.The associated analysis treats training and user loading as design variables.
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