ML Estimation and crbs for reverberation, speech, and noise psds in rank-deficient noise field

Speech communication systems are prone to performance degradation in reverberant and noisy acoustic environments. Dereverberation and noise reduction algorithms typically require several model parameters, e.g. The speech, reverberation and noise power spectral densities (PSDs). A commonly used assumption is that the noise PSD matrix is known. However, in practical acoustic scenarios, the noise PSD matrix is unknown and should be estimated along with the speech and reverberation PSDs. In this article, we consider the case of rank-deficient noise PSD matrix, which arises when the noise signal consists of multiple directional noise sources, whose number is less than the number of microphones. We derive two closed-form maximum likelihood estimators (MLEs). The first is a non-blocking-based estimator which jointly estimates the speech, reverberation and noise PSDs, and the second is a blocking-based estimator, which first blocks the speech signal and then jointly estimates the reverberation and noise PSDs. Both estimators are analytically compared and analyzed, and mean square errors (MSEs) expressions are derived. Furthermore, Cramér-Rao Bounds (CRBs) on the estimated PSDs are derived. The proposed estimators are examined using both simulation and real reverberant and noisy signals, demonstrating the advantage of the proposed method compared to competing estimators.