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


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.

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  1. 1.

    Boll, S.F. (1979). Suppression of acoustic noise in speech using spectral subtraction. IEEE Transactions ASSP, 27(2), 113–120.

    Article  Google Scholar 

  2. 2.

    Martin, R. (1994). Spectral subtraction based on minimum statistics. In Proclamation EUSIPCO, (pp. 1982–1185).

  3. 3.

    Ephraim, Y., & Malah, D. (1984). Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator. IEEE Transactions ASSP, 32(6), 1109–1121.

    Article  Google Scholar 

  4. 4.

    Dudgeon, D.E., & Johnson, D.H. (1993). Array signal processing: concepts and techniques. Prentice Hall, Englewood Cliffs.

  5. 5.

    Brandstein, M., & Ward, D. (2001). Microphone arrays: signal processing techniques and applications. Berlin, Heidelberg: Springer.

  6. 6.

    Itakura, F., & Saito, S. (1968). Analysis synthesis telephony based on the maximum likelihood method. In Report of 6th International Congress on Acoustics, (pp. 17–20).

  7. 7.

    Duong, N.Q.K., Vincent, E., Gribonval, R. (2010). Under-determined reverberant audio source separation using a full-rank spatial covariance model. IEEE Transactions ASLP, 18(7), 1830–1840.

    Google Scholar 

  8. 8.

    Nakatani, T., Yoshioka, T., Kinoshita, K., Miyoshi, M., Juang, B.-H. (2010). Speech dereverberation based on variance-normalized delayed linear prediction. IEEE Transactions ASLP, 18(7), 1717–1731.

    Google Scholar 

  9. 9.

    Vincent, E., Bertin, N., Gribonval, R., Bimbot, F. (2014). From blind to guided audio source separation. IEEE Signal Proclamation Magazine, 31(3).

  10. 10.

    Sawada, H., Kameoka, H., Araki, S., Ueda, N. (2013). Multichannel extensions of non-negative matrix factorization with complex-valued data. IEEE Transactions ASLP, 21(5), 971–982.

    Google Scholar 

  11. 11.

    Simmer, K.U., Bitzer, J., Marro, C. (2001). Post-filtering techniques, In M. Brandstein & D. Ward (Eds.), Microphone Arrays (pp. 39–60). Berlin, Heidelberg: Springer.

  12. 12.

    Doclo, S., & Moonen, M. (2002). GSVD-based optimal filtering for single and multimicrophone speech enhancement. IEEE Transactions SP, 50(9), 2230–2244.

    Article  Google Scholar 

  13. 13.

    Ito, N. (2012). Robust microphone array signal processing against diffuse noise, Ph.D. thesis, The University of Tokyo.

  14. 14.

    Ito, N., Vincent, E., Ono, N., Sagayama, S. (2013). General algorithms for estimating spectrogram and transfer functions of target signal for blind suppression of diffuse noise. In Proceedings of the IEEE international workshop on machine learning for signal processing (MLSP).

  15. 15.

    Ito, N., Shimizu, H., Ono, N., Sagayama, S. (2011). Diffuse noise suppression using crystal-shaped microphone arrays. IEEE Transactions ASLP, 19(7), 2101–2110.

    Google Scholar 

  16. 16.

    Zelinski, R. (1988). A microphone array with adaptive post-filtering for noise reduction in reverberant rooms. In Proclamation ICASSP (pp. 2578–2581).

  17. 17.

    McCowan, I.A., & Bourlard, H. (2003). Microphone array post-filter based on noise field coherence. IEEE Transactions SAP, 11(6), 709–716.

    Google Scholar 

  18. 18.

    Ito, N., Ono, N., Sagayama, S (2010). Designing the Wiener post-filter for diffuse noise suppression using imaginary parts of inter-channel cross-spectra, In Proclamation ICASSP (pp. 2818–2821).

  19. 19.

    Ito, N., Vincent, E., Ono, N., Gribonval, R., Sagayama, S. (2010). Crystal-MUSIC: Accurate localization of multiple sources in diffuse noise environments using crystal-shaped microphone arrays. In Proclamation of LVA/ICA, lecture notes in computer science (Vol. , pp. 81–88).

  20. 20.

    Srebro, N., & Jaakkola, T. (2003). Weighted low-rank approximations. In Proceedings of the international conference on machine learning (ICML) (pp. 720–727). AAAI Press.

  21. 21.

    Toh, K., & Yun, S. (2010). An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems. Pacific Journal of Optimization, 6(3), 615–640.

    MATH  MathSciNet  Google Scholar 

  22. 22.

    Pham, D.-T., & Cardoso, J.-F. (2001). Blind separation of instantaneous mixtures of non stationary sources. IEEE Transactions SP, 49(9), 1837–1848.

    Article  MathSciNet  Google Scholar 

  23. 23.

    Ozerov, A., & Févotte, C. (2010). Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation. IEEE Transactions ASLP, 18(3), 550–563.

    Google Scholar 

  24. 24.

    Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). “Maximum likelihood from incomplete data via the EM algorithm,”. Journal of the Royal Statistical Society: Series B (Methodological), 39(1), 1–38.

    MATH  MathSciNet  Google Scholar 

  25. 25.

    Kurematsu, A., Takeda, K., Sagisaka, Y., Katagiri, S., Kuwabara, H., Shikano, K. (1990). ATR Japanese speech database as a tool of speech recognition and synthesis. Speech Communications, 9(4), 357–363.

    Article  Google Scholar 

  26. 26.

    Ono, N. (2011). Stable and fast update rules for independent vector analysis based on auxiliary function technique. In Proceedings of IEEE workshop applications of signal processing audio acoustics (WASPAA) (pp. 189–192).

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Correspondence to Nobutaka Ito.

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

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

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

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

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

Here, we defined

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

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

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

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

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

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

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

we can derive a unified update rule. Clearly, the subspaces 𝓥uncor, 𝓥BND, 𝓥real defined in Section 3 belong to the class. We then derive the update rule for 𝓥coh, which does not belong to the class.

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

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

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

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

Here, {Q d }1 ≤ dD 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

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


$$ \hat{{\Phi}}^{{v}}_{t}\triangleq\bigl\langle({y}_{t}-{x}_{t})({y}_{t}-{x}_{t})^{\textsf{H}}\bigr\rangle_{p({x}_{t}|{y}_{t};{\Theta}^{\prime})}\\ $$
$$ \kern1.3pc=\bigl({\Phi}^{{x}|{y}}_{t}\bigr)^{\prime}+\bigl\{{y}_{t}-\bigl({\mu}_{t}^{{x}|{y}}\bigr)^{\prime}\bigr\}\bigl\{{y}_{t}-\bigl({\mu}_{t}^{{x}|{y}}\bigr)^{\prime}\bigr\}^{\textsf{H}}. $$

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:

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

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 = 𝓥coh. Substituting

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

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

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

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Ito, N., Vincent, E., Nakatani, T. et al. Blind Suppression of Nonstationary Diffuse Acoustic Noise Based on Spatial Covariance Matrix Decomposition. J Sign Process Syst 79, 145–157 (2015).

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  • Noise suppression
  • Diffuse noise
  • Spatial covariance matrix
  • Maximum likelihood estimation
  • Least squares estimation