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In this paper, a novel low-complexity scheme is proposed for cyclic-prefix based OFDM systems to reduce inter-carrier interference (ICI) caused by time-varying channel. Under the assumption that channel delay spread is smaller than the guard interval (GI) of an OFDM symbol, the suitably selected inter-symbol interference (ISI) free received signals are exploited through compensating the corresponding linear phases and taking the average in the frequency domain. Therefore, some ICI terms can be automatically cancelled with less number of FFT operations. Moreover, the theoretical results of bit error rate performance are derived. The simulation results are shown that the ICI effect can be significantly removed by the proposed scheme for high-mobility OFDM systems when the ISI-free region q is adequate. Meanwhile, the proposed scheme has the advantages of increasing the mobile speed up to N/(N-</) times of that of the OFDM systems with N sub-carriers. For example, the proposed scheme can give two times mobile speed improvement when q =N/2 and GI=N/4, the extended length of GI is only required extra N/4.