⚡ electrical engineering

Pilot Allocation for Multi-Hop Over-the-Air Neural Inference under Imperfect CSI

This paper investigates the impact of imperfect channel state information on multi-hop over-the-air neural inference and proposes five heuristic pilot allocation schemes that demonstrate a trade-off between training overhead and classification accuracy, showing that balanced resource allocation can achieve performance comparable to digital baselines.

Original authors: Tolga Girici, Meng Hua, Deniz Gündüz

Published 2026-04-09

📖 4 min read☕ Coffee break read

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, handwritten recipe from a chef in a kitchen (the Base Station) to a head chef in a distant restaurant (the Receiver).

In the old digital way, you would write the recipe down, put it in an envelope, mail it, and hope it arrives perfectly. But in this paper, the authors are exploring a magical, futuristic way called "Over-the-Air" (OTA) computing.

The Magic Trick: The "Whisper Chain"

Instead of writing the recipe down, the first chef shouts the ingredients into the air. A chain of relay helpers (the relays) stands in between.

The Goal: The final head chef needs to hear a specific mix of voices that, when combined, sounds exactly like the original recipe.
The Problem: The air is noisy. The helpers are far apart. And the helpers don't know exactly how loud or quiet the wind is blowing between them.

This is where the paper comes in. It's about how to get the best result when the helpers are guessing the conditions of the air (the "Channel State Information" or CSI).

The Core Challenge: The "Training" Phase

Before the chefs can start cooking (inference), the helpers need to practice. They need to shout test words ("Pilot signals") to figure out how the sound travels.

The Dilemma: You have a limited amount of time to practice.
- If you practice too little, the helpers guess wrong, and the final recipe sounds garbled.
- If you practice too much, you waste time that could be used for actual cooking.
- The Big Question: How should you split this limited practice time among the different helpers in the chain? Should everyone practice the same amount? Should the first helper practice more? Should the weakest link get the most practice?

The Five Strategies (The "Heuristics")

The authors tried five different ways to split the practice time:

The "Fairness" Approach (Uniform): Everyone gets the exact same amount of practice time. Like giving every student in a class the same number of study hours.
The "Proportional" Approach: You give more practice time to the links that have more people involved. Like giving a bigger team more rehearsal time.
The "Front-Loaded" Approach: You give the first helper in the chain the most practice time. Why? Because if the first person messes up the signal, everyone downstream gets the wrong message. It's like making sure the foundation of a house is perfect before building the walls.
The "All-In" Approach: You dump all the extra practice time on just the first link. (The paper shows this is usually a bad idea).
The "Weak Link" Approach: You look at who is struggling the most (the weakest signal) and give them extra practice time. Like a teacher spending extra time with the student who is having the hardest time.

What Did They Find? (The Results)

The researchers ran thousands of computer simulations (like running a million cooking tests in a virtual kitchen) to see which strategy worked best.

Practice Makes Perfect (to a point): If you give the helpers enough time to practice, the system works almost as well as if they knew the air conditions perfectly. The "magic" works!
Balance is Key: The best results came from balanced strategies (like Uniform or Front-Loaded).
- If you give all the practice time to just one person, the others are left guessing, and the whole chain breaks.
- If the signal is very weak (low power), you need more practice time overall, and it becomes even more important to help the early links in the chain.
More Helpers, Better Results: Surprisingly, having more relay groups (more hops) actually helped! It's like having a chain of many short whispers instead of one giant shout. Short whispers are easier to hear clearly than one loud shout across a noisy stadium.

The Big Picture

This paper is about efficiency. It tells us that we don't need perfect knowledge of the world to make these wireless neural networks work. We just need to be smart about how we spend our "practice time."

The Takeaway:
If you want to build a super-fast, low-energy AI system that runs on wireless signals, don't just throw money at the problem. Instead, carefully decide who gets to practice and for how long. If you balance the training across the chain, you can get near-perfect results even when the signal is imperfect.

It's the difference between a chaotic relay race where everyone runs at their own pace, and a well-coordinated team where everyone knows exactly when to run and how hard to push.

1. Problem Statement

The paper addresses the challenge of implementing Over-the-Air (OTA) neural inference in multi-hop wireless networks under Imperfect Channel State Information (CSI).

Context: OTA computation allows a multi-hop Amplify-and-Forward (AF) relay network to emulate a Fully Connected (FC) neural network layer by exploiting the superposition property of wireless channels. This avoids the latency and overhead of digital transmission.
The Gap: Existing literature often assumes perfect CSI or focuses on single-hop scenarios. In reality, CSI must be acquired via pilot-based estimation, which introduces errors.
The Core Challenge: In a multi-hop network, channel estimation errors accumulate across hops. Since the total time available for channel training (pilots) is limited, there is a critical trade-off: how should this training time be allocated across different hops to maximize classification accuracy? Allocating too little time to a specific hop degrades its CSI, which propagates errors through the entire network.

