Blind Suppression of Nonstationary Diffuse Acoustic Noise Based on Spatial Covariance Matrix Decomposition

Abstract

We propose methods for blind suppression of nonstationary diffuse noise based on decomposition of the observed spatial covariance matrix into signal and noise parts. In modeling noise to regularize the ill-posed decomposition problem, we exploit spatial invariance (isotropy) instead of temporal invariance (stationarity). The isotropy assumption is that the spatial cross-spectrum of noise is dependent on the distance between microphones and independent of the direction between them. We propose methods for spatial covariance matrix decomposition based on least squares and maximum likelihood estimation. The methods are validated on real-world data.

Appendix: Derivation of the Update Rules in the M-Step of Maximum Likelihood Estimation

By setting the partial derivative of the Q-function (38) to zero, we have

$$ -M\frac{1}{{\phi^{x}_{t}}}+\text{tr}\biggl[({B}^{{x}})^{-1}\bigl\langle{x}_{t}{x}_{t}^{\textsf{H}}\bigr\rangle\biggr]\frac{1}{({\phi^{x}_{t}})^{2}}=0. $$

(43)

By solving this w.r.t. ${\phi ^{x}_{t}}$, we get [7]

$$ {\phi^{x}_{t}}=\frac{1}{M}\text{tr}\bigl[({B}^{{x}})^{-1}\hat{{\Phi}}^{{x}}_{t}\bigr]. $$

(44)

Here, we defined

$$ \hat{\Phi}^{x}_{t} \triangleq\bigl\langle{x}_{t}{x}_{t}^{\textsf{H}}\bigr\rangle_{p({x}_{t}|{y}_{t};{\Theta}^{\prime})}$$

(45)

$$ =\bigl({\Phi}^{{x}|{y}}_{t}\bigr)^{\prime}+\bigl({\mu}_{t}^{{x}|{y}}\bigr)^{\prime}\bigl({\mu}_{t}^{{x}|{y}}\bigr)^{\prime\textsf{H}}. $$

(46)

Next, partial differentiation w.r.t. B ^x gives

$$ -T({B}^{{x}})^{-1}+({B}^{{x}})^{-1}\Biggl(\sum\limits_{t=1}^{T}\frac{1}{{\phi^{x}_{t}}}\bigl\langle{x}_{t}{x}_{t}^{\textsf{H}}\bigr\rangle \Biggr)({B}^{{x}})^{-1}=0. $$

(47)

Solving this w.r.t. B ^x, we have [7]

$$ {B}^{{x}}=\frac{1}{T}\sum\limits_{t=1}^{T}\frac{1}{{\phi^{x}_{t}}}\hat{{\Phi}}^{{x}}_{t}. $$

(48)

The update rule for ${\Phi }^{{v}}_{t}$ depends on the explicit form of the matrix subspace 𝓥. In the following, we first show that for the class of 𝓥 satisfying

$$ {\Phi}^{{v}}_{t}\in\mathcal{V}\text{: positive definite} \Rightarrow ({\Phi}^{{v}}_{t})^{-1}\in\mathcal{V}, $$

(49)

we can derive a unified update rule. Clearly, the subspaces 𝓥_uncor, 𝓥_BND, 𝓥_real defined in Section 3 belong to the class. We then derive the update rule for 𝓥_coh, which does not belong to the class.

When 𝓥 satisfies (49), the terms of (38) depending on ${\Phi }^{{v}}_{t}$ can be rewritten as

$$ -U\log\det{\Phi}^{{v}}_{t}\\ $$

$$ -\text{tr}\biggl\{({\Phi}^{{v}}_{t})^{-1}\mathcal{P}\biggl[\bigl\langle({y}_{t}-{x}_{t})({y}_{t}-{x}_{t})^{\textsf{H}}\bigr\rangle\biggr]\biggr\}. $$

(50)

Here, 𝓟[⋅] denotes the orthogonal projection onto 𝓥 defined using the standard inner product $\langle {A},{B}\rangle \triangleq \text {tr}[{AB}]$ of ℋ:

$$ \mathcal{P}\bigl[{A}\bigr]=\sum\limits_{d=1}^{D}\text{tr}\bigl[{A}{Q}_{d}\bigr]{Q}_{d}. $$

