⚡ electrical engineering

Realization of a Fully Connected Neural Layer Over-the-Air through Multi-hop Amplify-and-Forward Relays

This paper proposes and validates an alternating optimization framework for implementing a fully connected neural network layer over-the-air using multi-hop amplify-and-forward relays, demonstrating that this approach achieves near-perfect classification accuracy under power constraints.

Original authors: Tolga Girici, Meng Hua, Deniz Gündüz

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tolga Girici, Meng Hua, Deniz Gündüz

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

Imagine you are trying to send a complex recipe (a neural network) from a master chef (the transmitter) to a sous-chef (the receiver) in a massive, noisy stadium. Usually, you'd write the recipe down, put it in an envelope, and mail it. But mailing takes time, costs money (energy), and the paper might get smudged (data loss).

This paper proposes a clever trick called "Over-the-Air (OTA) Computing." Instead of mailing the recipe, the chef shouts the instructions, and the stadium's acoustics naturally mix the sounds so the sous-chef hears the final dish instructions directly.

Here is the breakdown of their new invention, explained simply:

1. The Problem: The "Rank-Deficient" Hallway

In a normal wireless system, if the chef and the sous-chef are far apart, or if there are obstacles (like walls or bad weather), the signal gets weak or distorted. Sometimes, the "hallway" between them is so narrow that you can't send all the necessary instructions at once. It's like trying to push a wide sofa through a narrow door; you have to break it apart, which ruins the recipe.

2. The Solution: The "Whisper Chain" (Multi-hop Relays)

The authors realized that instead of shouting directly across the stadium, they could use a chain of volunteers (relays) standing between the chef and the sous-chef.

The Setup: Imagine 5 groups of volunteers standing in a line.
The Action: The chef shouts to Group 1. Group 1 listens, amplifies the voice slightly, and shouts it to Group 2. Group 2 does the same, passing it down the line until it reaches the sous-chef.
The Magic: Each volunteer doesn't just repeat the words; they adjust their volume and tone (using math called "Amplify-and-Forward") to ensure that by the time the message reaches the end, it looks exactly like the original recipe the chef intended to send.

3. The "Neural Network" Twist

Usually, these volunteers just pass messages. But here, the authors wanted the entire chain to act like a Fully Connected Neural Network Layer.

Think of a neural network layer as a giant calculator that mixes inputs to create outputs.

Digital Way: You send the data to a supercomputer, it does the math, and sends the result back.
This Paper's Way: The air itself and the volunteers do the math. As the signal bounces from volunteer to volunteer, the physics of the wireless waves naturally perform the complex multiplication and addition required by the neural network. The volunteers are essentially "programming" the airwaves to do the calculation for them.

4. The "Conductor" (Optimization)

You can't just have random volunteers shouting; they would create a mess of noise. The authors created a smart "Conductor" (an algorithm) that tells every single volunteer exactly how loud to shout and what tone to use.

The Goal: The Conductor adjusts the system so that the final sound the sous-chef hears is a perfect imitation of the digital recipe, even though it traveled through a noisy, multi-hop chain.
The Challenge: If a volunteer shouts too loud, they create static (noise) that drowns out the next person. If they shout too soft, the signal dies. The Conductor balances this perfectly.

5. What Did They Find? (The Results)

They ran simulations (computer tests) to see how well this works:

Short Distances: If the stadium is small, even a few volunteers get the job done with near-perfect accuracy.
Long Distances: If the stadium is huge, a single line of volunteers isn't enough. But if you break them into 2 or 3 groups (hops), the accuracy skyrockets. It's like having a relay race where the runners are closer together, so they don't get tired (signal loss) or lose the baton (noise).
Power Matters: The volunteers need enough battery power. If they are too weak (low power), the signal gets lost in the noise. But with decent power, they can achieve 84.5% accuracy, which is almost as good as the traditional digital method.

The Big Picture

This paper shows that we might not need to send massive amounts of data to a cloud server to do AI calculations. Instead, we can use the wireless network itself as a giant, distributed computer.

The Analogy:
Imagine you want to paint a masterpiece.

Old Way: You send your sketch to a factory, they print it, and mail it back.
New Way: You stand in a field with a group of friends. You whisper a color to Friend A, who whispers a slightly different color to Friend B, who whispers to Friend C. By the time the message reaches the last person, the sequence of whispers has naturally evolved into the perfect color mix you needed, without ever needing a factory.

This method saves energy, reduces delay (latency), and turns the wireless network into a powerful tool for Artificial Intelligence.

1. Problem Statement

The paper addresses the challenge of implementing a Fully Connected (FC) layer of a neural network using Over-the-Air (OTA) computation in a wireless environment.

Context: Modern edge applications (AR/VR, IoT) require low-latency inference with strict energy budgets. Traditional digital inference at the cloud introduces latency and energy overhead due to heavy matrix operations, quantization, and packetization.
Specific Challenge: While single-hop OTA computing (e.g., AirFC) has been explored, it faces limitations in large-scale deployments:
- Pathloss: Signal attenuation over long distances.
- Rank Deficiency: In millimeter-wave bands or specific geometries, the direct MIMO channel between the Base Station (BS) and Receiver (Rx) may be rank-deficient, preventing the implementation of a full-rank weight matrix.
- Direct Link Blockage: Scenarios where a direct BS-Rx link does not exist.
Goal: To utilize multi-hop Amplify-and-Forward (AF) relays to create an effective channel that mimics a target digital FC weight matrix ( $W$ ) while managing noise accumulation and power constraints.

