Digital Over-the-Air Aggregation

• Over-the-air digital aggregation is a physical-layer protocol that digitizes, modulates, and processes data to compute aggregate functions (e.g., sums or means) from distributed wireless devices.
• It employs techniques like bit-sliced modulation, MAP detection, and non-coherent decoding to achieve robust, low-latency, and energy-efficient aggregation even in noisy and dynamic channel conditions.
• Its applications in federated learning, distributed sensing, and IoT control demonstrate its practical impact on enabling scalable edge intelligence in next-generation wireless networks.

Over-the-air digital aggregation (also known as digital over-the-air computation or digital AirComp) is a physical-layer protocol that integrates digitization, modulation, and code-domain processing to directly compute aggregate functions (such as sums or means) of distributed data from multiple wireless devices by exploiting the superposition property of the wireless multiple-access channel (MAC). Digital AirComp generalizes traditional analog AirComp—which aggregates continuous-valued signals modulated directly onto analog waveforms—by incorporating quantization, digital modulation, and often channel or error correcting codes. This framework enables robust, low-latency, and resource-efficient distributed aggregation necessary for edge intelligence applications such as federated learning, distributed sensing, and IoT control, particularly under the constraints and reliability requirements of 6G and beyond networks.

1. System Models and Fundamental Principles

At its core, over-the-air digital aggregation departs from the analog paradigm by replacing uncoded linear analog modulation with digitization and digital modulation. Each device quantizes its local observation or model update (e.g., a vector $x_k \in \mathbb{R}^d$) into a digital representation (e.g., via bit slicing, vector quantization, or balanced numeral encoding), modulates it using a digital constellation (e.g., QAM, FSK), and transmits the symbols simultaneously with other devices over a shared MAC. The receiver observes a superposed signal of all user transmissions plus additive noise and channel effects: $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$ where $s_{k,n,\ell}$ denote modulated symbols, $h_k$ the channel coefficients (with power control or pre-equalization), and $z_{n,\ell}$ additive white Gaussian noise (Liu et al., 2024).

A key technique—bit-sliced digital AirComp—partitions the digital representation (e.g., B-bit quantized index) into $L$ bit slices, and maps each slice to a QAM symbol of corresponding granularity. This allows adaptation of reliability to bit significance. Over-the-air digital aggregation also accommodates variants such as non-coherent (amplitude-only, phase-agnostic) transmission (Dahl et al., 27 May 2026), non-orthogonal multiple access (NOMA) for multi-user OFDM (You et al., 2022), as well as joint digital-analog hybrid schemes for compression and aggregation (Sun et al., 2022).

2. Detection, Decoding, and Aggregation Algorithms

At the receiver, the primary task is to estimate the aggregated function—often the sum or mean of user data—from the superposed digital signal. Key algorithms and detection strategies include:

• Maximum a posteriori (MAP) Aggregate Detection: The optimal digital AirComp boundary for each possible sum-aggregate is computed as (Liu et al., 2024):

$\hat s = \arg\max_{s_j \in S} p_j \cdot f(r|s_j)$

where $p_j$ is the a priori probability of the aggregate, and $f(r|s_j)$ the channel likelihood.

• Majority-voting and Sign Decisions: In schemes with one-bit digital aggregation, the superposed sum is mapped to the sign of the aggregate via $\hat g = \mathrm{sign}(\Re\{Y\})$ (Zhu et al., 2020, Jiang et al., 2020). This is robust to Gaussian noise and scales in reliability with the number of aggregated nodes.
• Combinatorial Decoding and Message Passing: For structured quantization/codebook approaches (e.g., MD-AirComp, UMA-based GD-OAC, learned digital codes), the receiver applies approximate message passing (AMP) or belief-propagation on a sparsity-regularized count-reconstruction problem, estimating the number of devices transmitting each codeword (Qiao et al., 2024, Qiao et al., 2023, Tarizzo et al., 20 Sep 2025, Tarizzo et al., 22 Dec 2025).
• Non-Coherent and Balanced Numeral Decoding: Non-coherent schemes avoid symbol-level phase tracking by encoding data in the energy across OFDM subcarriers, decoded via maximum-likelihood energy detection (Sahin et al., 2022). For functions requiring permutation-invariance, as in majority voting or trimmed mean, multiset decoding is performed on the reconstructed quantization centroids (Tarizzo et al., 22 Dec 2025).
• Channel-Aware Demodulation: Adaptive constellations leverage channel randomness to prevent aggregate-symbol overlap, enabling the reliable computation of arbitrary symmetric and asymmetric functions (Li et al., 24 Jan 2025).

