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Microwave Linear Analog Computer (MiLAC)-aided Multiuser MISO: Fundamental Limits and Beamforming Design

Zheyu Wu, Matteo Nerini, Bruno Clerckx

arXiv:2601.10060v1eess.SP

TL;DR

Gigantic MIMO makes fully digital beamforming difficult to scale, motivating MiLAC-based analog processing. The paper characterizes lossless reciprocal MiLAC flexibility, develops LMI- and WMMSE-based design methods, and finds near-digital performance with lower hardware and computational complexity.

  • Problem

    Gigantic MIMO makes fully digital beamforming costly and energy-intensive, while the flexibility limits and optimization of physically constrained MiLAC beamforming in multiuser systems remain insufficiently understood.

  • Method

    The paper characterizes achievable MiLAC beamforming matrices, reformulates constraints as convex LMIs, exploits a low-dimensional subspace property, and develops WMMSE-based algorithms.

  • Results

    MiLAC-aided beamforming achieves performance close to fully digital beamforming and comparable or superior performance to conventional hybrid beamforming in gigantic MIMO systems.

  • Takeaways & Limitations

    Hybrid digital–MiLAC achieves full digital beamforming flexibility with K RF chains instead of 2K, while MiLAC also avoids symbol-level digital processing and supports low-resolution DACs.

Abstract

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As wireless communication systems evolve toward the 6G era, ultra-massive/gigantic MIMO is envisioned as a key enabling technology. Recently, microwave linear analog computer (MiLAC) has emerged as a promising approach to realize beamforming entirely in the analog domain, thereby alleviating the scalability challenges associated with gigantic MIMO. In this paper, we investigate the fundamental beamforming flexibility and design of lossless and reciprocal MiLAC-aided beamforming for MU-MISO systems. We first provide a rigorous characterization of the set of beamforming matrices achievable by MiLAC. Based on this characterization, we prove that MiLAC-aided beamforming does not generally achieve the full flexibility of digital beamforming, while offering greater flexibility than conventional phase-shifter-based analog beamforming. Furthermore, we propose a hybrid digital-MiLAC architecture and show that it achieves digital beamforming flexibility when the number of radio frequency (RF) chains equals the number of data streams, halving that required by conventional hybrid beamforming. We then formulate the MiLAC-aided sum-rate maximization problem for MU-MISO systems. To solve the problem efficiently, we reformulate the MiLAC-related constraints as a convex linear matrix inequality and establish a low-dimensional subspace property that significantly reduces the problem dimension. Leveraging these results, we propose WMMSE-based algorithms for solving the resulting problem. Simulation results demonstrate that MiLAC-aided beamforming achieves performance close to that of digital beamforming in gigantic MIMO systems. Compared with hybrid beamforming, it achieves comparable or superior performance with lower hardware and computational complexity by avoiding symbol-level digital processing and enabling low-resolution digital-to-analog converters (DACs).

I. INTRODUCTION

Gigantic MIMO makes fully digital beamforming costly, motivating MiLAC as an analog alternative. The paper characterizes MiLAC flexibility, compares architectures, develops algorithms, and reports performance close to digital beamforming with lower complexity.

  • Gigantic MIMO can require hundreds or thousands of antennas, making dedicated RF chains with high-resolution DACs and power amplifiers prohibitively costly and energy-intensive.
  • MiLAC implements broad linear transformations and beamforming directly in the analog domain through a reconfigurable microwave network connected between RF chains and transmit antennas.
  • MiLAC-aided beamforming uses one RF chain per transmitted data symbol, supports low-resolution DACs, and avoids per-symbol digital beamforming operations.
  • The paper formulates MU-MISO sum-rate maximization, uses convex LMIs and low-dimensional structure, and proposes WMMSE-based algorithms; simulations show near-digital performance and comparable or superior performance to hybrid beamforming.
  • MiLAC-aided beamforming generally lacks digital beamforming’s full flexibility but exceeds conventional phase-shifter analog beamforming, while becoming comparable in gigantic MIMO as user channels approach orthogonality.
  • Hybrid digital–MiLAC beamforming achieves digital beamforming flexibility with K RF chains, compared with 2K required by conventional hybrid beamforming.

B. MiLAC Modeling

The paper models MiLAC through a reconfigurable admittance matrix and equivalent scattering matrix under lossless and reciprocal constraints. It defines the achievable beamforming set and uses this model to study flexibility and capacity implications.

