Maximum Likelihood Estimation Based Complex-Valued Robust Chinese Remainder Theorem and Its Fast Algorithm

In this paper, we investigate complex-valued Chinese remainder theorem (C-CRT) with erroneous remainders, where the moduli are Gaussian integers and the errors follow wrapped complex Gaussian distributions. Based on the existing real-valued CRT utilizing maximum likelihood estimation (MLE), we propose a fast MLE-based C-CRT (MLE C-CRT). The proposed algorithm requires only 2L searches to obtain the optimal estimate of the common remainder, where L is the number of moduli. Once the common remainder is estimated, the complex number can be determined using the C-CRT. Furthermore, we obtain a necessary and sufficient condition for the fast MLE C-CRT to achieve robust estimation. Finally, we apply the proposed algorithm to a multi-channel self-reset analog-to-digital converter (ADC) system with Gaussian integers as moduli, which enables the recovery of high dynamic range complex-valued bandlimited signals at the Nyquist sampling rate. The results demonstrate that the proposed algorithm outperforms the existing methods.