#
Dual-Channel Speech Enhancement Based on Extended Kalman Filter Relative Transfer Function Estimation^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. System Overview

## 3. Extended Kalman Filter-Based Relative Transfer Function Estimation

- Dynamic model for the RTF ${\mathbf{a}}_{21,t}$: We assume that the state vector ${\mathbf{a}}_{21,t}$ is a random walk stochastic process which can be expressed as$${\mathbf{a}}_{21,t}={\mathbf{a}}_{21,t-1}+{\mathbf{w}}_{t},$$$${\mathbf{w}}_{t}\sim \mathcal{N}\left(\mathbf{0},\mathbf{Q}\right)$$
- Observation model for the noisy speech at the secondary microphone, ${\mathbf{y}}_{2,t}$: It is defined using the distortion model in (8) as$$\begin{array}{cc}{\mathbf{y}}_{2,t}\hfill & =\mathbf{h}\left({\mathbf{a}}_{21,t},{\mathbf{n}}_{1,t};{\mathbf{y}}_{1,t}\right)+{\mathbf{n}}_{2,t}\hfill \\ \hfill & =\left(\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left({Y}_{1,t}^{r}-{N}_{1,t}^{r}\right)+\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left({Y}_{1,t}^{i}-{N}_{1,t}^{i}\right)\right){\mathbf{a}}_{21,t}+{\mathbf{n}}_{2,t},\hfill \end{array}$$$$\left[\begin{array}{c}{\mathbf{n}}_{1,t}\\ {\mathbf{n}}_{2,t}\end{array}\right]\sim \mathcal{N}\left(\mathbf{0},\left[\begin{array}{cc}{\mathsf{\Sigma}}_{{n}_{11},t}& {\mathsf{\Sigma}}_{{n}_{12},t}\\ {\mathsf{\Sigma}}_{{n}_{21},t}& {\mathsf{\Sigma}}_{{n}_{22},t}\end{array}\right]\right),$$

- The prediction step, using the model (12), is applied for every frame $t>0$,$${\widehat{\mathbf{a}}}_{21,t|t-1}={\widehat{\mathbf{a}}}_{21,t-1},$$$${\mathbf{P}}_{t|t-1}={\mathbf{P}}_{t-1}+\mathbf{Q},$$$${\mathbf{P}}_{t}=E\left[\left({\mathbf{a}}_{21,t}-{\widehat{\mathbf{a}}}_{21,t}\right){\left({\mathbf{a}}_{21,t}-{\widehat{\mathbf{a}}}_{21,t}\right)}^{\top}\right],$$$${\mathbf{P}}_{t|t-1}=E\left[\left({\mathbf{a}}_{21,t}-{\widehat{\mathbf{a}}}_{21,t|t-1}\right){\left({\mathbf{a}}_{21,t}-{\widehat{\mathbf{a}}}_{21,t|t-1}\right)}^{\top}\right]$$
- The updating step is applied to correct the previous estimation with the observations ${\mathbf{y}}_{1,t}$ and ${\mathbf{y}}_{2,t}$ (whose relationship is given by Equation (14)),$${\widehat{\mathbf{a}}}_{21,t}={\widehat{\mathbf{a}}}_{21,t|t-1}+{\mathbf{K}}_{t}\left({\mathbf{y}}_{2,t}-{\mathit{\mu}}_{y,t}\right),$$$${\mathbf{P}}_{t}={\mathbf{P}}_{t|t-1}-{\mathbf{K}}_{t}{\mathbf{S}}_{y,t}{\mathbf{K}}_{t}^{\top},$$$${\mathbf{K}}_{t}={\mathbf{C}}_{ay,t}{\mathbf{S}}_{y,t}^{-1}$$$${\mathit{\mu}}_{y,t}=E\left[{\mathbf{y}}_{2,t}\right],$$$${\mathbf{S}}_{y,t}=E\left[\left({\mathbf{y}}_{2,t}-{\mathit{\mu}}_{y,t}\right){\left({\mathbf{y}}_{2,t}-{\mathit{\mu}}_{y,t}\right)}^{\top}\right],$$$${\mathbf{C}}_{ay,t}=E\left[\left({\mathbf{a}}_{21,t}-{\widehat{\mathbf{a}}}_{21,t|t-1}\right){\left({\mathbf{y}}_{2,t}-{\mathit{\mu}}_{y,t}\right)}^{\top}\right]$$

#### 3.1. Vector Taylor Series Approximation

#### 3.2. A Priori RTF Statistics

## 4. Speech Presence Probability-Based Noise Statistics Estimation

#### 4.1. A Posteriori SPP Estimation

- The estimation of the RTF presented in the previous section is only accurate in time-frequency bins where speech is present. The a posteriori SPP indicates those bins where speech presence is more likely. Therefore, in our implementation we only update the eKF in those bins where ${p}_{x}(t,f)>{p}_{\mathrm{thr}}$, with ${p}_{\mathrm{thr}}$ being a predefined probability threshold. Otherwise, the previous values are preserved.
- The postfiltering performance can be improved if additional information about SPP is provided, as shown later in Section 5.

