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On the Performance of Lossless Reciprocal MiLAC Architectures in Multi-User Networks
Tianyu Fang, Xiaohua Zhou, Yijie Mao
TL;DR
The paper asks whether lossless and reciprocal MiLAC can match digital beamforming in general multi-user networks. It analyzes a MU-MISO transmitter, derives the performance limitation, and jointly optimizes power allocation and the scattering matrix; simulations show that the gap narrows at large array scales and can reverse against hybrid beamforming.
Problem
Whether lossless and reciprocal MiLAC can achieve digital-beamforming performance in general multi-user networks remains unknown.
Source: §I · p. 1Method
The paper analyzes a fully-connected lossless reciprocal MiLAC MU-MISO transmitter and jointly optimizes RF-chain power allocation with its scattering matrix.
Source: §I · p. 1, 1) Signal Model · p. 2, C. Scattering Matrix Updates · p. 4Results
MiLAC cannot generally match fully digital beamforming, but its performance gap narrows as array size grows and it can outperform hybrid beamforming at large scales.
Source: 3) Relationship between the… · p. 2, §IV · p. 5, §V · p. 5Takeaways & Limitations
MiLAC remains promising for extremely large-scale MIMO, particularly when channels become asymptotically orthogonal.
Source: §IV · p. 5, §V · p. 5
Abstract
from arXiv · showhide
Microwave linear analog computer (MiLAC)-aided beamforming, which processes the transmitted symbols fully in the analog domain, has recently emerged as a promising alternative to fully digital and hybrid beamforming architectures for multiple-input multiple-output (MIMO) systems. While prior studies have shown that lossless and reciprocal MiLAC can achieve the same capacity as digital beamforming in a single-user MIMO network, its performance in multi-user scenarios remains unknown. To answer this question, in this work, we establish a downlink multi-user multiple-input single-output (MU-MISO) network with a MiLAC-aided transmitter, and investigate its sum-rate performance. Based on the microwave network theory, we first prove that lossless and reciprocal MiLAC cannot achieve the same performance as digital beamforming in a general MU-MISO network. Then, we formulate a sum-rate maximization problem and develop an efficient optimization framework to jointly optimize the power allocation and the scattering matrix for MiLAC. Numerical results validate our theoretical analysis and demonstrate that MiLAC is a promising architecture for future extremely large-scale MIMO systems.
I. INTRODUCTION
MiLAC is an analog microwave-network approach for MIMO beamforming, but its ability to match digital beamforming in multi-user systems remains unresolved. This work formulates a downlink MU-MISO model and investigates MiLAC's performance through theoretical analysis and optimization.
Motivation
- MiLAC directly manipulates transmitted signals in the electromagnetic domain through a reconfigurable multiport microwave network.Signals enter input ports, undergo analog microwave interactions, and exit output ports for antenna radiation.Source: §I · p. 1
- Prior work showed that ideal MiLAC can match digital zero-forcing beamforming at 1/15000th of the computational cost.This result motivates MiLAC as a lower-complexity alternative for large-scale MIMO beamforming.Source: §I · p. 1
Open Question
- Lossless and reciprocal MiLAC achieves single-user MIMO capacity, but its performance in general multi-user networks remained unknown.Fully-connected designs provide flexibility but require substantial circuit complexity and control overhead.Source: §I · p. 1
System and Contributions
- The study examines a downlink MU-MISO system with a fully-connected, lossless, and reciprocal MiLAC transmitter serving K single-antenna users.The base station has L transmit antennas and K RF chains.Source: §I · p. 1, A. System Model · p. 1
- The proposed framework jointly optimizes RF-chain power allocation and the MiLAC scattering matrix for sum-rate maximization.The optimization uses iterative updates with closed-form solutions, while the signal model maps RF-chain outputs through the MiLAC beamforming matrix.Source: §I · p. 1, 1) Signal Model · pp. 1–2
2) MiLAC Model:
The MiLAC beamforming matrix is constrained by the scattering matrix of a lossless, reciprocal network. These constraints cause either transmit-power loss or orthogonality restrictions, so digital-beamforming performance is generally unattainable except in two special cases.
