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Tri-Hybrid Cell-Free Massive MIMO Systems

Janith Dassanayake, Gayan Aruma Baduge

arXiv:2610.12290v1eess.SP

TL;DR

Cell-free massive MIMO tri-hybrid precoding has not been explored as a GEE-maximization problem, despite prior tri-hybrid work. This paper combines precoder and power-allocation optimization, and reports higher GEE than fully digital and classical hybrid baselines at 20 dBm.

  • Problem

    Tri-hybrid precoding for cell-free massive MIMO has not been explored, and prior tri-hybrid work largely lacks an explicit GEE objective.

    Source: §I · p. 1, 3) Existing Tri-Hybrid precoder… · p. 3

  • Method

    The precoder stage uses WMMSE-based sum-SE optimization with manifold and gradient updates, while Dinkelbach iterations address the power-allocation subproblem.

    Source: 1) Sub-routine 1: Fix η and optimize F · pp. 7–8

  • Results

    At 20 dBm, the proposed tri-hybrid precoder achieves 87.32% and 10.68% higher GEE than fully-digital and classical hybrid precoders, respectively.

    Source: §VI · p. 11

  • Takeaways & Limitations

    The jointly optimized tri-hybrid architecture achieves the highest GEE among the compared designs, and adaptive power allocation sustains efficiency above its optimal power level.

    Source: §VI · p. 11

Abstract

from arXiv · show

This paper proposes a tri-hybrid precoder design, cascading digital, analog, and electromagnetic processing layers, for cell-free massive multiple-input multiple-output (CFmMIMO) systems. We jointly optimize the three-layer precoders and access point (AP) transmit power to maximize global energy efficiency (GEE) under per-AP power and fronthaul constraints. To solve this, we develop an alternating optimization algorithm. The first sub-routine maximizes sum spectral efficiency (SE) for precoder design using weighted minimum mean square error, Riemannian manifold optimization, and gradient ascent over Lorentzian phases. The second optimizes power allocation via Dinkelbach's fractional programming and successive convex approximation. Convergence and computational complexity are also analyzed. Simulation results show the proposed design improves GEE by 87% and 11% over fully digital and classical hybrid precoders, respectively, at 20dBm transmit power. Moreover, joint power optimization yields 22% SE and 16% GEE gains over equal-power baselines, while outperforming colocated architectures in SE. Finally, we identify GEE-optimal regimes for AP and RF chain counts, and demonstrate that increasing dynamic metasurface antenna sizes enhances GEE in low-power regimes with negligible active power overhead.

I. INTRODUCTION

The introduction motivates energy-efficient precoding for 6G and identifies a gap: tri-hybrid designs have not addressed GEE maximization in cell-free massive MIMO. It positions this work against energy-efficient hybrid and tri-hybrid precoding research.

I. INTRODUCTION

  • 6G targets motivate improving both spectral efficiency and energy efficiency, with RF chains and power amplifiers identified as major contributors to network energy consumption.The introduction cites a 100 bps/Hz SE target and a goal of improving physical-layer EE by 10 to 100 times over 5G.Source: §I · p. 1

1) Energy-Efficient Hybrid Precoding for Co-located mMIMO:

  • Co-located hybrid-precoding studies address EE and hardware trade-offs, including insertion-loss effects and RF-chain selection.The cited work reports that partially connected structures can improve EE at the cost of weaker SE gains.Source: 1) Energy-Efficient Hybrid Precoding… · p. 2

2) Energy-Efficient Hybrid Precoding for CFmMIMO:

  • Cell-free hybrid-precoding studies have explored GEE, wideband beam-squint compensation, adaptive RF-chain activation, and heterogeneous AP deployment.The approaches include fronthaul-aware GEE optimization and decentralized or semi-centralized uplink designs –.Source: 2) Energy-Efficient Hybrid Precoding… · p. 2

3) Existing Tri-Hybrid precoder Architectures:

