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arxiv: 2604.24538 · v1 · submitted 2026-04-27 · 📡 eess.SP

Quantization-Aware EE Optimization and SE-EE Tradeoff for MiLAC-Aided MU-MISO Beamforming

Yuchen Zhang , Pinjun Zheng , Tareq Y. Al-Naffouri This is my paper

Pith reviewed 2026-05-08 01:49 UTC · model grok-4.3

classification 📡 eess.SP
keywords MiLAC-aided beamformingenergy efficiencyquantization noiseMU-MISOSE-EE tradeoffbeamforming optimizationDinkelbach algorithm
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The pith

MiLAC-aided MU-MISO beamforming improves energy efficiency over digital and hybrid methods at moderate spectral efficiency cost while expanding the achievable tradeoff region.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework to maximize energy efficiency for downlink multiuser MISO beamforming when the front end uses a microwave linear analog computer to cut the number of active radio chains. It incorporates the effects of digital-to-analog converter quantization noise into the power model and proves that the effective beamformer can be optimized in a reduced space whose size depends on the number of streams rather than the number of antennas. Algorithms are given to solve the resulting non-convex problem to a stationary point and to trace the boundary between spectral efficiency and energy efficiency using weighted sums and successive convex approximations. A reader would care because large antenna arrays face power consumption as the dominant limit, and this method shows how to operate with lower energy use while keeping data rates close to those of full digital systems.

Core claim

MiLAC-aided beamforming for MU-MISO downlink admits a row-space optimality property for its effective beamformer when the front end is a passive reciprocal stream-to-antenna network. This property yields an equivalent reduced-dimension reformulation of the quantization-aware energy-efficiency maximization problem. The reformulated problem is solved by a Dinkelbach-weighted minimum mean-square error algorithm combined with projected gradient descent that converges to a stationary point. The spectral-efficiency versus energy-efficiency tradeoff is characterized by tracing its Pareto boundary through a weighted-sum formulation that employs an alternative reduced-dimension coordinate and an aux-

What carries the argument

Row-space optimality property of the effective MiLAC-aided beamformer, which reduces the optimization dimension from the antenna count to the stream count while preserving the post-quantized transmit power model.

If this is right

  • Energy-efficiency maximization reduces to an optimization whose complexity scales with the number of streams.
  • The Dinkelbach-WMMSE algorithm with projected gradient descent converges to a stationary point of the non-convex objective.
  • The SE-EE Pareto boundary is traced by convex per-iteration subproblems obtained via weighted sums and successive convex approximation.
  • Numerical results on realistic deployments confirm substantial EE gains over digital and hybrid benchmarks at moderate SE loss.
  • The achievable SE-EE operating region is significantly larger than that of conventional beamforming architectures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduced-dimension reformulation may be useful for other performance metrics such as sum-rate maximization under hardware constraints.
  • Extending the model to include additional impairments like phase noise could produce more accurate EE predictions for practical hardware.
  • The Pareto-tracing method could be adapted to online adaptation when channels vary over time.
  • Similar quantization-aware techniques may apply to uplink or cell-free massive MIMO scenarios.

Load-bearing premise

The MiLAC front-end can be modeled as a passive reciprocal stream-to-antenna network whose effective beamformer satisfies a row-space optimality property and whose quantization noise is captured exactly by the post-quantized transmit power used in the energy-efficiency objective.

What would settle it

A numerical evaluation or hardware measurement on the same channel data where MiLAC-aided beamforming yields no energy-efficiency gain over digital beamforming once the same quantization noise model is applied.

Figures

Figures reproduced from arXiv: 2604.24538 by Pinjun Zheng, Tareq Y. Al-Naffouri, Yuchen Zhang.

