# Single-Lead Fetal ECG Extraction Based on a Parallel Marginalized Particle Filter

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## Abstract

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## 1. Introduction

## 2. ECG Dynamical Model

## 3. Proposed Methodology for FECG Extraction

#### 3.1. Modified Abdominal Signal Dynamical Model

#### 3.2. Observation Equation of the Dynamical Model

#### 3.3. Parallel-Marginalized Particle Filter

- Step 1: Initialize the M particles:$$\{\begin{array}{l}{\left\{{x}_{0|-1}^{NL,(i)}\right\}}_{i=1}^{M}~p({x}_{0}^{NL})\\ {\left\{{x}_{0|-1}^{L,(i)}\right\}}_{i=1}^{M}~p({x}_{0}^{L})\end{array},$$
- Step 2: Calculate the importance:weight:$${w}_{k}^{(i)}=p({Y}_{k}|{x}_{k|k-1}^{NL,(i)})\text{}i=1,2,\cdots ,M;$$$${w}_{k}^{(i)}={w}_{k}^{(i)}/{\displaystyle \sum _{i=1}^{M}{w}_{k}^{(i)}}.$$
- Step 3: Resample the M particles:$$p({x}_{k|k}^{NL,(i)}={x}_{k|k-1}^{NL,(j)})={w}_{k}^{(j)}$$
- Step 4: Update the Kalman filter measurement:$$\{\begin{array}{l}{\tilde{\mathsf{\alpha}}}_{k|k}^{f/m,(\mathrm{i})}={\tilde{\mathsf{\alpha}}}_{k|k-1}^{f/m,(\mathrm{i})}\\ {P}_{k|k}^{f/m}={P}_{k|k-1}^{f/m}\end{array},$$
- Step 5: Update the particle filter time:$$p({x}_{k+1|k}^{f/mNL,(i)}|{x}_{k|k}^{f/mNL,(i)})=N({\mathsf{\mu}}_{k}^{f/m},{\Sigma}_{k}^{f/m})$$$$\{\begin{array}{l}{\mathsf{\mu}}_{k}^{f/m}={[{\theta}_{k}^{f/m,(i)}+{w}^{f/m}\delta ,-{g}^{T}({x}_{k}^{f/mNL,(i)}){\mathsf{\alpha}}_{k}^{f/m,(i)}+{z}_{k}^{f/m,(i)},{b}_{k}^{f/m,(i)},{\mathsf{\xi}}_{k}^{f/m,(i)}]}^{T}\\ {\Sigma}_{k}^{f/m}=diag({\sigma}_{{\theta}^{f/m}}^{2},{\sigma}_{{z}^{f/m}}^{2},{\sigma}_{{b}^{f/m}}^{2},{\sigma}_{{\mathsf{\xi}}^{f/m}}^{2})\\ g({x}_{k}^{NL})=[g({x}_{k,1}^{NL}),g({x}_{k,2}^{NL}),g({x}_{k,3}^{NL}),g({x}_{k,4}^{NL}),g({x}_{k,5}^{NL})]\\ g({x}_{k,j}^{NL})=\delta \frac{{\alpha}_{j,k}\omega}{{b}_{j,k}^{2}}\Delta {\theta}_{j,k}\mathrm{exp}(-\frac{\Delta {\theta}_{j,k}^{2}}{2{b}_{j,k}^{2}})\end{array}$$
- Step 6: Update the Kalman filter time:$${\tilde{\mathsf{\alpha}}}_{k+1|k}^{f/m,(\mathrm{i})}={\tilde{\mathsf{\alpha}}}_{k|k}^{f/m,(\mathrm{i})}+{L}_{k}^{f/m}({\tilde{z}}_{k+1|k}^{f/m,(i)}-{z}_{k}^{f/m,(i)}-{g}^{T}({x}_{k+1|k}^{f/mNL,(i)}){\tilde{\mathsf{\alpha}}}_{k|k}^{f/m,(\mathrm{i})})$$$$\{\begin{array}{l}{F}_{k}^{f/m}={g}^{T}({\tilde{x}}_{k+1|k}^{f/mNL,(i)}){P}_{k|k}^{f/m,(i)}g({\tilde{x}}_{k+1|k}^{f/mNL,(i)})+{Q}_{k}^{f/mNL}\\ {L}_{k}^{f/m}={P}_{k|k}^{f/m,(i)}g({\tilde{x}}_{k+1|k}^{f/mNL,(i)}){({F}_{k}^{f/m})}^{-1}\end{array},$$
- Step 7:$$k=k+1,$$

## 4. Performance Index of the FECG Extraction

#### 4.1. Performance Index of the Simulated Data

#### 4.2. Performance Index of the Clinical Data

- Step 1: The extracted FECG is divided into N pieces crossing to the R peak, and each piece includes M samples and one QRS complex.
- Step 2: All of the pieces are stored in columns of an M by N matrix $U$, and the vectors $u(k)$ are the zero mean and are normalized to the unit length; that is,$${u}^{\mathrm{T}}(k)u(k)=1.$$
- Step 3: A signal-to-noise ratio based on eigenvalues can be calculated as:$$SN{R}_{eig}=\frac{{\lambda}_{\mathrm{max}}}{N-{\lambda}_{\mathrm{max}}}$$
- Step 4: A signal-to-noise ratio based on the cross-correlation coefficients can be calculated as$$SN{R}_{cor}=\frac{\eta}{1-\eta}$$$$\eta =\frac{2}{M(M-1)}{\displaystyle \sum _{i=0}^{M-2}{\displaystyle \sum _{k=i+1}^{M-1}f{(i)}^{\mathrm{T}}f(k)}},$$

