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Massive MIMO Systems with Non-Ideal Hardware: Energy Efficiency, Estimation, and Capacity Limits

Emil Björnson, Jakob Hoydis, Marios Kountouris, Mérouane Debbah

arXiv:1307.2584v3cs.IT

TL;DR

Massive MIMO asymptotic results often assume ideal transceiver hardware, motivating analysis of whether those models remain valid with practical impairments. The paper models residual impairments at BSs and UEs and shows that estimation and capacity have finite ceilings, while massive MIMO retains array and energy-efficiency benefits.

  • Problem

    The paper addresses limited understanding of how practical transceiver impairments affect the performance and asymptotic properties of massive MIMO systems.

  • Method

    The paper models residual hardware impairments at each antenna as additive distortion noise proportional to the signal power and analyzes estimation, capacity, and asymptotic behavior.

  • Results

    Hardware impairments create non-zero channel-estimation error floors and finite uplink and downlink capacity ceilings, while UE impairments limit asymptotic capacities and interference can become negligible.

  • Takeaways & Limitations

    Massive MIMO can retain substantial spectral and energy efficiency and permit reduced BS hardware quality as the antenna count grows.

Abstract

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The use of large-scale antenna arrays can bring substantial improvements in energy and/or spectral efficiency to wireless systems due to the greatly improved spatial resolution and array gain. Recent works in the field of massive multiple-input multiple-output (MIMO) show that the user channels decorrelate when the number of antennas at the base stations (BSs) increases, thus strong signal gains are achievable with little inter-user interference. Since these results rely on asymptotics, it is important to investigate whether the conventional system models are reasonable in this asymptotic regime. This paper considers a new system model that incorporates general transceiver hardware impairments at both the BSs (equipped with large antenna arrays) and the single-antenna user equipments (UEs). As opposed to the conventional case of ideal hardware, we show that hardware impairments create finite ceilings on the channel estimation accuracy and on the downlink/uplink capacity of each UE. Surprisingly, the capacity is mainly limited by the hardware at the UE, while the impact of impairments in the large-scale arrays vanishes asymptotically and inter-user interference (in particular, pilot contamination) becomes negligible. Furthermore, we prove that the huge degrees of freedom offered by massive MIMO can be used to reduce the transmit power and/or to tolerate larger hardware impairments, which allows for the use of inexpensive and energy-efficient antenna elements.

I. INTRODUCTION

Massive MIMO uses large antenna arrays and TDD reciprocity to improve array gain, spatial resolution, and interference suppression, but practical deployment requires analyzing residual hardware impairments and implementation trade-offs.

  • Motivation: Massive MIMO uses many BS antennas and TDD channel reciprocity to improve spectral efficiency while relaxing some implementation requirements.Large arrays provide coherent beamforming/combining gains and can reduce interference leakage from channel-estimation errors asymptotically.
  • Motivation: Hardware impairments receive limited attention despite their importance for deploying large arrays with inexpensive antenna elements.Residual impairments include amplifier nonlinearities, I/Q imbalance, phase noise, and quantization errors.
  • Approach: The paper analyzes aggregate residual impairments at transmitters and receivers using additive distortion noises proportional to signal power.It derives a pilot-based channel estimator and capacity results under this non-ideal hardware model.
  • System model: The study focuses analytically on a single link between an N-antenna BS and a single-antenna UE under arbitrary interference conditions.The system uses flat-fading TDD with uplink and downlink pilot and data phases within each coherence period.
  • System model: The channel covariance assumptions allow spatial correlation and rank deficiency while requiring normalized trace and rank to scale appropriately with N.These assumptions model propagation environment and array geometry rather than a scaled identity covariance.
  • Design trade-offs: The paper studies spectral-efficiency limits alongside hardware-quality choices, including simulations with κ-parameters from 0 to 0.152.Smaller κ-values represent more accurate and expensive transceiver hardware.

C. Uplink System Model

The uplink model represents residual hardware impairments as signal-dependent distortion and derives an LMMSE channel estimator, revealing estimation floors and limits on estimator optimality.