Figure 1
Figure 1. Figure 1: Downlink MU-MISO MiLAC-aided beamforming architecture. The active front end, consisting of view at source ↗
Figure 3
Figure 3. Figure 3: EE and achieved SE versus transmit-power budget across architectures. view at source ↗
Figure 2
Figure 2. Figure 2: Two-panel illustration of the adopted DeepMIMO simulation setting. view at source ↗
Figure 5
Figure 5. Figure 5: EE and achieved SE versus antenna number across architectures. view at source ↗
Figure 7
Figure 7. Figure 7: EE versus the scaling MiLAC static-power proxy view at source ↗
Figure 8
Figure 8. Figure 8: SE-EE tradeoff boundaries across architectures. Each architecture is optimized with its own endpoint normalization via the weighted-sum method, but view at source ↗
read the original abstract

In large antenna arrays, hardware power consumption becomes a dominant design constraint, making energy efficiency (EE) a first-class objective alongside spectral efficiency (SE). Microwave linear analog computer (MiLAC)-aided beamforming, whose front end is a passive reciprocal stream-to-antenna network, addresses this tension by reducing the active radio-frequency chain count to the stream number, at a moderate SE cost. Despite this promise, no EE optimization framework has been established for MiLAC-aided beamforming that accounts for digital-to-analog converter quantization noise and post-quantized transmit power. We fill this gap for downlink multiuser multiple-input single-output (MU-MISO) systems by formulating quantization-aware EE maximization over the MiLAC-feasible beamformer and characterizing the resulting SE-EE tradeoff. Three contributions follow. First, we prove a row-space optimality property of the effective MiLAC-aided beamformer, yielding an equivalent reduced-dimension reformulation whose complexity scales with the stream number rather than the antenna number. Second, we develop a low-complexity Dinkelbach-weighted minimum mean-square error algorithm aided by projected gradient descent that is guaranteed to converge to a stationary point. Third, we cast the SE-EE tradeoff as a multi-objective problem and trace its Pareto boundary via a weighted-sum method that combines an alternative reduced-dimension coordinate with auxiliary-variable successive convex approximation, yielding convex per-iteration subproblems with guaranteed convergence. Numerical results on a DeepMIMO v4 deployment show MiLAC-aided beamforming substantially improves EE over digital and hybrid benchmarks at a moderate SE cost and significantly expands the achievable SE-EE operating region.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a quantization-aware energy efficiency (EE) maximization framework for MiLAC-aided downlink MU-MISO beamforming. It proves a row-space optimality property of the effective beamformer to obtain an equivalent reduced-dimension reformulation (complexity scaling with stream count), proposes a Dinkelbach-WMMSE algorithm augmented by projected gradient descent that converges to a stationary point, and traces the SE-EE Pareto boundary via weighted-sum optimization combined with auxiliary-variable successive convex approximation. Numerical results on DeepMIMO v4 are reported to show substantial EE gains over digital and hybrid benchmarks at moderate SE cost and an expanded SE-EE operating region.

Significance. If the row-space optimality property holds under the full quantization-aware EE objective and the algorithms are correctly formulated, the work offers a concrete algorithmic pathway for EE optimization in hardware-constrained large-array systems. The reduced-dimension reformulation and guaranteed convergence properties would be useful for practical deployment, and the numerical demonstration on realistic channel data strengthens the case for MiLAC as a viable alternative to conventional architectures.

major comments (2)
  1. [Section on row-space optimality property and equivalent reformulation] The row-space optimality property (abstract and the derivation leading to the reduced-dimension reformulation) must be shown to incorporate the DAC quantization noise and post-quantized transmit power terms that appear in the EE objective. If this property is established only for an unquantized linear model and quantization is introduced afterward, the Dinkelbach-WMMSE algorithm solves a non-equivalent problem; this would invalidate both the claimed complexity reduction and the reported EE gains.
  2. [Algorithm section (Dinkelbach-WMMSE with PGD)] The convergence guarantee for the Dinkelbach-WMMSE + projected gradient descent algorithm (Section on algorithm development) needs explicit verification that the quantization-aware objective and the projection onto the MiLAC-feasible set preserve stationarity; any mismatch between the reduced problem and the original quantization-inclusive EE function would affect both the EE maximization and the subsequent SE-EE Pareto tracing.
minor comments (2)
  1. [Numerical results section] Provide more detail on the DeepMIMO v4 data preprocessing, any exclusion rules, and whether results are averaged over multiple realizations with error bars or confidence intervals.
  2. [System model section] Clarify the exact definition of the MiLAC effective beamformer and the quantization noise model in the EE objective to ensure consistency with the row-space derivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important aspects of the row-space optimality derivation and algorithm convergence that we address point by point below. We have revised the manuscript to strengthen the explicit incorporation of quantization effects and to add formal verification steps.

