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Energy efficiency of mmWave massive MIMO precoding with low-resolution DACs

Lucas N. Ribeiro, Stefan Schwarz, Markus Rupp, André L. F. de Almeida

arXiv:1709.05139v2cs.IT

TL;DR

The paper addresses how to reduce power consumption in mmWave massive MIMO while maintaining spectral efficiency. It evaluates quantized hybrid and digital precoders using an AQN-based SU-MIMO model with realistic power-consumption and RF-loss models, finding that partially-connected hybrid precoders can be more energy-efficient than digital precoders, whereas fully-connected designs generally perform poorly energetically.

  • Problem

    The paper investigates the insufficiently evaluated energy efficiency of quantized hybrid transmitters with fully- and partially-connected phase-shifting networks under realistic RF power and loss models.

  • Method

    The paper uses an AQN-based quantized SU-MIMO model and realistic models of device power, phase-shifter topology, insertion loss, and precoding computation to compare hybrid and digital transmitters.

  • Results

    Partially-connected hybrid precoders can be more energy-efficient than digital precoders, while fully-connected hybrid precoders generally exhibit poor energy efficiency; active phase-shifters favor data rates and passive phase-shifters favor energy efficiency.

  • Takeaways & Limitations

    Partially-connected hybrid precoding offers the strongest energy-efficiency result among the studied architectures, while phase-shifter implementation determines the trade-off between throughput and energy efficiency.

Abstract

from arXiv · show

With the congestion of the sub-6 GHz spectrum, the interest in massive multiple-input multiple-output (MIMO) systems operating on millimeter wave spectrum grows. In order to reduce the power consumption of such massive MIMO systems, hybrid analog/digital transceivers and application of low-resolution digital-to-analog/analog-to-digital converters have been recently proposed. In this work, we investigate the energy efficiency of quantized hybrid transmitters equipped with a fully/partially-connected phase-shifting network composed of active/passive phase-shifters and compare it to that of quantized digital precoders. We introduce a quantized single-user MIMO system model based on an additive quantization noise approximation considering realistic power consumption and loss models to evaluate the spectral and energy efficiencies of the transmit precoding methods. Simulation results show that partially-connected hybrid precoders can be more energy-efficient compared to digital precoders, while fully-connected hybrid precoders exhibit poor energy efficiency in general. Also, the topology of phase-shifting components offers an energy-spectral efficiency trade-off: active phase-shifters provide higher data rates, while passive phase-shifters maintain better energy efficiency.

I. INTRODUCTION

mmWave massive MIMO addresses sub-6 GHz congestion but requires energy-conscious transceiver designs because large arrays and RF components increase power consumption and losses. This work evaluates quantized hybrid and digital precoding under realistic quantization, power, and RF-loss models.

  • Motivation: mmWave massive MIMO uses highly directional antenna arrays to compensate for severe path loss at elevated frequencies.For a given physical aperture, antenna directional gain can offset the increased free-space path loss.
  • Motivation: Fully-connected phase-shifting networks consume substantial power because each RF chain connects to many active phase-shifters, dividers, and combiners.Partially-connected sub-array beamforming reduces the number of phase-shifters and associated power consumption, but RF insertion losses can reduce practical spectral efficiency.
  • Motivation: Low-resolution DACs/ADCs are proposed to reduce transceiver power demands, with low-resolution DACs also relaxing power-amplifier linearity requirements.Transmit-side power is typically dominated by power amplifiers, whereas receive-side power is strongly affected by high-resolution ADC chains.
  • Research gap: Prior work has separately examined energy-efficient full-resolution hybrid systems and low-resolution fully-digital transceivers, leaving quantized hybrid transmitters under realistic RF modeling insufficiently evaluated.The stated gap concerns fully- and partially-connected phase-shifting networks and their energy performance.
  • Contributions: The paper introduces an AQN-based quantized SU-MIMO model, derives achievable-rate expressions, and assesses DAC-quantization effects on spectral efficiency.The model combines AWGN with RF-filtered quantization noise and treats quantized hybrid precoding as a generalization of quantized digital and classical hybrid precoding.
  • Contributions: The study models active and passive phase-shifters, insertion losses, and computational power, and evaluates partially-connected beamforming using maximum eigenmode transmission.The resulting comparison investigates the energy-spectral-efficiency trade-off across transmitter architectures.

C. Notation

The paper defines notation for scalar, vector, and matrix quantities, then distinguishes digital and hybrid precoding through their baseband and analog matrices. Quantization and RF-loss factors are incorporated into the transmitted signal model.

