Over-the-Air Aggregation Overview

• Over-the-air aggregation is a communication-computation paradigm that uses waveform superposition to compute aggregate functions directly from simultaneous transmissions.
• It enables efficient data aggregation in applications such as federated learning, distributed consensus, and wireless control, reducing latency and resource use.
• Physical-layer realizations vary from coherent analog designs to digital non-coherent schemes, each addressing tradeoffs in synchronization, noise, and channel fading.

Over-the-air aggregation, also called over-the-air computation (AirComp), over-the-air computation (OAC), or functional computation in the air, is a communication-computation paradigm in which distributed nodes transmit simultaneously over a shared wireless multiple-access channel so that the channel’s waveform superposition directly realizes a target aggregate such as a sum, average, weighted sum, or related nomographic function. In contrast to separated communication-and-computation pipelines, over-the-air aggregation treats interference as a functional primitive rather than a nuisance, and has been developed for wireless data aggregation, federated learning, reinforcement learning, average consensus, and distributed control (Zhu et al., 2020, Yang et al., 2023, Charalambous et al., 30 Jul 2025).

1. Foundational model and function class

The canonical signal model is the multiple-access superposition

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

where $h_k$ is the uplink channel, $x_k$ is the transmitted symbol carrying local data or a pre-processed local statistic, and $n$ is additive noise. The central design problem is not stream separation, as in conventional multiuser communications, but the recovery of a desired function of distributed data from the superposed waveform. In the simplest case, one seeks a sum or average; more generally, the target can be a nomographic function of the form

$$h(x_1,\dots,x_K)= f\!\Big(\sum_{k=1}^K g_k(x_k)\Big),$$

with transmitter-side pre-processing $g_k(\cdot)$ and receiver-side post-processing $f(\cdot)$ (Li et al., 2018, Zhu et al., 2020).

This formulation extends naturally to vector and MIMO settings. A representative MIMO AirComp model is

$$\hat{\mathbf{s}} = \mathbf{A}^H \sum_{k=1}^K \mathbf{H}_k \mathbf{B}_k \mathbf{s}_k + \mathbf{A}^H \mathbf{n},$$

where $\mathbf{B}_k$ is a sensor precoder and $\mathbf{A}$ is an aggregation beamformer. The objective is typically the minimization of function distortion, often measured by MSE between the desired aggregate and the received estimate, rather than sum-rate or interference suppression. This shifts the physical-layer design criterion from user separation to channel equalization and superposition alignment (Li et al., 2018).

A recurrent distinction in the literature is between conventional orthogonal wireless data aggregation and AirComp-based aggregation. In orthogonal schemes, communication resources scale with the number of devices because each node must be individually decoded. In AirComp, all devices can use the same time-frequency resource block, so aggregation latency is largely decoupled from the number of participating nodes. The 2020 overview characterizes this as turning the air into a computer for computing and communicating functions of distributed data at many devices (Zhu et al., 2020).

2. Physical-layer realizations

Over-the-air aggregation appears in several PHY realizations that differ in coherence assumptions, modulation, and synchronization requirements.

Variant	Key mechanism	Representative papers
Coherent analog AirComp	Channel inversion or beamforming aligns effective gains so received waveform approximates a sum	(Li et al., 2018, Guo et al., 2021)
Digital or non-coherent OAC	Encodes values into discrete structures such as signs, numerals, or subcarrier indices; receiver uses energy or majority rules	(Sahin et al., 2022, Jiang et al., 2020)
Receiver-normalized consensus OTA	Nodes transmit analog states; each receiver divides by an OTA estimate of summed channel gains	(Charalambous et al., 30 Jul 2025)

In coherent analog designs, transmitters pre-equalize channels so that the receiver observes an approximately common effective gain multiplying each local value. For OFDM-based federated learning, a practical obstacle is that perfect waveform superposition is disrupted by frame timing offset (TO) and carrier frequency offset (CFO). An OFDM prototype addresses this with a two-stage waveform pre-equalization technique and a customized multiple access protocol that estimate and mitigate TO and CFO; in experiments, the OTA solution achieved comparable performance to offline learning procedures in a real-world RSS prediction task with GPS information (Guo et al., 2021).

Digital and non-coherent realizations replace analog channel inversion with more robust symbol-domain aggregation. One approach represents real-valued gradients by balanced numerals, maps digit values to activated OFDM subcarriers, and uses a non-coherent receiver that estimates symbol counts from received energies, thereby avoiding precise sample-level time synchronization, channel estimation overhead, and transmitter-side pre-equalization (Sahin et al., 2022). Another line uses one-bit sign aggregation with phase correction and full-power transmission; its analysis introduces the normalized detection SNR, interpreted as an effective participation rate of users, and develops cluster-based relay selection to mitigate fading through local data fusion and spatial diversity (Jiang et al., 2020).

