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Optimal Design of Energy-Efficient Multi-User MIMO Systems: Is Massive MIMO the Answer?

Emil Björnson, Luca Sanguinetti, Jakob Hoydis, Mérouane Debbah

arXiv:1403.6150v2cs.ITcs.NI

TL;DR

The paper asks how to choose antennas, active users, and transmit power for maximum EE in a uniformly covered multi-user MIMO system. It jointly models uplink and downlink operation with realistic circuit power, derives ZF optima analytically in single-cell settings, and evaluates other settings numerically. The results favor massive MIMO with many antennas and users, while transmit power increases with antenna count; traffic and channel-correlation extensions remain outside the main scope.

  • Problem

    The paper addresses how to jointly choose BS antennas, active users, and transmit power to maximize EE under uniform coverage, where realistic power consumption is essential for reliable optimization.

  • Method

    The paper jointly analyzes uplink and downlink MIMO with several BS processing schemes, proposes a refined power model, and derives closed-form ZF optima for single-cell systems.

  • Results

    Numerical results identify 100–200 antennas serving a relatively large number of UEs as EE-optimal, while transmit power increases with M and similar behavior appears under imperfect CSI and symmetric multi-cell scenarios.

  • Takeaways & Limitations

    Energy-efficient operation favors massive MIMO with interference-suppressing processing such as ZF or MMSE rather than interference-ignoring MRT/MRC.

  • Takeaways & Limitations

    The analysis assumes spatially uncorrelated fading and leaves general multi-cell scenarios, traffic constraints, and multi-antenna UEs for future or extended analysis.

Abstract

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Assume that a multi-user multiple-input multiple-output (MIMO) system is designed from scratch to uniformly cover a given area with maximal energy efficiency (EE). What are the optimal number of antennas, active users, and transmit power? The aim of this paper is to answer this fundamental question. We consider jointly the uplink and downlink with different processing schemes at the base station and propose a new realistic power consumption model that reveals how the above parameters affect the EE. Closed-form expressions for the EE-optimal value of each parameter, when the other two are fixed, are provided for zero-forcing (ZF) processing in single-cell scenarios. These expressions prove how the parameters interact. For example, in sharp contrast to common belief, the transmit power is found to increase (not to decrease) with the number of antennas. This implies that energy-efficient systems can operate in high signal-to-noise ratio regimes in which interference-suppressing signal processing is mandatory. Numerical and analytical results show that the maximal EE is achieved by a massive MIMO setup wherein hundreds of antennas are deployed to serve a relatively large number of users using ZF processing. The numerical results show the same behavior under imperfect channel state information and in symmetric multi-cell scenarios.

I. INTRODUCTION

The paper asks how to jointly choose BS antennas, active users, and transmit power to maximize energy efficiency while uniformly covering an area. It motivates realistic power modeling and develops a joint uplink/downlink multi-user MIMO framework.

  • Research objective: The design objective is to maximize energy efficiency by jointly selecting M, K, and transmit power for uniform area coverage.Energy efficiency is measured in bits transferred per Joule.
  • Motivation: Existing constant-circuit-power models can predict unbounded energy efficiency as the antenna count grows.They omit power consumed by antenna-specific circuits and signal processing.
  • Contribution: The proposed power model accounts for consumption that scales faster than linearly with the numbers of antennas and users.The model is used to derive closed-form EE-optimal values under ZF in single-cell systems.
  • System model: The system jointly considers uplink and downlink TDD transmission, with fixed fractions of each coherence block assigned to the two directions.Orthogonal pilot signaling enables channel estimation for the users.
  • System model: Users are selected round-robin from a generic spatial distribution, while the channel model assumes non-line-of-sight propagation and uncorrelated Rayleigh fading.A circular-cell distribution with path-loss is used for simulations.
  • System model: The framework targets a uniform gross rate for every active UE through linear processing and power allocation.ZF, MRT/MRC, and MMSE processing are considered.

