## 1 Introduction

Thrust is the most important indicator of an aircraft engine. The indoor test station is an important platform for aero engine performance testing. Currently all engine thrust in research needs to be tested at the indoor test station, but signals from various sensors have large differences. Some signals have large fluctuation which almost obscures the true signal and make it very difficult for data processing, especially differential pressure signal. The error generated by aero engine’s thrust measure in upwind environment is a system error, which needs to be corrected, so the de-noising process is needed.

Wavelet transform has good time-frequency localization properties, low entropy, multi-resolution properties, decorrelation and wavelets selection diversity, so there are many popular de-noising methods, especially traditional wavelet threshold algorithms have been extensively studied and obtained good results in signal processing. Traditional wavelet transform includes soft threshold function and hard threshold function. The soft threshold function has good continuity, but it always has a constant deviation, which will lose some signal characteristic information. Although the hard threshold function does not have a constant deviation, the generated signal de-noised by it produces additional oscillation, and the result is not continuous [1–4].

Domain transform is a distance-preserving transform. By preserving the distance between two points on two-dimensional space curve to the one-dimensional real axis, the conversion of variable domain has come true [5–10].

## 2 The Wavelet Threshold De-noising Algorithm

### 2.1 Traditional wavelet threshold de-noising algorithm

The main theories of wavelet threshold de-noising algorithm: Wavelet transform, especially orthogonal wavelet transform can efficiently eliminate the correlation coefficients. It enables the signal energy to be concentrated in some large wavelet coefficients while the noisy energy to be distributed throughout the wavelet domain. Usually, the amplitude of signal wavelet coefficients is greater than the amplitude of noisy wavelet coefficients. Therefore, we can select an appropriate threshold function to keep the signal coefficients and reduce most of the noisy coefficients to zero.

Traditional wavelet threshold de-noising algorithm has three steps:

- Wavelet decomposition: Select the appropriate wavelet function and decomposition level, so we obtain the wavelet coefficients by decomposing the noisy signal.
- High-frequency coefficients thresholding: Select the appropriate threshold to process the high-frequency coefficients, then get new wavelet coefficients.
- Reconstruct the signal: The estimated signal is obtained by reconstructing the signal with all the low-frequency coefficients and the high-frequency coefficients obtained by the second step.

Traditional wavelet functions mainly include hard threshold function and soft threshold function as the following:

- Hard Threshold Function
${\stackrel{~}{w}}_{j,k}=\left(\begin{array}{cc}{w}_{j,k}& |{w}_{j,k}|\ge \lambda \\ 0& |{w}_{j,k}|<\lambda \end{array}\right)$ - Soft Threshold Function
${\stackrel{~}{w}}_{j,k}=\left(\begin{array}{cc}{w}_{j,k}-\lambda & |{w}_{j,k}|\ge \lambda \\ 0& |{w}_{j,k}|<\lambda \\ {w}_{j,k}+\lambda & {w}_{j,k}\le -\lambda \end{array}\right)$

where *λ* is the threshold, *w*_{j,k} is the wavelet coefficient.

There are many methods to determine the threshold, especially the global uniform threshold is one of the most famous methods among them. This method takes threshold
*σ* is the standard deviation of the signal, and *N* represents the length of the noisy signal.

The soft threshold function has good continuity, but it always has a constant deviation, which will lose some signal characteristic information that is very important for the signal processing. Although the hard threshold function does not have a constant deviation, it causes additional oscillation to the de-noised signal and discontinuouity [1–4].

### 2.2 Improved wavelet threshold algorithm

To overcome the disadvantages of hard threshold function and soft threshold function, we proposed an improved threshold function as follows:

where

Here *j* is the decomposition level, *λ*_{j} is an adaptive threshold, *σ* is the standard deviation of the signal, *N* represents the length of the noisy signal and *w*_{j,k} is the wavelet coefficient.

When |*w*_{j,k}| = *λ*_{j}, then *α* = 1,

When |*w*_{j,k}| → *λ*_{j}, then *α* → 1,
*w*_{j,k}| = *λ*_{j}.

When |*w*_{j,k}| → ∞, then *α*→ 0,
*w*_{j,k}. It overcomes the weakness that
*w*_{j,k} always have constant bias.

