Over-the-Air Computation (OAC)

• Over-the-Air Computation (OAC) is a paradigm that leverages the inherent linearity of wireless channels to compute functions directly from superposed signals, enabling low-latency data fusion.
• It utilizes both analog and digital regimes, where digital OAC optimizes modulation, quantization, and error correction to improve robustness in function estimation.
• OAC is key in federated edge learning and networked control systems, facilitating scalable gradient aggregation and privacy-preserving distributed computation.

Over-the-Air Computation (OAC) is a paradigm that exploits the physical superposition property of wireless multiple-access channels (MACs) to directly compute functions of distributed data during transmission. By leveraging the linearity of the electromagnetic channel, OAC enables the fusion center to recover target functions—typically arithmetic sums or nomographic functionals—without the need for decoding individual messages from each device. This approach yields a dramatic reduction in communication latency, facilitates scalable distributed learning and control, and is increasingly central in federated edge learning (FEEL), large-scale sensor fusion, and wireless control systems.

1. Principles and System Model

In the canonical OAC setup, $K$ devices transmit local measurements $x_k$ (or pre-processed values $\psi_k(x_k)$) simultaneously over a shared channel to a fusion center (FC). The received signal can be written as

$y = \sum_{k=1}^K h_k x_k + n$

where $h_k$ are the device-to-FC channel coefficients and $n$ is additive noise. By proper pre-processing (channel inversion, phase alignment, or digital quantization), the FC estimates $f(x_1, ..., x_K)$, most commonly with

$$f(x_1, ..., x_K) = \phi\left(\sum_{k=1}^K \psi_k(x_k)\right)$$

where $\phi$ is a post-processing function and $\psi_k$ are pre-processing functions chosen based on the desired aggregation (e.g., sum, weighted sum, mean, sign-majority, or polynomial).

OAC may operate in analog or digital regimes:

• Analog OAC: Directly modulates analog waveforms such that the desired sum is computed naturally by the channel superposition, subject to power and synchronization constraints.
• Digital OAC: Quantizes sources and utilizes digital modulation and (potentially) coding, permitting error correction, higher robustness, and integration with existing PHY layers.

A wide range of nomographic functions are supported, including arithmetic, max/min, polynomial, and histogram-type aggregations (Sahin et al., 2022).

2. Noise, Interference, and Channel Impairments

OAC performance is fundamentally determined by the interplay of noise, channel state information (CSI), synchronization, and interference sources. Analyses typically focus on the mean squared error (MSE) of the computed function, arising from three components:

• Noise-induced error: Additive channel noise impacts both analog and digital OAC. In digital schemes, noise-aware constellation design optimizes minimum distances between superposed points to minimize pairwise misclassification (Razavikia et al., 19 Jun 2025, Razavikia et al., 9 Nov 2025).
• Channel misalignment: Imperfect channel inversion or synchronization induces a gain mismatch and phase misalignment across terms in the superposition. This misalignment produces an SNR-independent error floor, particularly in high-mobility or asynchronous environments (Shao et al., 2021, Hellström et al., 2023).
• Inter-symbol/inter-link interference (ISI/ILI): Multipath propagation, especially under time-varying fading, introduces shifts in the delay-Doppler (DD) domain signal model. With orthogonal time frequency space (OTFS) modulation, ISI and ILI must be explicitly quantified and suppressed to avoid computation error saturation at high SNR (Huang et al., 2024).

Mitigation strategies include Bayesian estimation with device-provided sample statistics (Shao et al., 2021), optimal filter design via regularized Hankel systems to achieve unbiased function estimates in the presence of delay/phase uncertainties (Hellström et al., 2023), and OTFS-based interference cancellation using structured zero-padding and successive interference cancellation (Huang et al., 2024).

