Single-carrier frequency domain equalization (SC-FDE) has drawn increasing researches because of its lower computational complexity and better performance of mitigating intersymbol interference (ISI) as compared with the time domain equalization. The practical SC-FDE systems, however, require the knowledge of the channel frequency domain response and the noise power to calculate filter coefficients of the frequency domain equalizer. Improper channel and noise estimations have been shown to result in error floor at high signal-to-noise ratio (SNR), which may impede the utilization of SC-FDE in the high-rate underwater acoustic (UWA) communications. In this paper, we discuss the possible influence of inaccurate noise estimation on the error floor at high SNR, and propose a joint estimation scheme for the channel time domain and frequency domain responses and also the noise power based on Chu sequence. The channel energy concentration region is determined by the threshold and the sparse channel impulse response (CIR) of UWA channels, used in denoising process of the CIR and the noise power computation. The bite error ratio performance of the SC-FDE variation versus the threshold is simulated via the UWA models, and accurate noise power estimation is shown to be desirable to improve the SC-FDE performance. Numerical examples also indicate that our joint estimation scheme can eliminate the error floor and improve the performance of SC-FDE at high SNR.