## 3 Domain Transform

For deriving an isometric 1D transform, let *I* : *Ω* → *R*, *Ω* = [0, +∞) ⊂ *R* be a 1D signal, which defines a curve C in **R**^{2} by the graph (*x*, *I*(*x*)), for *x* ∈ *Ω*. Our goal is to find a transform *t* : *R*^{2} → *R* which preserves, in **R**, the original distances between points on C, given by some metric. Thus, let *S* = {*x*_{0}, *x*_{1}, *L*, *x*_{n}} be a sampling of *Ω*, where *x*_{i1+} = *x*_{i} + *h*, for some sampling interval *h*. We seek a transform *t* : *R*^{2} → *R* that satisfies

*x*

_{i},

*x*

_{j}∈

*S*, |⋅| is the absolute value operator, and ∥⋅∥ is some chosen metric. For simplicity, we use the nearest neighbor

*l*

_{1}norm; thus,

*t*only needs to preserve the distances between neighboring samples

*x*

_{i}and

*x*

_{i+1}. Finally, let

*l*

_{1}norm)

**R**) must equal the

*l*

_{1}distance between them in the original domain (

**R**

^{2}). To avoid the need for the absolute value operator on the left of (2), we constrain

*ct*to be monotonically increasing—i.e.

*ct*(

*x*+

*h*) ≥

*ct*(

*x*). Dividing both sides of (2) by

*h*and taking the limit as

*h*→ 0, we obtain

*ct*

^{′}(

*x*) denotes the derivative of

*ct*(

*x*) with respect to

*x*. Integrating (3) on both sides and letting

*ct*(0) = 0, we get

Intuitively, *ct* is “unfolding” the curve *C* defined in **R**^{2} into **R**, while preserving the distances among neighboring samples. Moreover, for any two points *u* and *w* in *Ω*, *w* > *u*, the distance between them in the new domain is given by

Which is the arc length of curve *C* in the interval [*u*, *w*] under the *l*_{1} norm. As such, the transformation given by Equation (4) preserves the geodesic distance between all points on the curve. A similar derivation is possible for the *l*_{2} norm or perhaps other metrics [5].

## 4 Domain-Wavelet Algorithm

In this section, we derive the proposed algorithm by combining domain transform with wavelet threshold algorithm, naming it here as the Domain-Wavelet algorithm, which has the following steps:

- Calculating domain transformation
*t*We suppose that the original signal*I*:*Ω*→*R*.{*x*_{0},*x*_{1},*L*,*x*_{n}} is a division of*Ω*with the step size*h*. LetObviously,$t\left({x}_{i}\right)=\left(\begin{array}{c}\phantom{\rule{0ex}{0ex}}I\left({x}_{0}\right)+{x}_{0}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}i=0\\ \phantom{\rule{0ex}{0ex}}t\left({x}_{i-1}\right)+|I\left({x}_{1}\right)-I\left({x}_{i-1}\right)|+h\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}i=1,\phantom{\rule{0ex}{0ex}}2,\phantom{\rule{0ex}{0ex}}L,\phantom{\rule{0ex}{0ex}}n\end{array}\right)$ *t*is a distance-preserving transform based on*l*_{1}norm and*t*is also an increasing function. Let*I*^{′}(*t*_{i})=*I*^{′}(*t*(*x*_{i})) =*I*(*x*_{i}),*i*= 0, 1,*L*,*n*. - ResamplingSince the value of
*t*_{i+1}−*t*_{i}which is related to*i*is not a constant, we need to resample. Take the interval [*t*(*x*_{i}),*t*(*x*_{i+1}] as an example.Let$m=\underset{i}{min}t\left({x}_{i}\right)=t\left({x}_{0}\right),\phantom{\rule{0ex}{0ex}}M\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\underset{i}{max}t\left({x}_{i}\right)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}t\left({x}_{n}\right),$ *h*^{′}= (*M*−*m*)/6*n*,*h*^{′}is a new step size. So the interval [*t*(*x*_{i}),*t*(*x*_{i+1})] contains the set of points after resampling.Here, we let$\{{t}_{k}^{i},k\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0,L,\frac{{t}_{i+1}-{t}_{i}}{h}\}$ So the resampled signal is$\left(\begin{array}{c}\phantom{\rule{0ex}{0ex}}{x}_{k}^{i}=\frac{{t}_{k}^{i}-{t}_{i}}{{t}_{i+1}-{t}_{i}}\cdot ({x}_{i+1}-{x}_{i})+{x}_{i}\\ \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}i=0,L,n-1;k=0,L,\frac{{t}_{i+1}-{t}_{i}}{h}\\ \phantom{\rule{0ex}{0ex}}{I}^{\prime}\left({t}_{k}^{i}\right)=I\left({x}_{k}^{i}\right)=\frac{{t}_{k}^{i}-{t}_{i}}{{t}_{i+1}-{t}_{i}}\cdot \left(I\right({x}_{i+1})-I({x}_{i}\left)\right)+I\left({x}_{i}\right)\end{array}\right)$ $\left\{\right({t}_{k}^{i},I\left({t}_{k}^{i}\right)),i=0,L,n-1;k=0,L,\frac{{t}_{i+1}-{t}_{i}}{h}\}.$ - De-noisingWe use the new wavelet threshold algorithm proposed in section 2.2 to process the signal
Let the number of decomposition level is equal to$\left\{\right({t}_{k}^{i},I\left({t}_{k}^{i}\right)),i=0,L,n-1;k=0,L,\frac{{t}_{i+1}-{t}_{i}}{h}\}.$ *J*and de-noising results is$\stackrel{~}{I}\left({t}_{k}\right).$ - Restoring samplesTo reconstruct the original signal, we just take points
Apparently, the de-noising signal is$({t}_{0}^{i},\stackrel{~}{I}({t}_{0}^{i}\left)\right),i=0,\phantom{\rule{0ex}{0ex}}1,\phantom{\rule{0ex}{0ex}}L,\phantom{\rule{0ex}{0ex}}n.$ $(x,\stackrel{~}{I}(t\left({x}_{i}\right)\left)\right)),i=0,\phantom{\rule{0ex}{0ex}}1,\phantom{\rule{0ex}{0ex}}L,\phantom{\rule{0ex}{0ex}}n.$ - Second De-noisingDuring the resampling process at step 2, it generates some redundant information for the signal, so we choose the wavelet threshold algorithms to de-noise the signal
again. Let the decomposition level is equal to$(x,\stackrel{~}{I}(t\left({x}_{i}\right)\left)\right)$ *J*^{′}.