  • Notation: Scalars, vectors, and matrices are denoted by x, x, and X, respectively, with matrix entries written as [X]i,j.The notation also defines common matrix operators, norms, expectations, and quantization operators.
  • Precoding architectures: Digital precoding uses FBB ∈ C^{Nt×Ns} to feed Nt DAC/RF-chain pairs, whereas hybrid precoding uses an analog stage after fewer RF chains.Hybrid systems employ Ns ≤ Lt < Nt DAC/RF-chain pairs.
  • Precoding architectures: Hybrid analog beamforming uses FRF ∈ C^{Nt×Lt}, whose entries have constant modulus and discrete phase resolution.The phase-shift set contains nPS = 2^bPS supported phases.
  • Signal model: DAC quantization and analog-beamforming insertion loss are modeled explicitly through Qb(·) and the multiplicative factor 1/√LRF.Digital precoding is recovered with FRF = INt and LRF = 1.

B. Quantized Signal Model

The quantized signal model approximates DAC nonlinearities with additive quantization noise and propagates both quantization effects and RF losses into an equivalent MIMO channel. Noise whitening then enables tractable achievable-rate analysis under simplifying assumptions.

  • Quantizer and AQN model: The uniform scalar quantizer Qb(·) independently quantizes real and imaginary components using Nb = 2^b levels per component.Gaussian-optimized quantization codes are used for the Gaussian baseband signals.
  • Quantizer and AQN model: The AQN approximation writes p = Qb(u) ≈ √(1−ρb)u + e, with quantization noise uncorrelated with the Gaussian input.The distortion factor ρb decreases as resolution increases and equals the inverse SQNR.
  • MIMO extension: For MIMO inputs, u and p are quantizer input and output vectors, while e is the quantization-noise vector and Υb contains per-quantizer scaling factors.The quantization-error covariance depends on the input covariance Ruu.
  • Received signal model: The received model combines the physical channel and analog precoder into Heq, scales it by Υb, and combines AWGN with RF-filtered quantization noise in n′.Low-resolution quantization reduces total channel power through the factor √(1−ρb).
  • Rate analysis: Because the resulting noise can be correlated, non-Gaussian, and precoder-dependent, the analysis assumes a Gaussian approximation and applies noise whitening to obtain a tractable lower bound.After whitening, the noise covariance is the identity matrix.

C. Channel Model

The channel model represents mmWave propagation as a narrow-band clustered channel with a small number of dominant paths and normalized path gains. Uniform linear transmit and receive arrays use half-wavelength element spacing.

  • Clustered channel model: MmWave massive MIMO channels are modeled with L propagation paths in a narrow-band clustered channel model.The model reflects spatial and angular sparsity with only a few dominant multipaths.
  • Path parameters: Each path gain αℓ is circularly symmetric Gaussian with zero mean and unit variance, and all paths have the same average power.Channel realizations are normalized so their expected squared Frobenius norm equals NrNt.
  • Path parameters: Transmit and receive elevation and azimuth angles are uniformly distributed over the specified angular intervals.The model separately assigns elevation and azimuth distributions at both link ends.
  • Array model: Both arrays are linear and uniformly spaced by λ/2, where λ is the carrier wavelength, producing Vandermonde-structured steering and array-response vectors.The methods can also be applied to arbitrary array geometries and element beam patterns.

D. Power Consumption and Loss Models

The power model accounts for static RF-front-end consumption, computational precoding cost, and insertion losses in analog beamforming. It compares digital and hybrid topologies while exposing the active-versus-passive phase-shifter trade-off.

  • Model objectives: Energy efficiency is defined as spectral efficiency divided by power consumption, motivating joint consideration of data rate and hardware energy use.Large arrays compensate mmWave path loss but increase RF-front-end power consumption and losses.
  • RF losses: Hybrid architectures reduce RF-chain count but incur losses from phase-shifters, dividers, and combiners that must be included in realistic evaluations.Compensating losses with high-gain, high-efficiency power amplifiers is not generally straightforward.
  • Phase-shifter implementations: Active phase-shifters consume nonnegligible power and provide moderate gain, whereas passive phase-shifters consume little power but introduce considerable insertion loss.The choice creates a trade-off between transmit power and power consumption.
  • Static power consumption: Static consumption includes DACs, RF chains, phase-shifters, and power amplifiers, with digital transmitters using Nt RF-chain/DAC pairs and fully connected hybrids using Lt pairs.A fully connected network contains NtLt phase-shifters, while a partially connected network contains NaLt.
  • Static power consumption: Partially connected PSNs have static consumption PLO + PPA + Lt[2PDAC(bDAC, Fs) + PRF] + NaLtPPS(bPS).Divider and combiner power is treated as negligible in the model.
  • Component assumptions: The model adopts optimistic component parameters, sets PA efficiency to η = 27%, and represents K-way divider or combiner loss using ⌈log2(K)⌉ two-way stages.Two-way dividers lose 0.6 dB, while combiners lose 0.6 dB plus an additional 3 dB.
  • Computational power: Computational power is proportional to optimization flops and coherence blocks per second, and inversely proportional to computational efficiency Ec.The update rate is C = B/(BCTC).