B. Uplink

The uplink model imposes equal gross rates across users and derives the corresponding power allocation and average power consumption for linear detectors, with a closed-form ZF parameterization.

  • Uplink model: The uplink rate is defined under Gaussian codebooks, linear processing, and perfect CSI.The model supports MRC, ZF, and MMSE detection.
  • Power allocation: Equal gross rates are enforced across all UEs through the uplink power-allocation vector.The allocation is direct for MRC and ZF but requires a fixed-point equation for MMSE.
  • Power consumption: Average uplink PA power includes both radiated transmit power and PA dissipation.The UE PA efficiency satisfies 0 < η^(ul) ≤ 1.
  • Feasibility: Some target rates may be infeasible for every transmit-power choice in interference-limited cases.The spectral radius of the interference matrix can detect this condition; perfect-CSI ZF avoids it.
  • ZF processing: For ZF with M ≥ K + 1, the gross rate is parameterized using ρ, which is proportional to received SINR.The required uplink PA power is then available in closed form and ρ can be optimized later.

C. Downlink

The downlink assigns powers to achieve equal gross rates and derives the average PA power, with a ZF expression parallel to the uplink analysis and an uplink-downlink duality relation.

  • Downlink model: The downlink model defines each UE’s achievable rate and average PA power under linear precoding.The BS PA efficiency satisfies 0 < η^(dl) ≤ 1.
  • Power allocation: Equal downlink gross rates require a power-allocation vector obtained from the downlink interference matrix.The resulting average downlink PA power follows by substituting this allocation into the PA-power expression.
  • Uplink-downlink relation: Using the same processing scheme for precoding and combining makes the downlink interference matrix the transpose of the uplink matrix.Total uplink and downlink PA powers then agree apart from transmission-fraction and PA-efficiency factors.
  • ZF processing: For ZF precoding with M ≥ K + 1, the average downlink PA power required for each UE’s target gross rate is given in closed form.The expression uses the same propagation-environment parameter as the uplink result.
  • Rate guarantee: With equal ZF power allocation, Jensen’s inequality shows that the target gross rate lower-bounds the average uplink and downlink gross rates.The expectations cover user locations and channel realizations.

III. PROBLEM STATEMENT

The paper formulates total EE as a joint uplink/downlink optimization over antennas, users, and rates, using a circuit-power model that prevents misleading unbounded-efficiency predictions. It solves ZF analytically and other schemes numerically.

  • EE formulation: Total EE is the average sum rate divided by average total power consumption for both uplink and downlink.The metric is measured in bit/Joule.
  • Circuit-power model: The total power model includes fixed consumption and parameter-dependent circuit power in addition to uplink and downlink transmit power.This formulation is used to define the EE-optimal setup.
  • Circuit-power model: A constant circuit-power model can falsely suggest unbounded EE as more antennas are added.The proposed model makes circuit power depend on M, K, and the user gross rate.
  • Solution approach: The optimization is solved analytically for ZF processing and numerically for other processing schemes.The paper’s problem statement specifies these two solution approaches.
  • EE formulation: The optimization jointly maximizes EE over the system design parameters rather than optimizing only one transmission direction.Uplink-only and downlink-only optimization are special cases of the joint formulation.
  • Interpretation: Increasing EE can involve increasing both sum rate and total power rather than simply minimizing power consumption.The goal is to choose and use a suitable power level effectively.

IV. REALISTIC CIRCUIT POWER CONSUMPTION MODEL

The paper proposes a refined circuit-power model that accounts for fixed, hardware, processing, coding, backhaul, and channel-estimation costs, including nonlinear dependence on system parameters. It then models the major uplink, downlink, and base-station processing contributions.