- Initialization: Estimate the noisy SCM with a recursive updating,$${\widehat{\mathsf{\Phi}}}_{Y}(t,f)=\tilde{\alpha}{\widehat{\mathsf{\Phi}}}_{Y}(t-1,f)+\left(1-\tilde{\alpha}\right)\mathbf{Y}(t,f){\mathbf{Y}}^{H}(t,f),$$
- 2nd iteration: Re-estimate ${p}_{x}(t,f)$ using now ${\widehat{\mathsf{\Phi}}}_{N}(t,f)$ in (49). Finally, re-estimate ${\widehat{\mathsf{\Phi}}}_{N}(t,f)$ using ${p}_{x}(t,f)$.

#### 4.2. A Priori SAP Estimation

## 5. Postfiltering Approaches for Dual-Microphone Smartphones

#### 5.1. Parametric Wiener Filtering

#### 5.2. Optimally Modified Log-Spectral Amplitude Estimator

#### 5.3. Single-Channel Speech and Noise PSD Estimators

#### 5.3.1. Power Level Difference-Based Estimation

#### 5.3.2. Minimum Variance Distortionless Response-Based Estimation

## 6. Experimental Evaluation

- The Perceptual Evaluation Speech Quality (PESQ) [37] metric is utilized to evaluate the speech quality of the enhanced speech signal. This metric gives a mean opinion score between one and five. The higher the PESQ values, the better the speech quality.
- The Short-Time Objective Intelligibility (STOI) [38] metric is intended to evaluate the speech intelligibility of the enhanced speech signal. The resulting score is a value between zero and one. The higher the STOI value, the better the speech intelligibility.

#### 6.1. Experimental Results: Performance of SAP Estimators

#### 6.2. Experimental Results: Performance of RTF Estimators

#### 6.3. Experimental Results: Performance of Single-Channel Clean Speech PSD Estimators

#### 6.4. Experimental Results: Performance of Postfiltering Approaches

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Predefined acoustic environments: each environment combines a reverberation environment with a given noise.

Reverberation | Noise (Test Only) |
---|---|

(A) No reverb. | Car, Street, Pedestrian street |

(B) Low | Bus, Cafe |

(C) Medium | Babble, Bus station |

(D) High | Mall |

N° Utterances | N° Speakers | |
---|---|---|

Training set | 700 | 440 |

Test set | 150 | 93 |

**Table 3.**Perceptual Evaluation Speech Quality (PESQ) and Short-Time Objective Intelligibility (STOI) results for different speech absence probability (SAP) estimators when combined with speech presence probability (SPP)-based extended Kalman filter - relative transfer function (eKF-RTF) estimation for Minimum Variance Distortionless Response (MDVR) beamforming. Results are broken down by both signal-to-noise ratio (SNR) and device placement.

Place. | Method | PESQ | STOI (%) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SNR (dB) | SNR (dB) | ||||||||||||

20 | 15 | 10 | 5 | 0 | −5 | 20 | 15 | 10 | 5 | 0 | −5 | ||

CT | Noisy | 2.26 | 1.80 | 1.46 | 1.23 | 1.11 | 1.07 | 95.36 | 91.01 | 83.82 | 74.05 | 62.48 | 51.46 |

eKF-MCRA | 2.28 | 1.84 | 1.49 | 1.26 | 1.13 | 1.08 | 95.58 | 91.91 | 85.37 | 75.57 | 63.65 | 52.20 | |

eKF-CDR | 2.42 | 2.00 | 1.63 | 1.35 | 1.18 | 1.12 | 93.37 | 89.66 | 83.13 | 73.80 | 62.11 | 50.90 | |

eKF-PLDn | 2.60 | 2.09 | 1.67 | 1.38 | 1.20 | 1.11 | 96.99 | 93.77 | 87.96 | 79.12 | 67.56 | 55.61 | |

eKF-P&C | 2.59 | 2.07 | 1.66 | 1.37 | 1.19 | 1.11 | 96.90 | 93.59 | 87.60 | 78.56 | 66.85 | 54.84 | |

eKF-OracleN | 2.76 | 2.21 | 1.76 | 1.44 | 1.23 | 1.12 | 97.76 | 95.18 | 90.26 | 82.35 | 71.43 | 59.49 | |

