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Deploying Dense Networks for Maximal Energy Efficiency: Small Cells Meet Massive MIMO

Emil Björnson, Luca Sanguinetti, Marios Kountouris

arXiv:1505.01181v2cs.IT

TL;DR

The paper asks how to design dense cellular networks for maximal uplink energy efficiency while meeting service-quality requirements. Using stochastic-geometry modeling, a tractable average-SE lower bound, and a realistic power model, it analytically optimizes deployment and transmission parameters. The analysis finds that densification improves EE but saturates, while an optimized massive-MIMO configuration can provide a further gain.

  • Problem

    The paper seeks an energy-efficient dense cellular-network design that balances increasing throughput demands against rising ICT power consumption.

  • Method

    It models multi-cell uplink networks with stochastic geometry, derives a tractable average-SE lower bound, and analytically optimizes BS density, transmit power, antennas, users, and pilot reuse using a realistic power model.

  • Results

    10.156 Mbit/J is achieved with (M⋆, K⋆) = (91, 10) and β⋆ = 7.08 in the reported optimized massive-MIMO configuration.

  • Takeaways & Limitations

    EE improves with smaller cells but saturates as BS density increases, while extra antennas and multiplexed users can produce a further massive-MIMO EE gain.

Abstract

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How would a cellular network designed for maximal energy efficiency look like? To answer this fundamental question, tools from stochastic geometry are used in this paper to model future cellular networks and obtain a new lower bound on the average uplink spectral efficiency. This enables us to formulate a tractable uplink energy efficiency (EE) maximization problem and solve it analytically with respect to the density of base stations (BSs), the transmit power levels, the number of BS antennas and users per cell, and the pilot reuse factor. The closed-form expressions obtained from this general EE maximization framework provide valuable insights on the interplay between the optimization variables, hardware characteristics, and propagation environment. Small cells are proved to give high EE, but the EE improvement saturates quickly with the BS density. Interestingly, the maximal EE is achieved by also equipping the BSs with multiple antennas and operate in a "massive MIMO" fashion, where the array gain from coherent detection mitigates interference and the multiplexing of many users reduces the energy cost per user.

I. INTRODUCTION

The paper addresses the tension between rapidly increasing throughput demands and energy consumption by developing an analytical framework for energy-efficient dense cellular networks. It models heterogeneous multi-cell deployments and jointly considers densification, antennas, users, spectral efficiency, and power costs.

  • Motivation: 1000× higher network area throughput is targeted over the next 10–15 years, alongside a 1000× energy-efficiency improvement without increasing ICT footprint.
  • Densification technologies: Small-cell networks bring BSs closer to UEs, increasing area throughput and reducing radiated power, but add circuit-power costs from additional hardware.
  • Densification technologies: Massive MIMO can improve throughput and reduce radiated power, while its additional antennas increase infrastructure and circuit-power costs.
  • Paper objective: The paper develops an analytical EE-design framework that balances spectral efficiency, radiated power, BS density, antennas, users, channel estimation, and circuit consumption.
  • Analytical approach: Stochastic geometry provides tractable models for dense random deployments and performance metrics such as coverage probability and average spectral efficiency.
  • System model: The network model uses BSs from a homogeneous PPP, with each BS equipped with M antennas and serving K single-antenna UEs.

A. Channel Model and Power-Control Policy

The channel model combines stochastic geometry, block-fading propagation, statistical channel inversion, and practical transceiver impairments. Power control equalizes average effective channel gain across UEs without requiring instantaneous CSI feedback.

  • Channel model: BSs and UEs communicate over block-fading Rayleigh channels with pathloss exponent α > 2 and reference-loss parameter ω.
  • Power control: Statistical channel inversion is used to prevent near-far blockage caused by the finite dynamic range of BS analog-to-digital converters.
  • Power control: The power-control coefficient ρ is designed, and the resulting policy gives every UE the same average effective channel gain Mρ regardless of location.
  • Power control: The equalized average gain does not imply equal SINR because interference varies across the network.
  • Hardware impairments: Hardware impairments reduce desired signal power by 1 − ϵ^2 and add Gaussian distortion noise with variance ϵ^2.
  • Hardware impairments: The ideal-hardware case is obtained by setting ϵ = 0, while impairments become especially relevant at high SNRs.

C. Channel Acquisition

Channel acquisition uses randomly assigned uplink pilots reused across cells, followed by MMSE channel estimation and MRC detection. The resulting imperfect-CSI setting motivates a tractable lower bound on ergodic capacity.

