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Orthogonal frequency division multiplexing (OFDM) systems require the knowledge of signal-to-noise ratio (SNR) at the receiver in order to optimize the system performance. In general, the noise variance is required by the SNR estimator and the knowledge of noise variance also improves the performance of carrier frequency offset (CFO) and channel state estimations. This paper commences by investigating the maximum likelihood noise variance estimator (ML-NVE) in OFDM systems with imperfect synchronization. Theoretical derivations show that both the mean and the variance of the ML-NVE are functions of the CFO and the advanced timing offset (ATO). In particular, it is demonstrated that the mean of the ML-NVE converges to a CFO-independent value as the ATO increases, while the variance converges to a CFO-dependent value. By exploiting the analytical results of the ML-NVE, we further investigate the SNR estimator. It is mathematically demonstrated that the SNR estimate is biased in the presence of CFO and ATO. Finally, the validity of the theoretical observations is confirmed by simulation experiments.