⚡ electrical engineering

Channel Estimation and Beamforming for Microwave Linear Analog Computers (MiLACs)-Aided Multiuser MISO Systems

This paper proposes computationally efficient channel estimation and beamforming schemes for MiLAC-aided multiuser MISO systems that exploit channel rank deficiency to compress signals in the analog domain, achieving massive reductions in computational complexity while maintaining performance comparable to digital baselines.

Original authors: Qiaosen Zhang, Matteo Nerini, Bruno Clerckx

Published 2026-07-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Qiaosen Zhang, Matteo Nerini, Bruno Clerckx

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Super-Connected" City

Imagine a future city where thousands of people are trying to talk to a central tower (the Base Station) at the same time. To make sure everyone is heard clearly, the tower needs to use a massive array of antennas—think of it as having thousands of microphones.

In the old way of doing this (Digital Beamforming), the tower would need a dedicated, expensive, and power-hungry computer processor for every single microphone. If you have 256 antennas, you need 256 processors. This is like hiring 256 secretaries just to listen to 256 people; it's incredibly expensive, slow, and uses too much electricity.

Microwave Linear Analog Computers (MiLACs) are a new invention that acts like a "smart filter" or a "magic lens." Instead of using a computer to process the sound, the MiLAC uses the physics of microwave waves traveling through a special network of pipes to do the work instantly. It can focus signals without needing a computer for every single antenna.

The Problem: The "Blind" Tower

The paper points out a major snag: While MiLACs are great at sending signals (beamforming), they struggle to listen and figure out where the users are (channel estimation).

To know where to send the signal, the tower needs to know the "map" of the air between it and the users.

The Old Way: The tower listens to everyone with all 256 microphones, records the massive amount of data, and then uses a super-computer to calculate the map. This defeats the purpose of using MiLACs because it brings back the expensive processors and high power usage.
The MiLAC Way: The tower only has a few "ears" (RF chains)—maybe just 16 for 16 users. It can't hear the full 256-dimensional sound. It's like trying to understand a symphony by only listening to 16 instruments out of 256.

The Solution: The "Smart Compression" Trick

The authors propose a clever two-step solution to solve this "blind" problem without needing expensive computers.

1. The "Grouping" Analogy (Channel Estimation)

Imagine the users aren't just random individuals; they are sitting in groups based on where they are standing. People in the same group are likely looking at the tower from similar angles.

The Trick: Instead of trying to listen to every single person individually with high precision, the MiLAC acts like a smart compression lens. It takes the full, messy sound from all 256 antennas and squashes it down into a smaller, cleaner version that fits into the tower's few "ears."
How it works: It uses the fact that people in a group share similar "patterns." The MiLAC filters out the noise and redundancy, keeping only the essential "shape" of the signal.
The Result: The tower can now calculate the map of the users using very little computing power.
- Analogy: Instead of taking a high-resolution photo of a whole crowd (which requires a massive computer to process), the MiLAC takes a stylized, low-resolution sketch that still captures exactly where the people are. The computer only has to draw the sketch, not the photo.

They tested two versions:

Small Groups: If there are few groups, they use a "Virtual Channel" method (focusing on the group patterns).
Large Groups: If there are many groups, they use a "Global Virtual Channel" method (a slightly more complex sketch that still works).

The Win: They found this method is up to 1,540 times faster and uses far less computing power than the old digital way, while still getting the map just as accurate (or even better in some cases).

2. The "Double-Lens" Trick (Beamforming)

Once the tower knows the map, it needs to send the data back. The old digital way requires a massive calculation to figure out how to aim the signal.

The Problem: A single MiLAC is great at simple tasks, but it can't easily perform the complex math needed to aim the signal perfectly for everyone at once.
The Solution: The authors built a Cascade MiLAC. Imagine connecting two magic lenses together in a row.
- The first lens does the first part of the math.
- The second lens finishes the job.
- Together, they act like a super-lens that can do the complex aiming work instantly, just by letting the waves flow through them. No computer needed during the actual transmission.

The Win: This "Double-Lens" system is up to 16,108 times faster than the digital computer method. It achieves the same speed of data delivery (sum rate) but with almost zero computing cost.

The Trade-Off: The "Cost of Learning"

There is one small catch. Because the tower has fewer "ears," it needs to spend a little extra time listening to the users to build its map (training overhead).

The Analogy: It's like a teacher who has fewer students to listen to but needs to ask a few extra questions to make sure they understand everyone.
The Result: This extra listening time slightly reduces the time available for sending actual data. In scenarios with many user groups, this "learning time" eats into the total speed a tiny bit. However, the massive savings in hardware cost and energy make it a worthwhile trade-off for future giant networks.

Summary of Claims

What they did: They created a new way to listen to users and aim signals for giant antenna systems using "Magic Microwave Lenses" (MiLACs) instead of super-computers.
How it works:
1. Listening: They use the MiLAC to compress the sound of thousands of antennas into a small, manageable size that fits the limited hardware, then do a simple calculation.
2. Sending: They connect two MiLACs in a row to instantly aim the signals without digital math.
The Results:
- Speed: The new method is roughly 1,500 times faster at figuring out the map and 16,000 times faster at aiming the signal compared to standard digital methods.
- Performance: It delivers data just as fast as the expensive digital way, provided you account for the small extra time needed to "listen" first.
- Hardware: It allows the system to work with very few expensive radio parts, saving huge amounts of money and electricity.