  • Pilot signaling: Each coherence block reserves B of S symbols for pilots, with K ≤ B pilot symbols randomly selected by each BS.
  • Pilot signaling: The pilot reuse factor β satisfies βK = B ≤ S, and cells reusing the typical UE’s pilot create pilot contamination.
  • Channel estimation: The typical UE’s channel is estimated from its pilot observation using the MMSE estimator, with a corresponding estimation-error covariance matrix.
  • Data detection: MRC detects each UE by correlating the received signal with its MMSE channel estimate.
  • Spectral-efficiency analysis: Because exact ergodic capacity is unavailable under imperfect CSI and shot-noise interference, the paper uses achievable lower bounds from massive-MIMO analysis.

where the pre-log factor

The paper turns a tractable lower bound on average uplink spectral efficiency into an energy-efficiency optimization problem. It accounts for pilot overhead, radiated power, hardware consumption, and a minimum average spectral-efficiency requirement.

  • Average spectral efficiency: The effective SINR accounts for pilot overhead through the pre-log factor and is used in the average-SE analysis.
  • Average spectral efficiency: A tractable and tight lower bound on average SE replaces heavy numerical evaluations of expectations over BS and UE locations.
  • SINR behavior: Coherent processing makes the SINR numerator scale with M, while hardware impairments impose the multiplicative loss (1 − ϵ^2)^2.
  • Energy-efficiency formulation: The EE metric is defined as area spectral efficiency divided by area power consumption, measured in bit/J.
  • Energy-efficiency formulation: Area power consumption includes radiated power, analog and digital hardware, backhaul, signal processing, cooling, and power-supply losses.
  • Optimization problem: The optimization jointly selects β, ρ, λ, K, and M under positivity, pilot-overhead, and average-SE constraints.

A. Feasibility

The EE optimization is feasible only over a SINR-target range constrained by inter-cell interference, hardware impairments, and pilot contamination. The optimal pilot reuse factor increases with user load but decreases with transmit power, antenna count, and pathloss severity.

  • Inter-cell interference restricts the SINR targets for which the EE optimization is feasible.
  • The maximal SINR is limited by hardware impairments and pilot contamination determined by α and coherence-block length S.
  • With ϵ = 0.05, α = 3, and S = 200, the upper limit on average SE per UE is approximately 7.65.This makes the optimization feasible in most practically relevant cases.
  • Increasing β improves channel-estimation accuracy and reduces coherent pilot contamination by assigning pilots to fewer cells.
  • β⋆ increases with K but decreases with ρ, M, and α because more users increase interference while higher power, array gain, and pathloss attenuation reduce sensitivity to interference.

C. Optimal BS Density and Radiated Power

Under the assumed power-control policy, EE increases with BS density and is maximized asymptotically as density grows while average UE transmit power vanishes. Infinite density is impractical, but the asymptotic limit is nearly reached at λ = 10 BS/km2.

  • EE(β⋆) increases monotonically with λ and is maximized as λ →∞, while average UE transmit power approaches zero.
  • Increasing λ reduces the transmit-power term proportionally to ρ/λ^(α/2), improving the EE objective.
  • At high BS density, power control resolves the increased nearby-UE interference by progressively reducing UE transmit power, making circuit power dominant.
  • λ = 10 BS/km2 nearly achieves the asymptotic EE limit, although equipment dimensions and UE density constrain practical BS deployment.In ultra-dense networks, load balancing and BS sleep modes can limit the number of active BSs.

D. Optimal Number of Antennas and UEs per BS

The paper optimizes users and antennas through an integer-relaxed formulation using antennas per UE, then alternates between optimizing each variable before searching nearby integer points. The resulting quasi-concave structure yields the global optimum of the relaxed problem and exposes hardware-dependent scaling laws.

  • D. Optimal Number of Antennas and UEs per BS: The optimization first relaxes M and K to real values and replaces M with c̄ = M/K, the number of BS antennas per UE.
  • D. Optimal Number of Antennas and UEs per BS: For fixed c̄, EE is quasi-concave in K, so the maximizing user count is obtained analytically from the stationary point.
  • D. Optimal Number of Antennas and UEs per BS: For fixed K, EE is quasi-concave in c̄, with the optimum given by an interior stationary point or the active constraint boundary.
  • D. Optimal Number of Antennas and UEs per BS: The optimal K decreases approximately as 1/c̄ for large c̄, whereas the optimal antenna count M increases with K to control interference.
  • D. Optimal Number of Antennas and UEs per BS: Static power C0 favors more UEs and antennas, while coefficients increasing with K or M favor fewer of the corresponding resources.
  • D. Optimal Number of Antennas and UEs per BS: Alternating optimization converges to the global optimum of the integer-relaxed problem, after which nearby integer points are searched for the integer solution.

V. NUMERICAL EXAMPLES

Numerical results validate the spectral-efficiency bound and show that energy efficiency benefits from densification but saturates, while a massive-MIMO configuration achieves the global optimum under the studied constraints.