3. Digital OAC and Modulation Design

Digital OAC enables robust, spectrally efficient computation by mapping quantized device symbols to modulation constellations. Modern frameworks optimize modulation for computational objectives, not just communication rate:

• ChannelComp methodology: For symmetric finite-valued functions, each device implements a lookup-table encoder $\mathcal{E}(s_k)$ such that the noiseless sum $r = \sum_k x_k$ is uniquely mapped to function outputs at the receiver. The constellation is optimized to maximize minimum distance between codewords associated with different function values, accounting for the actual noise distribution (Gaussian, Laplacian, heavy-tailed) via a max–min optimization (Razavikia et al., 19 Jun 2025, Razavikia et al., 2023, Razavikia et al., 9 Nov 2025).
• Dimension reduction and scalability: The exponential scaling of $q^K$ superposition points (with $q$ quantization levels and $K$ users) is addressed by pyramid/histogram sampling and majority-based aggregation. For symmetric functions, sampling the histogram space reduces constraints from $O(q^K)$ to $O(q^{K-p+1})$ for sampling order $p$. Majority-based sampling ($p=K$) collapses the design to $q$ points, permitting standard digital modulations without bespoke constellations (Razavikia et al., 19 Jun 2025).
• Error correction: Non-binary LDPC codes and nested lattice coding can be integrated with digital OAC to realize modulo-sum computation with error correction over Gaussian MACs (Xie et al., 2023).

Computational MSE is minimized by joint selection of modulation, power allocation, and, where possible, adaptive quantization. Notably, digital OAC can outperform analog AirComp, especially for non-additive functions (e.g., product, max) and in the presence of channel impairments (Razavikia et al., 2023, Razavikia et al., 9 Nov 2025).

4. Protocols for Federated Edge Learning and Clustering

OAC is frequently deployed in FEEL to accelerate aggregation of stochastic gradients or model updates:

• FEEL via OAC: Both analog and digital OAC have been engineered for one-shot gradient aggregation. Digital schemes based on balanced numeral systems and over-the-air voting enable aggregation of real-valued gradients without CSI or tight synchronization, relying on energy aggregating non-coherent receivers (Sahin, 2022, Sahin et al., 2022, Sahin, 2023).
• Adaptive quantization: The adaptive absolute maximum (AAM) approach dynamically adjusts quantization range in digital OAC to minimize quantization error as gradient norms decrease during learning (Sahin, 2022).
• Latency and scalability: OAC achieves constant per-round resource usage, independent of the number of devices $K$, in contrast to conventional orthogonal access protocols, which scale linearly in $K$ (Sahin, 2023).
• Structured consensus: In decentralized learning, OAC enables direct over-the-air averaging of local models or gradients in consensus steps, thereby enhancing scalability and noise robustness over point-to-point scheduling (Ozfatura et al., 2020).

Performance analyses confirm that carefully designed digital OAC can match or exceed analog schemes in test accuracy and convergence rate, especially under non-IID data distributions (Sahin, 2022, Qiao et al., 2023).

5. Extensions: Control, Security, and Reconfigurable Environments

OAC's scope extends well beyond learning, driving advancements in wireless control, security, and channel engineering:

• Networked control systems: Multi-sender/multi-receiver OAC architectures co-design static control laws and OAC-precode/-decode matrices to ensure closed-loop stability and robust performance under network and power constraints. Convergent algorithms based on iterative convexification and constrained matrix factorization optimize both the plant dynamics and OAC parameters (Hussein et al., 3 May 2025).
• Physical-layer security: OAC is inherently vulnerable to eavesdropping, but can be secured via artificial noise injection orthogonal to the legitimate receiver’s channel—referred to as zero-forced noise. Linear programs allocate unused transmit power to maximize eavesdropper MSE while maintaining target approximation accuracy at the FC (Maßny et al., 2022).
• RIS-assisted OAC: Reconfigurable intelligent surfaces enable channel reconfiguration to boost signal power and mitigate "worst-link" bottlenecks, thereby minimizing OAC distortion. Jointly optimizing RIS phases and OAC beamforming/precoding using alternating minimization paired with convex–concave saddle-point solvers yields substantial performance and efficiency gains (Fang et al., 2021).

