# A Frequency Estimation Scheme Based on Gaussian Average Filtering Decomposition and Hilbert Transform: With Estimation of Respiratory Rate as an Example

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.1.1. Data Resources and Conducted Experiments

^{®}Systems, Inc., Goleta, CA, USA). A dedicated cable SS60L (BIOPAC

^{®}Systems, Inc., Goleta, CA, USA) for SS39L was used to convey the signals to the multifunction physiological data acquisition (DAQ) system MP30 (BIOPAC

^{®}Systems, Inc., Goleta, CA, USA). The respiration signal was measured synchronously by the respiration belt transducer SS5LB (BIOPAC

^{®}Systems, Inc., Goleta, CA, USA). Both signals were simultaneously acquired by Biopac Student Lab Pro analysis software with band-pass filtering of 0.05 to 35 Hz and the sampling frequency was 100 Hz for each signal. Five healthy subjects (3 males and 2 females, aged 23 ± 1 years) without a history of cardiopulmonary diseases were recruited from Feng Chia university. The conducted experiments were approved by the Institutional Review Board of Changhua Christian Hospital, Taiwan. The wrist PPG signal has been shown to be easily modulated by respiration [35]. To investigate the influence of respiration on the PPG pattern, the two-minute experiments were conducted five times for each subject. During the experiments, the subjects were requested to control their breathing speed at one fixed frequency in the first minute and change to another speed in the second minute. The controlled respiration frequencies range from 0.1 Hz (6 bpm) to 0.25 Hz (15 bpm), which were randomly selected in each experiment. In summary, there are a total of 25 data points collected in the wrist PPG database, and each signal is of 2 min length.

^{®}Systems, Inc., Goleta, CA, USA) and channel 4 is SCG (bandwidth of 0.5 Hz to 100 Hz, by LIS344ALH, STMicroelectronics). The data were collected by the multifunction physiological data acquisition (DAQ) system MP36 (BIOPAC

^{®}Systems, Inc., Goleta, CA, USA) and the sampling frequency is 5000 Hz for each signal in the original database. With the help of online data viewing tool LightWAVE [42], it can be observed that the respiration signals in some records (such as 006, 007, 009, 010, 015, 016 and 018) are not qualified enough. The signals may contain artefacts (perhaps due to motion or other unknown reasons) or very irregular patterns, rendering the rate of respiration difficult to identify, even by visual inspection. The criterion for data selection is that the respiration signal must be regular for a one-minute duration and the rate of respiration must be able to be clearly determined by visual inspection during this interval. Finally, 500 segments of one-minute length screened from 13 subjects were collected in the SCG database.

#### 2.1.2. Computing Resources

#### 2.2. Methods

#### 2.2.1. Hilbert–Gaussian Transform (HGT)

#### 2.2.2. RR Estimation Algorithm

#### 2.2.3. Statistical Analysis

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

^{©}for providing the ASUS VivoWatch platform in this research.

## Conflicts of Interest

## References

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**Figure 1.**Decomposition results for (

**a**) finger PPG, (

**b**) wrist PPG, (

**c**) SCG, (

**d**) finger PPG with an SNR of 5 dB, (

**e**) wrist PPG with an SNR of 5 dB and (

**f**) SCG with an SNR of 5 dB. The illustrations are arranged in order of signal type (PPG or SCG), respiration signal and the respiration-related IMF decomposed by EMD, EEMD and GAFD (from top to bottom).

**Figure 2.**Hilbert spectrum of finger PPG (data numbered 03900006 from MIMIC database) by (

**a**) EMD, (

**b**) EEMD and (

**c**) GAFD. The illustrations in each subfigure are the Hilbert spectrum for all IMFs on the left, whereas the Hilbert spectrum for the respiration-related IMF is on the right. In the Hilbert spectrum of respiration (right subfigure), the dash-dotted line (blue) denotes the respiration frequency estimated by wavelet transform from respiration signal (ground truth), whereas the dotted line (dark red) represents the respiration frequency estimated from the Hilbert spectrum according to the rule introduced in Section 2.2.2.