  • Uplink model: The reciprocal uplink supports pilot-based channel estimation and data transmission at the BS.The received BS signal includes receiver noise, interference, and residual distortion from uplink transceiver hardware.
  • Uplink model: Uplink distortion noises are independent of the data signal but depend on channel realizations and have channel-dependent covariance matrices.The model includes transmitter and receiver distortion processes and interference whose statistics may differ between pilot and data phases.
  • Estimation challenge: The effective distortion can have a complex double Gaussian distribution because it is formed from products of Gaussian channel and impairment variables.This dependence prevents direct application of standard independent-Gaussian-noise estimation results.
  • Estimation method: The paper derives the LMMSE estimator of the current channel realization from the received uplink pilot observation.The estimator uses channel covariance information and the covariance of the received signal.
  • Estimation method: The LMMSE estimate and estimation error are uncorrelated with zero mean but are neither independent nor jointly complex Gaussian.Their covariance matrices are E{ĥĥH}=R−C and E{ϵϵH}=C.
  • Estimation limits: Nonlinear estimators may achieve smaller MSEs than the LMMSE estimator, unlike conventional estimation with independent Gaussian noise.The paper expects the difference to be small because the dependent distortion noises are relatively weak.
  • Estimation limits: Increasing pilot power cannot eliminate the estimation error floor caused by residual hardware impairments.For R=λI and S=0, the high-power error floor is strictly positive and depends on κUE and κBS.

A. Impact of the Pilot Length

Increasing pilot length can average out temporally uncorrelated distortion noise and reduce estimation error, but coherence-time and temporal-correlation limits prevent eliminating the floor in practice.

  • A. Impact of the Pilot Length: MSE decreases as 1/B when pilot length increases, although each pilot channel use retains a non-zero hardware-induced error floor.Averaging B independent observations mitigates distortion noise.
  • A. Impact of the Pilot Length: B ≤ Tcoher limits the improvement to at most a factor 1/Tcoher, so non-ideal hardware always leaves an estimation error floor.The bound assumes a coherence period that limits the available pilot length.
  • B. Numerical Illustrations: At 20–30 dB uplink SNR, estimation error is close to its floor, so further SNR increases provide only minor improvement.The impairment-ignoring MMSE estimator is only slightly worse than the proposed LMMSE estimator.
  • B. Numerical Illustrations: At high SNR, uncorrelated distortion noise is mitigated by longer pilots, whereas fully correlated distortion permits only small improvements.The actual performance is expected to lie between these temporal-correlation extremes.
  • B. Numerical Illustrations: At low SNR, hardware impairments have little impact and longer pilots provide a small gain because total pilot energy increases as BpUE.The comparison uses a fixed per-symbol energy pUE.
  • B. Numerical Illustrations: The covariance model strongly affects estimation accuracy: greater spatial correlation yields smaller errors, although hardware impairments still create error floors.The comparison covers uncorrelated, exponential-correlation, and one-ring covariance models.

IV. DOWNLINK AND UPLINK DATA TRANSMISSION

The paper bounds downlink and uplink capacities using perfect-CSI upper bounds and pilot-based LMMSE lower bounds. Both links exhibit finite asymptotic ceilings dominated by UE impairments.

  • IV. DOWNLINK AND UPLINK DATA TRANSMISSION: Capacity bounds use perfect CSI for upper bounds and imperfect pilot-based LMMSE channel estimates for lower bounds.The bounds therefore cover the two CSI extremes considered in the analysis.
  • IV. DOWNLINK AND UPLINK DATA TRANSMISSION: The upper bounds assume perfect interference suppression, whereas the lower bounds treat interference as Gaussian noise.Interference statistics may vary between coherence periods.
  • IV. DOWNLINK AND UPLINK DATA TRANSMISSION: Gaussian signaling and single-stream transmission are sufficient for the perfect-CSI upper bounds, with rank(W)=1.The result follows under circularly symmetric Gaussian receiver and distortion noises.
  • IV. DOWNLINK AND UPLINK DATA TRANSMISSION: Distortion noise acts as an interferer through the same channel as the data signal, so filtering cannot reduce it.This mechanism explains why capacity remains fundamentally limited by hardware impairments.
  • IV. DOWNLINK AND UPLINK DATA TRANSMISSION: Downlink capacity has finite ceilings as either transmit power or the number of BS antennas grows large.The ceilings depend on BS and UE impairment parameters.
  • IV. DOWNLINK AND UPLINK DATA TRANSMISSION: UE impairments are N times more influential than BS impairments in the downlink and dominate asymptotically in the uplink.The uplink also has finite ceilings as transmit power or antenna number grows large.

