MiLAC-Aided Beamforming

• MiLAC-Aided Beamforming is a microwave linear analog computing framework that implements large-scale matrix transformations and beamforming in massive MIMO systems.
• It leverages programmable scattering matrices and optimized circuit topologies to achieve low-latency, high-capacity wireless communication with reduced hardware complexity.
• The design utilizes advanced optimization techniques and scalable topologies, such as stem-connected networks, to realize capacity-achieving performance in next-generation networks.

A microwave linear analog computer (MiLAC) is a passive, multiport microwave network architecture that directly implements large-scale linear transformations in the analog domain via programmable admittance or scattering matrices. Leveraging a lossless, reciprocal configuration, MiLAC enables analog-domain computation such as matrix–vector products, beamforming, regularized inversions, and statistical estimators with extremely low computational latency and hardware complexity. In contemporary wireless communications, MiLAC is recognized as a viable hardware alternative to digital and hybrid beamforming techniques, particularly for gigantic (thousand-plus antenna) multiple-input multiple-output (MIMO) systems, where scaling bottlenecks in digital processing and RF-chain proliferation are most severe (Nerini et al., 10 Apr 2025, Nerini et al., 6 Jun 2025, Nerini et al., 18 Jun 2025, Fang et al., 5 Jan 2026, Wu et al., 15 Jan 2026).

1. Physical Model and Core Scattering-Matrix Formalism

A MiLAC is realized as an $N$-port network wherein each port corresponds to an RF-chain or antenna element. The network is constructed as a lattice of tunable passive elements (e.g., varactors, switchable inductors/capacitors, MEMS components), creating a programmable admittance matrix $\mathbf Y\in\mathbb C^{N\times N}$ that is symmetric and purely imaginary for lossless, reciprocal operation. The associated scattering (S-) parameter matrix is

$$\Theta = (\mathbf I_N + Z_0\mathbf Y)^{-1}(\mathbf I_N - Z_0\mathbf Y), \qquad Z_0 = Y_0^{-1}$$

for system reference admittance $Y_0$. The unitary and symmetric property $\Theta^H\Theta=\mathbf I_N$ and $\Theta=\Theta^T$ is pivotal; it enables MiLAC to realize arbitrary $N\times N$ symmetric unitary transformations between supported port pairs.

Transmit-side MiLAC implements an analog beamforming/precoding matrix $\mathbf F\in\mathbb C^{L\times K}$, extracted as a sub-block: $$\mathbf F = \frac{1}{2}\left[\Theta\right]_{K+1:K+L,\;1:K}$$ with $\mathbf c = \mathbf P^{1/2}\mathbf s$ ($\mathbf P$ power allocation, $\mathbf s$ baseband symbols), transmitting as $\mathbf x = \mathbf F\mathbf c$ directly via the microwave network, bypassing per-symbol digital matrix multiplications (Fang et al., 5 Jan 2026, Nerini et al., 6 Jun 2025).

2. MiLAC for MIMO Beamforming and Capacity-Achieving Designs

MiLAC architecture provides a physical implementation for analog-domain beamforming, where the joint transmit and receive scattering matrices $\Theta_F$ and $\Theta_G$ are synthesized to diagonalize the wireless channel $\mathbf H$ into its dominant eigenmodes. Under perfect channel state information (CSI) and water-filling, the MiLAC-aided system capacity for $N_S$ streams and $N_T$ antennas is (Nerini et al., 6 Jun 2025, Nerini et al., 18 Jun 2025): $$C_{\text{MiLAC}} = \sum_{s=1}^{N_S} \log_2\left(1+\frac{P_T p_s^\star \lambda_s}{4\,\sigma^2}\right)$$ where $\lambda_s$ are singular values from the SVD of $\mathbf H$, $p_s^\star$ optimal powers, and the necessary MiLAC scattering matrices are constructed in closed-form to realize the required right and left singular vector subspaces.

MiLAC thus achieves the same information-theoretic capacity as fully digital beamforming in single-user and decorrelated multi-user scenarios, while drastically reducing the number of required RF chains to the number of streams, $N_S$, compared to $N_T$ digitally (Nerini et al., 6 Jun 2025, Nerini et al., 18 Jun 2025).

3. Beamforming Flexibility, Algorithmic Design, and Joint Optimization

MiLAC’s physically implementable beamforming matrices are constrained compared to fully digital architectures. The set of feasible precoders $\mathbf F$ satisfies semi-unitary constraints, enforcing mutual orthogonality under full transmit power. For general multi-user MISO scenarios, especially with correlated channels, these restrictions result in performance loss relative to unconstrained digital beamforming due to limited orthogonality and reduced radiated power for fixed $\{p_k\}$ (Fang et al., 5 Jan 2026, Wu et al., 15 Jan 2026).

