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Frequency-Dependent Hybrid Digital and Microwave Linear Analog Computer (MiLAC) Beamforming for Multiuser MIMO-OFDM Systems

Yiyang Peng, Zheyu Wu, Bruno Clerckx

arXiv:2610.34603v1eess.SP

TL;DR

Wideband MiLAC beamforming is challenging because practical circuits have frequency-dependent responses, and one shared circuit generally cannot match desired directions across all subcarriers. The paper models these coupled responses and jointly optimizes digital and MiLAC beamformers; simulations report near-digital sum-rate with fewer RF chains and gains over conventional hybrid beamforming in highly frequency-selective channels.

  • Problem

    Practical MiLAC responses vary with frequency, while one shared circuit cannot independently realize desired beamforming directions across subcarriers in wideband systems.

    Source: §I · p. 2, 2) Wideband MiLAC-aided beamforming… · p. 2

  • Method

    The paper models frequency-coupled MiLAC responses and uses a BCD-ADMM approach to jointly optimize subcarrier-dependent digital precoders and the MiLAC circuit.

    Source: E. ADMM-Based MiLAC Beamformer… · p. 8, §VI · p. 13

  • Results

    With 64 transmit antennas and 4 streams, the design achieves 96.3% of the fully-digital sum-rate using 12.5% of the RF chains; with 4 RF chains in highly frequency-selective channels, it gives a 48% sum-rate gain over conventional hybrid beamforming.

    Source: 4) Performance evaluation and… · p. 2

  • Takeaways & Limitations

    The reported advantage over conventional hybrid digital-analog beamforming is more pronounced with fewer RF chains and greater channel frequency selectivity.

    Source: 4) Performance evaluation and… · p. 2

    A single MiLAC circuit generally cannot simultaneously realize the desired beamforming directions across the entire bandwidth.

    Source: 2) Wideband MiLAC-aided beamforming… · p. 2

Abstract

from arXiv · show

Microwave linear analog computing (MiLAC) has recently emerged as a promising approach for reducing the hardware and digital-processing requirements of large-scale multiple-input multiple-output (MIMO) systems. In narrowband multiuser systems, hybrid digital-MiLAC beamforming can achieve digital beamforming flexibility with only as many radio-frequency (RF) chains as data streams. However, extending this advantage to wideband systems is nontrivial because practical MiLAC circuits exhibit frequency-dependent responses. In this paper, we investigate hybrid digital-MiLAC beamforming for multiuser MISO-OFDM systems with a frequency-dependent MiLAC circuit model. We characterize the frequency dependence of MiLAC by relating the susceptance of each tunable component across subcarriers to its center-frequency value. Under this frequency coupling, we show that a single MiLAC circuit generally cannot simultaneously match the desired beamforming directions across all subcarriers, highlighting the essential role of subcarrier-dependent digital precoding. We then develop a block coordinate descent (BCD) algorithm to jointly optimize the digital and MiLAC beamformers for average sum-rate maximization. In particular, an efficient alternating direction method of multipliers (ADMM) algorithm is custom-designed to handle the frequency-coupled circuit constraints in the MiLAC beamformer subproblem. Simulation results show that the proposed hybrid digital-MiLAC design outperforms conventional hybrid digital-analog beamforming, particularly when fewer RF chains are available and the channel is more frequency-selective. With $64$ transmit antennas and $4$ data streams, it attains $96.3\%$ of the digital sum-rate using only $8$ RF chains. Furthermore, under highly frequency-selective channels and using $4$ RF chains, it provides a $48\%$ sum-rate gain over the conventional hybrid design.

I. INTRODUCTION

The paper models wideband hybrid digital-MiLAC beamforming with frequency-coupled circuit responses, then examines its flexibility and jointly optimizes the digital and MiLAC beamformers. Its simulations report near-digital sum-rate with fewer RF chains and gains over conventional hybrid beamforming under highly frequency-selective channels.

