This paper explores the performance of time-domain synchronous orthogonal frequency-division multiplexing (TDS-OFDM) systems operated under rapidly time-varying mobile channels. Since a rapidly time-varying channel contains more unknown channel coefficients than the number of observations, the mobile channel can conveniently be modeled with the discrete Legendre polynomial basis expansion model to reduce the number of unknowns. The linear minimum mean square error (LMMSE) estimate can be exploited for channel estimation on inter-block-interference-free received signal samples owing to transmitting pseudo-noise (PN) sequences. In conventional TDS-OFDM systems, the channel estimation performance is limited due to estimating channel responses only from the beginning part of the channel. Therefore, a new system model named "partitioned TDS-OFDM system" is proposed to improve the system performance by inserting multiple PN sequences to the middle and end parts of the channel as well. In addition to providing the reconstruction error performance, Bayesian Cramer-Rao lower hound is derived analytically. Also, the LMMSE-based symbol detection is employed. To alleviate the negative effects of inter-carrier-interference (ICI) occuring in mobile channels, ICI cancellation is applied to enhance the detection performance. The simulation results demonstrate that the proposed TDS-OFDM system is superior to the conventional system and its corresponding performance is able to approach the achievable lower performance bound.