  • The fully connected MiLAC architecture connects every port to ground through a tunable admittance and interconnects every pair of ports through another tunable admittance.
  • The MiLAC input–output relationship is determined by a reconfigurable admittance matrix Y controlled by tuning the network admittances.
  • The microwave network can equivalently be represented by a scattering matrix Θ related to Y, providing a clearer formulation of the MiLAC beamforming matrix.
  • Without physical constraints, arbitrary complex admittances would allow the MiLAC to implement any beamforming matrix, but this idealization is not practically realistic.
  • For a lossless and reciprocal MiLAC, admittances are purely imaginary and reciprocal, making Y purely imaginary and symmetric and Θ unitary and symmetric.
  • The paper defines the MiLAC-aided beamforming matrix set under maximum transmit power PT, notes that it contains matrices with unitary columns, and concludes that MiLAC is capacity-achieving for point-to-point MIMO.

III. CHARACTERIZATION OF MILAC-AIDED BEAMFORMING AND COMPARISON WITH CLASSICAL

The section derives an explicit characterization of lossless, reciprocal MiLAC beamforming matrices and uses it to compare MiLAC with classical schemes.

  • A. Beamforming Matrices of MiLAC-Aided Beamforming: MiLAC beamforming is characterized through the submatrix linking input ports to transmit antennas, translating unitary and symmetric network constraints into constraints on F.For any F satisfying ∥F∥2 ≤ 1, compatible symmetric and unitary network blocks can be constructed.
  • A. Beamforming Matrices of MiLAC-Aided Beamforming: Corollary 1 gives a compact equivalent description of the achievable lossless and reciprocal MiLAC beamforming matrix set.This form generalizes an earlier point-to-point result and makes the set easier to compare with other schemes.
  • A. Beamforming Matrices of MiLAC-Aided Beamforming: The resulting characterization represents each achievable beamforming matrix using F and a diagonal power-allocation factor.The factorization is stated with F satisfying ∥F∥2 = 1 and total power satisfying 1^T p = P_T in the full-power case.

B. Comparison with Digital Beamforming

MiLAC-achievable beamforming is a strict subset of digital beamforming under the physical constraints considered, although the gap narrows for nearly orthogonal user channels.

  • Digital Beamforming: Digital beamforming uses W ∈ C^N×K to map the data symbols into the N RF-chain signals.The source signal is c = Ws under a transmit-power constraint.
  • Digital Beamforming: Every MiLAC-achievable beamforming matrix is digitally achievable, but digital beamforming can realize additional matrices.The relation is W_MiLAC ⊆ W_digital with strict containment.
  • Digital Beamforming: Passive-network constraints, rather than losslessness and reciprocity specifically, account for the reduced MiLAC flexibility.Relaxing the latter constraints to passivity leaves the achievable matrix set unchanged.
  • Digital Beamforming: MiLAC imposes a spectral-norm constraint on F that is generally more restrictive than separately constraining each column norm.The difference accounts for correlations among columns and excludes some highly correlated digital beamforming matrices.
  • Digital Beamforming: MiLAC can approach digital-beamforming performance when user channels are nearly orthogonal, a condition associated with many-antenna systems.In that regime, optimal digital beamforming columns tend to be nearly orthogonal and are better represented within the MiLAC-achievable set.

C. Comparison with Phase-Shifter-Based Analog Beamforming

MiLAC provides more flexible analog beamforming than phase shifters, and a hybrid digital–MiLAC architecture can achieve digital flexibility with fewer RF chains.

  • Phase-Shifter-Based Analog Beamforming: Phase-shifter analog beamforming uses a constant-modulus, power-normalized matrix.The normalization prevents the phase-shifter network from amplifying its input.
  • Phase-Shifter-Based Analog Beamforming: The phase-shifter-achievable matrix class is strictly contained within the MiLAC-achievable analog class.Thus MiLAC offers greater analog beamforming flexibility than conventional phase shifters.
  • Digital-Achieving Hybrid Digital–MiLAC Architecture: A hybrid digital–MiLAC architecture replaces phase shifters with MiLAC and provides greater flexibility than traditional hybrid beamforming with the same RF-chain count.This architecture achieves any digitally achievable matrix with K RF chains.
  • Digital-Achieving Hybrid Digital–MiLAC Architecture: The proposed hybrid digital–MiLAC architecture reduces the RF-chain requirement for fully digital flexibility from 2K to K.The reduction follows from MiLAC’s greater flexibility relative to phase shifters.