A distinct implementation style appears in consensus over wireless graphs. There, simultaneous analog transmissions compute weighted sums of neighbors’ states, and each node normalizes by the OTA estimate of the sum of incoming channel gains. In time-varying channels this normalization must be refreshed every iteration, typically by having all neighbors also transmit a constant $h_k$0 pilot over the air (Charalambous et al., 30 Jul 2025).

3. Federated and reinforcement learning

In distributed learning, over-the-air aggregation is used to replace digital uplink collection of gradients or model updates by single-shot analog aggregation over the MAC. The learning objective remains algorithmic, but the aggregation operator becomes a noisy channel-dependent estimator.

For policy-gradient reinforcement learning, the aggregated quantity is not a static supervised-learning gradient but a policy-gradient estimator built from sampled trajectories. In the over-the-air federated policy gradient algorithm, all agents broadcast analog signals proportional to local mini-batch policy-gradient estimates, and the controller updates

$h_k$1

The analysis separates RL sampling noise, multi-agent averaging, and wireless fading/noise, proves communication complexity $h_k$2, and gives sampling complexity $h_k$3, yielding an $h_k$4-fold linear speedup when channel statistics are favorable (Yang et al., 2023).

Within federated learning proper, several OTA-specific designs have emerged. Time-correlated sparsification with hybrid aggregation partitions updates into a global sparse part, transmitted over the air because all devices share the same mask, and a local sparse part, transmitted digitally with orthogonal access. Its convergence bound includes an explicit OAC-noise term $h_k$5, and on CIFAR-10 it reported 68% fewer communication resources than the fully digital TCS variant and, under a strict resource budget in the non-i.i.d. setting, final accuracies of 81.1% for TCS-H, 77.7% for TCS-D, and 74.5% for Top-$h_k$6 (Sun et al., 2022).

Beamforming and device selection can also be treated as learning variables rather than purely communication variables. For single-antenna devices and a multi-antenna base station, the OTA aggregation distortion enters a convergence upper bound through a per-round objective that trades off device exclusion against analog aggregation error. The receive-beamforming subproblem is equivalent to downlink single-group multicast beamforming, and the resulting GSDS and ADSBF schemes achieved faster convergence than alternatives in real-world image classification experiments; ADSBF showed marginally inferior performance to GSDS but lower computational complexity when the number of devices is large (Kalarde et al., 2023).

A structurally different approach makes model compression itself OTA-compatible. In Riemannian low-rank model compression, each local and global model is constrained to the fixed-rank manifold $h_k$7, a consensus penalty is added to preserve OTA compatibility, and low-rank factors are transmitted with random linear coding precoders so that cross terms vanish in expectation. In the reported experiments, the proposed GBMA and CI variants required about 1.42% and 1.44% of the benchmark communication overhead, respectively, to reach 70% test accuracy (Xue et al., 2023).

Weighted aggregation further generalizes OTA-FL by treating aggregation coefficients as optimization variables rather than fixed dataset weights. A 2024 scheme derives aggregation cost metrics and algorithms for weights that account jointly for wireless channels and computational heterogeneity without CSIT, and reports accuracy improvements of 15% over a scheme using CSIT and 30% over a scheme without CSIT (Azimi-Abarghouyi et al., 2024).

4. Consensus and distributed optimization

Average consensus is a natural application because the target function is itself an average. In a wireless multi-agent system, if all in-neighbors of node $h_k$8 transmit simultaneously, the receiver observes

$h_k$9

A ratio-consensus construction uses two states $x_k$0 and $x_k$1, aggregated over the air and then locally normalized by the sum of incoming channel gains. For time-invariant channels,

$x_k$2

and the ratio $x_k$3 converges to the exact average under strong connectivity, reciprocal positive real channel gains, and negligible noise. For time-varying channels, the normalization factor $x_k$4 must be recomputed every iteration via an additional OTA pilot transmission (Charalambous et al., 30 Jul 2025).

When non-coherent aggregation and non-negligible interference are allowed, linear consensus iterations can become biased because the effective mixing matrix is no longer column-stochastic. A 2025 alternative formulates average consensus as distributed optimization with local objectives $x_k$5 and applies decentralized projected gradient descent. With a two-symbol non-coherent coding scheme, the conditional expectation of the OTA statistic equals a weighted sum of neighbor differences, and the update

$x_k$6

achieves unbiased mean-square average consensus. The same work introduces transmit power control and receive scaling to accelerate convergence by shaping the expected mixing matrix (Deng et al., 8 Apr 2025).

Taken together, these consensus works suggest two distinct OTA design philosophies. One normalizes the physical sum into a stochastic matrix and then exploits ratio cancellation; the other treats consensus as optimization and uses a gradient term to eliminate the bias caused by non-coherent interference. Both retain the central AirComp primitive—simultaneous analog addition over the MAC—but differ in how they neutralize channel-induced asymmetries.

