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On the Total Energy Efficiency of Cell-Free Massive MIMO

Hien Quoc Ngo, Le-Nam Tran, Trung Q. Duong, Michail Matthaiou, Erik G. Larsson

arXiv:1702.07601v2cs.IT

TL;DR

The paper examines how cell-free massive MIMO can achieve energy efficiency when channel estimation, hardware, power control, and backhaul consumption are considered. It derives closed-form spectral efficiency, optimizes power allocation, and selects serving AP subsets; the resulting schemes substantially improve energy efficiency, while cell-free operation outperforms colocated massive MIMO under uniform service requirements.

  • Problem

    The energy efficiency of cell-free massive MIMO was unclear because distributed AP cooperation may increase backhaul power consumption, particularly with multiple-antenna APs.

  • Method

    The paper derives closed-form downlink spectral efficiency and uses constrained total-energy-efficiency power control together with two AP selection schemes.

  • Results

    Cell-free massive MIMO achieves significantly higher energy efficiency than colocated massive MIMO under a 1 bit/s/Hz requirement for every user, with factors of about 7.4 and 2.2 for MN = 128 and MN = 256, respectively.

  • Takeaways & Limitations

    Power allocation and AP selection can significantly improve energy efficiency, while only a small number of APs need participate in serving a given user.

Abstract

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We consider the cell-free massive multiple-input multiple-output (MIMO) downlink, where a very large number of distributed multiple-antenna access points (APs) serve many single-antenna users in the same time-frequency resource. A simple (distributed) conjugate beamforming scheme is applied at each AP via the use of local channel state information (CSI). This CSI is acquired through time-division duplex operation and the reception of uplink training signals transmitted by the users. We derive a closed-form expression for the spectral efficiency taking into account the effects of channel estimation errors and power control. This closed-form result enables us to analyze the effects of backhaul power consumption, the number of APs, and the number of antennas per AP on the total energy efficiency, as well as, to design an optimal power allocation algorithm. The optimal power allocation algorithm aims at maximizing the total energy efficiency, subject to a per-user spectral efficiency constraint and a per-AP power constraint. Compared with the equal power control, our proposed power allocation scheme can double the total energy efficiency. Furthermore, we propose AP selections schemes, in which each user chooses a subset of APs, to reduce the power consumption caused by the backhaul links. With our proposed AP selection schemes, the total energy efficiency increases significantly, especially for large numbers of APs. Moreover, under a requirement of good quality-of-service for all users, cell-free massive MIMO outperforms the colocated counterpart in terms of energy efficiency.

I. INTRODUCTION

The paper studies the energy efficiency of cell-free massive MIMO with multiple-antenna APs, TDD-based CSI acquisition, and distributed conjugate beamforming. It develops spectral-efficiency and power-control analyses while accounting for backhaul consumption and proposes AP selection to improve efficiency.

  • Motivation: Cell-free massive MIMO uses distributed APs to coherently serve many users in the same time-frequency resource, mitigating inter-cell interference through cooperation.The considered system has multiple antennas at each AP, with APs connected to a central processing unit through a backhaul network.
  • Motivation: The study addresses whether cell-free massive MIMO remains energy-efficient when backhaul power consumption and multiple-antenna AP hardware are included.The paper notes that the energy efficiency of cell-free massive MIMO was not yet clear because distributed operation may require more backhaul.
  • Contributions: A closed-form downlink spectral-efficiency expression is derived for finite AP and user numbers with arbitrary pilot assignments.The result accounts for channel estimation effects and generalizes earlier colocated and single-antenna cell-free massive MIMO results.
  • Contributions: The paper proposes total-energy-efficiency power control subject to per-user spectral-efficiency and per-AP power constraints, approximately solved through sequential SOCPs.The optimization incorporates hardware and backhaul power consumption.
  • Contributions: Two AP selection schemes—received-power-based and largest-large-scale-fading-based—are proposed to reduce backhaul consumption, especially when many APs are deployed.The schemes are evaluated alongside quantitative comparisons between cell-free and colocated massive MIMO.

2) Downlink Payload Data Transmission:

Each AP forms its transmitted signal by power-scaling user symbols and applying conjugate channel estimates, while users detect desired signals using effective channel statistics. The resulting spectral-efficiency expression is available in closed form, but pilot nonorthogonality can bound performance as the number of APs grows.

  • Downlink transmission: Each AP generates its transmitted signal by scaling the K user symbols with power-control coefficients and multiplying them by conjugate channel estimates.The coefficients are selected subject to a per-AP power constraint.
  • Signal detection: The k-th user detects q_k from its received signal using knowledge of the effective channel gain and channel statistics rather than downlink pilots.This avoids downlink training and supports a closed-form spectral-efficiency expression.
  • Spectral-efficiency analysis: The received-signal model separates the desired signal, beamforming uncertainty, and inter-user interference to obtain the k-th user’s spectral efficiency.The paper then presents an exact closed-form representation of this spectral efficiency.
  • Pilot effects: With sufficiently long coherence intervals, pairwise orthogonal pilot sequences can eliminate pilot contamination and allow spectral efficiency to increase without bound as the number of APs grows.This condition corresponds to choosing τ_p ≥ K in low-mobility environments.
  • Pilot effects: When the coherence interval does not support orthogonal pilots, pilot contamination bounds spectral efficiency even as M approaches infinity.The expression also reduces to known special cases for single-antenna APs and colocated massive MIMO after the stated substitutions.

