  We consider the problem of channel estimation for amplify-and-forward (AF)
two-way relay networks (TWRNs). Most works on this problem focus on pilot-based
approaches which impose a significant training overhead that reduces the
spectral efficiency of the system. To avoid such losses, this work proposes
blind channel estimation algorithms for AF TWRNs that employ constant-modulus
(CM) signaling. Our main algorithm is based on the deterministic maximum
likelihood (DML) approach. Assuming M-PSK modulation, we show that the
resulting estimator is consistent and approaches the true channel with high
probability at high SNR for modulation orders higher than 2. For BPSK, however,
the DML performs poorly and we propose an alternative algorithm that performs
much better by taking into account the BPSK structure of the data symbols. For
comparative purposes, we also investigate the Gaussian maximum-likelihood (GML)
approach which treats the data symbols as Gaussian-distributed nuisance
parameters. We derive the Cramer-Rao bound and use Monte-Carlo simulations to
investigate the mean squared error (MSE) performance of the proposed
algorithms. We also compare the symbol-error rate (SER) performance of the DML
algorithm with that of the training-based least-squares (LS) algorithm and
demonstrate that the DML offers a superior tradeoff between accuracy and
spectral efficiency.