read point-by-point responses

  1. Referee: [Section on row-space optimality property and equivalent reformulation] The row-space optimality property (abstract and the derivation leading to the reduced-dimension reformulation) must be shown to incorporate the DAC quantization noise and post-quantized transmit power terms that appear in the EE objective. If this property is established only for an unquantized linear model and quantization is introduced afterward, the Dinkelbach-WMMSE algorithm solves a non-equivalent problem; this would invalidate both the claimed complexity reduction and the reported EE gains.

    Authors: We thank the referee for raising this point. The row-space optimality property is derived directly on the quantization-aware EE objective. In the system model, the quantization noise variance is proportional to the per-antenna transmit power (itself a function of the squared norm of the effective beamformer), and the post-quantized power consumption is modeled as a monotonic function of the same norm. The proof shows that any component of the beamformer lying outside the row space of the effective channel increases both the numerator (power) and the quantization noise term in the denominator without improving the achievable rate, thereby strictly decreasing EE. Consequently, the optimum can be restricted to the reduced row-space without loss of optimality, and the reformulation remains equivalent under the full quantization-inclusive objective. We have expanded the proof in Section III with an additional remark and intermediate steps that explicitly track the quantization noise and power terms. revision: partial

  2. Referee: [Algorithm section (Dinkelbach-WMMSE with PGD)] The convergence guarantee for the Dinkelbach-WMMSE + projected gradient descent algorithm (Section on algorithm development) needs explicit verification that the quantization-aware objective and the projection onto the MiLAC-feasible set preserve stationarity; any mismatch between the reduced problem and the original quantization-inclusive EE function would affect both the EE maximization and the subsequent SE-EE Pareto tracing.

    Authors: We appreciate the request for explicit verification. Because the reduced-dimension problem is equivalent to the original quantization-aware EE maximization (via the row-space property that already accounts for quantization noise and post-quantized power), any stationary point of the reduced problem is also stationary for the original objective. The Dinkelbach-WMMSE iterations solve the fractional program in the reduced space, while the projected gradient descent step enforces the MiLAC constraint set. We have added a new lemma in the algorithm section that formally establishes that the composite mapping (Dinkelbach-WMMSE update followed by projection) preserves first-order stationarity with respect to the quantization-inclusive objective, thereby ensuring the reported convergence properties and the validity of the subsequent SE-EE Pareto tracing. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations follow from system model and standard optimization

full rationale

The paper establishes a row-space optimality property for the MiLAC effective beamformer directly from the passive reciprocal network model, then applies it to obtain a reduced-dimension reformulation of the quantization-aware EE maximization problem. The Dinkelbach-WMMSE algorithm with projected gradient descent and the weighted-sum Pareto tracing via auxiliary-variable SCA are presented as convergent procedures operating on this reformulation, with the EE objective explicitly incorporating DAC quantization noise and post-quantized power. No step reduces a claimed result to a fitted parameter or prior self-citation by construction; the central claims are obtained by applying standard convex optimization techniques to the stated system equations. The numerical results on DeepMIMO are presented as validation rather than as the source of the optimality property itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; the central claim rests on standard wireless channel and quantization models plus the stated MiLAC passivity assumption.

axioms (2)
  • domain assumption MiLAC front-end is a passive reciprocal stream-to-antenna network
    Invoked to justify reduction of active RF chains to stream count and the row-space optimality property.
  • domain assumption DAC quantization noise is captured by the post-quantized transmit power model
    Used to make the EE objective quantization-aware.

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discussion (0)

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