PaperlayerView exact revision on arXiv

Source-linked AI summary

Energy Efficiency in Microwave Linear Analog Computer (MiLAC)-Enabled Communications

Ahmed Magbool, Marco Di Renzo

arXiv:2608.30639v1eess.SP

TL;DR

The paper addresses limited EE analysis for MiLAC beamforming, especially under minimum-rate constraints and across multiple architectures. It formulates constrained EE optimization and develops reduced-dimensional and closed-form approaches. MiLAC architectures achieve higher EE over practical antenna sizes, although their asymptotic scaling decays faster than DBF.

  • Problem

    Existing MiLAC EE analysis omits minimum user-rate constraints and does not cover TLM, HDM, or their asymptotic EE scaling.

  • Method

    The paper formulates constrained EE maximization, develops reduced-dimensional SCA solutions, and derives a K+1-candidate closed-form search.

  • Results

    MiLAC-based architectures substantially outperform DBF and HAD over practically relevant antenna-array sizes.

  • Takeaways & Limitations

    MiLAC EE scales as ln(N)/N^2 versus ln(N)/N for DBF, while MiLAC remains superior over practical finite antenna regimes.

Abstract

from arXiv · show

Microwave linear analog computers (MiLACs) have emerged as a promising architecture for energy-efficient wireless communications by shifting signal processing from the digital to the analog domain using tunable impedance networks. Several MiLAC architectures have recently been proposed, including the single-layer MiLAC (SLM), two-layer MiLAC (TLM), and hybrid digital-MiLAC (HDM). In this paper, we investigate the energy efficiency (EE) of these architectures. Specifically, we formulate EE maximization problems for the SLM, TLM, and HDM architectures under transmit power, user rate, and architecture-specific constraints, and develop a dimensionality reduction technique with successive convex approximation (SCA)-based algorithms to efficiently solve the resulting non-convex problems. We further derive a computationally efficient solution for EE maximization based on a search over only $(K+1)$ closed-form candidate solutions, where $K$ is the number of users, and analyze the asymptotic EE in the large-antenna regime under negligible quantization noise. Our analysis shows that the EE of the SLM, TLM, and HDM architectures scales as $\ln(N)/N^2$, whereas conventional digital beamforming (DBF) scales as $\ln(N)/N$, where $N$ denotes the number of transmit antennas. Despite these different scaling laws, the faster EE decay of MiLAC-based architectures becomes relevant only at very large antenna dimensions, typically involving thousands of antennas, while they maintain superior EE over practically relevant finite antenna regimes. Finally, we derive an approximation of the number of antennas required to achieve the maximum EE for each architecture and the corresponding EE. Simulation results show that, over practically relevant antenna regimes, MiLAC-based architectures achieve substantially higher EE than conventional fully digital and hybrid analog-digital beamforming.

I. INTRODUCTION

MiLACs shift linear signal processing into programmable microwave circuits to reduce beamforming power, but their energy-efficiency benefits and constraints require systematic study across architectures.

  • Motivation: DBF assigns one RF chain per antenna, creating substantial hardware power consumption and an energy-efficiency bottleneck for massive MIMO.
  • MiLAC architectures: MiLACs use tunable impedance networks to implement configurable linear transformations for analog beamforming.
  • MiLAC architectures: SLM uses power allocation and RF chains before a passive MiLAC network, while TLM adds a MiLAC layer and HDM adds a digital beamformer.
  • Research gap: Prior EE analysis omitted minimum user-rate constraints and did not study TLM, HDM, or asymptotic EE scaling.
  • Contributions: The paper develops a unified model and EE maximization formulations covering transmit, receive, power-consumption, quantization, and architecture-specific constraints.
  • Contributions: The proposed low-complexity analysis searches only K+1 closed-form candidates and derives MiLAC asymptotic scaling as ln(N)/N^2 versus ln(N)/N for DBF.

1) SLM:

The three architectures differ in how active beamforming and MiLAC layers are arranged, producing architecture-dependent circuit-power models.