Technical Summary: Channel Estimation and Beamforming for MiLAC-Aided Multiuser MISO Systems

Problem Statement
Future gigantic multiple-input multiple-output (MIMO) systems promise ultra-high data rates and massive connectivity but face prohibitive hardware costs and computational complexity when employing conventional digital beamforming, which requires one radio-frequency (RF) chain per antenna. Microwave Linear Analog Computers (MiLACs) have emerged as a solution to offload beamforming computations to the analog domain, drastically reducing the number of RF chains and computational load. However, a critical gap remains: existing MiLAC research focuses on beamforming design and modeling but lacks efficient channel estimation schemes for multiuser (MU) systems. Conventional channel estimation requires acquiring full-dimensional received signals and performing massive digital processing, which negates the hardware and complexity advantages of MiLACs. Furthermore, the limited number of RF chains in MiLAC-aided systems prevents direct access to full-dimensional signals, making accurate Channel State Information (CSI) acquisition challenging.

Methodology
The paper proposes a framework for MiLAC-aided Multiuser Multiple-Input Single-Output (MU-MISO) systems operating under Time Division Duplex (TDD) with a limited number of RF chains ( $L=K$ , where $K$ is the number of users). The methodology is built on three core pillars:

System Model and User Grouping: The system models users partitioned into $G$ disjoint groups based on the similarity of their channel correlation matrices (Joint Spatial Division and Multiplexing principle). Within each group, users share a common correlation structure, allowing the channel to be represented in a low-dimensional subspace spanned by the group's eigenvectors.
Analog Compression for Channel Estimation: To address the RF chain limitation, the authors propose exploiting the rank deficiency of channel correlation matrices.
- Small- $G$ Regime: When the number of groups is small and group subspaces do not overlap significantly, a Virtual Channel Estimation scheme is used. The MiLAC performs analog subspace projection using the group eigenvectors ( $U_g$ ) to compress the full-dimensional received signals into low-dimensional observations. Digital processing then estimates only the low-dimensional virtual channels.
- Large- $G$ Regime: When group subspaces overlap (large $G$ ), a Global Virtual Channel Estimation scheme is employed. The system computes a full-column-rank matrix spanning the union of all group subspaces via Singular Value Decomposition (SVD). The MiLAC projects signals onto this global subspace, enabling the estimation of global virtual channels.
- Both schemes utilize Least Squares (LS) and Minimum Mean Square Error (MMSE) estimators on the compressed observations, avoiding the need to store or process full-dimensional signal matrices.
Cascade MiLAC for Beamforming: To realize Regularized Zero-Forcing Beamforming (R-ZFBF) from low-dimensional channel estimates without reconstructing the full-dimensional channel, the authors introduce a Cascade MiLAC architecture. This consists of two MiLACs connected in series. The first MiLAC handles the inversion and regularization operations, while the second handles the projection back to the antenna domain. This structure allows the R-ZFBF precoder to be realized directly from the virtual channel estimates with low computational complexity.

Key Contributions

System Model: A comprehensive model integrating uplink training and downlink beamforming for MiLAC-aided MU-MISO systems, explicitly accounting for limited RF chains and varying channel correlation matrices across user groups.
Efficient Channel Estimation Schemes: Two novel estimation strategies (Virtual and Global Virtual) that leverage MiLAC for analog compression. These schemes reduce the dimensionality of the estimation problem from $M$ (antennas) to $r$ or $\tilde{r}$ (subspace dimensions), significantly lowering digital computational costs.
Cascade MiLAC Architecture: The proposal of a two-stage cascade MiLAC to implement R-ZFBF directly from low-dimensional estimates. This overcomes the limitation of a single MiLAC, which cannot efficiently realize R-ZFBF from virtual channels without reconstructing the full channel.
Theoretical and Numerical Validation: Analytical proofs and numerical simulations demonstrating that the proposed schemes achieve Mean Square Error (MSE) performance comparable to or better than digital baselines while drastically reducing computational complexity.

Results
Numerical results validate the proposed schemes against conventional digital baselines (LS and MMSE) and digital R-ZFBF:

Channel Estimation: The proposed MiLAC-aided schemes achieve lower MSE for LS estimation and identical MSE for MMSE estimation compared to digital counterparts. Crucially, they reduce computational complexity by up to 1540 $\times$ (for $G=4$ in the small- $G$ regime).
Beamforming: The Cascade MiLAC-based R-ZFBF achieves the same sum rate as digital R-ZFBF when training overhead is ignored. When training overhead is included, a minor reduction in effective sum rate is observed, particularly in the large- $G$ regime, due to the extra symbols required for training.
Complexity Reduction: The beamforming schemes reduce computational complexity by up to 16,108 $\times$ compared to digital R-ZFBF.
Trade-offs: The paper highlights a fundamental trade-off: while the schemes significantly reduce hardware complexity (RF chains) and computational load, they incur extra training overhead. This overhead is more pronounced in the large- $G$ regime, leading to a slight degradation in effective sum rate compared to fully digital systems.

Significance
The paper claims that these contributions represent a necessary step toward realizing the potential of MiLAC-aided gigantic MIMO systems. By solving the channel estimation bottleneck and providing a mechanism for efficient beamforming from compressed data, the work enables the deployment of systems with minimal RF chains and low computational complexity without sacrificing performance. The introduction of the Cascade MiLAC specifically addresses the challenge of implementing complex precoding matrices (like R-ZFBF) in the analog domain using only low-dimensional channel knowledge. The authors note that their analysis assumes fully reconfigurable MiLACs to characterize fundamental limits, with extensions to lossless and reciprocal constraints left for future work.