In this note, a moving horizon estimation (MHE) strategy for detectable linear systems is proposed. Like the idea of "pre-stabilizing" model-predictive control, the states are estimated by a forward simulation with a pre-estimating observer in the MHE formulation. Compared with standard linear MHE approaches, it has more degrees of freedom to optimize the noise filtering. Tuning parameters are chosen to minimize the effects of measurement noise and model errors, which is implemented by finding tightest estimation error bounds. The performance of the proposed observer is demonstrated on one linear discrete-time example.