MiLAC-Aided Beamforming for MIMO Over-the-Air Computation

Abstract: Over-the-air computation (AirComp) enables low-latency wireless data aggregation, but its accuracy is limited by imperfect signal alignment over fading channels and receiver noise. Fully digital beamforming improves aggregation accuracy in multiple-input multiple-output (MIMO) AirComp systems but requires one radio-frequency (RF) chain per antenna. To reduce this hardware burden, we investigate microwave linear analog computer (MiLAC)-aided beamforming for MIMO AirComp. Under a lossless and reciprocal MiLAC model, we jointly optimize the transmit digital precoding matrices and the receive-side MiLAC aggregation matrix to minimize the mean squared error (MSE). An alternating optimization algorithm is developed, in which the precoding matrices are optimally updated using the Karush--Kuhn--Tucker conditions and bisection, while the resulting convex aggregation matrix subproblem is solved globally using projected gradient descent. Numerical results verify the algorithm's convergence and demonstrate that MiLAC-aided beamforming approaches the MSE performance of fully digital beamforming with substantially fewer RF chains and outperforms phase-shifter-based hybrid beamforming under the same RF-chain budget.

• The paper introduces MiLAC-aided beamforming that models physically realizable aggregation matrices with the simple spectral-norm constraint ||F||₂ ≤ 1, enabling computation-focused signal alignment under lossless and reciprocal circuit assumptions.
• The paper develops an alternating optimization method that solves edge-device precoding optimally with bisection and the AP aggregation update globally over a convex spectral-norm set using projected gradient descent, while convergence of the overall nonconvex design is only guaranteed to a stationary point.
• The paper shows that with L RF chains, MiLAC closely approaches fully digital beamforming using M RF chains, outperforms phase-shifter hybrid beamforming at equal RF-chain budgets, and scales effectively to large AP arrays, although fully connected hardware may require costly quadratic interconnections.

Motivation and contribution

Over-the-air computation (AirComp) exploits waveform superposition to aggregate concurrently transmitted data from edge devices (EDs) in a single time–frequency resource, but aggregation accuracy is limited by imperfect signal alignment over fading channels and by receiver noise. Fully digital beamforming at a multi-antenna access point (AP) mitigates both impairments in multiple-input multiple-output (MIMO) AirComp, yet it requires one radio-frequency (RF) chain per antenna. Phase-shifter-based hybrid beamforming reduces the RF-chain count but imposes constant-modulus constraints that limit the analog-domain transformations available for multi-stream alignment. This paper introduces microwave linear analog computer (MiLAC)-aided beamforming for MIMO AirComp, a combination the authors identify as previously unexplored, since existing MiLAC designs target communication objectives such as rate and capacity rather than computation-oriented signal alignment (Nerini et al., 6 Jun 2025, Wu et al., 15 Jan 2026).

The system comprises $K$ EDs, each with $N$ antennas and $N$ RF chains transmitting an $L$-dimensional symbol vector, and an $M$-antenna AP ($M \gg N$) equipped with an $(M+L)$-port, fully connected, lossless and reciprocal MiLAC cascaded with only $L$ RF chains. The target function is the arithmetic mean of the EDs' symbol vectors, and the design goal is to minimize the mean squared error (MSE) by jointly optimizing the ED-side digital precoding matrices $\{\bm W_k\}$ and the AP-side MiLAC aggregation matrix $\bm F_{\rm MiLAC}$.

Feasible-set characterization of the MiLAC aggregation matrix

A key modeling step is the elimination of the physical scattering matrix $N$0 from the optimization. Under the lossless (skew-Hermitian admittance) and reciprocal (symmetric admittance) assumptions, the Cayley transform relating the admittance matrix $N$1 to $N$2 implies that $N$3 is unitary and symmetric. Consequently, the scaled aggregation matrix $N$4 is exactly the input–output block of such a matrix. Invoking a unitary-completion result from (Wu et al., 15 Jan 2026), the paper establishes that a matrix $N$5 is physically realizable by a lossless, reciprocal MiLAC if and only if $N$6. This reduces the circuit-feasibility constraint to a single spectral-norm condition, yielding a compact joint optimization over $N$7 and $N$8 with per-ED power constraints $N$9.