MiLAC network model
- Lossless reciprocal operation makes the admittance components purely imaginary and imposes unitary constraints on the scattering matrix.The susceptance matrix contains the real coefficients multiplying the imaginary unit.Source: 2) MiLAC Model · p. 2
Beamforming relationship
- MiLAC and digital beamforming differ through the constrained scattering submatrix Θ21 versus arbitrary normalized digital beamforming directions.The digital beamforming matrix can use any normalized beamforming vectors, unlike the structurally constrained MiLAC submatrix.Source: 3) Relationship between the… · pp. 2–3
MiLAC network model
- A fully connected MiLAC models beamforming through a symmetric scattering matrix partitioned into input and output submatrices.The beamforming matrix is determined by the network’s admittance and scattering representations.Source: 3) Relationship between the… · p. 2
Performance limitation
- Lossless reciprocal MiLAC cannot match fully digital performance except for a single user or mutually orthogonal multi-user channel directions.These are the two degenerate cases stated in Proposition 1.Source: 3) Relationship between the… · p. 2
- When Θ11 ≠ 0, input power is reflected, yielding strict transmit-power reduction and preventing Θ21 from matching the digital beamformer.The reflected power causes the MiLAC to radiate less power than the digital design.Source: 3) Relationship between the… · p. 3
- When Θ11 = 0, MiLAC beamforming columns must be orthogonal, whereas generic digital beamforming columns need not be.This restriction permits exact reproduction only for single-user systems or multi-user systems with mutually orthogonal channel directions.Source: 3) Relationship between the… · p. 3
B. Problem Formulation
The paper formulates MiLAC sum-rate maximization by jointly optimizing user power allocation and the scattering matrix. The resulting problem incorporates physical beamforming constraints and source-power limits but is non-convex and NP-hard.
Problem formulation
- The achievable SINR for each user is expressed using the corresponding column of the MiLAC beamforming matrix.The k-th column fk determines the beamforming associated with user k.Source: B. Problem Formulation · p. 3
- The optimization jointly chooses the power allocation matrix and scattering matrix to maximize the multi-user sum rate.The formulation is based on the user SINRs and the MiLAC beamforming structure.Source: B. Problem Formulation · p. 3
- MiLAC physical constraints govern the beamforming matrix, while separate constraints enforce the source power budget.The formulation explicitly distinguishes network-induced beamforming limitations from power-allocation limits.Source: B. Problem Formulation · p. 3
- The sum-rate problem is non-convex and NP-hard because fractional SINR expressions are coupled with a highly non-convex unitary scattering constraint.The paper therefore develops an efficient optimization framework to address these challenges.Source: B. Problem Formulation · p. 3
III. PROPOSED OPTIMIZATION FRAMEWORK
The framework transforms the original sum-rate problem into a tractable equivalent form using fractional programming, then solves the resulting multi-block problem by block coordinate descent.
- Fractional programming introduces auxiliary variables α and β to reformulate the original optimization problem equivalently.Source: §III · p. 3
- The transformed problem is a typical multi-block optimization problem that can be solved using block coordinate descent.Source: §III · p. 3
- The algorithm updates each block by fixing the remaining variables in turn.Source: §III · p. 3
A. Auxiliary Variable Updates
With the remaining variables fixed, the auxiliary-variable subproblems are unconstrained and convex, enabling direct updates from first-order optimality conditions.
- The subproblems for α and β are unconstrained and convex when the other variables are fixed.Source: A. Auxiliary Variable Updates · p. 3
- The updates for α and β are obtained by applying their first-order optimality conditions.Source: A. Auxiliary Variable Updates · p. 4
B. Power Allocation Matrix Updates
With α, β, and Θ fixed, the power-allocation subproblem is reformulated in terms of nonnegative square-root power variables and solved convexly.
