*4.3. Optimal Decomposition Level*

To optimize the performance of the denoising method using the wavelet transform and thresholding, determining the proper decomposition level is also necessary. However, the optimal decomposition level was first estimated in Section 4.1 based on the dominant frequency range and the sampling ratio of the signal. In this subsection, we will cover the results of the quantitative analysis for the optimal decomposition level. Figure 8 represents *e f f*(*j*, *k*) of the denoised signal. The decomposition level in range from 1 to 10 is denoted as the *x*-axis, the *e f f*(*j*, *k*) from the optimal mother wavelet "db9" is indicated as the *y*-axis. Figure 9 shows the *e f f*(*j*, *k*) of the six wavelet families at decomposition level 1 to 10. Each wavelet function is fixed to a wavelet length of 18, which is the most efficient wavelet length obtained in the previous subsection. Through the experimental result analysis of Figures 8 and 9, the denoising process with decomposition level seven achieves the best performance among all candidates but a higher decomposition level apparently gives less *e f f*(*j*, *k*) performance. Although Figure 9 shows that the "bior" graph has a maximum value at level six, still the maximum *e f f*(*j*, *k*) of the "bior" family is far less than the maximum *e f f*(*j*, *k*) of other families, the wavelet functions from the "bior" family are out of consideration in this study. This incident happens when the wavelet length of the mother wavelet function is too short for the length of the input signal. (, ) (, ) " " (, ) (, ) " " (, ) " " (, ) " "

(, ) **Figure 8.** *e f f*(*j*, *k*) values of 10 decomposition levels (1 to 10) for db9.

The approximation coefficients at the optimal decomposition level is *cA*7, which are composed of a partial signal whose frequency range is approximately 0~4 Hz. This frequency range is the subset of the dominant frequency band (0~5 Hz) of the original signal. Therefore, the decomposition level, which can extract the dominant frequency band of the original signal at approximation coefficients, is the optimal decomposition level for signal denoising. To prove this, the same denoising process was performed to the DCG signals with different sampling ratios. The DCG signals with a sampling frequency of 1000 Hz were sampled to different sampling ratios to provide identical DCG characteristics but different frequency features. The *e f f*(*j*, *k*) values of these sampled signals with four different sampling ratios (500 Hz, 250 Hz, 125 Hz, and 62 Hz) are described in Figure 10. As shown in Figure 10a, in case of a sampling ratio of 500 Hz, the most effective decomposition level lowered 1 step to level 6 as the sampling ratio decreased to half of the original. At this time, the denoising performance also decreased. Likewise, the plot of the signal with the sampling frequency of 125 Hz in Figure 10c represents that the optimal decomposition level is four, which is three levels lower than the original. Moreover, the maximum *e f f*(*j*, *k*) of the denoised signal decreases as the sampling ratio falls off (Figure 11). To summarize, the optimal decomposition level is determined by the dominant frequency band and the sampling ratio of the original signal. It is also shown that the sampling frequency of the

signal affects to the denoising performance because the higher sampling rate can obtain more useful information components than the signal with a lower sampling rate.

(, ) **Figure 9.** *e f f*(*j*, *k*) values of 10 decomposition levels (1 to 10) for six mother wavelet functions (db9, sym9, coif3, fk18, bior, and rbio).

<sup>7</sup>

–

(, ) (, ) (, ) (, ) (, ) **Figure 10.** *e f f*(*j*, *k*) values of 10 decomposition levels (1 to 10) for db9. (**a**) *e f f*(*j*, *k*) values for a sampling frequency of 500 Hz; (**b**) *e f f*(*j*, *k*) value for a sampling frequency of 250 Hz; (**c**) *e f f*(*j*, *k*) values for a sampling frequency of 125 Hz; and (**d**) *e f f*(*j*, *k*) values for a sampling frequency of 62 Hz.

(, )

author's knowledge,

(, )

(, ) **Figure 11.** Maximum *e f f*(*j*, *k*) values of signals with five sampling ratios (1000 Hz, 500 Hz, 250 Hz, 125 Hz, and 62 Hz).

