Orthogonal frequency-division multiplexing (OFD-M) and multi-cell architecture are widely adopted in current high speed train (HST) systems for providing high data rate wireless communications. In this paper, a typical multi-antenna OFDM HST communication system with multi-cell architecture is considered, where the inter-carrier interference (ICI) caused by high mobility and multi-cell interference (MCI) are both taken into consideration. By exploiting the train position information, a new position-based interference elimination method is proposed to eliminate both the MCI and ICI for a general basis expansion model (BEM). We show that the MCI and ICI can be completely eliminated by the proposed method to get the ICI-free pilots at each receive antenna. In addition, for the considered multi-cell HST system, we develop a low-complexity compressed channel estimation method and consider the optimal pilot pattern design. Both the proposed interference elimination method and the optimal pilot pattern are robust to the train speed and position,as well as the multi-cell multi-antenna system. Simulation results demonstrate the benefits and robustness of the proposed method in the multi-cell HST system.