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A new carrier frequency offset estimation scheme in orthogonal frequency-division multiplexing (OFDM) systems is proposed in this paper. Both the carrier frequency offset acquisition and tracking are based on a fixed-length training-symbol-block, which consists of multiple small identical training symbols. When each training symbol is shortened, the number of training symbols in the training-symbol-block should be increased accordingly to keep the total training-symbol-block length fixed. The proposed scheme extends Moose's estimator, where the estimation error is only dependent on total training symbol energy and cannot be reduced any more, once the total training symbol energy is determined. The proposed scheme can shorten each training symbol in a training-symbol block and select an appropriate estimator simultaneously, which can lead to further reduction of estimation error and increase of acquisition range, even with the total training-symbol-block energy being fixed. Performance analyzes for the proposed scheme in both the additive white Gaussian noise channel (AWGN) and the multipath channel are also presented in this paper. All estimators in the proposed scheme are conditionally unbiased, and simulation results demonstrate that they can work well both in the multipath channel and in the AWGN channel.