FT | Noisy | 2.38 | 1.89 | 1.51 | 1.26 | 1.11 | 1.07 | 94.65 | 89.91 | 82.52 | 72.69 | 61.09 | 50.09 |

eKF-MCRA | 2.35 | 1.90 | 1.52 | 1.27 | 1.13 | 1.07 | 94.47 | 90.48 | 83.71 | 73.65 | 61.11 | 49.69 | |

eKF-CDR | 2.57 | 2.08 | 1.66 | 1.36 | 1.16 | 1.08 | 94.80 | 90.79 | 83.77 | 73.75 | 61.24 | 49.41 | |

eKF-PLDn | 2.43 | 2.03 | 1.65 | 1.37 | 1.19 | 1.10 | 92.62 | 89.34 | 83.41 | 74.64 | 63.46 | 52.20 | |

eKF-P&C | 2.65 | 2.11 | 1.67 | 1.37 | 1.18 | 1.09 | 95.78 | 91.96 | 85.45 | 76.01 | 64.01 | 52.03 | |

eKF-OracleN | 2.99 | 2.41 | 1.88 | 1.51 | 1.26 | 1.13 | 97.25 | 94.68 | 89.88 | 82.29 | 71.64 | 59.85 |

**Table 4.**Speech distortion (SD) and STOI results for different RTF estimators when used for MVDR beamforming. Results are broken down by both SNR and device placement.

Place. | Method | SD (%) | STOI (%) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SNR (dB) | SNR (dB) | ||||||||||||

20 | 15 | 10 | 5 | 0 | −5 | 20 | 15 | 10 | 5 | 0 | −5 | ||

CT | EVD-PLDn | 0.52 | 0.66 | 0.98 | 1.53 | 2.46 | 3.64 | 96.96 | 93.72 | 87.76 | 78.70 | 66.89 | 54.88 |

CW-PLDn | 0.52 | 0.63 | 0.92 | 1.41 | 2.27 | 3.43 | 97.00 | 93.77 | 87.87 | 78.89 | 67.10 | 55.08 | |

eKF-PLDn | 0.52 | 0.58 | 0.72 | 0.90 | 1.16 | 1.44 | 96.99 | 93.77 | 87.96 | 79.12 | 67.56 | 55.61 | |

OracleC-PLDn | 0.07 | 0.11 | 0.16 | 0.22 | 0.28 | 0.34 | 97.33 | 94.32 | 88.72 | 80.05 | 68.57 | 56.60 | |

FT | EVD-P&C | 3.64 | 3.56 | 4.03 | 5.12 | 7.11 | 10.04 | 95.56 | 91.84 | 85.38 | 75.80 | 63.69 | 51.69 |

CW-P&C | 3.96 | 3.79 | 4.19 | 5.22 | 7.15 | 10.01 | 95.54 | 91.88 | 85.46 | 75.94 | 63.85 | 51.78 | |

eKF-P&C | 2.09 | 2.63 | 3.32 | 4.23 | 5.57 | 7.49 | 95.78 | 91.96 | 85.45 | 76.01 | 64.01 | 52.03 | |

OracleC-P&C | 0.24 | 0.45 | 0.81 | 1.26 | 1.82 | 2.43 | 97.05 | 93.89 | 88.23 | 79.66 | 68.18 | 56.17 |

**Table 5.**SD results for different RTF estimators when used for MVDR beamforming. Results are broken down by both reverberation environment and device placement. The noise environments are grouped in terms of the reverberant environment as in Table 1: A (Car, Street, Pedestrian street), B (Bus, Cafe), C (Bus station, Babble) and D (Mall).

Place. | Method | SD (%) | |||
---|---|---|---|---|---|

Environment | |||||

A | B | C | D | ||

CT | EVD-PLDn | 1.16 | 1.97 | 1.73 | 2.17 |

CW-PLDn | 1.08 | 1.87 | 1.59 | 2.08 | |

eKF-PLDn | 0.59 | 1.24 | 0.87 | 1.09 | |

OracleC-PLDn | 0.03 | 0.39 | 0.17 | 0.36 | |

FT | EVD-P&C | 3.62 | 7.29 | 5.54 | 8.13 |

CW-P&C | 3.62 | 7.78 | 5.55 | 8.26 | |

eKF-P&C | 2.92 | 5.12 | 4.32 | 6.13 | |

OracleC-P&C | 0.45 | 1.80 | 1.06 | 2.30 |

**Table 6.**PESQ and STOI results for different clean speech power spectral density (PSD) estimators when combined with Wiener postfiltering applied to the MVDR beamformer output. Results are broken down by SNR and device placement.