B. Lower Bounds on Channel Capacities

The paper derives lower capacity bounds for downlink and uplink transmission under non-ideal hardware, then characterizes their large-array behavior. As N grows, BS impairments become negligible while UE impairments determine the capacity limits.

  • Capacity-bound construction: The lower bounds use Gaussian signaling, linear processing, pilot-based channel estimation, and Gaussian modeling of CSI uncertainty.Linear processing is motivated by asymptotic optimality as N →∞.
  • Capacity-bound construction: The derived bounds can be computed numerically for arbitrary channel distributions and beamforming or combining choices when the conditional channel distribution is characterized.
  • Asymptotic behavior: As N →∞, interference terms vanish under sublinear scaling, while the remaining terms stay strictly positive.The stated conditions require interference growth slower than linear in N.
  • Asymptotic behavior: Both downlink capacity bounds become independent of BS impairment levels asymptotically, so BS transmitter impairments have little impact in massive MIMO.The lower-bound analysis and numerical results indicate that this impact is negligible.
  • Asymptotic behavior: UE hardware impairments remain in the asymptotic downlink and uplink bounds and mainly determine the capacity limits.The paper concludes that high-quality UE transceivers are more important than high-quality BS transceivers in this regime.
  • Downlink and uplink differences: The downlink is affected by both downlink and uplink impairments through reverse-link channel estimation, whereas the uplink is affected only by uplink impairments.

C. Numerical Illustrations

Numerical results illustrate finite capacity limits under hardware impairments and show that increasing BS antennas suppresses BS-impairment effects. Capacity also depends on SNR and channel covariance structure.

  • Section C. Numerical Illustrations: Under hardware impairments, lower and upper capacity bounds converge to finite limits as N grows, unlike ideal-hardware capacity, which grows without bound.The simulations use a spatially uncorrelated channel and impairment levels including κUE ∈ {0, 0.052, 0.152}.
  • Section C. Numerical Illustrations: Under non-ideal hardware, the gap between lower and upper bounds is small because distortion noise imposes a finite limit and channel hardening makes effective inner products increasingly deterministic.The associated estimation errors therefore have only a minor capacity impact in the large-N regime.
  • Section C. Numerical Illustrations: At 20 dB SNR, only minor capacity improvements occur beyond N = 100, whereas 0 dB requires many more antennas for convergence.The paper attributes the slower low-SNR convergence to the larger array gain needed to compensate for the lower SNR.
  • Section C. Numerical Illustrations: With κUE = 0.052 fixed, changing κBS affects the bounds mainly at small N; the curves converge to virtually the same value as N →∞.
  • Section C. Numerical Illustrations: For four covariance models with κBS = κUE = 0.05, the uncorrelated model has the highest lower-bound performance and the strongly correlated one-ring model the lowest.The upper bound is identical across models because it uses only the diagonal elements of R.

V. IMPROVING ENERGY EFFICIENCY AND REDUCING HARDWARE QUALITY

The paper extends massive-MIMO power-scaling laws to non-ideal hardware and analyzes overall energy efficiency with circuit-power costs. Large arrays can reduce transmit power and tolerate greater BS impairments, but per-antenna circuit power creates a finite optimal array size.

