**1. Introduction**

Cardiovascular diseases (CVDs) are the leading and rapidly increasing cause of death in modern society. They caused 32.1% of annual global deaths in 2015 and the occurrence is growing [1]. According to this trend, research on heartbeat detection and heart rate variability analysis has been gaining increasing attention. As the basic method of the heart signal recording, the electrocardiogram (ECG) test is used for measuring the electrophysiological signal of the heart activity to diagnose symptoms of cardiovascular problems [2]. The ECG test can improve the diagnosis and therapy of the most prevalent cardiac diseases if it does not interfere with daily activities [3]. However, some requirements of the ECG test make subjects uncomfortable. For example, subjects are required to attach multiple electrodes on their skin and remain in static state for quite some time [4]. In addition as the abnormal heartbeat signal appears for a very short time, the ECG test could miss these short and temporary symptoms because this test only records in limited circumstances.

Doppler radar sensors detect the electrophysiological signal of the heart and the variation of blood vessel movement by differences of the electromagnetic signal that is a few dozens of centimeters away from a subject [5–8]. Due to the non-contactable and flexible measurement condition, as opposed to ECG sensing, the Doppler radar sensors are studied for the alternative of ECG electrodes [9]. However, the noise occurs in the output signal obtained by the radar sensors due to the limitation of the non-contact measurement. Since the noise has a negative effect on data acquisition and processing, it leads to a

**Citation:** Jang, Y.I.; Sim, J.Y.; Yang, J.-R.; Kwon, N.K. The Optimal Selection of Mother Wavelet Function and Decomposition Level for Denoising of DCG Signal. *Sensors* **2021**, *21*, 1851. https://doi.org/ 10.3390/s21051851

Academic Editor: Manuel Blanco-Velasco

Received: 28 January 2021 Accepted: 3 March 2021 Published: 6 March 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

decrease in information accuracy. Therefore, the method for signal denoising is essential and a considerable amount of research related to denoising has been studied [10–14]. Among them, the noise reduction methods using the wavelet analysis are widely studied not only for the radar signal but also for various multi-dimensional signals [15–20].

Noise reduction methods using the wavelet transform and thresholding can improve the quality of the Doppler cardiogram (DCG) and overcome the limits of remote sensors. To successfully enhance signal quality, the noise reduction technique should satisfy the performance of removing the noise components and maintaining useful signal components. In comparison to other noise removal methods such as average filtering or frequency smoothing using the fast Fourier transform, the wavelet denoising method shows a superior performance for preserving the features of the original signal when removing noise components. In particular, it is effective in regards to the problem of non-stationary signals such as a heart signal, respiration signal, and ocular artifacts [21–25]. The multiresolution decomposition of the wavelet transform makes it possible to maintain the original information during the process of removing noise components. Meanwhile, since various scaling factors and features of the wavelet functions are available, the optimal selection procedure of the mother wavelet function is a significant step to the wavelet denoising method [26,27]. Traditionally, the mother wavelet function is selected to represent the characteristics of the signal by empirical research [28,29]. Since this method does not provide the actual optimal selection of the mother wavelet [30], studies of the mother wavelet selection for various signals such as the electroencephalography (EEG) signal, vibration signal of turbine, human voice signal, and many kinds of radar signal have been occurring [26,28,31–36]. However, to the best of the author's knowledge, studies on the optimal selection of the mother wavelet function and the decomposition level for the denoising of the DCG signal are rarely reported. Though selection of the mother wavelet function and the decomposition level on other radar signals were presented [35,36], distinctive features of each kind of radar signal require an individualized selection respectively, which is the motivation behind this study.

In this study, the optimal selection of the mother wavelet function and the decomposition level for the cardiac signal obtained by the Doppler radar sensors is proposed with quantitative analysis results. The result of the denoising process is defined with the signal-to-noise-ratio (SNR) efficiency of the denoised signal. The candidates for the optimal mother wavelet function are composed of 115 functions from six different wavelet families and the decomposition level has 10 candidates from 1 to 10. To determine the optimal mother wavelet function and the optimal decomposition level, the denoising process is repeated to examine all candidates and then all results are accumulated. The optimal choice of the mother wavelet function and the decomposition level is selected among the accumulated results which perform the highest SNR efficiency. The experiment is executed in two steps to find out the optimal selection of the mother wavelet function and the decomposition level. For the first step, the optimal mother wavelet function is determined by using the arbitrary wavelet decomposition level. Then the optimal decomposition level is obtained by using the optimal mother wavelet function which is selected at the first step. In this study, the arbitrary decomposition level in the first step is predicted based on the distribution of the signal components in the frequency domain. With these experiments, it is found that the length of the mother wavelet function is one of the key determinants for the optimal selection. The noisy signals were generated by adding white Gaussian noise to the original signals. The DCG dataset has 28 recordings that have been recorded for 160 s with a sampling frequency of 1000 Hz.

From the experimental results, it can be seen that the optimal mother wavelet function is found as db9 and sym9, and the optimal wavelet decomposition level is seven, respectively. The main contributions of this study can be described as follows:


• The new criterion is suggested to evaluate both the denoising performance and the denoising efficiency in once. 

