# Over-the-Air Computation Based on Balanced Number Systems for Federated Edge Learning

arXiv: https://arxiv.org/abs/2210.07012 · HTML: https://mlsanity.com/paper/2210.07012/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 49 references that are in the map.
- First posted on arXiv 13 Oct 2022.

## Builds on (most cited first, 25 of 25)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [A Joint Learning and Communications Framework for Federated Learning over Wireless Networks](https://mlsanity.com/paper/1909.07972/) (2019)
- [Federated Learning via Over-the-Air Computation](https://mlsanity.com/paper/1812.11750/) (2018)
- [Broadband Analog Aggregation for Low-Latency Federated Edge Learning (Extended Version)](https://mlsanity.com/paper/1812.11494/) (2018)
- [Federated Learning over Wireless Fading Channels](https://mlsanity.com/paper/1907.09769/) (2019)
- [Distributed Learning in Wireless Networks: Recent Progress and Future Challenges](https://mlsanity.com/paper/2104.02151/) (2021)
- [One-Bit Over-the-Air Aggregation for Communication-Efficient Federated Edge Learning: Design and Convergence Analysis](https://mlsanity.com/paper/2001.05713/) (2020)
- [Over-the-Air Federated Learning from Heterogeneous Data](https://mlsanity.com/paper/2009.12787/) (2020)
- [Ternary Compression for Communication-Efficient Federated Learning](https://mlsanity.com/paper/2003.03564/) (2020)
- [Blind Federated Edge Learning](https://mlsanity.com/paper/2010.10030/) (2020)
- [Over-the-Air Computation Systems: Optimization, Analysis and Scaling Laws](https://mlsanity.com/paper/1909.00329/) (2019)
- [5G New Radio: Unveiling the Essentials of the Next Generation Wireless Access Technology](https://mlsanity.com/paper/1806.06898/) (2018)
- [Random Orthogonalization for Federated Learning in Massive MIMO Systems](https://mlsanity.com/paper/2201.12490/) (2022)
- [Distributed Learning over a Wireless Network with FSK-Based Majority Vote](https://mlsanity.com/paper/2111.01850/) (2021)
- [On the Tradeoff between Energy, Precision, and Accuracy in Federated Quantized Neural Networks](https://mlsanity.com/paper/2111.07911/) (2021)
- [Collaborative Machine Learning at the Wireless Edge with Blind Transmitters](https://mlsanity.com/paper/1907.03909/) (2019)
- [Over-the-Air Computation with DFT-spread OFDM for Federated Edge Learning](https://mlsanity.com/paper/2112.13439/) (2021)
- [Distributed Learning over a Wireless Network with Non-coherent Majority Vote Computation](https://mlsanity.com/paper/2209.04692/) (2022)
- [Over-the-Air Computation Systems: Optimal Design with Sum-Power Constraint](https://mlsanity.com/paper/2005.09139/) (2020)
- [Cluster-Based Cooperative Digital Over-the-Air Aggregation for Wireless Federated Edge Learning](https://mlsanity.com/paper/2008.00994/) (2020)
- [A Demonstration of Over-the-Air Computation for Federated Edge Learning](https://mlsanity.com/paper/2209.09954/) (2022)
- [Over-the-Air Computation over Balanced Numerals](https://mlsanity.com/paper/2209.11004/) (2022)
- [Wyner-Ziv Gradient Compression for Federated Learning](https://mlsanity.com/paper/2111.08277/) (2021)
- [Chirp-Based Over-the-Air Computation for Long-Range Federated Edge Learning](https://mlsanity.com/paper/2206.10784/) (2022)

## Built on by (most cited first, 10 of 10)

- [A Survey on Over-the-Air Computation](https://mlsanity.com/paper/2210.11350/) (2022)
- [Federated Learning and Meta Learning: Approaches, Applications, and Directions](https://mlsanity.com/paper/2210.13111/) (2022)
- [Training Machine Learning models at the Edge: A Survey](https://mlsanity.com/paper/2403.02619/) (2024)
- [Blind Federated Learning via Over-the-Air q-QAM](https://mlsanity.com/paper/2311.04253/) (2023)
- [Private Collaborative Edge Inference via Over-the-Air Computation](https://mlsanity.com/paper/2407.21151/) (2024)
- [Adaptive Federated Learning Over the Air](https://mlsanity.com/paper/2403.06528/) (2024)
- [Scalable and Reliable Over-the-Air Federated Edge Learning](https://mlsanity.com/paper/2407.11807/) (2024)
- [VecComp: Vector Computing via MIMO Digital Over-the-Air Computation](https://mlsanity.com/paper/2511.02765/) (2025)
- [Wireless Federated k-Means Clustering with Non-coherent Over-the-Air Computation](https://mlsanity.com/paper/2308.06371/) (2023)
- [MD-AirComp+: Adaptive Quantization for Blind Massive Digital Over-the-Air Computation](https://mlsanity.com/paper/2602.18332/) (2026)

## Abstract (arXiv)

In this study, we propose a digital over-the-air computation (OAC) scheme for achieving continuous-valued (analog) aggregation for federated edge learning (FEEL). We show that the average of a set of real-valued parameters can be calculated approximately by using the average of the corresponding numerals, where the numerals are obtained based on a balanced number system. By exploiting this key property, the proposed scheme encodes the local stochastic gradients into a set of numerals. Next, it determines the positions of the activated orthogonal frequency division multiplexing (OFDM) subcarriers by using the values of the numerals. To eliminate the need for precise sample-level time synchronization, channel estimation overhead, and channel inversion, the proposed scheme also uses a non-coherent receiver at the edge server (ES) and does not utilize a pre-equalization at the edge devices (EDs). We theoretically analyze the MSE performance of the proposed scheme and the convergence rate for a non-convex loss function. To improve the test accuracy of FEEL with the proposed scheme, we introduce the concept of adaptive absolute maximum (AAM). Our numerical results show that when the proposed scheme is used with AAM for FEEL, the test accuracy can reach up to 98% for heterogeneous data distribution.