  • Energy-efficiency model: The overall energy-efficiency model includes amplifier power and baseband circuit power Nρ + ζ, where ρ scales with antennas and ζ is static.
  • Power scaling: Transmit powers can decrease with N while downlink and uplink spectral efficiencies converge to non-zero limits under the stated power-scaling conditions.The lower bounds use LMMSE channel estimation and linear MRT/MRC processing.
  • Energy-efficiency optimization: If circuit power does not scale with N, energy efficiency can approach its upper bounds as the antenna count grows; with ρ > 0, maximal energy efficiency occurs at finite N.Per-antenna circuit consumption causes the energy-efficiency denominator to grow while capacity remains bounded.
  • Reducing hardware quality: BS impairment levels can increase roughly proportionally to N while preserving non-zero asymptotic capacity, with only minor lower-bound degradation in simulations.
  • Reducing hardware quality: Because EVM equals the square root of the impairment parameters, tolerable EVM can increase proportionally to N^1/4.The paper gives an example replacing one EVM 0.03 antenna element with 256 EVM 0.12 elements with negligible capacity loss.
  • Reducing hardware quality: The practical joint optimization of transmit power and circuit power depends on how increased EVM maps to reduced hardware dissipation and remains outside the paper’s scope.

A. Numerical Illustrations

The numerical results examine how circuit power, transmit-power scaling, and hardware-impairment scaling affect energy efficiency and capacity as the antenna count grows. They show that energy efficiency can remain close to ideal-hardware performance, while excessive impairment growth eventually collapses capacity.

  • Energy efficiency: Energy efficiency is nearly identical for optimized, fixed, and decreasing transmit-power policies, but depends strongly on the circuit-power parameters ζ and ρ.With ρ = 0, energy efficiency increases and converges; with ρ > 0, it has a unique maximum before decreasing.
  • Energy efficiency: The energy-efficiency gap between ideal and non-ideal hardware is small at reasonable antenna counts because the performance loss from hardware impairments is relatively small.The cited numerical discussion attributes this observation to the limited impact of impairments in the considered range.
  • Transmit-power scaling: The transmit power can decrease monotonically with N, but this generally does not maximize energy efficiency when ρ > 0.For ρ = 0, optimal power decreases more slowly than 1/N; for ρ > 0, it decreases only until the energy-efficiency maximum and then increases.
  • Transmit-power scaling: Keeping total power fixed across antenna counts is presented as a simple design rule because decreasing power causes only a small energy-efficiency loss but a larger spectral-efficiency loss.The comparison concerns fixed, 1/N^t-scaled, and energy-efficiency-optimized transmit powers.
  • Impairment scaling: Capacity degradation remains small when BS impairment scaling follows the proposed law, such as τ = 1/4 or τ = 1/2.For τ = 1, the curve bends downward around N ≈ 350; for τ = 2, the lower bound quickly reaches zero.
  • Impairment scaling: When τ > 1, both the lower and upper capacity bounds converge to zero asymptotically.The lower-bound behavior is illustrated numerically, while the upper-bound convergence follows from the cited corollaries.

VI. EXTENSIONS TO MULTI-CELL SCENARIOS

The multi-cell extension analyzes regular and pilot-contaminated interference under large-array scaling. Channel decorrelation suppresses interference from users absent during the target pilot, whereas pilot-overlapping users can retain non-vanishing interference.

  • Interference model: The multi-cell analysis relaxes the bounded-interference assumptions used in the single-link asymptotic capacity results, including cases where pilot contamination makes interference scale linearly with N.The section models co-users scheduled by the serving or neighboring base stations.
  • Uplink interference: Interference from co-users that were silent during the target UE’s pilot vanishes asymptotically because user channels decorrelate as N grows.These users form the set U⊥ in the analysis.
  • Uplink interference: Pilot-contaminated interference remains and can scale with N because the channel estimate is correlated with co-user channels active during pilot transmission.These users form U∥, whose signals appear in the pilot-phase interference term.
  • Receive combining: The baseline theorem uses MRC, while MMSE combining can actively suppress interference in multi-cell multiuser scenarios.The theorem nevertheless identifies pilot-contaminated interference as the component that may substantially matter for large N.

B. Inter-User Interference in the Downlink

The downlink extension shows that only pilot-parallel users retain non-vanishing interference as the antenna count grows. Numerical multi-cell results further indicate that UE distortion can mask regular interference and pilot contamination.