6. Asynchronous, Multipath, and Mobility-Tolerant OAC

Emerging work addresses OAC under non-ideal synchronization, multipath fading, and mobility:

• OTFS-based OAC: Orthogonal time frequency space modulation provides a nearly time-invariant DD-domain representation in time-varying, Doppler-dispersive channels. Power–scaling optimization and zero-padding plus structured successive interference cancellation are effective in mitigating ISI and ILI, restoring MSE performance to nearly the noise-limited regime (Huang et al., 2024).
• Optimal receive filters: When time- and phase-misalignment is unknown or uncorrected, optimal receive filters can be designed to ensure unbiased aggregation. Tikhonov regularization provides a bias–variance trade-off, with closed-form feasibility determined by pulse-shaping and maximum anticipated delay (Hellström et al., 2023).

These methodologies expand the operational regime of OAC to high-mobility, dense, and asynchronous wireless networks.

7. OAC in Federated Privacy and Multi-Cell Systems

OAC functionalities extend naturally to privacy and multi-cell operation:

• Anonymous OAC for privacy: OAC inherently anonymizes device transmissions. Integration with distributed differential privacy ensures rigorous privacy budgets while leveraging channel noise to reduce artificial noise injection, thus improving learning utility (Hasircioglu et al., 2020).
• Multi-cell non-coherent OAC: Frequency-shift keying-based majority-vote OAC, operated sequentially in uplink and downlink across cells, enables scalable FEEL with no CSI and relaxed synchronization. Inter-cell interference is reinterpreted as computational signal, and functions with provable convergence under stochastic gradient descent are maintained (Adeli et al., 2022).

• OTFS-based OAC and advanced interference cancellation: "Interference Cancellation for OTFS-Based Over-the-Air Computation" (Huang et al., 2024)
• Noise-aware and scalable digital modulation design: "On Designing Modulation for Over-the-Air Computation -- Part I: Noise-Aware Design" (Razavikia et al., 19 Jun 2025); "On Designing Modulation for Over-the-Air Computation -- Part II: Pyramid Sampling" (Razavikia et al., 19 Jun 2025)
• OAC for federated edge learning: "Unsourced Massive Access-Based Digital Over-the-Air Computation for Efficient Federated Edge Learning" (Qiao et al., 2023); "Over-the-Air Computation Based on Balanced Number Systems for Federated Edge Learning" (Sahin, 2022)
• Hierarchical coding and optimality: "Function Computation Over Multiple Access Channels via Hierarchical Constellations" (Razavikia et al., 17 Jan 2026)
• Secure OAC: "Secure Over-the-Air Computation using Zero-Forced Artificial Noise" (Maßny et al., 2022)
• OAC in distributed control: "Multiple Receiver Over-the-Air Computation for Wireless Networked Control Systems" (Hussein et al., 3 May 2025)
• Extending OAC via RIS: "Over-the-Air Computation via Reconfigurable Intelligent Surface" (Fang et al., 2021)
• Bayesian and optimal filter OAC: "Bayesian Over-The-Air Computation" (Shao et al., 2021); "Optimal Receive Filter Design for Misaligned Over-the-Air Computation" (Hellström et al., 2023)
• Digital OAC via ChannelComp and joint coding-design: "Computing Functions Over-the-Air Using Digital Modulations" (Razavikia et al., 2023); "Joint Design of Coding and Modulation for Digital Over-the-Air Computation" (Xie et al., 2023)
• Survey and foundational concepts: "A Survey on Over-the-Air Computation" (Sahin et al., 2022)

OAC thus constitutes a foundational technology for next-generation wireless learning, coordination, and sensing—uniting advanced channel modeling, optimization, coding, and computation in distributed information processing.