  In molecular communication (MC) systems, the \textit{expected} number of
molecules observed at the receiver over time after the instantaneous release of
molecules by the transmitter is referred to as the channel impulse response
(CIR). Knowledge of the CIR is needed for the design of detection and
equalization schemes. In this paper, we present a training-based CIR estimation
framework for MC systems which aims at estimating the CIR based on the
\textit{observed} number of molecules at the receiver due to emission of a
\textit{sequence} of known numbers of molecules by the transmitter. Thereby, we
distinguish two scenarios depending on whether or not statistical channel
knowledge is available. In particular, we derive maximum likelihood (ML) and
least sum of square errors (LSSE) estimators which do not require any knowledge
of the channel statistics. For the case, when statistical channel knowledge is
available, the corresponding maximum a posteriori (MAP) and linear minimum mean
square error (LMMSE) estimators are provided. As performance bound, we derive
the classical Cramer Rao (CR) lower bound, valid for any unbiased estimator,
which does not exploit statistical channel knowledge, and the Bayesian CR lower
bound, valid for any unbiased estimator, which exploits statistical channel
knowledge. Finally, we propose optimal and suboptimal training sequence designs
for the considered MC system. Simulation results confirm the analysis and
compare the performance of the proposed estimation techniques with the
respective CR lower bounds.
