New lower bounds for noncoherent channel estimation and ML performance

We consider the optimal performance of noncoherent channel estimation, that is, where the codebook is known to the receiver but the actual transmitted data is not. It is well known that when training data is known to the receiver, the minimum variance of the channel estimation error for unbiased channel estimation is bounded by the Cramer-Rao lower bound. However, in the noncoherent case, where joint estimation of both a continuous channel and discrete data is required, the Cramer-Rao bound is not applicable. We derive a new bound for this mixed multiple parameter estimation problem for flat fading channels, based on the Hammersley-Chapman-Robbins bound for restricted parameters. We show that the new noncoherent bound asymptotically approaches the Cramer-Rao bound with increasing SNR and sequence length. As an example we consider channel estimation for BPSK over a positive real-valued channel. We show that the noncoherent ML detector is asymptotically unbiased and achieves the lower bound with increasing SNR. We also observe that for moderate SNR the noncoherent ML estimator can actually outperform the optimal coherent ML estimator.