In many applications, it is very important to separate the noise from the true signal, which could reveal the physical mechanism under the signal. For the purpose of illustration, we consider both the synthetic signals and the differential pressure signals. These numerical examples demonstrate the effectiveness of the Domain-Wavelet algorithm, proposed in this paper. For the bump signal with length N=1024, we add the zero-mean Gaussian noise to it. Here the signal–to-noise ratio (SNR) is defined by
$$SNR=20\mathrm{l}\mathrm{g}\frac{\parallel f-\hat{f}{\parallel}_{2}}{\parallel n|{|}_{2}}$$ where *f* stands for the original signal, and
$\hat{f}$ represents the noisy signal. The input signals are depicted in Fig. 1.

Figure 1 Left: Original Signal. Right: noisy bump.

To verify domain-wavelet algorithm introduced above is superior in terms of de-noising synthetic signal, we choose wavelet threshold algorithm and domain-wavelet algorithm to process the synthetic signal (see Fig. 1). From Fig. 2, we know that the de-noised signals are much better when the domain-wavelet method is used, relative to those generatefd using the traditional wavelet threshold methods, including both hard threshold and soft threshold.

Figure 2 De-noising Results for the Bump signal. (a) Hard threshold (b) Soft threshold (c) Domain-Wavelet (Hard threshold) (d) Domain-Wavelet (Hard threshold).

Here in the experiments, we choose wavelet function as ‘db8’, the decomposition level of wavelet threshold algorithm is 5 and the decomposition level of domain-wavelet algorithm are J = 8 and J^{′} = 5 separately. We now demonstrate the de-noising results of the traditional wavelet threshold methods and domain-wavelet method in Fig. 2.

Table 1 Results of the de-noised Bump signal with four methods.

Fig. 3(a) shows the SNR curve with the decomposition levels using hard threshold and domain-wavelet algorithm, while (b) show the SNR curve with the decomposition level using the soft threshold and domain-wavelet algorithm.

The de-noising approaches based on wavelet threshold and domain-wavelet algorithm are denoted by Hard, Domain (hard), Soft, Domain (soft), respectively, for convenience. Fig. 3 show the SNRs of the four de-noised signals obtained by the different methods introduced in section 2 and section 4 separately, when the decomposition level equal to 2,3,4,5,6,7,8, and 9. We note that the SNR curves obtained by the domain-wavelet are larger than those obtained with the traditional wavelet threshold, either using soft or hard threshold. It indicates that the proposed domain-wavelet algorithm is more suitable for de-noising the synthetic signals so as to obtain much more efficient and accurate de-noised results in the future.

Figure 3 Comparison between wavelet threshold results and domain-wavelet results by using hard threshold (a) and soft threshold (b).

Subsequently, we apply both the wavelet threshold and Domain-Wavelet algorithm to process the true signal. Here we select a differential pressure signal that has 45855 samples, it is shown in Fig. 4.

Figure 4 Differential pressure signal.

Here we choose wavelet function as “db6”, the decomposition level of wavelet threshold algorithm is 8 and the decomposition level of domain-wavelet algorithm are J=13 and J^{′} =8, respectively. The results following de-noising with the four methods are illustrated in Fig. 5.

Figure 5 De-noised Results of the differential pressure signal.

Fig. 5 from (a) to (d) show the waveforms of de-noised differential pressure signals using the wavelet threshold and domain-wavelet algorithm. It is noted that the de-noised pressure signal using the domain-wavelet are much improved relative to the pressure signal de-noised by the wavelet threshold for both hard threshold and soft threshold. It manifests that domain-wavelet algorithm is superior to wavelet threshold in small waveform distortion and low energy losses of de-noised signals.

To give an objective evaluation criteria, we introduce the concept of smoothness index, which is defined as follows:
$$r=\frac{\sqrt{\sum _{n}[\hat{f}(n+1)-\hat{f}(n){]}^{2}}}{\sqrt{\sum _{n}[f(n+1)-f(n){]}^{2}}}$$ Where, *f*(*u*) is noisy signal,
$\hat{f}(n)$ is de-noised signal. The index reflects the smoothing degree of de-noised signal. Smoothing is an important index for judging the effect of de-noised signal. The results show that the smaller the smoothness index, the better the de-noising results.

shows the smoothness index of de-noised differential pressure signals obtained by hard threshold, soft threshold, domain-wavelet (hard) and domain-wavelet (soft). The de-noised differential pressure signals obtained by domain-wavelet have very small smoothness index, compared to those obtained by wavelet thresholds. Their smoothness indexes are less than 10% the smoothness indexes of the de-noised signals obtained by wavelet thresholds. The smaller the smoothness index, the more helpful for extracting features of the differential pressure signals, and thus, also helpful for obtaining successful pattern recognition of the differential pressure signals.

Table 2 The smoothness index of the de-noised differential pressure signals.

Further, we demonstrate the relation between decomposition level and the smoothness index by using wavelet threshold and domain-wavelet for hard threshold and soft threshold (see Fig. 6).

Figure 6 Comparison of The smoothness index between wavelet threshold results and domain-wavelet results by using hard threshold (a) and soft threshold (b).

Fig. 6 shows that the domain-wavelet algorithm is superior to wavelet threshold algorithm, the de-noising results of domain-wavelet algorithm are also relatively stable for different decomposition levels.

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