  • Circuit-power model: The refined model decomposes circuit power into fixed, transceiver, channel-estimation, coding/decoding, backhaul, and linear-processing components.These terms depend linearly or nonlinearly on M, K, and the gross rate.
  • Hardware consumption: Transceiver power includes per-antenna BS power, local-oscillator power, and per-UE circuit power.PBS applies to each BS antenna, PSYN is generally independent of M, and PUE covers each single-antenna UE.
  • Channel estimation: Channel-estimation power is computed from uplink and downlink pilot-processing operations performed once per coherence block.The model uses separate uplink and downlink computational terms at the BS and UEs.
  • Rate-dependent costs: Coding and decoding power is proportional to the number of transferred bits, while load-dependent backhaul power is proportional to the average sum rate.The uplink and downlink are modeled jointly for these rate-dependent terms.
  • Linear processing: Linear-processing power includes data-symbol matrix-vector multiplications and coherence-block computation of precoding and combining matrices.ZF requires matrix inversion operations, whereas MRT/MRC mainly requires column normalization; MMSE uses a fixed number Q of iterative computations.

V. ENERGY EFFICIENCY OPTIMIZATION WITH ZF PROCESSING

The paper specializes the EE optimization problem to ZF processing and derives closed-form coordinate-wise optima using a Lambert W-based analysis. The resulting expressions support both insight into parameter interactions and an alternating optimization procedure.

  • ZF optimization: ZF processing is selected for both uplink and downlink because it enables analytic treatment and is close to optimal in the numerical results.The paper notes that a similar MRC analysis was conducted separately.
  • ZF optimization: The ZF formulation incorporates the refined circuit-power terms into the total power expression used for EE optimization.The total power includes fixed, transceiver, channel-estimation, coding/decoding, backhaul, and ZF linear-processing components.
  • Coordinate-wise solutions: Closed-form EE-optimal values of M, K, or ρ are derived when the other two parameters are fixed.The coordinate-wise solutions also provide the basis for an alternating optimization algorithm.
  • Analytical tool: The Lambert W function is used to solve the scalar optimization subproblem underlying the coordinate-wise EE solutions.Its monotonicity and bounds are then used to analyze how the solutions behave.
  • Analytical tool: The analysis is generic in the power-model coefficients, while simulations use hardware-characterization values reported in the paper.The proofs are provided in the appendix.

B. Optimal Number of Users

For fixed antenna count and transmit-power parameter, the paper characterizes how the EE-optimal number of active users responds to power costs, noise, and coverage area. The resulting user count is obtained from a quartic-root condition and nearby integer comparison.

  • Optimal user count: The EE-optimal user count is obtained by solving a quartic polynomial and comparing the closest smaller and larger integers around its continuous root.This procedure determines the integer K⋆ with the higher EE.
  • Power-cost effects: K⋆ decreases with UE- and BS-dependent power costs that increase with K or M.This dependence is stated under the analytical assumptions used for the user-count result.
  • Power-cost effects: K⋆ increases with fixed power, synchronization power, PA power, and noise power.These quantities are independent of K and M in the relevant decomposition.
  • Rate-dependent costs: K⋆ is unaffected by coding, decoding, and load-dependent backhaul power when those terms scale with average sum rate.The result assumes negligible channel-estimation and ZF linear-processing power for the simplified corollary.
  • Coverage-area effects: A larger coverage area requires serving more UEs because the coverage-related parameter Sx increases with cell radius.The result is summarized directly as a monotonic increase in the required user count.

C. Optimal Number of BS Antennas

For fixed user count and transmit-power parameter, the paper derives an explicit EE-optimal antenna count and analyzes its dependence on power costs and coverage. It also shows that EE-optimal transmit power increases with antenna count rather than following the conventional decreasing-power strategy.