3. Performance Analysis: Error, Reliability, and SNR Scaling

Over-the-air digital aggregation fundamentally alters the error behavior and reliability profile compared to both analog AirComp and conventional orthogonal transmission:

• Error Scaling: The normalized mean squared error (MSE) for digital AirComp decays exponentially in the per-user SNR $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$0 down to a quantization floor, whereas analog AirComp decays only linearly with $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$1. For example, for $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$2, $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$3, $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$4, bit-sliced digital AirComp achieves MSE $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$5 dB at $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$6 dB, whereas analog repetition achieves only $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$7 dB under identical resources (Liu et al., 2024).
• Operational SNR Windows: Digital AirComp outperforms analog only within specific SNR intervals, widening as quantization resolution increases (e.g., $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$8 dB for $$r_{n,\ell} = \sum_{k=1}^K h_k \sqrt{p_k} s_{k,n,\ell} + z_{n,\ell}$$9; $s_{k,n,\ell}$0 dB for $s_{k,n,\ell}$1) (Liu et al., 2024).
• Reliability and Latency: Digital AirComp provides a constant-latency aggregation protocol independent of $s_{k,n,\ell}$2, whereas orthogonal schemes (e.g., TDMA/OFDMA) latency grows linearly with the number of devices (Liu et al., 2024, Qiao et al., 2024).
• MAP vs. ML Detection: Optimal MAP detection of aggregate sums yields lower error rates compared to conventional maximum-likelihood, especially given the non-uniform prior induced by quantized data (Liu et al., 2024).
• Non-coherent and Augmented-Affine Mapping: Non-coherent AirComp with amplitude-only (unknown phase) aggregation achieves unbiased sum estimation, with augmented-affine mappings reducing estimation variance by an order of magnitude over simple affine mappings under uniform data (Dahl et al., 27 May 2026).

4. Protocol Variants, Extensions, and Design Strategies

The digital AirComp landscape includes multiple specialized designs:

• One-Bit Digital AirComp (OBDA): Devices perform one-bit quantization (e.g., signSGD), map to BPSK or QAM symbols, and aggregate via over-the-air majority vote. Convergence is robust to noise, fading, and channel estimation errors, with scalability improving as the device population increases (Zhu et al., 2020, Jiang et al., 2020).
• Cluster-Based Cooperative Digital AirComp: Devices are grouped into clusters with local fusion and cluster-head relay selection, enhancing spatial diversity and improving effective SNR under fading (Jiang et al., 2020).
• Balanced numeral OAC: Encodes continuous-valued data into balanced-radix numerals across OFDM subcarriers, enabling non-coherent and CSI-free averaging with MSE scaling as $s_{k,n,\ell}$3, robust to synchronization and channel uncertainties (Sahin et al., 2022).
• Massive Digital AirComp (MD-AirComp): Employs shared vector quantization and unsourced random access (URA)-style codebooks to aggregate quantized model updates in federated learning. AMP-based decoders directly recover device-counts for each codeword, matching ideal FL accuracy at practical antenna and SNR settings (Qiao et al., 2024).
• Channel-Aware and Functional Digital OTA: Adaptive constellation design leverages the randomness of fading channels for scalable, function-agnostic aggregation, minimizing transmit power and decoding complexity (Li et al., 24 Jan 2025).
• Hybrid Digital-Analog Aggregation: Time-correlated sparsification with hybrid aggregation selects a global mask for analog AirComp and local refinements for digital-orthogonal aggregation, dramatically reducing communication overhead (Sun et al., 2022).
• End-to-End Learned Digital OTA Codes: Integrates unsourced random-access codebooks and unrolled neural AMP decoders, trained to maximize recovery and aggregation accuracy at low SNRs, with robust performance under device heterogeneity and Byzantine corruption (Tarizzo et al., 20 Sep 2025, Tarizzo et al., 22 Dec 2025).

