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.Comment: to be appeared in IEEE Transactions on Communications. arXiv admin note: text overlap with arXiv:1510.0861

The (p-R-phenyl)dimesitylboranes (R=Me(2)N, MeO, MeS, Br, I), (p-R-phenylethynyl)dimesitylboranes (R=Me(2)N, MeO, MeS, H), (E)-[2-(p-R-phenyl)ethenyl]dimesitylboranes (R=Me(2)N, H(2)N, MeO, MeS, H, CN, NO(2)), (E)-[2-(2-thienyl)ethenyl]dimesitylborane, and (E)-[2-(o-carboranyl)ethenyl]dimesitylborane have been prepared through the reaction of the appropriate p-R-phenyl- and p-R-phenylethynyllithium reagents with dimesitylboron fluoride and by hydroboration of the appropriate p-R-phenylacetylene, 2-ethynylthiophene, and o-ethynylcarborane with dimesitylborane. Their UV/Vis absorption and emission spectra have been recorded in a range of solvents with the fluorescence maxima of the donor-substituted compounds in particular exhibiting large bathochromic shifts in highly polar solvents, indicative of charge transfer leading to large dipole moments in the excited state. The molecular structures of the (p-R-phenyl)dimesitylboranes (R=Me(2)N, MeO, MeS, Br, I), the (E)-[2-(p-R-phenyl)ethenyl]dimesitylboranes (R=Me(2)N, H(2)N MeO, MeS, H), (p-R-phenylethynyl)dimesitylborane (R=Me(2)N), and (E)-[2-(2-thienyl)ethenyl]dimesitylborane, which have been determined from single-crystal X-ray diffraction measurements, offer evidence of increased conjugation in the ground state with increased donor strength of the R substituent. Their first- and second-order molecular hyperpolarizabilities have been obtained from EFISH and THG measurements, the first-order hyperpolarizabilities being largest for the strongest R-substituent donors. AM1 calculations have been performed on these compounds, showing reasonable agreement with the experimentally obtained bond lengths and hyperpolarizabilities, as well as on several related hypothetical compounds containing multiple C==C bonds, most of which are proposed to have even larger hyperpolarizabilities.