III. POWER CONSUMPTION MODEL AND ENERGY EFFICIENCY

The power model combines AP amplifier and circuit consumption with backhaul consumption. Backhaul power includes fixed and traffic-dependent components tied to the network’s sum spectral efficiency.

  • Power Consumption Model: Each AP’s power consumption includes amplifier consumption and circuit power for transceiver chains and signal processing.The amplifier and circuit terms are represented by P_m.
  • Power Consumption Model: Backhaul power accounts for links connecting the CPU and APs and transfers data between them.The backhaul term is denoted P_bh,m.
  • Power Consumption Model: Backhaul power is proportional to the sum spectral efficiency and contains fixed and traffic-dependent components.The fixed component may depend on AP–CPU distance and topology, while the traffic-dependent component is measured in Watt per bit/s.

B. Total Energy Efficiency

Total energy efficiency is optimized by allocating AP–user power coefficients under per-user spectral-efficiency and per-AP power constraints. Because the formulation is nonconvex, the proposed scalable method uses sequential convex approximations and SOCP reformulations.

  • B. Total Energy Efficiency: Total energy efficiency is defined as network sum throughput divided by total network power consumption.The resulting metric is measured in bit/Joule.
  • B. Total Energy Efficiency: The optimization allocates power coefficients to maximize total energy efficiency subject to minimum per-user spectral-efficiency and per-AP transmit-power constraints.The minimum spectral-efficiency target for user k is denoted S^o_k.
  • B. Total Energy Efficiency: The original optimization is nonconvex because the spectral-efficiency function is neither convex nor concave in the power coefficients.The reformulation introduces variables and constraints that enable tractable approximations.
  • B. Total Energy Efficiency: Sequential convex approximation replaces difficult constraints with convex approximations, including SOC-representable forms.One constraint is handled through a first-order Taylor approximation, while another convex constraint is approximated by an SOC constraint for scalability.
  • B. Total Energy Efficiency: Perspective transformation reformulates each iteration as an SOCP that modern convex solvers can solve for relatively large problem sizes.The approach is intended to reduce computational complexity compared with globally optimal monotonic optimization.

Convergence Analysis

The SCA algorithm preserves feasibility across iterations and produces a monotonically increasing objective sequence. Since the objective is bounded above, the algorithm is guaranteed to converge and is observed to do so quickly.

  • Convergence Analysis: A feasible solution at iteration n remains feasible for the reformulated original problem.This follows from the convex approximation’s lower-bound property.
  • Convergence Analysis: An optimal solution at iteration n is feasible for the next iteration’s problem.The approximation is exact at the current iterate, preserving the previous solution’s feasibility.
  • Convergence Analysis: The algorithm generates a monotonically increasing objective sequence.The objective is also bounded above by the total power constraint, so the sequence converges.
  • Convergence Analysis: About 10 iterations are observed to be sufficient for the algorithm to converge quickly in the numerical results.This is an empirical observation rather than a general iteration bound.

V. ACCESS POINT SELECTION

The section develops AP-selection schemes to reduce backhaul power consumption and improve total energy efficiency by serving each user through a selected subset of APs. This addresses the increasing impact of backhaul power as the AP count grows.

  • Backhaul power consumption significantly affects energy efficiency, especially as the number of APs increases.The total backhaul power is proportional to the sum spectral efficiency and the number of APs.
  • The proposed AP-selection schemes reduce backhaul power consumption and thereby increase total energy efficiency.
  • Selecting only APs serving particular users reduces the data transferred over each AP's backhaul link.If AP m serves users in U_m, its backhaul carries only those users' data.
  • Users should be served by selected AP groups because distant APs add little to overall spatial-diversity gains.The paper proposes received-power-based and largest-large-scale-fading-based selection methods.

A. Received-Power-Based Selection

Received-power-based selection chooses, for each user, the smallest group of APs whose useful received power reaches a target fraction. The resulting assignments modify the optimization problem and require a bound-based approximate solution.

  • A. Received-Power-Based Selection: Each user selects APs contributing at least δ% of its total desired-signal received power.The selected set contains the largest received-power contributions and is minimal under this threshold.
  • A. Received-Power-Based Selection: The selection procedure first obtains optimal power-control coefficients, computes received-power contributions, and then reruns power control after restricting AP-user associations.
  • A. Received-Power-Based Selection: The selected AP sets determine U_m, and η_mk is forced to zero when user k is not served by AP m.
  • A. Received-Power-Based Selection: The resulting energy-efficiency problem contains discrete AP-user sets and cannot be solved directly with convex optimization tools.A bound is introduced to obtain an approximate solution.

