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Channel Estimation in MIMO Systems Aided by Microwave Linear Analog Computers (MiLACs)
Qiaosen Zhang, Matteo Nerini, Bruno Clerckx
TL;DR
MiLAC-aided MIMO channel estimation remains challenging because conventional LS and MMSE methods depend on intensive digital processing. The paper designs MiLAC training precoders and combiners that perform both estimators fully in the analog domain, achieving digital-equivalent performance with lower implementation cost. Numerical results verify matching estimation performance and reduced computation, including a saving of up to 2.15 × 10^9 real operations in one MMSE configuration.
Problem
CSI acquisition remains open for MiLAC-aided systems because conventional LS and MMSE estimation require digital aggregation and matrix processing that undermine analog-domain benefits.
Method
The paper designs MiLAC-implemented training precoders and combiners that directly obtain LS and MMSE channel estimates in the analog domain.
Results
2.15 × 10^9 real operations are saved for MMSE estimation in a 2048×64 MIMO system, while MiLAC-aided LS and MMSE match their digital counterparts' estimation performance.
Takeaways & Limitations
MiLAC-based estimation reduces digital computation, transmit RF chains, ADC/DAC resolution requirements, and PAPR for future gigantic MIMO systems.
Abstract
from arXiv · showhide
Microwave linear analog computers (MiLACs) have recently emerged as a promising solution for future gigantic multiple-input multiple-output (MIMO) systems, enabling beamforming with greatly reduced hardware and computational cost. However, channel estimation for MiLAC-aided systems remains an open problem. Conventional least squares (LS) and minimum mean square error (MMSE) estimation rely on intensive digital computation, which undermines the benefits offered by MiLACs. In this letter, we propose efficient LS and MMSE channel estimation schemes for MiLAC-aided MIMO systems. By designing training precoders and combiners implemented by MiLACs, both LS and MMSE estimation are performed fully in the analog domain, achieving identical performance to their digital counterparts while significantly reducing computational complexity, transmit RF chains, analog-to-digital/digital-to-analog converters (ADCs/DACs) resolution requirements, and peak-to-average power ratio (PAPR). Numerical results verify the effectiveness and advantages of the proposed schemes.
I. INTRODUCTION
Gigantic MIMO makes conventional digital beamforming costly, while MiLACs shift beamforming into the analog domain. This paper addresses the open problem of acquiring CSI for MiLAC-aided MIMO through analog-domain LS and MMSE estimation.
- Gigantic MIMO increases hardware cost, computational complexity, and power consumption because conventional digital beamforming requires one RF chain per antenna.
- MiLACs are tunable multiport microwave networks that process signals in the analog domain and realize arbitrary beamforming.
- CSI acquisition remains open because conventional LS and MMSE estimation require multi-slot signal aggregation, post-multiplication, vectorization, digital storage, and computation.
- The proposed system uses transmitter- and receiver-side MiLACs to implement training precoders and combiners for point-to-point MIMO channel estimation.
- The setup uses NT transmit antennas, NR receive antennas, LT and LR RF chains, and τ = NT training time slots under LR = NR.
- Equal-power source signals and unconstrained MiLAC admittance matrices support arbitrary analog training vectors while maintaining unit PAPR per transmit RF chain.
III. MILAC-AIDED LS CHANNEL ESTIMATION
The LS section targets fully analog channel estimation by designing MiLAC training precoders and combiners so each estimated channel column is directly available at the receive RF chains.
- The goal is to perform LS channel estimation entirely in the analog domain without online digital computation.
- The proposed precoder and combiner designs directly produce the t-th column of the LS channel estimate at the receive RF chains.
A. Training Precoder and Combiner Design
The training design uses the received signal matrix and an LS-optimal training matrix to make each channel-estimate column directly readable after analog combining. The construction also preserves unit PAPR per RF chain.
- Collecting received signals across τ time slots forms the matrix Y from the per-slot relation yt = Hxt + nt.
- The LS estimator is obtained from the training matrix X and received signal matrix Y under a transmit-power constraint.
- The MSE-minimizing training matrix satisfies XXH = PT/NT INT.
- Choosing the identity-based training matrix makes each training vector proportional to a column of INT and enables direct analog recovery of the LS estimate.
- Fixing the source signal to equal power across transmit RF chains maintains unit PAPR while allowing the MiLAC to generate the required training vectors.
- The resulting training precoders and combiners allow the LS estimate to be obtained directly at the receive RF chains without online digital computation.
B. MiLAC Admittance Matrix Design
The proposed training precoders and combiners can be implemented by MiLACs through appropriately selected admittance matrices. Unconstrained admittance matrices provide the degrees of freedom needed for exact realization.
- The admittance design assumes LR = NR and τ = NT while constructing transmitter- and receiver-side MiLAC implementations.
- Auxiliary matrices QF,t and QG,t are chosen so their specified blocks equal the desired training precoder Ft and combiner Gt.