  • Existing tri-hybrid designs mainly optimize SE, and their GEE integration with cell-free massive MIMO remains unexplored.Prior work includes DMA, radiation-center reconfigurable, tri-timescale, and pinching-antenna architectures, –.Source: 3) Existing Tri-Hybrid precoder… · pp. 2–3

B. Motivation

  • The paper addresses this gap by jointly optimizing tri-hybrid precoders and distributed AP power allocation for GEE under per-AP power constraints.Unlike the cited co-located EE design, the proposed approach updates all three precoding layers and has a separate Dinkelbach/SCA power-allocation stage.Source: B. Motivation · p. 3

C. Our Distinct Contribution

The paper proposes joint GEE optimization of tri-hybrid precoders and per-AP power allocation in wideband mmWave cell-free massive MIMO. Its design combines an alternating optimization method with numerical comparisons and hardware-dimensioning insights.

C. Our Distinct Contribution

  • The proposed OFDM-based mmWave cell-free system jointly optimizes digital, analog, and DMA-based electromagnetic precoders with per-AP power allocation to maximize GEE.The objective accounts for sum rate across subcarriers and RF-chain, phase-shifter, DMA, and fronthaul power.Source: C. Our Distinct Contribution · p. 3

A. System Model

  • Each AP has cascaded EM, analog, and digital layers, with DMA subarrays driven by feed ports and an analog precoder connecting RF chains to feed ports.The EM precoder has a block-diagonal structure because each reconfigurable subarray is excited by one feed port.Source: C. Our Distinct Contribution · p. 3, A. System Model · p. 4

C. Our Distinct Contribution

  • An alternating framework updates precoders and power allocation, using WMMSE, Riemannian manifold optimization, projected gradient ascent, and Dinkelbach/SCA optimization.The power-allocation stage linearizes sum SE and solves a convex quadratic program at each iteration.Source: C. Our Distinct Contribution · p. 3
  • The numerical evaluation covers convergence, computational complexity, precoding baselines, cell-free versus co-located deployment, and system-parameter effects on SE and GEE.The comparisons include fully digital and conventional hybrid precoders.Source: C. Our Distinct Contribution · p. 3
  • The reported design insights include higher GEE for tri-hybrid precoding, higher SE for cell-free than co-located deployment, and optimal AP and RF-chain regimes.Increasing DMA element count can increase GEE in the low-transmit-power regime at infinitesimal additional active power cost.Source: C. Our Distinct Contribution · p. 4

A. System Model

  • The model uses distributed APs serving multiple users over OFDM subcarriers, with Nt feed ports, Ne radiating elements per port, and NRF RF chains.The total number of radiating elements per AP is N_rad = NtNe, and the system requires Ns ≤ NRF ≤ Nt.Source: A. System Model · p. 4

B. Channel Model

The channel model represents the mmWave link between each AP and user using multiple delay taps, then forms each subcarrier channel by a discrete Fourier transform.

B. Channel Model

  • The AP-to-user channel is modeled with a mmWave D-delay-tap channel model.The dth tap is described using propagation paths and their path parameters.Source: B. Channel Model · p. 5
  • The path description identifies the number of propagation paths, complex path gain, pulse-shaping function, and angles of arrival and departure.The text also defines frequency-dependent transmit and receive steering vectors.Source: B. Channel Model · p. 5
  • The frequency-domain channel on each subcarrier is obtained by an Nc-point discrete Fourier transform of the delay-tap channel.At the carrier frequency, the steering vector reduces to the standard half-wavelength-spaced vector.Source: B. Channel Model · p. 5

C. Signal Model

The signal model combines AP transmissions into each user's received signal, represents the effective channels and interference-plus-noise covariance, and uses these quantities to express achievable sum SE.

