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.
翻译:空中模拟( OAC ) 是一种高效的解决方案, 它可以解决多进入频道( MAC) 上行数据汇总任务类别, 接收者称聚变中心, 目的是重建在边缘设备而不是单个数据本身上分布的数据的函数。 现有的 OAC 完全依靠聚变中心的最大可能性估计值( ML) 来恢复从不同设备中传输的信号的计算总和。 但是, ML 估计极易受到噪音的影响。 特别是, 在多进入频道( MAC ) 上出现频道错位, ML 估计有严重错误的传播和噪音增强。 为了应对这些挑战, 本文通过让每个边缘设备向聚变中心传输两个先前的信息, 为 OAAC 提供了一种 Byees 方法。 三个 OAC 系统: 与 OAC 的 OAC 和 SP 信号的计算总和; 收到信号之间频道和时间错乱的 OAC 的不牢凑的 OAC 。 利用先前的信息, 我们为 O- SEM 最短的 O- mal mest IM 的 mest 校验算结果 。