III. PRECODING STRATEGIES

The section formulates quantized hybrid precoding under a system model with perfect, instantaneous channel state information and an achievable-rate objective.

  • The section presents analog strategies for fully- and partially-connected phase-shifting networks, baseband precoding, complexity analysis, and achievable-rate lower bounds.
  • The quantized hybrid precoding problem seeks FRF and FBB that maximize the achievable instantaneous rate R.

FRF, FBB

The quantized rate model accounts for RF loss and quantization effects, while encompassing hybrid and fully digital precoding as special cases.

  • RF loss reduces actual transmit power to Px = Pmax/LRF when the baseband precoder compensates according to the model.Without proper RF-loss compensation, the reduced transmit power decreases achievable rate.
  • Setting ρb = 0 removes quantization noise and yields the lossy-RF hybrid precoding problem.
  • With FBB ∈CNt×Ns, FRF = INt, and LRF = 1, the model reduces to classical single-user MIMO digital precoding.The corresponding quantized digital problem is obtained when ρb ≠ 0.
  • The proposed design decouples analog and baseband optimization: FRF forms an equivalent channel, then FBB is designed from its SVD.

A. Analog Precoding Strategies

Analog precoding uses different low-complexity constructions for fully- and partially-connected networks, followed by baseband SVD precoding with water-filling under an approximation that is sub-optimal at low resolution.

  • A. Analog Precoding Strategies: The analog design prioritizes low computational complexity because FRF can have very large dimensions.
  • A. Analog Precoding Strategies: Fully-connected analog precoding alternates between constant-modulus and semi-unitary projections until convergence, then applies phase quantization.
  • A. Analog Precoding Strategies: Partially-connected precoding partitions H into antenna-subarray subchannels Hℓ and designs one quantized beamformer per subarray.
  • A. Analog Precoding Strategies: Each partially-connected beamformer uses the dominant right singular vector of Hℓ, projected onto the phase-shifter constant-modulus constraint.
  • A. Analog Precoding Strategies: The power method computes each dominant singular vector with lower expense than full-SVD methods and converges under stated singular-value and initialization conditions.
  • A. Analog Precoding Strategies: For fixed FRF, optimal digital precoding is difficult because the total noise is structured and its covariance depends on FBB.
  • A. Analog Precoding Strategies: SVD precoding with water-filling is optimal in the infinite-resolution case but generally sub-optimal at low resolution, especially outside low-SNR conditions.The strategy is expected to remain close to optimal particularly at low SNR.
  • A. Analog Precoding Strategies: The baseband design uses the SVD of Heq and water-filling power allocation, with normalization enforcing the average power constraint.

C. Computational Complexity Analysis

The complexity analysis counts floating-point operations for digital and hybrid precoding stages, while the rate analysis supplies a lower bound and highlights high-SNR effects under low-resolution quantization.

  • C. Computational Complexity Analysis: Matrix multiplication complexity is modeled as 2MNR flops for multiplying A ∈CM×R by B ∈CR×N.
  • C. Computational Complexity Analysis: Fully-digital precoding is dominated by the SVD of H, whose full decomposition requires 4P^2Q + 22Q^3 flops for a P × Q matrix.
  • C. Computational Complexity Analysis: Hybrid precoding separates baseband and analog processing, with the baseband stage dominated by SVD of Heq and the cost of forming Heq.
  • C. Computational Complexity Analysis: The equivalent-channel SVD complexity includes the matrix-product term 2NrNtLt used to form Heq.
  • C. Computational Complexity Analysis: The achievable-rate equation provides a lower bound, while water-filling can be sub-optimal for low-resolution quantization at high SNR.

A. Spectral Efficiency

The spectral-efficiency comparison shows that quantized digital precoders retain higher data rates than hybrid precoders, while RF losses make phase-shifter type consequential. Active phase-shifters improve spectral efficiency relative to passive ones, but passive architectures reduce power consumption.