Joint optimization involves maximizing sum-rate under the MiLAC constraints: $$\max_{\{\mathbf p\},\,\Theta} \sum_{k=1}^K \log_2\left(1+\mathrm{SINR}_k\right)$$ where

$$\mathrm{SINR}_k = \frac{p_k|\mathbf h_k^H\mathbf f_k|^2}{\sum_{i\neq k}p_i|\mathbf h_k^H\mathbf f_i|^2+\sigma_k^2}$$

subject to $\Theta^H\Theta=\mathbf I_N$, $\Theta=\Theta^T$, $\sum_k p_k\le P_{\max}$. Nonconvexity is addressed via fractional programming and block-coordinate descent—for example, alternating auxiliary variable updates, KKT-based power allocation, and symmetric-polar decomposition for the scattering-matrix update (Fang et al., 5 Jan 2026). The complexity scales as $\mathcal O((K+L)^3)$ per inner iteration.

4. Circuit Complexity and Scalable Topologies

A key MiLAC system-level distinction is the scaling of the required tunable impedance components. In the fully-connected topology, every port pair interacts, leading to circuit complexity $O((N_S+N_T)^2)$. Stem-connected MiLACs, in contrast, organize port connectivity utilizing a reduced “center” set to maintain flexibility, lowering complexity to $O(N_S N_T)$ while preserving capacity-achieving performance (Nerini et al., 18 Jun 2025). This advancement enables practical implementation for gigantic-MIMO arrays (hundreds or thousands of antennas), facilitating analog-domain beamforming with linear hardware overhead, and no loss in achievable rate or flexibility in typical scenarios.

Architecture	Flexibility	Tunable Components
Fully-connected	Any symmetric unitary S-matrix	$O((N_S+N_T)^2)$
Stem-connected	Capacity-achieving for all $H$	$O(N_S N_T)$

5. Complexity Reduction and Channel Estimation

Analog-domain MiLAC computation eliminates all per-symbol digital arithmetic, moving complexity to precalculation of the matrix parameters. For zero-forcing and MMSE algorithms, this amounts to $O(N_TN_R)$ per coherence block, a reduction by $3$–$5$ orders of magnitude over digital approaches for large $N$ (Nerini et al., 10 Apr 2025, Zhang et al., 16 Jan 2026). Channel estimation can be implemented analogically via MiLAC, further reducing online computational burden and enabling channel estimation with only a single RF chain by encoding the training sequences in the MiLAC mapping, yielding online complexity of zero, optimal NMSE, and enabling ultra-low PAPR and resolution ADCs/DACs (Zhang et al., 16 Jan 2026).

Method	NMSE Performance	Online flops	Tx RF chains	ADC/DAC res.	PAPR
Digital LS	Digital-optimal	$8\tau N_R N_T$	$N_T$	High	High
MiLAC LS	Digital-optimal	0	1	Low	Unit
Digital MMSE	Digital-optimal	$8\tau N_R^2$	$N_T$	High	High
MiLAC MMSE	Digital-optimal	0	1	Low	Unit

6. Architectures Beyond Standard MiLAC: Metastructures and Chaotic Media

Experimental and theoretical work extends MiLAC concepts to metastructures and random-cavity implementations. Phase-programmable metasurfaces and programmable reflect-arrays can realize arbitrary linear operators, including matrix inversion and iterative algorithms (e.g., Newton’s method, Lagrange multiplier schemes), by controlling the phase and amplitude of wavefronts in chaotic microwave cavities. Component count, programmability, and error rates are subject to hardware constraints and physical modeling; feedback and ensemble averaging improve accuracy, and time-sequential approaches scale for large matrix operations (Tzarouchis et al., 2022, Hougne et al., 2018). These implementations highlight generality and flexibility, translating microwave analog computing principles to alternative electromagnetic platforms with similar computational primitives.

7. Practical Guidelines, Limitations, and Future Directions

MiLACs present unique opportunities for energy-efficient, scalable analog linear algebra in next-generation wireless base stations and other domains. Key practical guidelines include the choice of stem-connected topologies for linear scaling, RF-chain minimization, and use of low-resolution converters. Lossless reciprocal MiLACs are optimal in single-user or decorrelated multi-user contexts; for general multi-user networks, hybrid approaches or controlled loss/reciprocity breaks can mitigate performance gaps (Fang et al., 5 Jan 2026, Wu et al., 15 Jan 2026). Fundamental limitations stem from hardware tuning range, precision, analog calibration stability, and nonlinear effects. Emerging research directions involve integration into data-centre racks, photonic analog computing, and combined analog–digital workflows for tasks where low-latency linear transformations are essential.

In summary, MiLAC provides a rigorous framework for physically embedding large-scale matrix transformations, beamforming, and statistical estimation into microwave networks, offering substantial reductions in computational and hardware costs, especially for gigantic MIMO deployments (Nerini et al., 10 Apr 2025, Nerini et al., 6 Jun 2025, Fang et al., 5 Jan 2026, Wu et al., 15 Jan 2026, Nerini et al., 18 Jun 2025).