  • 1) SLM:: SLM uses power allocation followed by K RF chains, with each chain feeding one input port of a single-layer MiLAC network.
  • 2) TLM:: TLM places a first MiLAC layer before power allocation and K RF chains, then applies a second common MiLAC layer.
  • 3) HDM:: HDM combines a digital precoder with M ≥ K RF chains before the MiLAC transformation.
  • B. Power Consumption Model: Total power combines baseband-processing power with static circuit power from the underlying hardware components.
  • B. Power Consumption Model: Circuit power varies by architecture because configurations require different numbers of RF chains and tunable microwave components.
  • 1) SLM:: The SLM circuit model includes front-end, RF-chain, and MiLAC-network component power terms.
  • 2) TLM:: The TLM circuit model adds power for its additional MiLAC layer and associated tunable microwave components.
  • B. Power Consumption Model: Energy efficiency is expressed from total power consumption after combining the architecture-specific power model with the system sum-rate model.

III. PROBLEM FORMULATION

The SLM EE problem optimizes the effective beamforming matrix under a transmit-power budget, MiLAC feasibility constraints, and minimum user-rate requirements.

  • 1) SLM:: For SLM, the effective beamforming matrix is W ≜ FP, making it analogous to a digital beamforming matrix.
  • 1) SLM:: The reciprocal and lossless MiLAC network imposes WHW ≼ diag(p1, . . . , pK) on W.
  • 1) SLM:: After the transformation W = FP, SINR, user rate, sum rate, and EE become functions of W.
  • 1) SLM:: The baseband power-consumption expression can be rewritten using the power-allocation matrix P.
  • P SLM: The SLM EE maximization problem is formulated explicitly over the effective beamforming representation.
  • P SLM: The formulation imposes the transmit power budget PT, the feasible set of W, and minimum user rate r for every user.

2) TLM:

The TLM reformulation uses an effective beamforming matrix to simplify optimization while retaining fully digital beamforming capability. A reduced-dimensional representation substantially decreases the number of optimization variables when users are fewer than antennas.

  • TLM architecture: The TLM effective beamforming matrix is W = FPΞ, and its two-layer structure can realize any desired DBF matrix through SVD-based factorization.This removes constraint (14c) and enables DBF-equivalent beamforming.
  • TLM architecture: The TLM EE problem accounts for transmit, quantization-noise, power-consumption, and minimum-rate constraints.The formulation uses the effective matrix W and includes low-resolution RF-chain effects through received quantization noise.
  • Reduced-dimension reformulation: The reformulation writes W = UQ, where U spans the channel space and Q is the reduced-dimension optimization variable.Because W interacts with the channels through h_k^H w_i, restricting W to span(H) preserves optimality.
  • Reduced-dimension reformulation: The reduced formulation expresses effective channels, SINR, rates, EE, quantization noise, and the LMI constraint as functions of Q.The constraint becomes Q^HQ ≼ diag(p_1, ..., p_K).
  • Reduced-dimension reformulation: 86.8% fewer optimization variables are required for N = 64 and K = 8, decreasing the dimension from 1032 to 136.In general, the real-valued variable count falls from 2NK + K to 2K^2 + K.

B. Proposed Solution for (P4)

The SLM solution applies dimensionality reduction, Dinkelbach transformation, and SCA to construct convex subproblems for EE maximization. The resulting iterative algorithm updates the reduced variables until the EE improvement meets a tolerance.

  • SCA-based solution: Dinkelbach’s algorithm transforms the fractional EE objective before each SCA iteration.The transformed objective is optimized jointly over the reduced beamforming, power, and slack variables.
  • SCA-based solution: Slack variables and first-order Taylor surrogates convert the nonconvex rate-related expressions into concave or convex approximations.The surrogate rate function is tight at the current iterate, lower-bounds the original sum rate, and matches its first-order derivative there.
  • Algorithm 1: The method enforces the transmit-power, LMI, rate, and non-negativity constraints within the SCA subproblem.These constraints are represented using the reduced formulation and convex surrogate conditions.
  • Algorithm 1: The resulting SCA subproblem is convex and can be solved with numerical optimization methods such as CVX.Algorithm 1 repeatedly solves this subproblem and updates Q and P until the EE difference is at most ϵ.

