Analysis of Eigenvalue Decomposition-Based Late Reverberation Power Spectral Density Estimation

Abstract

Many speech dereverberation techniques require an estimate of the late reverberation power spectral density (PSD). State-of-the-art multi-channel methods for estimating the late reverberation PSD typically rely on 1) an estimate of the relative transfer functions (RTFs) of the target signal, 2) a model for the spatial coherence matrix of the late reverberation, and 3) an estimate of the reverberant speech or reverberant and noisy speech PSD matrix. The RTFs, the spatial coherence matrix, and the speech PSD matrix are all prone to modeling and estimation errors in practice, with the RTFs being particularly difficult to estimate accurately, especially in highly reverberant and noisy scenarios. Recently, we proposed an eigenvalue decomposition (EVD)-based late reverberation PSD estimator which does not require an estimate of the RTFs. In this paper, this EVD-based PSD estimator is further analyzed and its estimation accuracy and computational complexity is analytically compared to a state-of-the-art maximum likelihood (ML)-based PSD estimator. It is shown that for perfect knowledge of the RTFs, spatial coherence matrix, and reverberant speech PSD matrix, the ML-based and EVD-based PSD estimates are both equal to the true late reverberation PSD. In addition, it is shown that for erroneous RTFs but perfect knowledge of the spatial coherence matrix and reverberant speech PSD matrix, the MLbased PSD estimate is larger than or equal to the true late reverberation PSD, whereas the EVD-based PSD estimate is obviously still equal to the true late reverberation PSD. Finally, it is shown that when modeling and estimation errors occur in all quantities, the ML-based PSD estimate is larger than or equal to the EVD-based PSD estimate. Simulation results for several realistic acoustic scenarios demonstrate the advantages of using the EVD-based PSD estimator in a multi-channel Wiener filter, yielding a significantly better performance than the ML-based PSD estimator.

Integrated Sidelobe Cancellation and Linear Prediction Kalman Filter for Joint Multi-Microphone Speech Dereverberation, Interfering Speech Cancellation, and Noise Reduction

“…Based on the generalized eigenvectors and generalized eigenvalues, Ψ y (l) may be decomposed into a diffuse component, cf. also the diffuse PSD estimator in [39], and a factorized early rank-N component Ψ xe (l) = Ψ 1 /2 xe (l)Ψ H /2 xe (l). A temporally smooth estimateΨ y|sm (l) of Ψ y (l) itself is obtained from the microphone signals by recursively averaging y H (l)y(l).…”

Square Root-Based Multi-Source Early PSD Estimation and Recursive RETF Update in Reverberant Environments by Means of the Orthogonal Procrustes Problem

“…An alternative, but mathematically equivalent formulation to the GEVD in (51) is given by the EVD of the pre-whitened matrix Ψ [14], [18], [32], which is defined by Ψ x P = P Λ x . By comparison with (51), we find Λ x = Λ x and P = Γ H /2 P, provided that the respective (generalized) eigenvalues are sorted in the same order, and the (generalized) eigenvectors scaled accordingly.…”

“…Then, inserting Ψ x = Ψ xe + Ψ x with Ψ x = ϕ x Γ, cf. (9) and (14), into (54) while making use of (53) yields…”

“…2) Data Generation: LetΨ x be available and free of modeling and estimation errors, i.e. we haveΨ x = Ψ x with Ψ x adhering to (9)- (14). We generate Ψ x based on assumed geometric and physical properties.…”

“…In order to estimate these PSDs, some parameters of the correlation matrix model are assumed to be known or estimated beforehand, such as the direction(s) of arrival (DoA(s)) or the relative early transfer function(s) (RETF(s)) associated to the source(s) [5]- [13], or the spatial coherence matrix of the noise or the late reverberant component, where in particular the latter is commonly modeled as a spatially diffuse sound field [6]- [8], [10]- [16]. It should be noted that majority of these approaches consider a single source [5], [8], [10]- [14], while only some consider multiple sources [6], [7], [9], [15], which is the focus of this paper.…”