**3. Experiment**

#### *3.1. Additive White Gaussian Noise*

Additive white Gaussian noise is the fundamental noise model in the information signal processing with three important characteristics manifested in its term. First, additive means the way in which noise joins in the signal. The noisy signal *sN*[*n*] is generated by adding the noise components *N*[*n*] to the original signal *s*[*n*] [47]. This process is described in Figure 4 and the mathematical expression is described as below:

$$s\_N[n] = s[n] + N[n].\tag{9}$$

**Figure 4.** The block diagram of the denoising method.

– Second, white denotes the power spectrum density of the noise is a constant value across the frequency range and the noise exists at almost every frequency [48]. Finally, Gaussian represents that the distribution of the noise is the Gaussian random process. The additive white Gaussian noise consists of normal distribution in the time domain and independent and identical distribution (i.i.d) in the frequency domain [49]. In this study, the additive white Gaussian noise components follow the distribution of *N*(0, *σ*), which is copied to the electromagnetic noise distribution of the original signal. Since the distribution is identical, the generated noisy signal *sN*[*n*] can represent comparable features with the real world noisy signal [50]. In the experiment, the white Gaussian noise is added to each

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, Stein's Unbiased Risk Estimate

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original DCG signal with a SNR of 10 dB by the signal processing tool that is provided from Mathworks MATLAB.

#### *3.2. Denoising Process with Wavelet Transform and Thresholding*

To improve the quality of the noisy signal, the denoising process should hold the real data factors and remove the additive noise components [27]. In the previous studies, the noise reduction methods are generally based on the model simulation or spectral analysis, in which it is difficult to keep the complicated features of the signal [27,51–53]. On the other hand, the wavelet decomposition thresholding method can control the noisy signal to maintain the original signal components and separate the noise components more precisely. In this study, the denoising method is composed of three steps as follows [54,55]:


Figure 4 describes the whole denoising process as a block diagram with an illustration of the three steps of the denoising method. *j* denotes the decomposition level of 1 to 10, *m*[*k*] is the mother wavelet function, and *k* is the function index from 1 to 115. Meanwhile, at the thresholding step, the soft thresholding is used for its superior thresholding performance to the hard thresholding [56]. There are various threshold methods for noise elimination such as Empirical Bayes [57], Block James–Stein [58], False Discovery Rate [59], Minimax Estimation [60], Stein's Unbiased Risk Estimate [61], and Universal Threshold [62,63]. As the threshold method affects the denoising performance, studies on the powerful threshold determination method are kept reported [64,65]. In this study, the Universal Threshold is selected for its simple operation and powerful performance. The mathematical expressions for the soft thresholding are described in Equations (10)–(12) [51]:

$$y\_s(c) = \left\{ \begin{array}{ll} \text{sgn}(c) \cdot (|c| - T\_{\text{II}}) , & |c| > T\_{\text{II}} \\ 0 , & |c| \le \, T\_{\text{II}} \end{array} \right\} ,\tag{10}$$

$$T\_{\rm II} = \vartheta \sqrt{2 \log(N)}.\tag{11}$$

$$
\vartheta = \text{MAD} / 0.6745,\tag{12}
$$

where *c* is the wavelet coefficient of the decomposed noisy signal [65,66] and *T<sup>U</sup>* is the universal threshold value proposed by Donoho and Johnstone [51,67]. *N* is the number of samples in the signal and *MAD* is the median absolute deviation of the wavelet coefficients. The meaning of *σ*ˆ is the estimate of the standard deviation of the noise. Following these equations, the threshold value is produced with the median value of the wavelet coefficients and the number of the sampling ratio in the signal.
