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This paper considers the problem of the joint channel and carrier frequency offset (CFO) estimation for asynchronous amplify-and-forward (AF) two-way relay networks (TWRNs). This problem has not been solved to the best of our knowledge, and the presence of timing offsets between the two terminals, in addition to the frequency offsets, makes it necessary to develop signal sampling and joint estimation schemes that explicitly take into account both offsets. As potential solutions, we investigate two estimation strategies. The first strategy is pilot-based estimation, and in this context, we develop the maximum-likelihood joint CFO and channel estimator. Then, we consider the semi-blind estimation strategy, proposing a novel estimator based on expectation maximization (EM). As an alternative to the EM, we also investigate decision-directed (DD) semi-blind estimation. The semi-blind Cramer–Rao bound (CRB) is also obtained as a benchmark. Extensive experiments are used to compare the different estimation methods, showing that the proposed EM algorithm yields significantly lower mean-squared error than both the pilot-based estimator and the DD estimator and converges to the obtained CRB. It also yields much better symbol error rates. This is achieved at a reasonable computational complexity, and only a limited number of EM iterations are sufficient for convergence. The main contribution of our work is the development of the EM estimator, the corresponding CRB, and the use of simulations to demonstrate the superior performance of the proposed EM algorithm compared to other potential solutions.

Spectrum Efficient Joint Frequency Offset and Channel Estimation for Time-Asynchronous Amplify-and Forward Two-Way Relay Networks