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We propose intra-symbol carrier phase recovery (IS-CPR) for reduced-guard-interval (RGI) CO-OFDM in order to compensate for the intra-symbol phase shift (ISPS) between subcarriers that is caused by the dispersion-enhanced phase noise (DEPN). We begin by proposing a pre-emphasized pilot subcarrier (PEPS) approach to reduce the pilot subcarrier overhead for the following IS-CPR algorithms. Then, we show a statistical analysis of the DEPN-induced ISPS between subcarriers within one symbol, which is related to the accumulated chromatic dispersion (CD). Next, three algorithms are proposed for IS-CPR including maximum-likelihood (ML) phase estimation, digital phase-locked loop (DPLL), and feedforward carrier recovery (FFCR) employing either the Mth power scheme in case of QPSK modulation or the QPSK partitioning scheme for the 16-QAM case. The performance and complexity of these algorithms are compared. Through simulations, we show that in comparison to conventional common phase error (CPE) compensation, IS-CPR significantly improves the linewidth tolerance at 1 dB signal-to-noise ratio (SNR) penalty for a bit error rate (BER) = 10-3 from 300 kHz to 2 MHz for 112 Gb/s systems (28 Gbaud QPSK) at 3200 km transmission distance, and from 70 kHz to 550 kHz for 448 Gb/s (56 Gbaud 16-QAM) systems at 1600 km transmission distance.