IV. MILAC-AIDED BEAMFORMING DESIGN

The design section reformulates MiLAC beamforming optimization as a convexly constrained WMMSE problem and reduces its dimension through a channel-subspace property.

  • MiLAC-Aided Beamforming Design: The section develops algorithms because MiLAC cannot generally attain the full flexibility of digital beamforming.The design objective is effective and efficient MiLAC-aided beamforming.
  • A. A Convex Reformulation of W_MiLAC: The achievable MiLAC set is equivalently expressed as a convex set.This reformulation provides the basis for optimization.
  • A. A Convex Reformulation of W_MiLAC: WMMSE and block coordinate descent transform the sum-rate objective into a block-convex optimization with MiLAC-specific constraints.The new constraints replace the usual digital beamforming power constraint and include an LMI representation.
  • A. A Convex Reformulation of W_MiLAC: The resulting semidefinite program can be solved to global optimality with CVX, but its (N + K)-dimensional formulation is expensive for gigantic MIMO.This computational burden motivates the subsequent dimensionality reduction.

B. A Low-Dimensional Reformulation of (21)

The problem admits a lossless low-dimensional reformulation because optimal beamforming can be restricted to the channel-row subspace. This reduces the optimization variable from N×K to K×K and yields a lower-dimensional WMMSE semidefinite program.

  • Subspace property: The channel-induced subspace Ran(H^H) suffices for optimization because projecting any feasible W onto it preserves feasibility and objective value.The projection is Π_HH W = H^H(HH^H)^−1HW.
  • Subspace property: An optimal solution exists with W∗∈Ran(H^H), allowing the substitution W = H^H X.This restriction is without loss of optimality.
  • Dimension reduction: The reformulation replaces W∈C^N×K with X∈C^K×K, substantially reducing variable dimension when N≫K.The reduction is especially relevant to gigantic MIMO systems.
  • WMMSE reformulation: The WMMSE transformation converts the reduced problem into a weighted MSE minimization framework solved by alternating updates of auxiliary variables and (X,p).The (X,p) block is updated through a semidefinite program.
  • Computational reduction: The reduced LMI has dimension 2K, compared with N+K for the original formulation, lowering the semidefinite-programming burden.The reduced dimension improves WMMSE efficiency.
  • Computational reduction: The reduced WMMSE algorithm has complexity O(NK^2 + K^6 Σ_i I_inner^(i)), versus O((N+K)^6 Σ_i I_inner^(i)) for the original formulation.Both expressions include the total outer and inner iteration counts.

D. A Low-Complexity Algorithm for Solving (25)

The low-complexity method separates the complex LMI into simpler constraints on X and p, then updates these variables independently within a block-coordinate framework.

  • LMI decomposition: The complex LMI constraint is reformulated into separate simpler constraints on X and p.
  • Alternating updates: The algorithm updates X and p independently within the block-coordinate descent framework.
  • Computational strategy: This decomposition is designed to avoid solving the original coupled constraint directly.

2 Ydiag(p)

The low-complexity algorithm alternates updates of the auxiliary variables, power allocation, and beamforming matrix, solving the latter with projected gradient descent over a spectral-norm ball.

  • Block updates: With other blocks fixed, the algorithm updates u and ω using the standard WMMSE rules.
  • Power update: The p-subproblem is handled through its Karush–Kuhn–Tucker conditions under the total-power constraint 1^T p ≤ P_T.
  • Beamforming update: The Y-subproblem has a quadratic objective with spectral-norm constraint ∥Y∥_2≤1 and is solved using projected gradient descent.The feasible set is the spectral-norm ball.
  • Beamforming update: Projection onto the feasible set clips each singular value to at most 1.
  • Beamforming update: The projected-gradient stepsize is set to (∥Q∥_2 max_k{p_k})^−1 using a Lipschitz constant of the gradient.
  • Complexity and convergence: The resulting WMMSE-LC algorithm reduces per-iteration complexity from O(K^6) to O(K^3) without degrading performance relative to Algorithm 1.It achieves nearly the same performance while requiring significantly less CPU time.

V. SIMULATION RESULTS

The simulations compare the proposed WMMSE variants using sum-rate and CPU-time experiments, then evaluate MiLAC-aided beamforming against digital and hybrid beamforming. The proposed algorithms achieve approximately the same sum-rate, while the low-complexity implementation is more computationally efficient.

