How much coherent interval should be dedicated to non-redundant diagonal precoding for blind channel estimation in single-carrier block transmission?

Transmit precoding is a key technique for facilitating blind channel estimation at the receiver but the impact due to precoding on the channel capacity is scarcely addressed in the literature. In this paper we consider the single-carrier block transmission with cyclic prefix, in which a recently proposed diagonal-precoding assisted blind channel estimation scheme via covariance matching is adopted to acquire the channel information. It is shown that, when perfect channel knowledge is available at the receiver, the optimal noise resistant precoder proposed in the literature incurs the worst-case capacity penalty. When the coherent interval is finite, channel mismatch occurs due to finite-sample covariance matrix estimation. Thus, we aim to determine how much of the coherent interval should be dedicated to precoding in order to trade channel estimation accuracy for the maximal capacity. Toward this end, we leverage the matrix perturbation theory to derive a closed-form capacity measure which explicitly takes account of the channel uncertainty in the considered blind estimation setup. Such a capacity metric is seen to be a complicated function of the precoding interval. To facilitate analysis, an approximate formula for the derived capacity measure is further given. This allows us to find a closed-form estimate of the capacity-maximizing precoding time fraction, and can also provide insights into the optimal tradeoff between channel estimation accuracy and achievable capacity. Numerical simulations are used for evidencing the proposed analytic study.