## 5 Applications

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

*f*stands for the original signal, and

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.

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.

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

Algorithm | SNR/dB |
---|---|

Wavelet threshold (Hard threshold) | 26.7408 |

Wavelet threshold (Soft threshold) | 24.0098 |

Domain-wavelet(Hard threshold) | 26.9288 |

Domain-wavelet (Soft threshold) | 27.5421 |

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.

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.

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.

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:

*f*(

*u*) is noisy signal,

Table 2 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.

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

Algorithm | Smoothness Index |
---|---|

Wavelet threshold (Hard threshold) | 0.6517 |

Wavelet threshold (Soft threshold) | 0.2318 |

Domain-wavelet (Hard threshold) | 0.033 |

Domain-wavelet (Soft threshold) | 0.0251 |

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).

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.

## 6 Conclusions

Thrust is the most important indicator of an aircraft engine. The indoor test station is an important platform for aero engine performance testing. Currently all engine thrust in research needs to be tested at the indoor test station, but signals from various sensors have large differences. Some signals have large fluctuation which almost obscures the true signal and make it very difficult for data processing, especially for the differential pressure signal. The error generated by aero engine’s thrust measure in upwind environment is a system error, which needs to be corrected.

Based on domain transform and wavelet threshold algorithm, we propose a new de-noising method: Domain-Wavelet Algorithm. First, transformed the signal domain transform, then sampled the transformed signal, and finally the wavelet threshold algorithm was performed for the signal. By observing the de-noised signal and comparing the SNR values we conclude that domain-wavelet algorithm is superior to traditional wavelet threshold algorithms. For the differential pressure signal, there are no original signals, so we use the smoothness index which is similar to SNR. It also proves that the new method proposed in this paper can optimize de-noising results relative to traditional wavelet threshold methods.

As shown in Figure 3 and Figure 6, the domain-wavelet algorithm achieved higher SNR and lower smoothness index for both the synthetic signal and the differential pressure signal, in comparison with the traditional wavelet threshold algorithms. However, it’s running time is longer than the traditional wavelet threshold algorithms, because of the resampling procedure in within the domain-wavelet algorithm, which increases the size of data and affects the operational speed. In the future, the running time of domain-wavelet algorithm we proposed in this paper should be shortened and this algorithm should be tested for other signal processing.

The authors acknowledge the fund of National Basic Research Program of China (52850301) and 973 Grant(73026101).

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