This paper develops a linear minimum mean-square error (LMMSE) channel estimator for single and multicarrier systems that takes advantage of the mutual coupling in antenna arrays. We model the mutual coupling through multiport networks and express the single-user multiple-input multiple-output (MIMO) communication channel in terms of the impedance and scattering parameters of the antenna arrays. We put forward a novel scattering description of the communication channel which requires only the scattering parameters of the arrays as well as the terminated far-field embedded antenna patterns. In multi-antenna single-carrier systems under frequency-flat channels, we show that neglecting the mutual coupling effects leads to inaccurate characterization of the channel and noise correlations. We also extend the analysis to frequency-selective multicarrier channels wherein we further demonstrate that the coupling between the antenna elements within each array increases the number of resolvable channel taps. Standard LMMSE estimators based on existing inaccurate channel models become sub-optimal when applied to the new physically consistent model. We hence develop a new LMMSE estimator that calibrates the coupling and optimally estimates the MIMO channel. It is shown that appropriately accounting for mutual coupling through the developed physically consistent model leads to remarkable performance improvements both in terms of channel estimation accuracy and achievable rate. We demonstrate those gains in a rich-scattering environment using a connected array of slot antennas both at the transmitter and receiver sides.