I. INTRODUCTION

  • The introduction frames MiLAC as an analog-RF linear-transform network that can reduce RF-chain and digital-processing demands, while identifying practical frequency dependence as an unresolved wideband beamforming challenge.Conventional phase-shifter hybrids share an analog beamformer across frequencies, limiting adaptation to frequency-selective channels.Source: §I · pp. 1–2

1) Frequency-dependent wideband hybrid digital-MiLAC beamforming modeling:

2) Wideband MiLAC-aided beamforming flexibility and limitations:

  • A single MiLAC circuit generally cannot independently match desired beamforming directions across all subcarriers, motivating subcarrier-dependent digital precoders to provide additional flexibility.The flexibility analysis examines gain, magnitude efficiency, and directional alignment in a single-user example with two transmit antennas.Source: 2) Wideband MiLAC-aided beamforming… · p. 2, §I · p. 2

3) Joint wideband hybrid digital-MiLAC beamforming design:

  • The joint design maximizes average sum-rate under lossless, reciprocal, frequency-dependent circuit constraints and a radiated-power budget using a WMMSE reformulation and BCD.An ADMM framework handles the frequency-coupled MiLAC beamformer subproblem; structured updates reduce multiplier search to scalar evaluations and cast the susceptance update as box-constrained least squares solved by PGD.Source: 3) Joint wideband hybrid digital-MiLAC… · p. 2

4) Performance evaluation and practical considerations:

  • With 64 transmit antennas and 4 streams, the proposed design achieves 96.3% of the fully-digital sum-rate using 12.5% of its RF chains; under highly frequency-selective channels, it gives a 48% sum-rate gain over conventional hybrid beamforming with 4 RF chains.The paper also reports that OFDM cyclic-prefix overhead can be as low as 9.22%.Source: 4) Performance evaluation and… · p. 2

A. Hybrid Digital-MiLAC OFDM System Model

  • The model considers a wideband MU-MISO system with OFDM, where subcarrier-specific digital precoding precedes the frequency-dependent MiLAC and the resulting signals pass through the wireless channel.The CP is assumed to cover the finite memory of the cascaded MiLAC-channel response.Source: A. Hybrid Digital-MiLAC OFDM System… · p. 3

B. Frequency-Dependent MiLAC Modeling

The paper models MiLAC as a fully connected multiport network whose tunable components produce frequency-dependent admittance and scattering responses. Lossless, reciprocal operation imposes purely imaginary admittances and a symmetric susceptance matrix.

B. Frequency-Dependent MiLAC Modeling

  • The fully connected MiLAC has RF-chain ports and transmit-antenna ports, with tunable components connecting each port to ground and every other port.The network is treated as an (NRF + NT)-port circuit.Source: B. Frequency-Dependent MiLAC Modeling · pp. 3–4
  • At each subcarrier, the network admittance and scattering matrices determine the MiLAC input-to-antenna response, with Z0 = 50 Ω as the reference impedance.The paper extracts the input-to-antenna block of the scattering matrix to obtain the beamforming response.Source: B. Frequency-Dependent MiLAC Modeling · p. 4

1) Lossless and Reciprocal MiLAC:

  • Reciprocity makes the admittance matrix symmetric, while losslessness makes its components purely imaginary, so it can be written as j times a real susceptance matrix.The component connecting ports m and m′ has the same admittance in either direction.Source: 1) Lossless and Reciprocal MiLAC · p. 4

2) Frequency-Dependent Modeling of Tunable Admittance:

The tunable-admittance model uses a shared center-frequency susceptance to determine each component’s response across subcarriers. This frequency coupling means a fixed circuit controls the MiLAC responses jointly rather than independently at each frequency.

2) Frequency-Dependent Modeling of Tunable Admittance:

  • A lumped circuit model with two fixed inductors and one tunable capacitor specifies each component’s frequency-dependent admittance.Once the capacitance is selected, the model determines the component response at every subcarrier.Source: 2) Frequency-Dependent Modeling of… · p. 4, Figure 2 · p. 4
  • The center-frequency susceptance is the tunable variable, with its feasible range determined by the component’s minimum and maximum capacitance.The illustrated operating setup uses an SMV1231-079LF varactor model at 2.4 GHz over 2.35–2.45 GHz.Source: 2) Frequency-Dependent Modeling of… · p. 4

A. Frequency Coupling of the Wideband MiLAC Beamforming

B. Visualization of Frequency-Dependent MiLAC Beamforming Flexibility

A sampled-circuit example evaluates wideband MiLAC beamforming using gain ratio, magnitude efficiency, and directional alignment. The results indicate that one circuit cannot independently adapt its direction at every subcarrier, motivating hybrid digital-MiLAC beamforming.