C. Signal Model

  • The received signal includes the intended user's transmitted signal, contributions from other users, and additive white Gaussian noise.The data symbols are modeled as zero-mean with unit variance.Source: C. Signal Model · p. 5
  • The effective channel incorporates the EM and analog layers, while stacked AP channels and power-weighted digital precoders represent joint service across APs.The received signal is rewritten using the effective channel and digital precoders; the second signal term represents inter-user interference.Source: C. Signal Model · p. 5
  • Achievable sum SE is parameterized by the tri-hybrid precoder and AP-to-user power allocation, making their joint optimization relevant to GEE under per-AP power budgets.The cell-free setting has geographical heterogeneity in AP-user path losses, unlike a co-located system with one global power constraint.Source: C. Signal Model · p. 5

D. CSI Acquisition and the DMA Model

The section describes CSI assumptions and DMA-model scope, then specifies the GEE power accounting for the tri-hybrid CFmMIMO system.

D. CSI Acquisition and the DMA Model

  • Perfect CSI is assumed available at the CPU to characterize the proposed tri-hybrid optimization, while existing channel-estimation techniques may be adapted using TDD reciprocity and fronthaul.The tri-timescale approach can reduce training overhead by updating the EM and analog layers from channel statistics.Source: D. CSI Acquisition and the DMA Model · pp. 5–6
  • DMA channel estimation requires a dedicated design because Lorentzian-constrained weights restrict realizable training beams, so it is left for future work.Source: D. CSI Acquisition and the DMA Model · p. 6
  • The model applies the EM precoder across subcarriers, justified for the 28 GHz, 800 MHz setting, but leaves practical frequency-selective DMA characteristics for future work.The cited setting has a fractional bandwidth of about 2.9%; frequency selectivity is more pronounced at wider bandwidths and with longer waveguides.Source: D. CSI Acquisition and the DMA Model · p. 6

III. GLOBAL ENERGY EFFICIENCY

  • The GEE objective uses achievable sum SE and total power consumption, with the power model accounting for tri-hybrid component requirements and cell-free fronthaul.Source: §III · p. 6
  • Per-AP circuit power includes the local oscillator, RF chains, and phase shifters, while DMA varactor bias is approximately zero and PIN-diode bias is positive.Source: §III · p. 6

IV. GLOBAL ENERGY EFFICIENCY OPTIMIZATION

The section introduces the joint tri-hybrid precoder and transmit-power allocation optimization, along with the proposed solution and its algorithm.

IV. GLOBAL ENERGY EFFICIENCY OPTIMIZATION

  • The section formulates an optimization problem for joint tri-hybrid precoder design and transmit power allocation.Source: §IV · p. 6
  • The proposed solution addresses the joint design problem introduced in the section.Source: §IV · p. 6
  • The section also presents the underlying optimization algorithm.Source: §IV · p. 6

A. The proposed optimization problem formulation

The formulation maximizes GEE by jointly optimizing the tri-hybrid precoder and AP power allocation under system-specific constraints. Its non-convex fractional structure motivates an alternating-optimization solution.

A. The proposed optimization problem formulation

  • The design maximizes GEE by jointly optimizing tri-hybrid precoders and transmit-power allocation coefficients under system-specific constraints.Source: A. The proposed optimization problem… · p. 6
  • The constraints enforce analog constant modulus, Lorentzian EM structure, precoder normalization, per-AP transmit-power limits, and nonnegative power allocation.Normalization ensures unity gain through the EM and analog layers by scaling FBB.Source: A. The proposed optimization problem… · p. 6
  • The problem is a non-convex fractional program because its rate numerator and fronthaul-dependent denominator are coupled and its analog and EM feasible sets are non-convex.The authors therefore resort to alternating optimization.Source: A. The proposed optimization problem… · p. 6

B. The proposed optimization solution

The proposed solution uses two optimization sub-routines for joint precoder and transmit-power allocation, executing them alternately until convergence.

B. The proposed optimization solution

1) Sub-routine 1: Fix η and optimize F:

With η fixed, Sub-routine 1 optimizes the digital, analog, and electromagnetic precoders to maximize sum SE. Its layers use WMMSE, manifold optimization, and Lorentzian-phase gradient ascent.

