  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 paper, we consider the
case of rank-deficient noise PSD matrix, which arises when the noise signal
consists of multiple directional interference 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, Cramer-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.