5. Applications in Distributed Learning and Edge Inference

Over-the-air digital aggregation is a critical enabler for several next-generation edge intelligence scenarios:

• Federated Edge Learning (FEEL): Digital AirComp allows simultaneous aggregation of quantized stochastic gradients and model updates from massive devices, reducing uplink, computational, and latency bottlenecks. MD-AirComp and learned codebook approaches extend efficient learning to ultra-low SNR environments (reliable convergence at SNRs $s_{k,n,\ell}$4 dB) (Qiao et al., 2024, Tarizzo et al., 20 Sep 2025, Tarizzo et al., 22 Dec 2025).
• Edge Sensing and Inference: The HRD-AirComp framework integrates digital aggregation with hybrid reconfigurable intelligent surfaces, jointly optimizing quantization, transmission, beamforming, and RIS reflection to maximize inference accuracy for aggregated multi-view sensory features (Fu et al., 25 Sep 2025).
• Distributed Sensing and Control: Over-the-air digital aggregation facilitates low-latency computation of arithmetic and nomographic functions (mean, sum, max, consensus) in large-scale sensor and control networks (Zhu et al., 2020).
• Robust Aggregation Against Corruption: Learned digital codes enable aggregation of trimmed means or majority votes, enhancing robustness to faulty or adversarial devices without changing the codebook/decoding structure (Tarizzo et al., 22 Dec 2025).

6. Practical Considerations, Experimental Results, and Limitations

Digital AirComp performance and design hinge on several practical factors:

• Synchronization and Channel Knowledge: Tight timing and phase synchronization (via cyclic prefix, pilots, and preamble) remain necessary for coherent digital AirComp, while non-coherent and balanced numeral schemes relax sample-level synchronization and CSI requirements (Sahin et al., 2022, You et al., 2022).
• Power Control and Resource Allocation: Channel-inversion and power normalization are used to facilitate aggregate computation, balanced against power-explosion risks at deep fades. Hybrid analog-digital and cluster-based solutions further mitigate channel-induced unreliability (Liu et al., 2024, Sun et al., 2022, Jiang et al., 2020).
• Scalability: Digital AirComp supports device scaling to hundreds (and in theory, thousands) of devices, with transmission latency independent of network size for in-band aggregation (Liu et al., 2024, Qiao et al., 2024).
• Experimental Validation: Prototypical USRP SDR implementations and trace-driven simulations confirm that digital AirComp, especially with LDPC-coded joint decoders, achieves near-ideal federated learning performance at practical SNRs (e.g., SNR $s_{k,n,\ell}$5 dB with four users) and remains robust to phase and timing asynchrony (You et al., 2022, Zhao et al., 2021).
• Limitations: The achievable SNR window depends on quantization resolution, codebook size, and antenna resources. Complexity of combinatorial decoders can grow sharply with the number of users and quantization levels, but practical implementations (e.g., RSJD or AMP-DA) remain tractable for realistic problem sizes.

7. Outlook and Research Directions

Current research in over-the-air digital aggregation addresses several frontiers:

• Function Generalization: Extending aggregate computation capabilities beyond sums to arbitrary symmetric and asymmetric functions, enabled by advanced digital mapping and learned codebooks (Li et al., 24 Jan 2025, Tarizzo et al., 22 Dec 2025).
• Non-Coherent and Low-CSI Aggregation: Designing unbiased, low-variance mappings and transmission schemes for networks where phase or amplitude CSI is unavailable (Dahl et al., 27 May 2026).
• Task-Oriented Bit Allocation and Hybrid Architectures: Joint optimization of quantization levels, transmission coefficients, beamforming, and channel resources under strict edge-inference accuracy requirements (Fu et al., 25 Sep 2025).
• End-to-End Learnable Protocols: Leveraging deep learning for joint codebook and decoder design to achieve reliable aggregation in highly adversarial or heterogeneous network scenarios (Tarizzo et al., 20 Sep 2025, Tarizzo et al., 22 Dec 2025).
• Integration and Standardization: Realizing digital AirComp protocols that are compatible with 5G/6G standards, supporting ultra-reliable low-latency communication requirements, and energy-efficient massive IoT deployments (Zhu et al., 2020).

Over-the-air digital aggregation merges the fields of wireless communications, signal processing, and distributed optimization, providing the physical-layer foundation for scalable edge intelligence and distributed inference under stringent wireless resource constraints and reliability targets.