The outermost layer: Dinkelbach’s transform and its reduction to sum-SE maximization

  • Under normalization C3, the precoder stage reduces to sum-SE maximization; the Dinkelbach loop is redundant, and GEE enters through Sub-routine 2.This reduction would not hold if C3 were relaxed to permit a trade-off between precoder gain and radiated power.Source: 1) Sub-routine 1: Fix η and optimize F · p. 7

Inner Layer 1: Weighted sum MSE minimization-based Transformation for the Digital Precoder FBB

  • WMMSE updates the receive filters, weight matrices, and digital precoder in turn, with the weighted-MSE cost non-increasing across iterations.At optimal auxiliary variables, the weighted-MSE cost has the same stationary points as negative sum SE.Source: 1) Sub-routine 1: Fix η and optimize F · pp. 7–8

Inner Layer 2: Riemannian Manifold Optimization for the Analog Precoder FRF

  • The analog precoder is optimized on its constant-modulus manifold by projecting the Euclidean gradient onto the tangent space and retracting each entry to unit modulus.The retraction preserves the constant-modulus constraint at every iteration.Source: 1) Sub-routine 1: Fix η and optimize F · p. 8

Inner Layer 3: Projected Gradient Ascent for the EM Precoder FEM

  • The EM precoder is updated by gradient ascent over phase parameters, with Lorentzian weights recomputed from the phases to satisfy the Lorentzian constraint at every iteration.The phase update keeps each phase in [0, 2π).Source: 1) Sub-routine 1: Fix η and optimize F · pp. 8–9

2) Sub-routine 2: Fix F and optimize

With the precoders fixed, Sub-routine 2 optimizes power allocation using Dinkelbach iterations and successive convex approximation of the nonconvex sum-SE objective.

2) Sub-routine 2: Fix F and optimize

  • Because sum SE is nonconvex in η, each Dinkelbach iteration uses successive convex approximation, linearizing sum SE around the current power allocation.The resulting subproblem applies the per-AP power and fronthaul constraints.Source: 2) Sub-routine 2: Fix F and optimize · p. 9
  • A proximal regularizer limits step size and stabilizes SCA iterations, yielding a convex quadratic program solvable by standard convex solvers.The regularizer penalizes movement from the previous SCA iterate.Source: 2) Sub-routine 2: Fix F and optimize · p. 9
  • The algorithm retains its best solution, so GEE is non-decreasing and bounded; the authors state that convergence occurs within a few alternating iterations.The passage attributes the bound to finite R and positive total power under the stated power limits.Source: 2) Sub-routine 2: Fix F and optimize · p. 9

V. COMPUTATIONAL COMPLEXITY ANALYSIS

The complexity analysis accounts for the precoder and power-allocation sub-routines separately, then combines their per-iteration costs across alternating iterations.

V. COMPUTATIONAL COMPLEXITY ANALYSIS

  • Per alternating iteration, Sub-routine 1 has cost CB1 = CFBB + Cman + CEM for digital-precoder, manifold, and EM-gradient updates.The EM-gradient calculation exploits the fact that all NtNe phase parameters of an AP share the same Euclidean gradient.Source: §V · p. 9
  • Sub-routine 2 has cost CB2 = Cpre + NDKNSCA(Cinv + Cgrad + CCVX), accounting for precomputation and nested Dinkelbach and SCA iterations.The inner-iteration costs include matrix inversions, SCA gradient calculations, and the convex quadratic program.Source: §V · p. 9
  • Across Nalt alternating iterations, the overall cost is Ctotal = Nalt(CB1 + CB2).The total combines the two sub-routines’ per-iteration costs.Source: §V · p. 9

VI. NUMERICAL RESULTS

The numerical results compare spectral efficiency (SE) and global energy efficiency (GEE) across precoder architectures, power allocations, and system configurations. Jointly optimized tri-hybrid precoding achieves the strongest GEE, while the results also expose trade-offs involving SE, AP and RF-chain counts, and DMA size.

