# Over-the-Air Computation over Balanced Numerals

arXiv: https://arxiv.org/abs/2209.11004 · HTML: https://mlsanity.com/paper/2209.11004/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 21 references that are in the map.
- First posted on arXiv 22 Sep 2022.

## Builds on (most cited first, 11 of 11)

- [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)
- [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)
- [Over-the-Air Computation Systems: Optimization, Analysis and Scaling Laws](https://mlsanity.com/paper/1909.00329/) (2019)
- [Distributed Learning over a Wireless Network with FSK-Based Majority Vote](https://mlsanity.com/paper/2111.01850/) (2021)
- [Over-the-Air Computation with DFT-spread OFDM for Federated Edge Learning](https://mlsanity.com/paper/2112.13439/) (2021)

## Built on by (most cited first, 4 of 4)

- [A Survey on Over-the-Air Computation](https://mlsanity.com/paper/2210.11350/) (2022)
- [Over-the-Air Computation Based on Balanced Number Systems for Federated Edge Learning](https://mlsanity.com/paper/2210.07012/) (2022)
- [Graph Neural Networks over the Air for Decentralized Tasks in Wireless Networks](https://mlsanity.com/paper/2302.08447/) (2023)
- [OptiVote: Non-Coherent FSO Over-the-Air Majority Vote for Communication-Efficient Distributed Federated Learning in Space Data Centers](https://mlsanity.com/paper/2512.24334/) (2025)

## Abstract (arXiv)

In this study, a digital over-the-air computation (OAC) scheme for achieving continuous-valued gradient aggregation is proposed. It is shown 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 using this property, the proposed scheme encodes the local gradients into a set of numerals. It then determines the positions of the activated orthogonal frequency division multiplexing (OFDM) subcarriers by using the values of the numerals. To eliminate the need for a precise sample-level time synchronization, channel estimation overhead, and power instabilities due to the 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). Finally, the theoretical mean squared error (MSE) performance of the proposed scheme is derived and its performance for federated edge learning (FEEL) is demonstrated.