  • The lower and upper energy-efficiency bounds show only a small gap, validating the accuracy of the average spectral-efficiency expression in Proposition 1.
  • Increasing BS density greatly improves energy efficiency, but the gain saturates between λ = 10 and λ = 100 BS/km2.This range corresponds roughly to average inter-BS distances of 100–315 meters.
  • Energy efficiency decreases as the SINR constraint γ increases, making a target SINR necessary to avoid spectrally inefficient operating points.
  • 10.156 Mbit/J is the global energy-efficiency maximum at (M⋆, K⋆) = (91, 10) with pilot reuse factor β⋆ = 7.08 for γ = 3.This operating point belongs to the massive-MIMO class.
  • The alternating algorithm converges after three iterations to (M⋆⋆, K⋆⋆) = (91.6, 10.1), whose 10.157 Mbit/J is only 0.009% above the integer solution.The result indicates a flat energy-efficiency surface near the global optimum.
  • Static power C0 and BS transceiver-chain power D0M dominate the area power consumption at the optimum.The relative contributions can differ for other hardware parameters C0, C1, D0, and D1.

B. Impact of Transceiver Hardware Impairments

In the saturation regime, hardware impairments reduce EE, especially at higher SINR constraints, while dense small-cell deployment and massive multi-user MIMO jointly improve energy efficiency. For practical UE densities, massive MIMO can deliver higher EE with substantially fewer BSs than single-user transmission.

  • Hardware impairments: EE decreases as hardware impairment level ϵ increases because desired signal power decays as (1−ϵ^2)^2.The loss is marginal for γ = 1 but can be relatively large when γ increases.
  • Hardware impairments: For ϵ ≤0.1 and γ ∈{1, 3}, hardware-impairment EE loss is negligible in the investigated network.These SINR values correspond to operating points giving the highest EE.
  • Fixed UE density: For sufficiently large UE density, EE becomes independent of UE density, while BS density can be scaled linearly with UE density using fixed M and K.The saturation occurs for µ ≥100 in the optimal and fixed massive MIMO cases, and for µ ≥2 in single-user SIMO.
  • Fixed UE density: The fixed massive MIMO configuration (M, K) = (91, 10) achieves maximal EE in the saturation regime, whereas single-user SIMO saturates at an EE level 3.14× lower.Single-user transmission requires 10× higher BS density in the saturation regime.
  • Massive MIMO: Adding massive multi-user MIMO can increase EE by a few hundred percentages and reduce BS density by an order of magnitude.Multiplexing several UEs shares circuit-power costs, while additional antennas support interference suppression.

APPENDIX A: PROOF OF LEMMA 3

The proof constructs a tractable achievable uplink SE bound under imperfect CSI by modeling uncorrelated terms as worst-case Gaussian noise and evaluating the resulting expectations. Pilot contamination is isolated through channel-estimate correlations and pilot reuse probabilities.

  • Detection and estimation: The MRC detector is defined as v_00k = ν_00kĥ_00k before deriving the received-signal representation.The normalization factor ν_00k is selected using MMSE orthogonality and the adopted power-control policy.
  • SE lower bound: Evaluating all expectations exactly or by upper bounds yields the achievable lower bound in (10).The derivation combines the computed interference, noise, estimation, and impairment-related expectations.
  • SE lower bound: The proof obtains a rigorous mutual-information lower bound because exact ergodic capacity is unavailable under imperfect CSI.The desired term is retained, while uncorrelated terms are treated as worst-case Gaussian noise, yielding log2(1+SINR_0k).
  • Pilot contamination: Pilot contamination occurs when a user in another cell shares the typical user’s pilot, correlating ĥ_00k with the interfering channel h_0ji.Non-contaminating terms use independence, whereas the contaminating term is handled separately.
  • Pilot contamination: The contaminating term appears with probability E{χ_0kl} = 1/β under stochastic pilot selection.This probability determines how pilot reuse enters the interference calculation.

APPENDIX B: PROOF OF PROPOSITION 1

The proof derives the average-SE lower bound by expanding the expected inverse SINR and bounding PPP interference terms. Finite-region calculations, conditioning, and limiting arguments provide the expectations needed for Proposition 1.

  • Average-SE bound: The proof targets a tractable lower bound on average SE by applying Jensen’s inequality and expanding the expectation of the inverse SINR.The average is taken over the PPP and UE locations before the inverse-SINR terms are evaluated.
  • PPP calculations: A finite circular region with wrap-around is used to compute a finite number of interfering BSs, followed by the limit r →∞.The limiting construction recovers a PPP over R^2 without bias under the stated intra-cell distance assumption.
  • PPP calculations: PPP summations are evaluated by conditioning on interfering distances and averaging over uniformly distributed BS locations.The method applies for κ = 1 or κ = 2 to handle different powers in the interference expressions.
  • Expectation bounds: The remaining interference terms are decomposed into components whose expectations are computed separately or bounded using Hölder’s inequality.The resulting expressions are related to previously derived terms before being substituted back into the proof.
  • Average-SE bound: Combining all exact expectations and upper bounds produces the achievable lower bound in (10).This completes the proof’s derivation of the proposition’s tractable SE expression.
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