In this paper, an extended best linear unbiased estimator (EBLUE) based on a periodic training sequence is proposed and investigated for frequency offset estimation in orthogonal frequency division multiplexing (OFDM) systems. The structure of EBLUE is general and flexible so it adapts to different complexity constraints, and is attractive in practical implementation. Performance analysis and design strategy of EBLUE are provided to realize the best tradeoff between performance and complexity. Moreover, closed-form results of both weight and performance make EBLUE even more attractive in practical implementation. Both the performance and complexity of EBLUE are compared with other proposals and the Cramer-Rao lower bound (CRLB) to demonstrate the merit of EBLUE.