Over-the-air computation (OAC) is a promising technique to realize fast model aggregation in the uplink of federated edge learning. OAC, however, hinges on accurate channel-gain precoding and strict synchronization among the edge devices, which are challenging in practice. As such, how to design the maximum likelihood (ML) estimator in the presence of residual channel-gain mismatch and asynchronies is an open problem. To fill this gap, this paper formulates the problem of misaligned OAC for federated edge learning and puts forth a whitened matched filtering and sampling scheme to obtain oversampled, but independent, samples from the misaligned and overlapped signals. Given the whitened samples, a sum-product ML estimator and an aligned-sample estimator are devised to estimate the arithmetic sum of the transmitted symbols. In particular, the computational complexity of our sum-product ML estimator is linear in the packet length and hence is significantly lower than the conventional ML estimator. Extensive simulations on the test accuracy versus the average received energy per symbol to noise power spectral density ratio (EsN0) yield two main results: 1) In the low EsN0 regime, the aligned-sample estimator can achieve superior test accuracy provided that the phase misalignment is non-severe. In contrast, the ML estimator does not work well due to the error propagation and noise enhancement in the estimation process. 2) In the high EsN0 regime, the ML estimator attains the optimal learning performance regardless of the severity of phase misalignment. On the other hand, the aligned-sample estimator suffers from a test-accuracy loss caused by phase misalignment.

中文翻译：

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空中计算失调的联合边缘学习

空中计算（OAC）是一种有前途的技术，可在联邦边缘学习的上行链路中实现快速模型聚合。但是，OAC取决于准确的信道增益预编码和边缘设备之间的严格同步，这在实践中具有挑战性。因此，在存在残留信道增益失配和异步的情况下如何设计最大似然（ML）估计器是一个未解决的问题。为了填补这一空白，本文提出了用于联合边缘学习的OAC不对齐的问题，并提出了一种白化的匹配滤波和采样方案，以从未对齐和重叠的信号中获取过采样但独立的样本。给定白化的样本，设计了和积ML估计器和对齐样本估计器以估计发送符号的算术和。特别是，我们的和积ML估计器的计算复杂度在数据包长度上是线性的，因此比传统的ML估计器要低得多。关于测试精度与每个符号的平均接收能量与噪声功率谱密度比（EsN0）的广泛仿真，得出两个主要结果：1）在低EsN0体制下，如果相位失准，对准样本估计器可以实现更高的测试精度。不严重。相反，由于估计过程中的错误传播和噪声增强，ML估计器无法很好地工作。2）在高EsN0体制下，无论相位失准的严重程度如何，ML估计器都能获得最佳的学习性能。另一方面，对准样本估计器遭受由相位未对准引起的测试精度损失。