## 5. Results and Analysis

#### 5.1. FECG Extraction on the Simulated Data

#### 5.1.1. Simulated Data without Noise

#### 5.1.2. Simulated Data with Noise

#### 5.2. FECG Extraction on the Different Database.

#### 5.2.1. Database for the Identification of Systems

#### 5.2.2. Abdominal and Direct Fetal ECG Database

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**FECG estimation: (

**a**) FECG extracted by par-MPF; (

**b**) FECG extracted by EKS; (

**c**) FECG extracted by EKF.

**Figure 6.**FECG estimation: (

**a**) FECG extracted by par-MPF; (

**b**) FECG extracted by EKS; (

**c**) FECG extracted by EKF.

**Figure 9.**FECG estimation: (

**a**) FECG extracted by par-MPF; (

**b**) FECG extracted by EKS; (

**c**) FECG extracted by EKF; (

**d**) FECG extracted by ANFIS-EKS; (

**e**) FECG extracted by ANFIS-EKF.

**Figure 10.**FECG estimation: (

**a**) FECG extracted from lead_2; (

**b**) FECG extracted from lead_3; (

**c**) FECG extracted from lead_5.

**Figure 12.**FECG estimation: (

**a**) FECG extracted by par-MPF; (

**b**) FECG extracted by EKS; (

**c**) FECG extracted by EKF; (

**d**) FECG extracted by ANFIS-EKS; (

**e**) FECG extracted by ANFIS-EKF.

**Figure 13.**FECG estimation: (

**a**) FECG extracted from Abdomen_2; (

**b**) FECG extracted from Abdomen_3; (

**c**) FECG extracted from Abdomen_4.

Filtering Algorithm | Main Idea |
---|---|

par-MPF | Marginalized particle filter |

EKS [10] | Extended Kalman filter + Smooth |

EKF [11] | Extended Kalman filter |

ANFIS-EKS [12] | EKS + Adaptive neuro fuzzy inference system |

ANFIS-EKF [12] | EKF + Adaptive neuro fuzzy inference system |

Methods | Lead 1 | Lead 2 | Lead 3 | Lead 4 | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{e}\mathit{i}\mathit{g}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{c}\mathit{o}\mathit{r}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{e}\mathit{i}\mathit{g}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{c}\mathit{o}\mathit{r}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{e}\mathit{i}\mathit{g}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{c}\mathit{o}\mathit{r}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{e}\mathit{i}\mathit{g}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{c}\mathit{o}\mathit{r}}$ | |

par-MPF | 2.0864 | 1.6820 | 2.0849 | 1.7120 | 2.4119 | 2.1926 | 2.1033 | 1.7037 |

EKS [10] | 1.7497 | 1.5677 | 1.8250 | 1.3714 | 1.3903 | 0.7797 | 1.1414 | 0.9533 |

EKF [11] | 1.2245 | 1.1506 | 1.2367 | 0.8392 | 1.3590 | 0.7465 | 0.6258 | 0.4130 |

ANFIS-EKS [12] | 1.8334 | 1.6152 | 1.8709 | 1.4889 | 1.4452 | 1.4917 | 1.1837 | 0.9856 |

ANFIS-EKF [12] | 1.5097 | 1.3160 | 1.5491 | 1.2557 | 1.3767 | 0.9247 | 0.8465 | 0.6551 |

Method | Abdomen-1 | Abdomen-2 | Abdomen-3 | Abdomen-4 | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{e}\mathit{i}\mathit{g}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{c}\mathit{o}\mathit{r}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{e}\mathit{i}\mathit{g}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{c}\mathit{o}\mathit{r}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{e}\mathit{i}\mathit{g}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{c}\mathit{o}\mathit{r}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{e}\mathit{i}\mathit{g}}$ | $\mathit{S}\mathit{N}{\mathit{R}}_{\mathit{c}\mathit{o}\mathit{r}}$ | |

par-MPF | 12.8898 | 14.7332 | 12.1464 | 13.9624 | 11.4695 | 13.2144 | 13.352 | 15.3952 |

EKS [10] | 3.5025 | 3.6133 | 3.8609 | 4.1144 | 5.0389 | 5.1286 | 6.7488 | 6.9799 |

EKF [11] | 2.4222 | 2.3902 | 2.236 | 2.2596 | 3.5868 | 3.2583 | 4.2651 | 3.8168 |

ANFIS-EKS [12] | 4.2285 | 4.5879 | 4.1438 | 4.6256 | 5.5427 | 5.5549 | 7.2039 | 7.2104 |

ANFIS-EKF [12] | 3.0071 | 3.0873 | 2.5573 | 2.7417 | 3.9021 | 3.8444 | 5.5771 | 5.4508 |

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**MDPI and ACS Style**

Zhao, Z.; Tong, H.; Deng, Y.; Xu, W.; Zhang, Y.; Ye, H.
Single-Lead Fetal ECG Extraction Based on a Parallel Marginalized Particle Filter. *Sensors* **2017**, *17*, 1456.
https://doi.org/10.3390/s17061456

**AMA Style**

Zhao Z, Tong H, Deng Y, Xu W, Zhang Y, Ye H.
Single-Lead Fetal ECG Extraction Based on a Parallel Marginalized Particle Filter. *Sensors*. 2017; 17(6):1456.
https://doi.org/10.3390/s17061456

**Chicago/Turabian Style**

Zhao, Zhidong, Huiling Tong, Yanjun Deng, Wen Xu, Yefei Zhang, and Haihui Ye.
2017. "Single-Lead Fetal ECG Extraction Based on a Parallel Marginalized Particle Filter" *Sensors* 17, no. 6: 1456.
https://doi.org/10.3390/s17061456