  • Downlink interference: In the downlink, UEs with parallel uplink pilots cause non-vanishing interference, while other interfering transmissions vanish as N grows large.The behavior parallels the uplink pilot-contamination result.
  • Interference and distortion: With non-ideal hardware, regular interference from a channel 10 dB weaker than the useful channel has little impact on performance.The comparison is made against ideal hardware, which is more sensitive to both regular and pilot-contaminated interference.
  • Pilot contamination: Pilot-contaminated interference is negligible at large N when its relative channel strength is sufficiently below the useful channel, under the condition in Corollary 8.The condition depends on relative SNR differences rather than absolute SNRs.
  • Numerical multi-cell results: In a 16-cell scenario, pilot contamination substantially degrades ideal-hardware spectral efficiency, whereas unique and reused pilots perform almost identically with hardware impairments.The setup uses 400 m × 400 m wrap-around cells and six UEs per cell.
  • Numerical multi-cell results: UE distortion noise is the main limiting factor in the considered impaired-hardware scenario, causing regular interference and pilot contamination to drown in distortion.The paper describes this distortion as a fog that prevents the BS from seeing distant pilot interferers.
  • Numerical multi-cell results: More antennas can suppress regular interference, but convergence to the asymptotic limit is faster with non-ideal hardware because interference need only fall below distortion noise.In the single-user impaired-hardware case, gains from increasing N beyond 100 were small.
  • Model basis: The additive distortion-noise model is analytically tractable and can be motivated by Bussgang-based affine approximations of nonlinear distortion.The paper notes that the resulting noise model is similar to, but not identical with, the assumed model.
  • Model limitations: Time-varying impairments and refined distortion covariances are outside the analyzed model and would typically reduce the lower capacity bounds.The upper capacity bounds are described as typically remaining valid under these refinements.

A. Power Loss

Non-ideal hardware introduces power losses and changes how capacity and estimation accuracy scale with transmit power, antenna count, oscillator architecture, and calibration. The paper finds that UE impairments remain influential as BS impairments become asymptotically less harmful, while several high-power conclusions rely on constant impairment coefficients.

  • Power loss: As N grows, BS distortion noise can vanish, but a residual BS power loss of κBS_t/(1+κBS_t) remains.The power-loss factor must still be considered when designing massive MIMO systems.
  • Power scaling: Power-amplifier distortion can make capacity and estimation accuracy peak at finite transmit powers rather than increase monotonically.When proportionality coefficients grow with transmit power, distortion power increases faster than signal power.
  • Power scaling: The paper’s high-power limits are optimistic because constant impairment coefficients exclude the regime where nonlinearities increase rapidly.The authors state that these limits may instead correspond to reducing propagation distance rather than increasing emitted power.
  • Multiplicative distortions: With common oscillators, relative phase-noise distortion scales as O(t), whereas independent oscillators yield O(t/N).The useful signal power scales as O(N), producing the different relative-distortion scalings.
  • Asymptotic hardware impact: UE impairments are N times more influential on capacity, allowing BS hardware quality to degrade with N with only a minor performance loss.The conclusion supports inexpensive antenna elements and distributed deployments with independent oscillators.
  • Model: The model represents residual impairments as additive distortion noise proportional to per-antenna signal power after compensation algorithms are applied.The model is described as mathematically tractable and experimentally verified in prior work.
  • Interference: Inter-user interference and pilot contamination become negligible relative to distortion noise when a simple pilot allocation avoids the strongest contaminated interference.Additional antennas remain useful for suppressing inter-user interference in multicell scenarios.

APPENDIX A NEW AND OLD RESULTS ON RANDOM VECTORS

The appendix collects matrix and random-vector lemmas used to control moments, eigenvalues, channel-estimate decompositions, and asymptotic scaling. These results support boundedness and O(N)-type arguments under the paper’s covariance and rank assumptions.

  • Matrix bounds: The appendix provides matrix lemmas that bound quadratic forms using spectral norms and establish O(N) scaling.These bounds apply to uniformly bounded matrix families and positive semidefinite matrices.
  • Gaussian identities: Gaussian-channel lemmas evaluate expectations involving exponential distributions and exponential integrals.For h distributed as CN(0,r), |h|^2 has an exponential distribution with mean r.
  • Proof tools: The appendix includes elementary complex-number inequalities and channel-estimate decompositions used throughout the proofs.These steps separate error terms and bound sums of moments.
  • Random vectors: Random-vector lemmas assume independent zero-mean unit-variance entries with finite higher moments to control polynomial expectations.The resulting constants depend on the moment order rather than the antenna dimension.
  • Asymptotic assumptions: The covariance assumptions allow rank deficiency while requiring rank to grow linearly with N and the smallest nonzero eigenvalue to remain relevant.The analysis uses bounded spectral norms and positive nonzero eigenvalues to establish asymptotic bounds.