2. Methodology

System Model

The authors propose a system consisting of:

Transmitter: A multi-antenna Base Station (BS) with $N_t$ antennas.
Relays: $K$ single-antenna AF relays organized into $L$ sequential groups (hops).
Receiver: A multi-antenna Rx with $N_r$ antennas.
Channel: Includes direct BS-Rx links ( $H_0$ ) and cascaded links between hops ( $H_1, \dots, H_{L+1}$ ).
Operation: The BS precodes the input signal $x$ . Each relay group amplifies the received signal with a complex gain and forwards it to the next group. The Rx combines the final signal.

The effective channel ( $H_{eff}$ ) is the sum of the direct link and the cascaded multi-hop paths:
$H_{eff} = H_0 + H_{L+1}A_L H_L \cdots A_1 H_1$
where $A_l$ is the diagonal matrix of amplification gains for relay group $l$ .

Optimization Framework

The objective is to minimize the imitation error (difference between the OTA effective channel and the target weight matrix $W$ ) while penalizing noise propagation.

Objective Function:
$\min_{F_1, F_2, \{A_l\}} \| F_2 H_{eff} F_1 - W \|_F^2 + \text{tr}(F_2 R_{in}^n F_2^H)$

$F_1$ : BS precoding matrix.
$F_2$ : Rx combining matrix.
$A_l$ : Relay amplification gain matrices.
$R_{in}^n$ : Aggregate noise covariance at the receiver (accumulated noise from all hops).

Constraints:

Transmitter power constraint ( $\|F_1\|_F^2 \leq P_{max}$ ).
Per-relay power constraints (based on the signal variance incident on each relay).

Solution Algorithm: Alternating Optimization (AO)
The authors propose an iterative AO framework to solve the non-convex joint optimization problem by fixing two sets of variables while optimizing the third:

Optimize BS Precoder ( $F_1$ ): Solves a constrained least-squares problem using Lagrange multipliers.
Optimize Rx Combiner ( $F_2$ ): Derives a closed-form solution (similar to MMSE) that balances channel matching and noise suppression.
Optimize Relay Gains ( $A_l$ ):
- Isolates the contribution of the current hop.
- Solves a regularized least-squares problem using the Khatri-Rao product to vectorize the matrix equation.
- Noise-Aware Regularization: Introduces a diagonal regularizer to prevent ill-conditioning and exploding gains in low-rank or Line-of-Sight (LoS) scenarios.
- Power Projection: Enforces per-relay power constraints via magnitude projection after the gain calculation.

3. Key Contributions

Multi-hop OTA FC Implementation: First work to propose and analyze multi-hop AF relaying specifically for implementing a neural network FC layer, moving beyond single-hop or two-hop sensor fusion models.
Rank Enhancement: Demonstrates that multi-hop relays can improve the effective channel rank, allowing the implementation of full-rank weight matrices even when the direct BS-Rx link is rank-deficient or blocked.
Joint Optimization Framework: Developed a novel AO algorithm that jointly optimizes precoders, combiners, and relay gains while explicitly accounting for noise accumulation across hops.
Noise-Aware Regularization: Introduced a specific regularization technique for relay gain optimization to handle ill-conditioned channels and prevent noise explosion.

4. Simulation Results

The authors evaluated the framework using a Fashion-MNIST dataset with a custom neural network architecture (including complex-valued layers).

Accuracy vs. Relay Count: With sufficient relays per group ( $K > 12$ ) and moderate power ($1$ W), the system achieves near-perfect classification accuracy (matching the digital baseline of ~84.5%).
Impact of Hops ( $L$ ):
- In small areas ($100$m), a single hop ( $L=1$ ) is sufficient.
- In larger areas ($200$m), increasing hops ( $L=2, 3$ ) significantly improves accuracy by reducing pathloss and noise buildup. However, diminishing returns are observed beyond $L=3$ due to cumulative noise.
Power Sensitivity:
- High relay power ($1$ W) yields stable, high accuracy.
- Low relay power ($0.01$ W) results in poor, volatile performance, highlighting the need for sufficient power budgets to overcome noise.
Direct Link: The presence of a direct BS-Rx link provides only a marginal improvement (~1%), confirming that the multi-hop relay system is the primary driver of performance.
Robustness: The system is robust to channel variations, with small error bars in accuracy across different channel realizations.

5. Significance

Scalability: This approach enables the deployment of neural network inference in large-scale, distributed environments where direct communication is impossible or unreliable.
Hardware Efficiency: By using simple AF relays (which do not require complex decoding/encoding or CSI estimation at the relay), the system reduces hardware complexity and latency compared to decode-and-forward schemes.
Future Edge AI: It provides a viable pathway for "Analog Inference" in 6G and massive IoT networks, enabling millisecond-level inference with tight energy budgets by leveraging the physics of the wireless channel for computation.
Theoretical Insight: The work clarifies the trade-offs between the number of hops, relay density, and power constraints in OTA computing, establishing that while multi-hop solves distance and rank issues, noise accumulation remains the fundamental limiting factor.