This paper presents a novel preamble-aided method for joint estimation of timing, carrier frequency offset, and channel for orthogonal frequency division multiplexing systems that operate in high-mobility situations. Basis expansion modeling (BEM) that captures the time variations of the channel is used to reduce the number of unknown channel parameters. The BEM coefficients along with timing and frequency offsets are estimated by using a maximum likelihood approach. An efficient algorithm is then proposed for reducing the computational complexity of the joint estimation. The complexity of the new method is assessed in terms of the number of multiplications. The mean square estimation error of the proposed method is evaluated in comparison with previous methods and the hybrid Cramer-Rao lower bound, indicating a remarkable performance improvement by the new method.