B. Largest-Large-Scale-Fading-Based Selection

The largest-large-scale-fading-based method avoids the computationally expensive initial power-control step by assigning each user to APs with the strongest large-scale fading coefficients. Power control is then optimized under the resulting associations.

  • B. Largest-Large-Scale-Fading-Based Selection: The method selects AP sets without first implementing the power-control algorithm used by received-power-based selection.This makes it a simpler selection method.
  • B. Largest-Large-Scale-Fading-Based Selection: Each user is associated with M_0,k ≤ M APs having the M_0,k largest large-scale fading coefficients.
  • B. Largest-Large-Scale-Fading-Based Selection: After selecting the AP sets, the method determines U_m and reruns the power-control algorithm with non-serving AP-user coefficients replaced by zero.

VI. NUMERICAL RESULTS AND DISCUSSION

The numerical-results section evaluates cell-free massive MIMO's total energy efficiency and verifies the benefits of the proposed AP-selection schemes.

  • VI. NUMERICAL RESULTS AND DISCUSSION: The numerical results quantitatively study total energy efficiency and verify the benefits of the proposed AP-selection schemes.

A. Parameters and Setup

The evaluation uses a wrapped-square deployment with random pilot assignment and specified fading and power-consumption parameters. It examines convergence, computational complexity, and energy-efficiency gains from the proposed power-control method.

  • Simulation setup: Users randomly select pilots from τp orthogonal sequences, while APs and users are placed in a wrapped D×D km^2 square.Wrapping avoids boundary effects.
  • Channel model: The large-scale fading model combines path loss and log-normal shadowing, with σsh = 8 dB, d0 = 10 m, d1 = 50 m, and L = 140.7 dB.The path-loss parameters are chosen to resemble those in.
  • Power parameters: The simulations use B = 20 MHz, a 9 dB noise figure, ρd = 1 W, ρp = 0.2 W, and τc = 200 unless otherwise stated.Power-consumption values are taken from Table II and prior references.
  • Power allocation: Algorithm 1 is evaluated for total-energy-efficiency convergence, with NI = 10 and ǫ = 0.01 selected for subsequent experiments.Small configurations are benchmarked against a BRB-based optimal solution.
  • Computational evaluation: The proposed power-control runtime is measured for different M and K using MATLAB, YALMIP, and MOSEK on a 64-bit computer with 16 GB RAM.The implementation is intended to assess complexity rather than provide real-time execution.
  • Energy-efficiency results: Compared with equal power control (II), the proposed method improves total energy efficiency by more than 2.9× for (M = 100, K = 20) and 2.6× for (M = 100, K = 40).The spectral-efficiency target is set to the value obtained under equal power control (II).

2) AP Selection:

AP selection reduces backhaul-related power consumption while preserving energy efficiency, particularly when traffic-dependent power or AP count is large. The study also shows that antenna placement and AP distribution create an energy-efficiency trade-off, with cell-free deployment outperforming colocated massive MIMO under the reported QoS requirement.

  • AP Selection: AP selection schemes improve energy efficiency significantly, especially at high traffic-dependent power and large AP counts.The received-power-based scheme outperforms largest-large-scale-fading-based selection slightly, but has higher computational complexity.
  • AP Selection: Only about 10%–20% of APs participate in serving a given user on average.This supports selecting a small number of nearby APs to reduce backhaul requirements.
  • Effect of the Number of Antennas per AP: With a fixed total of 256 service antennas, increasing antennas per AP first raises, then optimizes, and finally lowers average total energy efficiency.Fewer APs reduce backhaul consumption, but larger AP spacing can reduce spectral efficiency.
  • Effect of the Number of Antennas per AP: The optimal number of antennas per AP depends strongly on traffic-dependent power, spectral-efficiency target, and area size.Higher Pbt,m, lower Sok, and smaller D favor fewer APs with more antennas per AP; the reverse conditions favor more APs.
  • Cell-Free Massive MIMO Versus Colocated Massive MIMO: 7.4 and 2.2 are the reported cell-free energy-efficiency improvement factors over colocated massive MIMO for MN = 128 and MN = 256, respectively, at Sok = 1 bit/s/Hz per user.The comparison uses received-power-based AP selection and the antenna count maximizing average energy efficiency.
  • Cell-Free Massive MIMO Versus Colocated Massive MIMO: With a 1 bit/s/Hz requirement for every user, cell-free massive MIMO improves energy efficiency by an order of magnitude compared with colocated massive MIMO.The conclusion attributes this result to the considered power control and AP selection schemes while maintaining uniformly good service.

APPENDIX

The appendix derives intermediate quantities for the spectral-efficiency expression and establishes equivalence between the original optimization problem and a reformulated problem. It then substitutes the derived expectations and identities into the target expressions.

  • Derivation: The appendix computes DSk using independence between the estimated and non-estimated channel components.The derivation begins from the relevant spectral-efficiency quantities and uses the stated independence relation.
  • Optimization Reformulation: The optimization problem (P) is shown to be equivalent to (P1) by analyzing the objective after dividing its numerator and denominator by B · Se({ηmk}).The appendix states that the objective increases as the reformulation is applied, yielding the equivalence.
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