2. System Model

The authors propose a system model consisting of:

Network Topology: A multi-antenna Base Station (BS) with $N_t$ antennas transmitting to a multi-antenna Receiver (Rx) with $N_r$ antennas. The signal passes through $L$ groups of single-antenna AF relays.
Signal Flow: The BS precodes the input signal. Each relay group amplifies and forwards the signal to the next group. The final group transmits to the Rx.
Effective Channel: The system emulates a linear transformation $Wx$. The effective channel is the product of the channel matrices and relay gain matrices across all hops, plus a direct link (if available).
Channel Estimation: The system uses Least Squares (LS) estimation based on orthogonal pilot sequences.
- Training is performed in Time-Division Multiple Access (TDMA) phases.
- The minimum pilot length for a hop is determined by the number of transmitting antennas/devices in that hop.
- Estimation errors are modeled as additive Gaussian noise, which accumulates as the signal traverses the multi-hop chain.

3. Methodology

The paper combines optimization theory with heuristic resource allocation strategies.

A. Optimization Framework (AirFC)

The goal is to design the BS precoder ( $F_1$ ), Rx combiner ( $F_2$ ), and relay amplification gains ( $A_l$ ) to minimize the difference between the effective OTA channel and the target neural network weight matrix ( $W$ ).

Objective Function: Minimize the Frobenius norm of the imitation error ( $\|F_2 \hat{H}_{eff} F_1 - W\|_F^2$ ) plus a noise propagation penalty term.
Constraints: Power constraints at the BS and individual relays.
Solution: An Alternating Optimization (AO) algorithm is used to iteratively solve for $F_1$ , $F_2$ , and $\{A_l\}$ , treating the estimated channel $\hat{H}_{eff}$ as fixed during each step.

B. Pilot Allocation Heuristics

The core contribution is the investigation of how to distribute the total channel training time ( $\tau_{tot}$ ) across the $L+1$ hops (BS-to-Group1, Inter-groups, GroupL-to-Rx). The authors propose five heuristic strategies to incrementally increase the repetition factor ( $m_l$ ) of pilots for specific hops:

Uniform Excess Allocation: Distributes extra training time evenly across all hops.
Proportional-to-Minimum Allocation: Allocates extra time proportional to the minimum required pilot length of each hop.
Front-Loaded Allocation: Prioritizes earlier hops (BS-to-Group1) to prevent early-stage errors from propagating and degrading downstream performance.
All-Excess-to-First-Hop: Assigns all available extra training time to the first hop only.
Channel-Strength-Aware Allocation: Allocates more time to "weaker" hops (those with lower average channel gain or higher noise) to balance the estimation quality across the network.

4. Key Results

The authors evaluated these strategies using simulations based on the Fashion-MNIST dataset with a 3GPP UMi Street Canyon pathloss model.

Trade-off between Overhead and Accuracy: There is a clear diminishing return curve. Increasing total training time improves accuracy, but the gains plateau once CSI quality is sufficient.
Impact of Pilot Power:
- High Pilot Power: With sufficient pilot power (e.g., 1W), most allocation heuristics (Uniform, Proportional, Channel-Aware) converge to near-perfect CSI performance if enough training time is provided.
- Low Pilot Power: When pilot power is low (e.g., 0.1W), performance is strictly limited by estimation quality. In this regime, balanced allocation (Uniform/Proportional) significantly outperforms asymmetric strategies. "Front-loaded" allocation becomes effective only as training duration increases.
Multi-Hop vs. Single-Hop:
- Systems with more relay groups (e.g., $L=3$ ) achieve higher accuracy and lower variance compared to single-hop ( $L=1$ ) systems, even with the same total number of relays. This is because multi-hop reduces the distance per link, improving channel estimation quality.
- Single-hop systems with many relays suffer from high variance and poor accuracy because the long link distance makes channel estimation difficult.
Heuristic Performance:
- Uniform and Proportional allocations generally perform best and are robust.
- Channel-Strength-Aware offers marginal improvements in heterogeneous channel environments.
- All-Excess-to-First-Hop and highly asymmetric allocations lead to suboptimal performance due to the accumulation of errors in poorly estimated downstream hops.

5. Significance and Contributions

Bridging Theory and Practice: The paper moves beyond the idealized assumption of perfect CSI, addressing a critical bottleneck for real-world OTA neural inference systems.
Resource Allocation Framework: It provides a concrete framework for managing the "training overhead vs. inference accuracy" trade-off in multi-hop networks, which is essential for energy-constrained edge devices.
Design Guidelines: The results offer actionable guidelines for system designers:
- Multi-hop architectures are superior for long-range OTA inference.
- Balanced pilot allocation is crucial, especially when transmit power is limited.
- Sufficient pilot power is a prerequisite for achieving digital-baseline accuracy.
Future Directions: The paper identifies synchronization errors, optimal relay selection, and energy harvesting as key areas for future research.

In conclusion, the study demonstrates that while imperfect CSI degrades OTA neural inference, careful management of pilot resources—specifically through balanced allocation across hops—can restore classification accuracy to levels comparable to fully digital baselines.