#### **2. Materials and Methods**

#### *2.1. Wavelet Transform*

The wavelet transform, a linear transformation with the mother wavelet function, is an efficient signal analysis method for both time and frequency resolution [22,23,37]. The wavelet transform has advantages over traditional Fourier transforms for accurately deconstructing and reconstructing finite, non-periodic, and non-stationary signals such as DCG signals. The wavelet transform decomposes the input signal into detail coefficients (*cD*) and approximation coefficients (*cA*) that can be defined as high frequency coefficients (*yhigh* [*n*]) and low frequency coefficients (*ylow*[*n*]), respectively [38]. The frequency band coefficients of the wavelet transform are expressed mathematically as follows [38]: ℎℎ[] []

$$y\_{high}[n] = \sum\_{i=-\infty}^{\infty} s[i]h[2n-i]\_\prime \tag{1}$$

$$y\_{low}[n] = \sum\_{i=-\infty}^{\infty} s[i]g[2n-i],\tag{2}$$

where *i* is a sampling data point; *n* is the number of the sampling data; *s*[*i*] is the discrete radar signal with noise; and *g*[2*n* − *i*] and *h*[2*n* − *i*] are low-pass and high-pass filters that vary depending on the mother wavelet function [39]. With frequency filters, the wavelet transform enables one to extract the particular frequency band from the original signal [40,41]. Figure 1 shows the *x* level decomposition process into the approximation coefficients *cA* and the detail coefficients *cD* of the *N* Hz signal: = −∞ [] [2 − ] ℎ[2 − ] 

**Figure 1.** Illustration of the level decomposition process. **Figure 1.** Illustration of the *x* level decomposition process.

() () The wavelet function is composed with the scaled and translated copies of the scaling function *φ*(*x*) and the mother wavelet function *ψ*(*x*) [26,38] which is a continuously differentiable function with compact support. Since there are many different types of mother wavelet functions, finding the optimal mother wavelet is essential for the best performance not only in denoising but also in other signal processing. The coefficients of the discrete wavelet transform represent the projection of the signal over a set of basis functions generated as the translation and dilation of the mother wavelet function and the

scaling function [42]. More specifically, the low-pass coefficients are related to the scaling function and the high-pass coefficients are related to the mother wavelet function as in Equations (3)–(5) [38]:

$$\lg(h) = (-1)^n h (1 - n),\tag{3}$$

$$\phi(\mathbf{x}) = \sum\_{n} h(n) \sqrt{2} \phi(2\mathbf{x} - n), \tag{4}$$

$$
\psi(\mathbf{x}) = \sum\_{n} g(n) \sqrt{2} \phi(2\mathbf{x} - n). \tag{5}
$$

The low-pass and high-pass filters are distinctive variables for each mother wavelet function. Since the selection of the mother wavelet function and the decomposition level affects the noise reduction performance, the suitable mother wavelet function and the decomposition level could be optimizing the performance of the processing methods based on the wavelet transform.

#### *2.2. Doppler Cardiogram Signal Dataset and Recording Procedure*

A total of 28 DCG signals from 11 subjects were examined in this study. The datasets were recorded from healthy control subjects composed of 6 females and 5 males aged 24.08 ± 2.35 years (mean ± standard deviation). The subjects were recruited from the Electronic Engineering department of Yeungnam University. The subjects have no history of CVDs. Each signal data was recorded for 160 s and 1 k samples per second. The measurement system was constructed with a Doppler radar sensor, which measured the cardiogram of the subject at a distance of 30 cm. Table 1 shows the socio-demographic and clinical data of the control subjects. Permission to use the DCG data was granted from all subjects.

**Table 1.** Sociodemographic data of the control subjects (Age in years, mean ± standard deviation, SD).


The DCG signals were detected by using the radar sensor module with two discrete patch antennas, which was implemented on an FR4 printed circuit board with a thickness of 0.6 mm, as shown in Figure 2 [43]. A total of 5.8 GHz of signals were generated by a voltage-controlled oscillator (HMC431LP4, Analog Devices) with an output power of 7.8 dBm radiating from the transmitting antenna with a directivity of 4.0 dBi. The received signals modulated by respiration and heartbeat were incident by using the receiving antenna, and they are down-converted to baseband signals by using quadrature mixers with low-pass filters with a cut-off frequency of 500 MHz. The overall noise figure in the receiver path was calculated to approximately 1.9 dB. The baseband signals were collected on a PC by using the data acquisition board with a sampling frequency of 1 kHz. Vital signs were obtained by filtering the baseband signals with a high-order band-pass filter in the digital domain. The multi-phase mixing in the radar module was used in the DC offset compensation in the received signals to improve the accuracy of the phase demodulation in signal processing [43].

**Figure 2.** A 5.8 GHz continuous-wave Doppler radar sensors for the measurement of vital signs [43].

Figure 3 shows the experimental setup for vital signs detection with a 5.8 GHz CW Doppler radar. The vital signs detected by the radar, which is located at a distance of 30 cm from the thorax of the subject and fixed on the bottom of the table above the subject, were obtained by using the data acquisition board (NI USB-6366, National Instruments, Austin, TX, USA) at a sampling rate of 1 kHz. A three-electrode ECG sensor (Vernier Software and Technology, Beaverton, OR, USA.), which was used as the reference sensor for heartbeat detection of the radar, was obtained at a sampling rate of 200 Hz. Heartbeat signals simultaneously obtained by the radar and ECG sensors were compared using NI LabVIEW and MATLAB on a computer. The vital signals of 11 subjects (5 males and 6 females in their 20 s) lying on the bed were continuously measured for 160 s. The movement of the subjects were limited to minimize the effect of the noise generated by motion on the ECG and radar sensors.

**Figure 3.** Block diagram of the experimental setup for vital signal detection using the 5.8 GHz CW radar. **Figure 3.** Block diagram of the experimental setup for vital signal detection using the 5.8 GHz CW radar.

Though the ECG signals were not used in this study, ECG and DCG signals were measured from each subject once due to the structure of the sensor system. The corresponding matches of the ECG and DCG were obtained as a dataset for the future work of cardiogram analysis.
