*3.3. The Process of the Optimal Selection of the Mother Wavelet Function and the Decomposition Level*

As the decomposition level and mother wavelet function have various candidates, the optimal selection not only evaluates a superior performance of denoising but also rates the execution efficiency to reduce unnecessary calculation. Table 2 and Figure 5 described the process of the optimal selection. First, the denoising process is repeated to accumulate the evaluation results for all candidates. Then, the optimal decomposition level and wavelet function index are obtained by finding the arguments of the maxima from the accumulated results. In Figure 5, *j* ∗ denotes the optimal decomposition level and *m*[*k* ∗ ] denotes the optimal mother wavelet function, where *k* ∗ is the function index of the optimal mother wavelet.

**Table 2.** The table of algorithm step for the analysis process of the optimal selection. <sup>∗</sup> [

#### **Algorithm Step** ∗

∗ ]

∗


]


[ ∗ ] ∗ **Figure 5.** The block diagram of the optimal selection process for the mother wavelet function *m*[*k* ∗ ] and decomposition level *j* ∗ .

#### **4. Result**

#### *4.1. Wavelet Decomposition Level Prediction*

To determine the optimal mother wavelet function, the arbitrary decomposition level should be estimated. In this study, the decomposition level is estimated based on the dominant frequency range of the original signal. At the data analysis step, the original DCG signal has the dominant frequency band at 0 to 5 Hz. The approximation coefficients of the decomposition level seven obtain a frequency band of about 0 to 4 Hz, depending on the frequency decomposition rule [68]. Tables 3–8 show the evaluation criterion *e f f*(*j*, *k*) values for the candidates of the wavelet families at decomposition levels 3 to 8. According to Tables 3–8, it can be shown that each wavelet family has the maximum *e f f*(*j*, *k*) at seven. Thus, the estimated optimal decomposition level is set to seven.


**Table 3.** The *e f f*(*j*, *k*) value of the Coiflets family (wavelet length 18–30).

**Table 4.** The *e f f*(*j*, *k*) value of the Daubechies family (wavelet length 18–22).


**Table 5.** The *e f f*(*j*, *k*) value of the Fejer–Korovkin family (wavelet length 18–22).


**Table 6.** The *e f f*(*j*, *k*) value of the Biorthogonal Spline family (wavelet length 18–20).


**Table 7.** The *e f f*(*j*, *k*) value of the Reverse Biorthogonal Spline family (wavelet length 18–20).


**Table 8.** The *e f f*(*j*, *k*) value of the Symlets family (wavelet length 18–20).


#### *4.2. Most Efficient Mother Wavelet Selection*

In this section, the analytic results of the denoising performance using 115 wavelet functions are presented. The examined wavelet functions are selected from the six wavelet families including Coiflets (coif1-coif5), Daubechies (db1-db45), Fejer–Korovkin (fk4-fk22), Biorthogonal Spline (bior1.1-bior6.8), Reverse Biorthogonal Spline (rbio1.1-rbio6.8), and Symlets (sym2-sym30) [69,70]. The length of the wavelet varies by the number of the mother wavelet function name in range of 2 (db1, coif1, etc.) to 90 (db45), only even numbers. Traditionally, the optimal mother wavelet for noise reduction should satisfy the properties of orthogonality, symmetry, regularity, similarity of the shape with the signal, and so on [29,49,71]. According to the experiment result, however, the wavelet length of the mother wavelet function was the key determinant for the noise elimination of the DCG signal. The blue plot of the Figure 6 shows that the *SNRj*, *<sup>k</sup>* of the denoised signal increases with the wavelet length of the mother wavelet function and is saturated at a wavelet length of 18. Although the values of the *SNRj*, *<sup>k</sup>* from the wavelet length of 22 to 90 are similar, execution complexity keeping up with the wavelet length makes the longer wavelet function excessive. As the evaluation criterion *e f f*(*j*, *k*) includes both

the denoising performance *SNRj*, *<sup>k</sup>* and the execution complexity *η*(*k*), the most efficient wavelet length can be selected as 18 from the red plot of Figure 6. Not only does the execution efficiency become worse but the required length of the signal also increases when the wavelet length grows. (, ) , ()

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, (, ) **Figure 6.** Denoising performance of the wavelet length (2 to 90), blue plot denotes *SNRj*, *<sup>k</sup>* values and red plot denotes *e f f*(*j*, *k*). The used decomposition level *j* is 7.

(, ) " 8" " 8" " 9" " 9" " " " " However, the wavelet length is not the only feature that represents the characteristic of the mother wavelet function. Though few numbers of the wavelet functions share the same wavelet length, the basis functions and basic features are distinctive. Therefore, even with the same wavelet length, the denoising performance *e f f*(*j*, *k*) will vary depending on the basis function of the mother wavelet function. At a wavelet length of 16, "db8" and "sym8" scored 5.28 and 5.31 which are higher than other wavelets and the average performance of a wavelet length of 16 is 4.92 (Figure 7a). In the case a wavelet length of 18, "db9" and "sym9" perform better again (Figure 7b). For a wavelet length of 20 (Figure 7c), "db" and "sym" are always the optimal wavelet functions. On the contrary with the Daubechies and Symlets families, the function from the Biorthogonal family shows the lowest performance at all three graphs.

(, ) **Figure 7.** *e f f*(*j*, *k*) values of six wavelet families (db, sym, coif, fk, bior, and rbio) for wavelet lengths of: (**a**) 16, (**b**) 18, and (**c**) 20. The used decomposition level *j* is 7.

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