B. Visualization of Frequency-Dependent MiLAC Beamforming Flexibility

  • The example samples six center-frequency susceptances at five values each, producing 15,625 fixed circuit configurations for a single-user, two-antenna system with 64 subcarriers.The circuit has one input port, two output ports, and six tunable components.Source: B. Visualization of… · p. 5
  • The MiLAC beamforming gain is normalized by the digital matched-beamforming gain to form the gain ratio used for comparison.The sampled circuits can have high gain ratios at some subcarriers but substantial reductions at others.Source: B. Visualization of… · p. 5
  • For the average-subcarrier circuit, magnitude efficiency stays near one while gain ratio largely follows directional alignment, making directional mismatch the main source of gain reduction.The worst-subcarrier circuit improves minimum gain but still cannot sustain a high gain ratio across the band.Source: B. Visualization of… · p. 5

IV. JOINT WIDEBAND HYBRID DIGITAL-MILAC BEAMFORMING OPTIMIZATION

The section explains why wideband hybrid digital-MiLAC has fewer design variables than fully digital beamforming, then formulates average sum-rate maximization under frequency-coupled circuit constraints. It uses a WMMSE reformulation to support alternating updates.

A. Design Degrees of Freedom of Wideband Hybrid Digital-MiLAC Beamforming

  • For NT = 64, K = 4, and N = 64, hybrid digital-MiLAC with NRF = K = 4 uses 4394 real variables, versus 32768 for digital beamforming.The MiLAC controls are shared across subcarriers, while reducing RF chains removes digital design variables at every subcarrier.Source: A. Design Degrees of Freedom of… · p. 6
  • Wideband hybrid digital-MiLAC combines subcarrier-specific digital precoders with shared MiLAC circuit controls, unlike unrestricted digital precoders at each subcarrier.One MiLAC configuration determines responses across all subcarriers, making the wideband characterization nontrivial.Source: A. Design Degrees of Freedom of… · p. 6

B. Problem Formulation and WMMSE Reformulation

  • The objective is average sum-rate maximization subject to lossless reciprocal MiLAC constraints and a total radiated-power budget.The problem couples digital precoders and auxiliary variables, includes a nonlinear matrix inverse, and links subcarriers through common center-frequency susceptances.Source: B. Problem Formulation and WMMSE… · pp. 6–7
  • The WMMSE reformulation supports block-coordinate updates of equalizers, weights, digital precoders, and MiLAC beamformers, with the MiLAC update handled by ADMM.The objective is convex in each variable block when the other blocks are fixed.Source: B. Problem Formulation and WMMSE… · p. 7

C. Auxiliary Variables Update

The auxiliary-variable block updates equalizers and weights from the WMMSE objective. The digital-precoder block solves a convex QCQP, using KKT conditions and a power-budget multiplier search.

C. Auxiliary Variables Update

  • For fixed other variable blocks, the equalizers and weights are updated by setting their objective partial derivatives to zero.The resulting updates are for the per-user, per-subcarrier equalizers and weights.Source: C. Auxiliary Variables Update · p. 7

D. Digital Precoder Optimization

  • The digital precoders are updated by solving a convex QCQP using the KKT conditions.The multiplier is set to zero when the unconstrained solution meets the power budget; otherwise, bisection finds the multiplier satisfying the budget.Source: D. Digital Precoder Optimization · pp. 7–8
  • When the MiLAC precoder lacks full column rank, optimization can be restricted without loss of optimality to its compact-SVD subspace.The reduced precoder is recovered after solving in that lower-dimensional subspace.Source: D. Digital Precoder Optimization · p. 8

E. ADMM-Based MiLAC Beamformer Optimization

The MiLAC beamformer update uses ADMM to handle coupled circuit variables and constraints. Its precoder subproblem is solved through spectral preprocessing and a scalar power search, while shared susceptances are optimized by projected gradient descent.