  • Optimal antenna count: The EE-optimal antenna count M⋆ has an explicit expression for given K and ρ.The result provides direct guidance for selecting the number of BS antennas.
  • Power-cost effects: M⋆ decreases with per-BS-antenna power and increases with fixed, synchronization, and UE-dependent power.It does not depend on coding, decoding, or load-dependent backhaul power.
  • Coverage-area effects: A larger coverage area requires more BS antennas, with M⋆ increasing almost linearly with the coverage-related parameter Sx.Sx increases with the cell radius.
  • Power-cost effects: The optimal transmit power increases with fixed, synchronization, UE-dependent, and BS-antenna power, but not with rate-dependent power.Higher transmit power can be afforded when fixed circuit costs dominate total consumption.
  • Optimal transmit power: The EE-optimal transmit power increases with M, opposing strategies that reduce power proportionally to 1/M while preserving non-zero rates.The paper attributes this behavior to circuit power growing with M, allowing additional transmit power to improve rates.
  • Optimal transmit power: For large M, the EE-maximizing total transmit power scales approximately as M/ln(M).Average transmit power per antenna, and per UE when K scales linearly with M, decreases as 1/ln(M).

E. Joint and Alternating Optimization of K, M, and ρ.

The paper combines closed-form single-cell analysis with numerical extensions to imperfect CSI and symmetric multi-cell settings. Joint optimization can use exhaustive search, while alternating optimization offers a lower-complexity route that converges to a local optimum.

  • Joint optimization: Closed-form expressions optimize K, M, or ρ separately under ZF when the other two parameters are fixed.The joint optimum can then be found by exhaustive search over reasonable integer pairs (K, M), using the optimal power for each pair.
  • Alternating optimization: Alternating optimization sequentially updates K, M, and ρ until convergence.The procedure is intended to accommodate changing system settings more practically than brute-force optimization.
  • Imperfect CSI: Under imperfect CSI, approximate ZF rates are obtained using pilot signaling and MMSE channel estimation.The achievable-rate expression includes a parameter ρ and depends on K, M, and pilot-sequence length.
  • Imperfect CSI: Imperfect CSI prevents closed-form optimization of K and ρ, although the optimal number of BS antennas can still be derived similarly to the perfect-CSI case.The difficulty arises because K and ρ appear in both the numerator and denominator of the SINRs.
  • Multi-cell extension: Numerical results indicate that the single-cell behaviors established analytically also apply under imperfect CSI.The paper extends its framework further to symmetric multi-cell scenarios with jointly optimized M, K, and average rate parameters.
  • Multi-cell extension: Symmetric multi-cell analysis uses common cell parameters, user distributions, propagation conditions, and pilot-reuse patterns across cells.Inter-cell interference is characterized through average attenuation ratios and pilot-contamination parameters.
  • Multi-cell extension: In symmetric multi-cell imperfect-CSI scenarios, all design parameters enter both SINR numerators and denominators, making closed-form EE optimization generally intractable.The paper therefore evaluates such scenarios numerically and reports behavior similar to the single-cell case.

VII. NUMERICAL RESULTS

Numerical results evaluate energy efficiency across processing schemes and channel conditions, confirming a massive-MIMO optimum for ZF and contrasting it with MRT/MRC. The simulations also quantify computational complexity, transmit-power behavior, and multi-cell effects.

  • A. Single-Cell Scenario: M = 165 and K = 104 maximize EE with ZF under perfect CSI, using ρ = 0.8747 and 5.7644 bit/symbol per UE.The optimum is a massive-MIMO setup, and the EE surface is concave and smooth around it.
  • A. Single-Cell Scenario: ZF achieves higher EE than MMSE, although their difference is small because MMSE has higher computational complexity.MMSE is optimal from a throughput perspective, but ZF is more energy efficient in these simulations.
  • A. Single-Cell Scenario: M = 81 and K = 77 maximize the much smaller MRT/MRC EE, where strong inter-user interference makes M and K nearly equal.Matching ZF rates with MRT/MRC would require M ≫ K, increasing computational and circuit power without improving EE.
  • A. Single-Cell Scenario: 710 Gflops for ZF, 239 Gflops for MRT/MRC, and 664 Gflops for MMSE are the respective complexities at their EE-optimal operating points.ZF requires only 3× more operations than MRT/MRC because matrix computation occurs once per coherence block, while data processing dominates total complexity.
  • A. Single-Cell Scenario: The EE-optimal transmit power increases with M, while transmit power per BS antenna decreases; ZF and MMSE use around 100 mW/antenna, versus 23 mW/antenna for MRT.The EE-optimal solution can therefore use low-power UE-like RF amplifiers, far below conventional macro-BS amplifier levels.
  • A. Single-Cell Scenario: ZF and MMSE provide a 3-fold EE improvement and an 8-fold area-throughput improvement over MRT/MRC at their respective optima.Most of the area-throughput gain remains under imperfect CSI, whereas MRT/MRC becomes severely limiting for both EE and area throughput.