(51)

Here, {Q _d }_{1 ≤ d ≤ D} is an orthonormal basis of 𝓥, and D denotes the dimension of 𝓥. The explicit form of Q _d depends on the choice of 𝓥, for which the readers are referred to [13, 14]. The term in 𝓟[⋅] in (50) generally has both components parallel and orthogonal to 𝓥. However, the latter vanishes owing to $({\Phi }^{{v}}_{t})^{-1}\in \mathcal {V}$, and hence (50). To derive ${\Phi }^{{v}}_{t}\in \mathcal {V}$ that maximizes (50), we forget the constraint ${\Phi }^{{v}}_{t}\in \mathcal {V}$ for the moment, and differentiate (50) w.r.t. ${\Phi }^{{v}}_{t}$. We have

$$ {\Phi}^{{v}}_{t}=\mathcal{P}\bigl[\hat{{\Phi}}^{{v}}_{t}\bigr], $$

(52)

where

$$ \hat{{\Phi}}^{{v}}_{t}\triangleq\bigl\langle({y}_{t}-{x}_{t})({y}_{t}-{x}_{t})^{\textsf{H}}\bigr\rangle_{p({x}_{t}|{y}_{t};{\Theta}^{\prime})}\\ $$

(53)

$$ \kern1.3pc=\bigl({\Phi}^{{x}|{y}}_{t}\bigr)^{\prime}+\bigl\{{y}_{t}-\bigl({\mu}_{t}^{{x}|{y}}\bigr)^{\prime}\bigr\}\bigl\{{y}_{t}-\bigl({\mu}_{t}^{{x}|{y}}\bigr)^{\prime}\bigr\}^{\textsf{H}}. $$

(54)

As is clear from the definition of 𝓟[⋅], (52) certainly satisfies ${\Phi }^{{v}}_{t}\in \mathcal {V}$.

Although we have derived (52) through partial differentiation, we can also derive it more intuitively as follows. Inverting the sign and ignoring a constant independent of ${\Phi }^{{v}}_{t}$, (50) becomes the following matrix Itakura-Saito divergence:

$$\begin{array}{@{}rcl@{}} D_{\text{IS}}\bigl(\mathcal{P}\bigl[\hat{{\Phi}}^{{v}}_{t}\bigr];{\Phi}^{{v}}_{t}\bigr)&\triangleq& \text{tr}\bigl\{\mathcal{P}\bigl[\hat{{\Phi}}^{{v}}_{t}\bigr]({\Phi}^{{v}}_{t})^{-1}\bigr\}\notag\\ &&-\log\det\bigl\{\mathcal{P}\bigl[\hat{{\Phi}}^{{v}}_{t}\bigr]({\Phi}^{{v}}_{t})^{-1}\bigr\}-M. \end{array} $$

(55)

Therefore, the maximization of (50) is equivalent to the minimization of (55). D _IS(⋅,⋅) is nonnegative, and equal to zero if and only if the two arguments are equal. Since $\mathcal {P}\bigl [\hat {{\Phi }}^{{v}}_{t}\bigr ]$ belong to the feasible set 𝓥 of ${\Phi }^{{v}}_{t}$, (55) is minimized when ${\Phi }^{{v}}_{t}=\mathcal {P}\bigl [\hat {{\Phi }}^{{v}}_{t}\bigr ]$.

Next we consider the case 𝓥 = 𝓥_coh. Substituting

$$ {\Phi}^{{v}}_{t}={\phi^{v}_{t}}{B}^{{v}} $$

(56)

into the Q-function (38), and differentiating it w.r.t. ${\phi ^{v}_{t}}$, we have, as in the derivation of (44),

$$ {\phi^{v}_{t}}=\frac{1}{M}\text{tr}\bigl[({B}^{{v}})^{-1}\hat{{\Phi}}^{{v}}_{t}\bigr]. $$

(57)