**Figure 3.**Hilbert spectrum of wrist PPG (measured from a male subject aged 23 in the sitting posture with a controlled respiration of 6 bpm in the first minute and changed to 10 bpm in the second minute) by (

**a**) EMD, (

**b**) EEMD and (

**c**) GAFD. The illustrations in each subfigure are the Hilbert spectrum for all IMFs on the left, whereas the Hilbert spectrum for the respiration-related IMF is on the right. In the Hilbert spectrum of respiration (right subfigure), the dash-dotted line (blue) denotes the respiration frequency estimated by wavelet transform from respiration signal (ground truth), whereas the dotted line (dark red) represents the respiration frequency estimated from the Hilbert spectrum according to the rule introduced in Section 2.2.2.

**Figure 4.**Hilbert spectrum of SCG (data numbered b017 from CEBS database) by (

**a**) EMD, (

**b**) EEMD and (

**c**) GAFD. The illustrations in each subfigure are the Hilbert spectrum for all IMFs on the left, whereas the Hilbert spectrum for the respiration-related IMF is on the right. In the Hilbert spectrum of respiration (right subfigure), the dash-dotted line (blue) denotes the respiration frequency estimated by wavelet transform from respiration signal (ground truth), whereas the dotted line (dark red) represents the respiration frequency estimated from the Hilbert spectrum according to the rule introduced in Section 2.2.2.

**Figure 5.**Bland–Altman plot for the RR estimation between the proposed method and the conventional approach.

Subject Number ^{1} | Age Range ^{2} | Posture | Record Number ^{3} | |
---|---|---|---|---|

Finger PPG | 50 (30M, 20F) | 21–92 years | Supine (ICU) | 3000 records |

Wrist PPG | 5 (3M, 2F) | 22–24 years | Sitting | 150 records |

SCG | 13 (6M, 7F) | 19–30 years | Supine | 1500 records |

**Subject Number indicates the total number of subjects included in the dataset, with ‘M’ representing male and F representing female.**

^{1}^{2}Age Range refers to the range of age range for the participated subjects.

^{3}Record Number is the number of records of 20 s length in the dataset.

**Table 2.**Summary of ICC and Bland–Altman agreement analysis for RR estimation under different signal conditions.

ICC ^{1} | 95% C.I. of ICC ^{2} | Bias ^{3} | Limit of Agreement ^{4} | |
---|---|---|---|---|

Finger PPG | $0.9680$ | $0.9655,0.9704$ | $-0.0048\pm 0.5493$ | $-1.1033,1.0937$ |

Wrist PPG | $0.9650$ | $0.9573,0.9714$ | $0.0018\pm 0.5315$ | $-1.0611,1.0647$ |

SCG | $0.9691$ | $0.9658,0.9720$ | $-0.0104\pm 0.5403$ | $-1.0910,1.0702$ |

Combined ^{5} | $0.9688$ | $0.9669,0.9706$ | $-0.0062\pm 0.5446$ | $-1.0954,1.0830$ |

^{1}ICC means the intraclass correlation coefficient.

^{2}C.I. is the abbreviation of confidence interval.

^{3}Bias denotes the difference between the estimated and the ground truth values, which is presented as mean ± SD (standard deviation) in the unit of bpm.

^{4}Limit of Agreement represents the lower and the upper limit of agreement for the Bland–Altman agreement analysis between the estimated values and the ground truth values derived by the conventional approach in the unit of bpm.

^{5}“Combined” refers to the fusion of finger PPG, wrist PPG and SCG signals for ICC and Bland–Altman agreement analysis.

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**MDPI and ACS Style**

Lin, Y.-D.; Tan, Y.-K.; Ku, T.; Tian, B.
A Frequency Estimation Scheme Based on Gaussian Average Filtering Decomposition and Hilbert Transform: With Estimation of Respiratory Rate as an Example. *Sensors* **2023**, *23*, 3785.
https://doi.org/10.3390/s23083785

**AMA Style**

Lin Y-D, Tan Y-K, Ku T, Tian B.
A Frequency Estimation Scheme Based on Gaussian Average Filtering Decomposition and Hilbert Transform: With Estimation of Respiratory Rate as an Example. *Sensors*. 2023; 23(8):3785.
https://doi.org/10.3390/s23083785

**Chicago/Turabian Style**

Lin, Yue-Der, Yong-Kok Tan, Tienhsiung Ku, and Baofeng Tian.
2023. "A Frequency Estimation Scheme Based on Gaussian Average Filtering Decomposition and Hilbert Transform: With Estimation of Respiratory Rate as an Example" *Sensors* 23, no. 8: 3785.
https://doi.org/10.3390/s23083785