APPENDIX C COLLECTION OF PROOFS

The appendix derives capacity bounds by optimizing SINR-based expressions under idealized CSI or interference-suppression assumptions. Generalized Rayleigh-quotient solutions provide the optimizing transmit and receive vectors for downlink and uplink bounds.

  • Downlink bound: The downlink capacity upper bound assumes perfect CSI, canceled interference, and optimal single-stream Gaussian signaling.The bound is obtained by maximizing the SINR over unit-norm beamforming vectors.
  • Downlink optimization: The optimal downlink beamforming vector solves a generalized Rayleigh quotient problem.Substitution of the optimizer yields the stated downlink capacity bound.
  • Uplink bound: The uplink capacity bound likewise assumes canceled interference and is achieved by the receiver-combining vector that maximizes the uplink SINR.The uplink and downlink derivations share the same SINR-optimization structure.
  • Closed forms: Closed-form capacity upper bounds are obtained by rewriting the SINR expressions and applying Jensen’s inequality and Gaussian expectation identities.The downlink derivation is followed analogously for the uplink.

C. Proof of Theorem 4

The proof establishes asymptotic capacity behavior by bounding normalized expectations and tracking how pilot power and interference terms scale with the antenna count. The key condition is that combined transmit-power scaling remains slower than linear in N.

  • Expectation bounds: The proof separates numerator and denominator expectations with Hölder’s inequality and invokes earlier moment lemmas to establish their asymptotic orders.These bounds yield the equivalence results used for the capacity analysis.
  • Capacity proofs: The downlink and uplink proofs use the same expectation structure, so the uplink argument follows analogously from the downlink bound.Both analyses rely on the capacity bounds having matching forms.
  • Pilot scaling: Pilot power scaling as pUE proportional to 1/N^tUE permits asymptotic limit arguments when N^tUE pUE converges to a finite positive constant.The proof applies dominated convergence to move the antenna-count limit inside expectations.
  • Interference scaling: The interference term vanishes asymptotically when the total transmit-power exponent satisfies tBS+tUE=tsum<1.This follows because the relevant covariance trace grows at least linearly with N.
  • Convergence condition: The downlink proof requires tUE<1/2 for the residual terms to vanish as N approaches infinity.The uplink proof is stated to be analogous because its capacity bound has the same structure.

E. Proof of Corollary 7

The proof establishes Corollary 7 by taking large-antenna limits in the downlink and uplink capacity bounds and showing that interference and noise terms vanish under the stated scaling conditions.

  • Limit evaluation: The dominated convergence theorem permits taking the limit N →∞ inside the expectations in the downlink and uplink capacity bounds.This transfers the asymptotic analysis directly to the expectations appearing in the bounds.
  • Interference terms: The downlink interference contribution vanishes when κBSThe passage states this condition corresponds to the corollary’s requirement.
  • Interference terms: The uplink receives an analogous scaling condition for the corresponding interference contribution.The uplink condition is obtained by the same asymptotic reasoning used for the downlink.
  • Noise terms: Noise and O(1/√N) terms behave as O(κBSThese terms therefore disappear under the corollary’s condition τr < 1/2.
  • Interference bounds: The interference bounds follow from Hölder’s, Cauchy–Schwarz, and auxiliary lemmas, with independence yielding an O(1) bound for the orthogonal-user case.The proof separately treats users in U∥ and U⊥ and bounds the resulting terms using the cited inequalities and lemmas.
  • Uplink denominator: The denominator noise term in the uplink capacity bound is related to the interference expression through the preceding bound and the trace relation tr(R −C).This connects the final noise-term evaluation to equation (61) and the covariance trace identity.
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