C. Proposed Solutions for (P2) and (P3)

The TLM problem uses a simplified SCA framework, while the HDM problem requires alternating optimization because its MiLAC and digital beamformers are coupled. The proposed procedures have convergence guarantees under stated regularity conditions and exploit reduced dimensions.

  • TLM solution: The TLM problem has one optimization variable, no LMI constraint, and is solved with a modified SCA framework.The factors F, Ξ, and P are recovered from W using SVD after optimization.
  • HDM solution: The HDM algorithm alternates between optimizing the reduced-dimensional MiLAC beamformer Z and digital beamformer D until convergence.Each block update uses the SCA procedure.
  • Convergence analysis: The SLM and TLM EE sequences converge to finite limits, and every accumulation point satisfies the KKT conditions under standard SCA-Dinkelbach regularity conditions.The guarantee follows from monotonic EE improvement, boundedness, and surrogate tightness and consistency.
  • Convergence analysis: The HDM alternating procedure converges to a block-wise stationary point of its optimization problem.Its EE is monotonically improved or preserved and remains upper bounded.
  • Computational complexity: The dominant computational cost comes from solving convex SCA subproblems with a primal-dual interior-point method.For SLM, the reduced problem has 3K^2 + 3K real variables and a 2K × 2K LMI.

NK2 +ISLM

The paper derives closed-form SLM EE candidates using asymptotic orthogonality and negligible quantization noise, then searches over rate-margin sets. It shows that MiLAC EE eventually decays faster than DBF because MiLAC circuit power grows quadratically with antenna count.

  • SLM asymptotic analysis: Under asymptotically orthogonal channels and negligible quantization noise, SLM beamforming decouples into independent single-user problems solved by MRT directions.The assumptions eliminate inter-user interference and make the beamforming direction align with each user’s channel.
  • Closed-form search: K + 1 closed-form candidate solutions are searched by varying the number K′ of users receiving positive rate margins.Users are sorted by effective channel gain, and candidate feasibility is checked against the transmit-power constraint.
  • Search validation: The candidate search exactly matches Algorithm 1 for orthogonal channels without quantization noise but gives an optimistic EE estimate for non-orthogonal channels with quantization noise.The optimism results from neglecting multi-user interference and quantization noise.
  • Asymptotic EE scaling: ln(N)/N^2 is the asymptotic EE scaling of SLM, caused primarily by quadratic circuit-power growth from tunable impedance elements.The scaling holds as N →∞ under the stated large-antenna assumptions.

C. Optimal Number of Transmit Antennas for SLM

The asymptotic EE analysis does not establish an optimum antenna dimension, so the paper derives a unique maximum for SLM under an approximately fixed nonzero rate-margin set and locates it by one-dimensional search.

  • A unique EE maximum exists for SLM when the nonzero rate-margin set K′ is approximately fixed.
  • The optimum antenna dimension N_opt is determined by solving the equation in Lemma 2.
  • The uniqueness follows because the Lambert-W term is strictly increasing while the opposing right-hand side is strictly decreasing in N.
  • A simple linear search over N suffices to determine N_opt because the crossing point is unique.

D. DBF, TLM and HDM

The analysis extends asymptotic EE characterization and antenna-dimension optimization from SLM to TLM, HDM, and DBF. It establishes a unified comparison of their beamforming gains, circuit-power effects, and large-antenna EE behavior.

  • TLM and HDM extend the SLM asymptotic analysis by substituting their architecture-specific circuit-power models and applying Algorithm 2.
  • All four architectures achieve the same asymptotic beamforming gain because none inherently limits beamforming gain under asymptotic channel orthogonality.TLM can realize any DBF beamformer, while HDM further generalizes TLM with a digital beamforming stage.
  • MiLAC architectures have EE scaling proportional to ln(N)/N^2, while DBF scales as ln(N)/N in the asymptotic large-antenna regime.The different scaling follows from quadratic circuit-power growth caused by the increasing number of tunable impedance elements in MiLAC architectures.
  • For TLM and HDM, the optimal antenna dimension is obtained by solving architecture-specific equations after replacing the generic circuit-power term with the corresponding model.
  • The unified framework develops low-dimensional closed-form search solutions alongside the asymptotic analysis for SLM, TLM, and HDM.