  • Digital and hybrid spectral efficiency: Quantized digital precoders lose little spectral efficiency at low SNR but saturate at high SNR because quantization distortion persists.With 1-bit DACs, saturation occurs at 20 dB; with 8-bit DACs, it occurs only above 40 dB.
  • Digital and hybrid spectral efficiency: Digital precoding generally provides higher data rates than hybrid precoding at the same DAC resolution and SNR.Analog precoding filters quantization distortion, increasing total noise power and causing earlier saturation.
  • RF hardware losses: Ignoring RF hardware losses, fully-connected hybrid precoders outperform partially-connected ones, and phase-shifter type does not affect spectral efficiency.The comparison uses lossless hybrid precoders with active and passive phase-shifters.
  • RF hardware losses: Considering RF hardware losses, active phase-shifters are more spectral-efficient than passive phase-shifters because passive components attenuate signals.For 8-bit DACs, the reported SNR losses are 10 dB for active PSNs and 20 dB for passive PSNs.
  • Power and spectral-efficiency trade-off: Passive phase-shifters drastically reduce hybrid-precoder power demand but incur spectral-efficiency degradation compared with active components.Passive partially-connected hybrid precoding can consume more power than passive fully-connected hybrid precoding because its power amplifiers receive more power.
  • Scaling with antenna count: Computational power is negligible up to 64 antennas but becomes substantial for fully-connected hybrid and digital precoders as antenna count grows.Partially-connected hybrid precoding is less power-hungry because its required flop count scales linearly with Nt, unlike the other schemes' quadratic and cubic terms.

C. Energy Efficiency

Considering realistic RF losses and static power, partially-connected hybrid precoding generally offers the strongest energy efficiency, while fully-connected hybrid precoding can be inefficient. Phase-shifter topology and DAC resolution create energy–spectral-efficiency trade-offs.

  • Evaluation setup: The energy efficiency is defined as spectral efficiency divided by static power consumption, with RF hardware losses included in the evaluation.The study examines energy-spectral-efficiency curves under different SNRs and DAC resolutions.
  • Topology comparison: Fully-connected hybrid precoding can be inefficient because phase-shifter and combiner insertion losses combine with high PSN power consumption.Ignoring these losses can incorrectly suggest that fully-connected topology is generally the most spectrally efficient hybrid option.
  • Topology comparison: Partially-connected hybrid precoding can exceed fully-connected hybrid precoding in both energy and spectral efficiency because it uses fewer phase-shifters and no power combiners.The reduced hardware count lowers power loss and consumption.
  • Topology trade-off: Active phase-shifters favor spectral efficiency, whereas passive phase-shifters favor energy efficiency.Active components amplify shifted signals, while passive components have negligible power consumption.
  • DAC resolution: DAC resolutions above 3 bits do not necessarily improve energy efficiency; passive hybrid methods and digital precoding can become less efficient at higher resolution.The simulations use bDAC = 3 for the antenna-scaling comparison because higher resolution does not consistently improve efficiency.
  • Array-size effects: At both high and low SNR, partially-connected hybrid precoding is generally most energy-efficient, followed by digital precoding and fully-connected hybrid precoding.For 32 antennas at 0 dB SNR, passive fully-connected hybrid precoding is most efficient, but its efficiency quickly drops as the antenna count increases.
  • Array-size effects: Increasing the transmit-array size asymptotically does not improve energy efficiency because static power grows with antenna count at fixed SNR faster than spectral efficiency.This conclusion is reported for the evaluated antenna tuples with bDAC = 3.
  • Conclusion: The conclusion identifies partially-connected PSNs as energy-efficient because they consume less power and have limited insertion loss, while DACs above 3 bits may degrade efficiency.The study also notes that computational power should be included in the budget of very large arrays.

APPENDIX

The appendix derives an approximation showing that the lossless transmitted-signal power under the DAC model satisfies the hybrid precoding average power constraint. The same result extends to digital precoding by setting the RF precoder to the identity.

  • Power derivation: The appendix evaluates the average power of the lossless transmitted signal under the DAC approximation for hybrid precoding.The derivation is framed around the transmitted-signal power P̃x and the average power constraint.
  • Power derivation: The diagonal of the baseband precoder covariance is approximated using equal per-chain power allocation based on its trace.The approximation assumes the diagonal elements are similar because of power allocation.
  • Power constraint: The resulting estimate gives P̃x ≈ [(1 − ρb) + ρb]Pmax = Pmax, satisfying the hybrid precoding average power constraint.The derivation uses water-filling power allocation with tr(QQ^H) = Pmax.
  • Digital extension: The power result extends to digital precoding by choosing FBB ∈ C^(Nt×Ns) and FRF = INt.This removes the separate RF beamforming transformation in the digital case.
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