Algorithm convergence

  • The alternating algorithm’s GEE increases over its first few iterations and then settles at Pmax values of 5, 15, and 25 dBm.The text attributes faster convergence at higher power to larger-magnitude GEE gradients with respect to power allocation in the interference-limited regime.Source: Figure 2 · p. 10, §VI · pp. 10–11

SE versus transmit power

  • At Pmax = 20 dBm, joint power optimization raises tri-hybrid SE by 21.86% over its equal-power baseline, but tri-hybrid SE remains below fully-digital and hybrid designs.For Pmax ≤ 20 dBm, adaptive and full-power designs coincide; at 30 dBm, adaptive tri-hybrid SE is 21.18% below full-power SE.Source: Figure 3 · p. 11, §VI · p. 11

GEE versus transmit power

  • At Pmax = 20 dBm, tri-hybrid precoding achieves 10.68% and 87.32% higher GEE than classical hybrid and fully-digital precoders, respectively, under joint power allocation.Joint optimization also yields GEE gains of 14.27%, 11.05%, and 16.40% over equal-power baselines for fully-digital, hybrid, and tri-hybrid designs.Source: Figure 4 · p. 11, §VI · p. 11
  • GEE versus transmit power is non-monotone for equal-power and full-power designs, peaking at moderate power before declining as transmit power rises.The reported explanation is that achievable rate scales logarithmically while amplifier power scales linearly.Source: §VI · p. 11

Cell-free versus co-located systems

  • Across the useful transmit-power regime, CFmMIMO with M = 8 and Ne = 4 achieves approximately 562.5% higher SE than co-located mMIMO with Nrad_t = 32.Each AP and the co-located BS use the same per-node Pmax, while CFmMIMO has aggregate transmit-power budget MPmax.Source: §VI · pp. 11–12
  • Increasing AP count from M = 8 to M = 16 raises SE by approximately 16.8% at Pmax = 20 dBm but lowers GEE because aggregate static circuit power doubles.The reported GEE comparison uses Ne = 4 for both AP counts.Source: §VI · pp. 11–12

GEE versus RF-chain count

  • Tri-hybrid GEE peaks at NRF = 2 for every evaluated K and Pmax; at Pmax = 15 dBm, increasing NRF from 2 to 8 reduces GEE by 31.6%, 27.4%, and 20.9% for K = 2, 3, and 5.At fixed NRF and Pmax, GEE increases with K; at NRF = 2 and Pmax = 15 dBm, values are 1.36, 1.63, and 1.93 bits/s/Hz/W for K = 2, 3, and 5.Source: Figure 6 · p. 12, §VI · p. 13

GEE versus DMA-element count

  • GEE increases monotonically with DMA element count in all four evaluated cases, with about 56% gain from Ne = 1 to Ne = 16 for K = 5 at Pmax = 5 dBm.The gains are upper bounds for large Ne: the power model excludes power dissipated inside the DMA, and PIN-diode power with Pvar > 0 would reduce them.Source: Figure 7 · p. 13, §VI · p. 13

VII. CONCLUSION

The paper jointly optimizes tri-hybrid precoding and AP transmit powers for CFmMIMO to maximize GEE under per-AP power constraints. Its evaluations report advantages over fully digital and hybrid benchmarks, equal-power allocation, and co-located deployments.

VII. CONCLUSION

  • The proposed design outperforms fully digital and hybrid benchmarks and co-located tri-hybrid deployments, while achieving GEE gains over equal power allocation.The paper also reports GEE gains without compromising SE gains attainable by more complicated digital precoders.Source: §VII · p. 14
  • The three precoding layers and AP transmit powers are jointly optimized to maximize GEE subject to per-AP power constraints.Transmit power optimization at each AP is embedded as a sub-routine.Source: §VII · p. 14
  • An alternating algorithm combines WMMSE, Riemannian manifold optimization, and Lorentzian-phase gradient ascent for precoding with Dinkelbach fractional programming and SCA for power optimization.The power sub-routine is solved iteratively through a convex solver.Source: §VII · p. 14
  • The paper evaluates algorithm convergence and computational complexity and analyzes the SE-GEE trade-off of the tri-hybrid architecture.The authors conclude that the architecture can provide GEE gains without compromising SE gains attainable by more complicated digital precoders.Source: §VII · p. 14
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