- Fixing α, β, and Θ yields a subproblem with respect to the power-allocation matrix P.Source: B. Power Allocation Matrix Updates · p. 4
- Defining z_k = √p_k reformulates the power update using a vector of square-root powers.Source: B. Power Allocation Matrix Updates · p. 4
- The reformulated problem is convex, with z constrained to be element-wise nonnegative, and its optimum is obtained by a Lagrangian method.Source: B. Power Allocation Matrix Updates · p. 4
- The dual variable associated with the transmit-power constraint is found efficiently by bisection search.Source: B. Power Allocation Matrix Updates · p. 4
C. Scattering Matrix Updates
The scattering-matrix update handles coupling between the scattering matrix and beamforming objective through selection-matrix representations, convexification, projection, and iterative approximation.
C. Scattering Matrix Updates
- With α, β, and P fixed, the method forms a subproblem for updating the scattering matrix Θ.Source: C. Scattering Matrix Updates · p. 4
- The beamforming matrix F is represented as S1ΘS2, exposing its coupling with the optimization objective.Source: C. Scattering Matrix Updates · p. 4
- Adding λTr(ΘΘ^HX2) preserves feasibility and makes the objective convex in Θ when λ = max_k{p_k}.Source: C. Scattering Matrix Updates · p. 4
- The proposed algorithm alternates the block updates, finds the power dual variable by bisection, and stops when the objective converges.Source: C. Scattering Matrix Updates · p. 4
- The feasible scattering-matrix set enforces symmetric-unitary structure through the projection operator ΠM(X).Source: C. Scattering Matrix Updates · p. 4
D. Overall Algorithm Development
The proposed algorithm alternates closed-form auxiliary updates, power allocation, and scattering-matrix refinement until the objective converges. Each iteration preserves feasibility and monotonically improves the objective.
Overall Algorithm Development
- The procedure repeats its outer and inner updates until the objective value converges.Source: D. Overall Algorithm Development · p. 4
- Each iteration produces a feasible sequence with a monotonically improving objective value.Source: D. Overall Algorithm Development · p. 4
E. Convergence and Complexity Analysis
The convergence analysis is omitted, while complexity is dominated by scattering-matrix updates and depends on the outer and inner iteration counts.
Convergence and Complexity Analysis
- The convergence analysis follows prior work and is omitted from the paper’s presentation.Source: E. Convergence and Complexity Analysis · p. 4
- Scattering-matrix updates dominate the computational complexity of Algorithm 1.Source: E. Convergence and Complexity Analysis · p. 4
- The overall complexity depends on I2 outer-loop iterations and I1 inner-loop iterations.Source: E. Convergence and Complexity Analysis · p. 4
IV. SIMULATION RESULTS
Simulations under Rayleigh and orthogonal channel models validate the algorithm and theoretical analysis. MiLAC converges rapidly, matches digital beamforming for orthogonal channels, and can outperform hybrid beamforming at large array sizes.
Simulation Setup
- Simulations use Rayleigh fading and synthetically generated orthogonal channels, averaging results over 100 independent realizations.The algorithm uses I1 = 50 inner iterations and a 10^-4 convergence tolerance.Source: §IV · pp. 4–5
Convergence Results
- The proposed algorithm monotonically increases the objective value and converges rapidly across transmit SNRs.Source: §IV · p. 5
Sum-Rate Results
- MiLAC has lower sum rate than digital and hybrid beamforming in general channels, with the gap widening as network load or transmit SNR increases.The reported loss is attributed to lossless reciprocal network constraints and the absence of digital beamforming capabilities.Source: §IV · p. 5
- MiLAC matches digital beamforming when desired beamforming directions are orthogonal.Source: §IV · p. 5
Array-Scaling Results
- At 128 transmit antennas, MiLAC outperforms hybrid beamforming, while its gap to digital beamforming narrows at large array sizes.The narrowing is associated with channels becoming asymptotically orthogonal as the array grows.Source: §IV · p. 5, §V · p. 5