Blind Channel Estimation in Full Duplex Systems: Identifiability Analysis, Bounds, and Estimators

We consider blind channel estimation in a single-input single-output full-duplex communication system, where both the self-interference and the communication channels need to be accurately estimated. In this context, blind estimators are attractive as they improve bandwidth efficiency but they suffer from the phase ambiguity problem. In this paper, we first formally define and analyse this ambiguity and then develop a general framework for testing and designing modulation sets for blind estimation of channel parameters. We mathematically show that simply shifting the mean of the $M$-PSK modulation resolves the ambiguity problem. We also show how this can be extended to more general modulation sets. Finally, we propose an expectation maximization (EM) iterative estimator and a closed form minimum mean square error (MMSE) estimator for use with the shifted modulation set. Since the non-data aided Cram\'er-Rao lower bound (CRLB) or the Bayesian CRLB (BCRLB) are intractable, we derive the data-aided CRLB and the data-aided BCRLB to assess the performance of these estimators. Simulations show that both estimators reach the performance of their corresponding bounds. The EM estimator has considerably lower computational complexity compared to the MMSE estimator for a large number of observations. The MMSE estimator performs well for number of observations as low as $N=6$, which is desirable for delay constrained systems. The simulation results also show the robustness of proposed estimators to increasing power of self-interference signal.