E. ADMM-Based MiLAC Beamformer Optimization

  • ADMM alternates updates of the transformed MiLAC precoders, susceptance matrices, and dual variables to handle their bilinear constraint.The precoder subproblem is a convex QCQP with a radiated-power constraint.Source: E. ADMM-Based MiLAC Beamformer… · p. 8
  • The proposed reduced-complexity update replaces repeated NtotNRF-dimensional matrix factorizations in the κ search with scalar power evaluations.After preprocessing, bisection determines κ using scalar power evaluations.Source: E. ADMM-Based MiLAC Beamformer… · pp. 8–9
  • The shared center-frequency susceptances are updated through a linearly constrained least-squares problem solved by projected gradient descent.The formulation incorporates symmetry and component-mapping constraints, with projection onto the feasible susceptance bounds.Source: E. ADMM-Based MiLAC Beamformer… · p. 9
  • The projected-gradient step size ζ = 1/LG ensures convergence to a global minimizer of the susceptance least-squares problem.The gradient Lipschitz constant is LG = 2∥Ã∥2^2.Source: E. ADMM-Based MiLAC Beamformer… · p. 9

V. SIMULATIONS

The section evaluates frequency-dependent hybrid digital-MiLAC beamforming through numerical simulations and compares it with benchmark schemes.

V. SIMULATIONS

  • Numerical simulations evaluate the frequency-dependent hybrid digital-MiLAC design against several benchmark schemes.Source: §V · p. 10

A. Simulation Settings

The simulations use a 64-subcarrier, 64-antenna, 4-user MU-MISO-OFDM system with 16 independent channel taps. Channel frequency selectivity is characterized by normalized RMS delay spread.

A. Simulation Settings

  • The baseline setup uses N = 64 subcarriers, NT = 64 transmit antennas, K = 4 users, and D = 16 independent channel taps.The tap powers follow a normalized exponentially decaying power-delay profile.Source: A. Simulation Settings · p. 10
  • Channel frequency selectivity is measured by the normalized RMS delay spread τrms/Ts, with Ts = 1/B and bandwidth B = 100 MHz.Source: A. Simulation Settings · p. 10
  • Noise power is normalized to σ^2 = 1 for every user and subcarrier.Source: A. Simulation Settings · p. 10

B. Benchmark Schemes

The benchmarks compare digital beamforming, frequency-independent MiLAC designs, and the proposed frequency-dependent hybrid design; digital precoders are optimized across subcarriers using WMMSE.

B. Benchmark Schemes

1) Digital Beamforming:

B. Benchmark Schemes

2) FI-HDM:

The comparisons show that directly optimizing frequency-dependent MiLAC responses outperforms frequency-independent MiLAC designs and conventional phase-shifter hybrid beamforming in the studied wideband settings. The gains are strongest with limited RF chains and greater channel frequency selectivity, while effective-rank results connect cross-subcarrier spatial variation to these trends.

2) FI-HDM:

  • FI-HDM fixes its MiLAC beamformer across subcarriers, whereas Proj-HDM first optimizes a frequency-independent MiLAC beamformer and then projects it onto physical MiLAC constraints.Source: 2) FI-HDM · p. 10, 3) Proj-HDM · p. 10

4) FD-MiLAC:

  • FD-MiLAC jointly optimizes its MiLAC beamformer and digital power allocation using NRF = K MiLAC input ports.Source: 4) FD-MiLAC · pp. 10–11

C. Comparison with Frequency-Independent MiLAC Designs

  • FD-HDM consistently achieves the highest sum-rate among the compared MiLAC designs, while projecting an FI design onto physical frequency-dependent constraints does not recover direct FD optimization's performance.Source: C. Comparison with… · p. 11, Figure 4 · p. 11

D. Comparison of Different Beamforming Architectures

E. Effective MiLAC Temporal Memory and CP Overhead

The proposed design approaches fully digital sum-rate with fewer RF chains, while its effective MiLAC response can be represented with a finite tap count. The reported tap and CP results quantify the response’s temporal memory and associated OFDM overhead.

E. Effective MiLAC Temporal Memory and CP Overhead

VI. CONCLUSIONS

  • In simulations, the proposed FD-HDM achieves 96.3% of fully-digital sum-rate using 12.5% of RF chains; with 6.25% of fully-digital RF chains, it gains 48% over FI-HDA under highly frequency-selective channels, with CP overhead as low as 9.22%.Source: §VI · p. 13

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