  • Simulation setup: The simulations evaluate the proposed algorithms and compare MiLAC-aided, digital, and hybrid beamforming through sum-rate experiments.
  • Algorithm comparison: The WMMSE full-dimensional, WMMSE reduced-dimensional, and WMMSE-LC algorithms are included in the algorithm comparison.
  • Algorithm comparison: Figures 4 and 5 report sum-rate and CPU time for Rayleigh-fading channels, respectively, with Fig. 4 using N=64 and K=4 and Fig. 5 using SNR=15 dB.
  • Performance: The three proposed algorithms achieve approximately the same performance, validating the low-dimensional subspace property and low-complexity method.
  • Computational efficiency: The reduced-dimensional WMMSE problem has dimension independent of N, and convergence can become faster as N grows when K is large.
  • Computational efficiency: The low-complexity implementation is significantly more efficient because it avoids solving any semidefinite program.

B. Comparison of Different Beamforming Schemes

The comparison evaluates digital, MiLAC-aided, and hybrid beamforming in MU-MISO systems across channel models, SNR, and antenna counts. MiLAC closely approaches digital performance and matches or exceeds hybrid beamforming under the reported settings, while offering broader architectural flexibility.

  • Simulation setup: The simulations compare digital, MiLAC-aided, and hybrid beamforming under i.i.d. Rayleigh and clustered geometric channels.The clustered geometric model uses L = 5 paths and randomly generated departure angles.
  • Simulation setup: MiLAC-aided beamforming uses K RF chains, while the hybrid baselines use NRF = K and NRF = K + 1 for fair comparison.The reported Fig. 6 setting has N = 64 transmit antennas and K = 4 users.
  • SNR comparison: All three schemes achieve comparable sum-rate performance versus SNR, with MiLAC only marginally below digital beamforming.The gap is especially small at low SNR because a single-user beamforming solution is achievable by MiLAC.
  • SNR comparison: MiLAC achieves nearly the same high-SNR performance as both hybrid baselines and outperforms them in the low-SNR regime.The reported explanation is that hybrid beamforming cannot generate arbitrary beamforming vectors with its limited flexibility.
  • Conclusions: Overall, MiLAC does not exactly match digital flexibility but delivers comparable gigantic-MIMO performance and comparable or superior performance to hybrid beamforming with the same RF-chain count.It also avoids symbol-level digital processing and supports low-resolution DACs.
  • Conclusions: A hybrid digital–MiLAC architecture offers greater flexibility than conventional hybrid beamforming and reaches full digital flexibility with K RF chains.The paper states that this halves the RF-chain requirement of conventional hybrid beamforming.

APPENDIX A PROOF OF PROPOSITION 4

The appendix proves equivalence between KKT conditions for two reformulated optimization problems and establishes convergence properties for the proposed block-coordinate algorithms. The proof uses gradient relations, null-space structure, compact feasibility, and uniqueness of block updates.

  • KKT equivalence: The proof begins by writing the KKT conditions for problems (21) and (25), including multiplier and power-budget conditions.Λ and µ are associated with WHW ⪯ diag(p) and 1^T p ≤ P_T, respectively.
  • Gradient relation: The gradient relation ∇_W R(W) = H^H ∇_Z R(Z) connects the rate objective derivatives used in the two formulations.The mapping G(·) expresses the gradient of R with respect to Z.
  • KKT equivalence: A KKT point of problem (21) is mapped to a KKT point of problem (25) using X = H̄^-1HW.The proof verifies the stationarity and complementary conditions through the relation Π_HH W = H^H X.
  • KKT equivalence: The reverse mapping sends a KKT point of problem (25) to (H^H X, p, Λ, µ), completing the equivalence of the relevant KKT conditions.Multiplication by H^H H̄^-1 yields the stationarity condition for problem (21).
  • Algorithm convergence: Both algorithms generate bounded iterates because their feasible regions are compact and their update rules keep the auxiliary variables in compact sets.This boundedness is one of the conditions used in the convergence argument.
  • Algorithm convergence: Any limit point generated by either algorithm is a stationary point because the objective is differentiable, most block subproblems have unique solutions, and the iterates remain bounded.Algorithm 2 additionally has a unique p-subproblem solution determined by λ ≥ 0.
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