VI. NUMERICAL SIMULATIONS

The simulations compare SLM, TLM, and HDM against DBF and HAD using a common mmWave communication setup. The evaluation measures achievable EE under specified antenna, user, RF-chain, hardware, power, rate, and channel parameters.

  • The numerical study evaluates SLM, TLM, and HDM against conventional DBF and HAD in terms of achievable EE.
  • The default setup uses N = 64 transmit antennas, K = 8 users, and M = 12 RF chains for HDM and HAD.
  • The experiments use fc = 28 GHz, BW = 20 MHz, fs = 1 GHz, and b = 4 DAC quantization bits.
  • The setup imposes PT = 35 dBm, σ_k^2 = −93 dBm, and a minimum rate requirement of r = 1 bit/sec/Hz for all users.
  • Channels follow a Saleh–Valenzuela mmWave model with L = 5 multipath components, random complex Gaussian path gains, and uniformly distributed departure angles.

A. Convergence Analysis

The proposed iterative algorithms converge rapidly and achieve substantial EE gains over DBF and HAD. The closed-form search remains useful as a low-complexity approximation even when channel realizations are non-orthogonal.

  • All proposed algorithms converge within fewer than eight iterations, demonstrating fast convergence of the optimization framework.The iteration-wise EE values are compared with the low-dimensional closed-form search results in Fig. 3.
  • 245%, 231%, and 185% EE improvements are achieved by TLM, SLM, and HDM, respectively, over DBF at convergence.
  • 67%, 60%, and 38% EE improvements are achieved by TLM, SLM, and HDM, respectively, over HAD at convergence.
  • TLM and SLM reduce RF chains to K = 8 from N = 64 in DBF and M = 12 in HAD, while HDM lowers circuit power by replacing per-antenna phase shifters with tunable impedance networks.
  • The low-dimensional closed-form search provides a useful low-complexity EE approximation without requiring iterative optimization to converge, even for non-orthogonal channels.It supports efficient evaluation of system configurations and design parameters.

B. Impact of Varying the Minimum Rate Constraint

MiLAC-based architectures maintain higher energy efficiency than DBF and HAD under practical rate, antenna, resolution, and user-load conditions, although their relative advantage varies with system requirements. The results also validate the proposed algorithms and asymptotic analysis across these regimes.

  • Minimum rate constraint: At high rate thresholds such as 4 bit/sec/Hz, TLM, SLM, and HDM retain approximately 1.2–1.45× higher EE than DBF.All systems lose EE as rate requirements increase, while HAD becomes infeasible above 3 bit/sec/Hz.
  • Number of antennas: MiLAC-based architectures peak at N = 256–512 antennas, compared with N = 32–64 for DBF and N = 128–256 for HAD.The approximate optimal antenna numbers and peak EE values from Lemma 2 closely track the simulated maxima under idealized orthogonal-channel and negligible-quantization-noise assumptions.
  • Asymptotic behavior: At very large N, MiLAC EE scales as ln(N)/N^2, whereas DBF scales as ln(N)/N.Normalized EE metrics converge toward their predicted asymptotic factors; these factors are 6K′P_IT ln 2 for MiLAC architectures and 2K′P_RF for DBF.
  • DAC resolution: Beyond four DAC bits, marginal quantization improvements no longer offset ADC/DAC power growth, causing system EE to decline.DAC resolution should balance quantization-noise reduction against additional converter power consumption.
  • Architecture and user effects: The TLM generally achieves the highest EE, while HDM becomes more efficient with very low-resolution RF chains and MiLAC EE depends on antenna and user counts.With fixed RF-chain counts, additional users can improve EE through multiuser diversity, but minimum-rate constraints and limited degrees of freedom can eventually reduce it.
Loading 2608.30639v1…