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This paper proposes using communication pipelining to enhance the wireless spectrum utilization efficiency and convergence speed of federated learning in mobile edge computing applications. Due to limited wireless sub-channels, a subset of the total clients is scheduled in each iteration of federated learning algorithms. On the other hand, the scheduled clients wait for the slowest client to finish its computation. We propose to first cluster the clients based on the time they need per iteration to compute the local gradients of the federated learning model. Then, we schedule a mixture of clients from all clusters to send their local updates in a pipelined manner. In this way, instead of just waiting for the slower clients to finish their computation, more clients can participate in each iteration. While the time duration of a single iteration does not change, the proposed method can significantly reduce the number of required iterations to achieve a target accuracy. We provide a generic formulation for optimal client clustering under different settings, and we analytically derive an efficient algorithm for obtaining the optimal solution. We also provide numerical results to demonstrate the gains of the proposed method for different datasets and deep learning architectures.