Alternating optimization algorithm

The joint problem is nonconvex because of the coupled terms $N$0, so the authors develop an alternating optimization (AO) scheme in which each block is solved optimally:

• Precoding update. For fixed $N$1, the problem decouples across the $N$2 EDs into convex QCQPs. The KKT stationarity condition yields $N$3 with $N$4. If the power constraint is inactive, $N$5; otherwise the unique $N$6 satisfying $N$7 is found by bisection, which is efficient because $N$8 is strictly decreasing in $N$9. The update is therefore globally optimal for each ED.
• Aggregation-matrix update. For fixed $L$0, the subproblem is strongly convex (since $L$1) over the convex spectral-norm ball. Projected gradient descent (PGD) with step size $L$2, using the Lipschitz constant of the gradient and singular-value clipping as the projection, converges to the unique global optimum.

The objective sequence is monotonically non-increasing and converges to a finite limit; the paper explicitly notes that because the joint problem is nonconvex, this guarantees only convergence to a stationary point, not global optimality. After convergence, the optimized $L$3 is mapped back to physical circuit parameters via SVD-based symmetric unitary completion, the inverse Cayley transform to the admittance matrix, and extraction of the tunable shunt and mutual admittances. This recovery is a one-time post-processing step. The overall complexity is $L$4, where $L$5, $L$6, $L$7 are the numbers of AO, bisection, and PGD iterations. An appendix also derives an equivalent SDP formulation of the $L$8-subproblem, used as a validation benchmark.

Numerical results

Simulations use i.i.d. Rayleigh channels, $L$9 for all EDs, and 1000 channel realizations, with per-stream MSE ($M$0) as the metric. Three findings stand out:

1. PGD accuracy. The AO-PGD and AO-SDP convergence curves are nearly overlapping and attain essentially identical objective values, providing numerical evidence that PGD reaches the global optimum of the convex $M$1-subproblem; AO-SDP is thereafter used only for validation.
2. Near-digital performance with $M$2 RF chains. With $M$3, $M$4, and $M$5, MiLAC-aided beamforming closely approaches the MSE of fully digital beamforming—which requires $M$6 RF chains—while using only $M$7 RF chains at the AP, and it consistently outperforms phase-shifter-based hybrid beamforming at the same RF-chain budget. The gap widens with increasing $M$8, indicating that multi-stream aggregation imposes stricter alignment requirements.
3. Scalability. Across $M$9 from small arrays to $M \gg N$0, the MiLAC design remains close to fully digital performance with its RF-chain count fixed at $M \gg N$1, so the hardware savings grow as the array scales.

Limitations and open questions

The paper concedes two main limitations. First, convergence of the AO objective does not certify global optimality of the nonconvex joint design, and no optimality gap bound is provided. Second, the fully connected MiLAC topology requires a quadratic number of tunable admittances in $M \gg N$2, which may dominate the hardware cost that the RF-chain reduction is meant to save; reduced-connectivity MiLAC architectures and practical component constraints (e.g., finite admittance ranges and losses) are left to future work. The analysis also assumes perfect impedance matching, lossless and reciprocal components, and full row rank of $M \gg N$3, which holds only almost surely under the continuous fading model.

Conclusion

This paper extends MiLAC-aided beamforming from communication-centric objectives to MIMO AirComp by characterizing the physically realizable aggregation matrices through the spectral-norm condition $M \gg N$4 and solving the resulting joint MSE minimization via an AO algorithm whose two blocks are each solved to global optimality. The design achieves MSE performance close to fully digital beamforming with $M \gg N$5 rather than $M \gg N$6 RF chains at the AP and outperforms phase-shifter-based hybrid beamforming under equal RF-chain budgets. The main open issue is whether these gains survive under reduced-connectivity MiLAC topologies and realistic circuit imperfections.