- Because the MiLAC admittance matrices are unconstrained, the remaining blocks can be selected arbitrarily provided the auxiliary matrices remain invertible.
- Block-matrix inversion yields the corresponding MiLAC admittance matrices, confirming exact analog realization of the proposed precoders and combiners.
C. Comparison with Digital LS Channel Estimation
The proposed MiLAC-aided LS estimator matches digital LS performance while moving estimation into the analog domain and reducing digital, RF-chain, and converter requirements.
- Digital LS uses a scaled DFT training matrix because the identity training matrix causes excessively high PAPR at each transmit RF chain.
- MiLAC-aided LS achieves identical estimation performance to digital LS because both training matrices satisfy the LS optimality condition.
- Computational Complexity: MiLAC-aided LS requires no online real operations, whereas digital LS is dominated by the matrix multiplication YXH requiring 8τNRNT real operations.
- Number of Transmit RF Chains: A single transmit RF chain suffices for MiLAC-aided LS, compared with NT transmit RF chains for digital LS.
- Low-Resolution ADCs/DACs: Analog generation of the training vector and LS estimate permits low-resolution DACs and ADCs without subsequent digital computation.
IV. MILAC-AIDED MMSE CHANNEL ESTIMATION
The MiLAC-aided MMSE scheme addresses the difficulty of analog-only MMSE estimation by exploiting channel correlation and the diagonal structure of the virtual channel covariance.
- MiLAC-aided MMSE improves over MiLAC-aided LS by exploiting channel correlation.
- Known transmit and receive correlation matrices enable MMSE estimation on a virtual channel with diagonal correlation structure.
- The proposed training precoders and combiners perform MMSE estimation fully in the analog domain without online digital computation.
A. Channel Model
The channel model represents the physical MIMO channel through a virtual eigen-domain channel whose correlation structure supports MMSE estimation without MSE loss.
- The canonical model uses unitary transmit and receive eigenvector matrices together with diagonal matrices of corresponding eigenvalues.
- The virtual channel Hv has zero-mean uncorrelated entries and a correlation matrix determined by the transmit and receive eigenvalue matrices.
- The virtual-channel correlation matrix Rv is assumed full rank.
- Because H and Hv are unitarily equivalent, MMSE estimation can be performed on Hv without loss in MSE.
B. Training Precoder and Combiner Design
Training design diagonalizes the virtual-channel estimation problem and configures MiLAC precoders and combiners so that MMSE channel-estimate columns are obtained directly at the receive RF chains.
- The design assumes LR = NR and τ = NT, then rewrites the channel model to construct blockwise training operations.
- The optimal virtual-domain training matrix is diagonal, with diagonal entries √p_t representing power allocated to transmit eigen-directions.
- The training vector is x_t = √p_tu_t, where u_t is the t-th transmit eigenvector, and the precoder is chosen to generate it.
- With diagonal Rv and Xv, the MMSE estimator becomes diagonal and its entries determine the combiner needed to obtain each estimate column.
- The resulting precoders and combiners directly produce the MMSE estimate at receive RF chains, using offline designs based on slowly varying correlation matrices and no online digital computation.
C. Comparison with Digital MMSE Channel Estimation
MiLAC-aided MMSE estimation achieves the same minimum MSE as digital MMSE while eliminating online digital computation and reducing PAPR.
- Performance and implementation: MiLAC-aided MMSE achieves the same minimum MSE as digital MMSE by implementing the training precoder and combiner in the analog domain.The corresponding admittance matrices can realize the proposed precoder and combiner directly.
- Power efficiency: MiLAC-aided MMSE has lower PAPR than digital MMSE.Its source signal ensures unit PAPR per RF chain, whereas digital MMSE concentrates power along dominant transmit eigen-directions.
V. NUMERICAL RESULTS
Numerical results show that MiLAC-aided LS and MMSE match their digital counterparts in estimation performance while avoiding online digital computation and substantially reducing operations.
- Estimation performance: MiLAC-aided LS and MMSE achieve the same estimation performance as their digital counterparts for N_T=N_R∈{16,64}.MMSE performs better than LS by exploiting channel statistics, while larger antenna numbers produce slightly higher NMSE under fixed total transmit power.
- Computational complexity: Digital MMSE complexity grows quadratically with N_R, whereas digital LS scales linearly with N_R and quadratically with N_T.The comparison uses τ=N_T.
- Computational complexity: MiLAC-aided LS and MMSE require no online digital computation.
- Computational complexity: 2.15 × 10^9 real operations are saved for MMSE estimation in a 2048×64 MIMO system.The reported saving implies lower processing latency and power consumption.
VI. CONCLUSION
The proposed MiLAC-aided LS and MMSE schemes perform channel estimation entirely in the analog domain while matching digital performance and reducing implementation costs and PAPR.
- MiLAC-aided LS and MMSE estimation achieves the same performance as digital counterparts while reducing computational complexity, RF chains, ADC/DAC resolution requirements, and PAPR.