5. Beamforming, power control, and structured channels

A large part of the OTA literature is concerned with the design of beamformers, precoders, and power-control laws that make the received superposition as close as possible to the desired function.

In wirelessly powered AirComp, the downlink WPT phase and the uplink AirComp phase are jointly optimized. The uplink estimate

$x_k$7

is coupled to harvested energy constraints

$x_k$8

The derived structure is explicit: optimal energy beams point to the dominant eigen-directions of the WPT channels, and optimal power allocation tends to equalize the close-loop effective channels of different sensors. In the MISO case, the sensor precoder performs channel inversion with respect to the effective beamformed uplink channel; in the MIMO case, the AP aggregation beamformer is obtained through semidefinite relaxation and the WPT beamformers align with dominant singular vectors (Li et al., 2018).

For clustered massive MIMO, the high-dimensional but low-rank channel structure motivates reduced-dimension AirComp. In separable clusters with non-overlapping AoA ranges, the optimal decomposed aggregation beamforming (DAB) has the form

$x_k$9

where $n$0 extracts the cluster eigen-subspace and $n$1 performs equalization in the reduced dimension. In inseparable clusters, the beamformer is factorized as $n$2, separating covariance-level dimension reduction from reduced-space AirComp equalization. The same work also develops rank optimization for DAB components and channel-feedback schemes that themselves exploit the AirComp principle, so feedback duration scales with the number of clusters rather than the number of devices (Wen et al., 2018).

These designs are unified by one repeated principle: effective channels should be equalized in the metric that matters for function distortion, not in the sense used by conventional rate-maximizing multiuser systems. This suggests why AirComp beamforming problems often resemble multicast or max-min formulations rather than interference-cancellation formulations.

6. Error sources, assumptions, and open directions

The analytical treatment of over-the-air aggregation is dominated by three impairment classes: wireless noise, channel variation, and structural mismatch between what the channel sums and what the learning or consensus algorithm requires.

In OTA policy-gradient RL, the mean-square deviation of the aggregated gradient contains three terms: an AWGN term $n$3, a channel-fading variance term $n$4, and a gradient-dependent channel term. Under the condition $n$5, standard non-convex SGD-like convergence is recovered; for arbitrary channel statistics, Theorem 2 yields irreducible error contributions from channel variance, showing that increasing $n$6 and $n$7 cannot eliminate all channel-induced error under harsh fading (Yang et al., 2023).

Several consensus formulations rely on strong assumptions. The OTA ratio-consensus construction assumes positive real reciprocal channels, negligible receiver noise, and, for notational simplicity, full-duplex communication with perfect self-interference cancellation. Its authors explicitly note that with non-negligible noise, repeated analog aggregation would cause noise accumulation and exact convergence would generally be lost without additional mechanisms such as step-size decay or filtering (Charalambous et al., 30 Jul 2025). By contrast, the non-coherent D-PGD approach tolerates interference and noise but requires diminishing step sizes and a projected-gradient structure (Deng et al., 8 Apr 2025).

Implementation-oriented work identifies synchronization as a decisive bottleneck. The OFDM prototype for federated learning shows that perfect waveform superposition is difficult because of TO and CFO, and addresses this with a customized two-stage pre-equalization protocol (Guo et al., 2021). Digital balanced-numeral OAC removes precise sample-level synchronization and pre-equalization requirements, but introduces a new bias-variance tradeoff between numeral quantization error and count-estimation error at the non-coherent receiver (Sahin et al., 2022).

The broader survey literature points to further open problems: robust power control under imperfect CSI, joint precoder-beamformer optimization in MIMO AirComp, multi-cell coordinated AirComp, adaptive modulation and quantization for digital AirComp, CSI-feedback reduction or CSI-free designs, and coexistence of AirComp traffic with conventional unicast communications (Zhu et al., 2020). More recent learning-oriented papers add fully decentralized OTA aggregation, multi-agent interaction beyond parallel single-agent RL, and wireless realism such as delayed CSI, synchronization and phase-alignment errors, nonlinear transmitter effects, and per-antenna power limits (Yang et al., 2023).

These results indicate that over-the-air aggregation is best understood not as a single protocol but as a family of computation-centric wireless mechanisms. Its mature forms already support exact average consensus under idealized normalization models, provably convergent policy-gradient and federated-learning algorithms under noisy fading, beamforming and power-control co-design in MIMO and WPT systems, and hardware validation in OFDM prototypes. Its unresolved questions remain concentrated at the interface between algorithmic invariants—consensus, unbiasedness, stationarity, low-rank structure—and the physical constraints of synchronization, fading, and noise.