B. Multi-Cell Scenario

The multi-cell analysis finds that inter-cell interference lowers throughput, transmit-power consumption, and energy efficiency, while massive MIMO remains EE-optimal under ZF processing.

  • B. Multi-Cell Scenario: Inter-cell interference reduces throughput, transmit-power consumption, and EE relative to the corresponding single-cell results.The multi-cell figures are reported as similar to the single-cell counterparts but with smaller values.
  • B. Multi-Cell Scenario: The largest pilot reuse factor, τ (ul) = 4, achieves the highest EE and area throughput.The paper links this result to the need for actively mitigating pilot contamination in multi-cell systems.
  • B. Multi-Cell Scenario: Transmit power still increases with M at the EE optimum, while radiated power per antenna decreases with M.This preserves the single-cell scaling direction but changes its per-antenna level in the multi-cell setting.
  • B. Multi-Cell Scenario: M = 123 and K = 40 give the smaller-system EE optimum with 1.94 bit/symbol per UE under pilot reuse 4.The reduced dimensions are mainly attributed to inter-cell interference and pilot-overhead constraints.
  • B. Multi-Cell Scenario: The paper concludes that massive MIMO is the EE-optimal architecture despite inter-cell interference.The multi-cell analysis uses ZF processing and focuses on a representative cell with interference from the two closest cells in each direction.
  • B. Multi-Cell Scenario: The broader analysis uses a nonlinear power-consumption model and derives closed-form EE-optimal parameter values under ZF processing.Results are verified under other processing schemes, imperfect CSI, and symmetric multi-cell scenarios.

APPENDIX: COLLECTION OF PROOFS

The appendix proves quasi-concavity and derives optimization conditions for ZF-based EE parameters, channel statistics, imperfect CSI, and symmetric multi-cell rates.

  • APPENDIX: COLLECTION OF PROOFS: ZF detection simplifies the uplink interference-related matrix to a diagonal form used in the lemma proofs.The derivation uses the kth column of the ZF channel inverse.
  • APPENDIX: COLLECTION OF PROOFS: The user-count objective is shown quasi-concave, so its optimum is obtained by solving a quartic stationarity equation and checking nearby integers.The candidate integer is either the closest smaller or closest larger integer than a real root.
  • APPENDIX: COLLECTION OF PROOFS: The antenna-count objective is optimized by relaxing M to a real variable and converting the quasi-concave solution to an integer satisfying M ≥ K + 1.The objective goes to zero at M = K and as M approaches infinity.
  • APPENDIX: COLLECTION OF PROOFS: The transmit-power objective is quasi-concave and has a positive optimum because it is zero at zero power and as power approaches infinity.The proof applies the same scalar optimization lemma used for the antenna-count result.
  • APPENDIX: COLLECTION OF PROOFS: The channel model averages over channel realizations and user locations, with H^H H following a complex Wishart distribution for fixed locations.The large-scale fading enters through the diagonal parameter matrix Λ.
  • APPENDIX: COLLECTION OF PROOFS: Under imperfect CSI, approximate ZF treats channel estimates as true channels and estimation errors as noise when computing average UE rates.The resulting uplink and downlink expressions account for pilot-related interference and channel uncertainty.
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