#### **5. Discussions**

(, ) (, )

– author's knowledge, In this study, the denoising process using wavelet decomposition and thresholding method is performed to improve the quality of the DCG signal. Although there are many studies of the denoising process using the wavelet function [35,72–74], to the best of the author's knowledge, previous studies have yet to provide the optimal mother wavelet selection for the DCG signal. The purpose of this study is proposing the optimal selection of the mother wavelet function and the decomposition level for the DCG signal to optimize the performance of the denoising process.

Xu et al. [35] and Srivastava et al. [36] proposed the selection of the mother wavelet function and the decomposition level of the object radar signal, respectively. However the proposed mother wavelet function and decomposition level for their object radar signal performed less powerfully on the DCG signal than the proposed selection from this study (Table 9). As a consequence, the optimal mother wavelet function and the decomposition level should be selected identically by analyzing the characteristics of the object signal.

**Table 9.** The table for the comparison of the *SNRj*, *<sup>k</sup>* performance between the Xu et al., Srivastava et al., and proposed selection.


The denoising process with the optimal selection for the DCG signal enhanced the quality of the DCG signals in the information field. As the DCG signals can be obtained in a more flexible condition than the ECG signals, the improvement of the DCG signals could enhance the diagnosis of the CVDs.

#### **6. Conclusions**

The wavelet decomposition thresholding is a powerful denoising method. The performance of the denoising method can be optimized by using the optimal set of the mother wavelet function and the decomposition level. For the DCG signal, this study suggested the optimal selection of the mother wavelet function and the decomposition level based on the signal analysis. To select the optimal mother wavelet function, the wavelet length of the mother wavelet function is an important element. In this study, the length of the examined signal was 160,000, with a 1000 Hz sampling rate signal recorded for 160 s and the most efficient wavelet length was 18. There are six wavelet families with wavelet length

18; "db9", "sym9", "coif3", "fk18", "bior", and "rbio". Most of these six functions recorded superior performance to other functions with a different wavelet length and "db9" and "sym9" were selected for the optimal mother wavelet functions among all 115 wavelet candidates. The optimal decomposition level for the DCG signal was determined as seven, which shows that the estimation of the decomposition level based on the length of the signal was precise.

Since the noise reduction method based on the wavelet decomposition and the thresholding with optimal parameters successfully removes the noise from the DCG signal, the quality of the heart rate signal obtained by Doppler radar sensors was improved in the information field. Therefore, analyzing a DCG signal using artificial neural networks for the diagnosis of CVDs is conducted as the aim of the future work following to this study. The major findings in this study is denoted as follows:

• The wavelet length of the mother wavelet function was the important element for the selection of the most efficient mother wavelet. The longer mother wavelet function did not provide a better denoising performance. As the longer wavelet function requires more performance complexity, the optimal wavelet length for the performance efficiency should be considered;


**Author Contributions:** Conceptualization, Y.I.J., J.-R.Y. and N.K.K.; Data curation, J.Y.S.; Formal analysis, Y.I.J. and N.K.K.; Funding acquisition, J.-R.Y. and N.K.K.; Investigation, Y.I.J.; Methodology, Y.I.J. and N.K.K.; Project administration, J.-R.Y. and N.K.K.; Resources, J.Y.S.; Software, Y.I.J. and J.Y.S.; Supervision, J.-R.Y. and N.K.K.; Validation, Y.I.J.; Visualization, Y.I.J.; Writing—original draft, Y.I.J.; Writing—review & editing, N.K.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in part by the National Research Foundation of Korea (NRF) through the Basic Science Research Program funded by the Ministry of Education under Grant 2020R1F1A1064330, and in part by Yeungnam University, through a research grant, in 2019.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** The data are not publicly available due to the company security policy and personal protection of subjects.

**Acknowledgments:** The authors wish to represent proper appreciations to all researchers who presented their studies and efforts for this study.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


*Article*