Place. | Method | PESQ | STOI (%) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SNR (dB) | SNR (dB) | ||||||||||||

20 | 15 | 10 | 5 | 0 | −5 | 20 | 15 | 10 | 5 | 0 | −5 | ||

CT | eKF-PLDn | 2.60 | 2.09 | 1.67 | 1.38 | 1.20 | 1.11 | 96.99 | 93.77 | 87.96 | 79.12 | 67.56 | 55.61 |

WF-Ps-eKF-PLDn | 2.79 | 2.32 | 1.91 | 1.56 | 1.31 | 1.16 | 97.17 | 94.18 | 88.66 | 80.02 | 67.95 | 54.88 | |

WF-Ms-eKF-PLDn | 2.81 | 2.34 | 1.95 | 1.62 | 1.36 | 1.20 | 97.18 | 94.20 | 88.72 | 80.11 | 67.97 | 54.51 | |

FT | eKF-P&C | 2.65 | 2.11 | 1.67 | 1.37 | 1.18 | 1.09 | 95.78 | 91.96 | 85.45 | 76.01 | 64.01 | 52.03 |

WF-Ps-eKF-P&C | 2.59 | 2.22 | 1.85 | 1.54 | 1.31 | 1.17 | 92.59 | 89.56 | 83.75 | 74.60 | 62.70 | 50.25 | |

WF-Ms-eKF-P&C | 2.95 | 2.45 | 1.99 | 1.64 | 1.36 | 1.21 | 96.10 | 92.64 | 86.51 | 76.91 | 64.20 | 50.80 |

**Table 7.**PESQ and STOI results for different postfilters applied to the MVDR beamformer output and for other related state-of-the-art approaches. Results are broken down by SNR and device placement.

Place. | Method | PESQ | STOI (%) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SNR (dB) | SNR (dB) | ||||||||||||

20 | 15 | 10 | 5 | 0 | −5 | 20 | 15 | 10 | 5 | 0 | −5 | ||

CT | PLDwf | 2.81 | 2.38 | 1.98 | 1.64 | 1.36 | 1.20 | 95.94 | 92.21 | 85.70 | 76.11 | 63.53 | 50.42 |

WF-Ms-eKF-PLDn | 2.81 | 2.34 | 1.95 | 1.62 | 1.36 | 1.20 | 97.18 | 94.20 | 88.72 | 80.11 | 67.97 | 54.51 | |

pWF-Ms-eKF-PLDn | 2.86 | 2.40 | 2.00 | 1.64 | 1.37 | 1.20 | 97.24 | 94.35 | 89.00 | 80.53 | 68.41 | 54.54 | |

OMLSA-Ms-eKF-PLDn | 2.96 | 2.49 | 2.06 | 1.68 | 1.40 | 1.23 | 97.25 | 94.45 | 89.24 | 80.86 | 68.72 | 55.11 | |

FT | SPPCwf | 2.74 | 2.26 | 1.81 | 1.48 | 1.25 | 1.12 | 94.43 | 90.26 | 83.27 | 73.28 | 61.16 | 49.34 |

WF-Ms-eKF-P&C | 2.95 | 2.45 | 1.99 | 1.64 | 1.36 | 1.21 | 96.10 | 92.64 | 86.51 | 76.91 | 64.20 | 50.80 | |

pWF-Ms-eKF-P&C | 2.99 | 2.49 | 2.01 | 1.63 | 1.36 | 1.21 | 96.12 | 92.73 | 86.68 | 77.08 | 64.18 | 50.36 | |

OMLSA-Ms-eKF-P&C | 2.85 | 2.38 | 1.94 | 1.60 | 1.35 | 1.20 | 95.98 | 92.64 | 86.63 | 77.03 | 64.13 | 50.54 |

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**MDPI and ACS Style**

Martín-Doñas, J.M.; Peinado, A.M.; López-Espejo, I.; Gomez, A. Dual-Channel Speech Enhancement Based on Extended Kalman Filter Relative Transfer Function Estimation. *Appl. Sci.* **2019**, *9*, 2520.
https://doi.org/10.3390/app9122520

**AMA Style**

Martín-Doñas JM, Peinado AM, López-Espejo I, Gomez A. Dual-Channel Speech Enhancement Based on Extended Kalman Filter Relative Transfer Function Estimation. *Applied Sciences*. 2019; 9(12):2520.
https://doi.org/10.3390/app9122520

**Chicago/Turabian Style**

Martín-Doñas, Juan M., Antonio M. Peinado, Iván López-Espejo, and Angel Gomez. 2019. "Dual-Channel Speech Enhancement Based on Extended Kalman Filter Relative Transfer Function Estimation" *Applied Sciences* 9, no. 12: 2520.
https://doi.org/10.3390/app9122520