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Federated learning (FL) is a popular collaborative distributed machine learning paradigm across mobile devices. However, practical FL over resource constrained mobile devices confronts multiple challenges, e.g., the local on-device training and model updates in FL are power hungry and radio resource intensive for mobile devices. To address these challenges, in this paper, we attempt to take FL into the design of future wireless networks and develop a novel joint design of wireless transmission and weight quantization for energy efficient FL over mobile devices. Specifically, we develop flexible weight quantization schemes to facilitate on-device local training over heterogeneous mobile devices. Based on the observation that the energy consumption of local computing is comparable to that of model updates, we formulate the energy efficient FL problem into a mixed-integer programming problem where the quantization and spectrum resource allocation strategies are jointly determined for heterogeneous mobile devices to minimize the overall FL energy consumption (computation + transmissions) while guaranteeing model performance and training latency. Since the optimization variables of the problem are strongly coupled, an efficient iterative algorithm is proposed, where the bandwidth allocation and weight quantization levels are derived. Extensive simulations are conducted to verify the effectiveness of the proposed scheme.