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Analog over-the-air computation (OAC) is an efficient solution to a class of uplink data aggregation tasks over a multiple-access channel (MAC), wherein the receiver, dubbed the fusion center, aims to reconstruct a function of the data distributed at edge devices rather than the individual data themselves. Existing OAC relies exclusively on the maximum likelihood (ML) estimation at the fusion center to recover the arithmetic sum of the transmitted signals from different devices. ML estimation, however, is much susceptible to noise. In particular, in the misaligned OAC where there are channel misalignments among transmitted signals, ML estimation suffers from severe error propagation and noise enhancement. To address these challenges, this paper puts forth a Bayesian approach for OAC by letting each edge device transmit two pieces of prior information to the fusion center. Three OAC systems are studied: the aligned OAC with perfectly-aligned signals; the synchronous OAC with misaligned channel gains among the received signals; and the asynchronous OAC with both channel-gain and time misalignments. Using the prior information, we devise linear minimum mean squared error (LMMSE) estimators and a sum-product maximum a posteriori (SP-MAP) estimator for the three OAC systems. Numerical results verify that, 1) For the aligned and synchronous OAC, our LMMSE estimator significantly outperforms the ML estimator. In the low signal-to-noise ratio (SNR) regime, the LMMSE estimator reduces the mean squared error (MSE) by at least 6 dB; in the high SNR regime, the LMMSE estimator lowers the error floor on the MSE by 86.4%; 2) For the asynchronous OAC, our LMMSE and SP-MAP estimators are on an equal footing in terms of the MSE performance, and are significantly better than the ML estimator.
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Over-the-air computation (AirComp) is a disruptive technique for fast wireless data aggregation in Internet of Things (IoT) networks via exploiting the waveform superposition property of multiple-access channels. However, the performance of AirComp is bottlenecked by the worst channel condition among all links between the IoT devices and the access point. In this paper, a reconfigurable intelligent surface (RIS) assisted AirComp system is proposed to boost the received signal power and thus mitigate the performance bottleneck by reconfiguring the propagation channels. With an objective to minimize the AirComp distortion, we propose a joint design of AirComp transceivers and RIS phase-shifts, which however turns out to be a highly intractable non-convex programming problem. To this end, we develop a novel alternating minimization framework in conjunction with the successive convex approximation technique, which is proved to converge monotonically. To reduce the computational complexity, we transform the subproblem in each alternation as a smooth convex-concave saddle point problem, which is then tackled by proposing a Mirror-Prox method that only involves a sequence of closed-form updates. Simulations show that the computation time of the proposed algorithm can be two orders of magnitude smaller than that of the state-of-the-art algorithms, while achieving a similar distortion performance.
Reconfigurable intelligent surface (RIS)-empowered communications is on the rise and is a promising technology envisioned to aid in 6G and beyond wireless communication networks. RISs can manipulate impinging waves through their electromagnetic elements enabling some sort of a control over the wireless channel. In this paper, the potential of RIS technology is explored to perform equalization over-the-air for frequency-selective channels whereas, equalization is generally conducted at either the transmitter or receiver in conventional communication systems. Specifically, with the aid of an RIS, the frequency-selective channel from the transmitter to the RIS is transformed to a frequency-flat channel through elimination of inter-symbol interference (ISI) components at the receiver. ISI is eliminated by adjusting the phases of impinging signals particularly to maximize the incoming signal of the strongest tap. First, a general end-to-end system model is provided and a continuous to discrete-time signal model is presented. Subsequently, a probabilistic analysis for the elimination of ISI terms is conducted and reinforced with computer simulations. Furthermore, a theoretical error probability analysis is performed along with computer simulations. It is demonstrated that with the proposed method, ISI can successfully be eliminated and the RIS-aided communication channel can be converted from frequency-selective to frequency-flat.
Over-the-air computation (AirComp) has been recognized as a promising technique in Internet-of-Things (IoT) networks for fast data aggregation from a large number of wireless devices. However, as the number of devices becomes large, the computational accuracy of AirComp would seriously degrade due to the vanishing signal-to-noise ratio (SNR). To address this issue, we exploit the massive multiple-input multiple-output (MIMO) with hybrid beamforming, in order to enhance the computational accuracy of AirComp in a cost-effective manner. In particular, we consider the scenario with a large number of multi-antenna devices simultaneously sending data to an access point (AP) equipped with massive antennas for functional computation over the air. Under this setup, we jointly optimize the transmit digital beamforming at the wireless devices and the receive hybrid beamforming at the AP, with the objective of minimizing the computational mean-squared error (MSE) subject to the individual transmit power constraints at the wireless devices. To solve the non-convex hybrid beamforming design optimization problem, we propose an alternating-optimization-based approach. In particular, we propose two computationally efficient algorithms to handle the challenging receive analog beamforming problem, by exploiting the techniques of successive convex approximation (SCA) and block coordinate descent (BCD), respectively. It is shown that for the special case with a fully-digital receiver at the AP, the achieved MSE of the massive MIMO AirComp system is inversely proportional to the number of receive antennas. Furthermore, numerical results show that the proposed hybrid beamforming design substantially enhances the computation MSE performance as compared to other benchmark schemes, while the SCA-based algorithm performs closely to the performance upper bound achieved by the fully-digital beamforming.
Over-the-air computation (AirComp) has been recognized as a low-latency solution for wireless sensor data fusion, where multiple sensors send their measurement signals to a receiver simultaneously for computation. Most existing work only considered performing AirComp over a single frequency channel. However, for a sensor network with a massive number of nodes, a single frequency channel may not be sufficient to accommodate the large number of sensors, and the AirComp performance will be very limited. So it is highly desirable to have more frequency channels for large-scale AirComp systems to benefit from multi-channel diversity. In this letter, we propose an $M$-frequency AirComp system, where each sensor selects a subset of the $M$ frequencies and broadcasts its signal over these channels under a certain power constraint. We derive the optimal sensors transmission and receivers signal processing methods separately, and develop an algorithm for joint design to achieve the best AirComp performance. Numerical results show that increasing one frequency channel can improve the AirComp performance by threefold compared to the single-frequency case.
Federated learning (FL) is a promising technique that enables many edge devices to train a machine learning model collaboratively in wireless networks. By exploiting the superposition nature of wireless waveforms, over-the-air computation (AirComp) can accelerate model aggregation and hence facilitate communication-efficient FL. Due to channel fading, power control is crucial in AirComp. Prior works assume that the signals to be aggregated from each device, i.e., local gradients have identical statistics. In FL, however, gradient statistics vary over both training iterations and feature dimensions, and are unknown in advance. This paper studies the power control problem for over-the-air FL by taking gradient statistics into account. The goal is to minimize the aggregation error by optimizing the transmit power at each device subject to peak power constraints. We obtain the optimal policy in closed form when gradient statistics are given. Notably, we show that the optimal transmit power is continuous and monotonically decreases with the squared multivariate coefficient of variation (SMCV) of gradient vectors. We then propose a method to estimate gradient statistics with negligible communication cost. Experimental results demonstrate that the proposed gradient-statistics-aware power control achieves higher test accuracy than the existing schemes for a wide range of scenarios.
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