# Read e-book The lifting scheme a construction of second generation wavelets

In addition to generation of first generation wavelets with lifting, the lifting method also enables you to design second generation wavelets, which cannot be designed using Fourier-based methods. With lifting, you can design wavelets that address the shortcomings of the first generation wavelets. Lifting Background. Lifting Functions. Two main properties of interest are. To illustrate the perfect reconstruction property, the following filter bank contains two decomposition filters and two synthesis filters.

The decomposition and synthesis filters may constitute a pair of biorthogonal bases or an orthogonal basis. The capital letters denote the Z-transforms of the filters.. The first condition is usually incorrectly called the perfect reconstruction condition and the second is the anti-aliasing condition. This results if the analysis filters are shifted to be causal. Lifting designs perfect reconstruction filter banks by beginning from the basic nature of the wavelet transform. Wavelet transforms build sparse representations by exploiting the correlation inherent in most real world data. For example, plot the example of electricity consumption over a 3-day period. The data do not exhibit arbitrary changes from sample to sample. Neighboring samples exhibit correlation. This implies that if you have only the odd or even samples from the data, you can predict the even or odd samples.

How accurate your prediction is obviously depends on the nature of the correlation between adjacent samples and how closely your predictor approximates that correlation. The polyphase representation of a signal is an important concept in lifting. You can view each signal as consisting of phases , which consist of taking every N -th sample beginning with some index.

These are the even and odd polyphase components of the data. Because your increment between samples is 2, there are only two phases. If you increased your increment to 4, you can extract 4 phases. For lifting, it is sufficient to concentrate on the even and odd polyphase components. The following diagram illustrates this operation for an input signal.

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In the language of lifting, the operation of separating an input signal into even and odd components is known as the split operation, or the lazy wavelet. To understand lifting mathematically, it is necessary to understand the z-domain representation of the even and odd polyphase components. You can write the z-transform of the input signal as the sum of dilated versions of the z-transforms of the polyphase components.

Split — the signal into disjoint components. A common way to do this is to extract the even and odd polyphase components explained in Polyphase Representation. This is also known as the lazy wavelet. Generally speaking, traditional approaches entail collecting sufficient failure samples to estimate the general probability of the system or component failures and the distribution of the time-to-failure. It is usually difficult to use probability and statistics for turbo generator safety analysis due to the lack of failure samples and time-to-failure data.

The failure rate of a generator includes all the failures which cause the generator to shut down and also depends on the maintenance and operating policy of utilities. In fact, turbo generators are usually set on different operating parameters and conditions e. The variations of the operating parameters can affect operational safety whenever a single parameter or condition is out of limit and failures can also be caused by the interaction of operating parameters.

## The lifting scheme: A construction of second generation wavelets

It has been realized from the real-time operation that a component will experience more failures during heavy loading conditions than during light loading conditions, which means that the failure rate of a component in real-time operation is not constant and varies with operating parameters [ 7 ]. Depending on the operating parameters and conditions, the constitutive components of a turbo generator will go through a series of degradation states evolving from functioning to failure.

Therefore, there is a great demand of ways of assessing the operational safety of turbo generators with time-varying operational parameters and conditions during their whole life span, which is beneficial for implementing optimal condition-based maintenance schedules with low failure risk. When condition monitoring is performed during plant operational transients, the intrinsically dynamic behavior of the monitored time-varying signals should be taken into account [ 8 , 9 , 10 ]. Monitoring the condition of a component is typically based on several sensors that estimate the values of some measurable parameters signals and triggering a fault alarm when the measured signal is out of limit.

To this purpose, Baraldi et al. Lu et al. Li et al.

Hua et al. The above operational safety diagnosis and assessment methods have mainly utilized dynamic monitored information, so how to process the monitored information and associate it with operational safety is very essential. On the basis of the information entropy theory, the most uncertain probability distribution such as the equal probability distribution has the largest entropy, and the most certain probability distribution has the smallest entropy.

On this basis, the use of information entropy is widespread in engineering applications. Different types of information entropy have been defined in accordance with their own usage, such as topological entropy of a given interval map [ 14 ], spatial entropy of pixels [ 15 ], weighted multiscale permutation entropy of nonlinear time series [ 16 ], Shannon differential entropy for distributions [ 17 ], min-and max-entropies [ 18 ], collision entropy [ 19 ], permutation entropy [ 20 ], time entropy [ 21 ], multiscale entropy [ 22 ], wavelet entropy [ 23 ] and so on.

Entropy is well used in machinery fault diagnosis. Sawalhi et al. Tafreshi et al. He et al. Wu et al. Firstly, the sensor-dependent vibration signals reflecting the time-varying characteristic of an individual turbo generator are acquired by professional sensors and then analyzed by the second generation wavelet package since the wavelet transform excels in analyzing unsteady signals in both the time domain and frequency domain. Finally, the proposed method is applied in a 50 MW turbo generator to validate the effectiveness of the proposed method for operation and maintenance.

A case study of a 50 MW turbo generator is conducted in Section 4. General conclusions are drawn in Section 5. The second generation wavelet inherits the good multi-resolution ratio characteristics and the time—frequency localization properties of the first generation wavelet transform, which is a more effective and faster implementation of the wavelet transform and has the advantages of simple construction, small amount of calculations and in-place calculation [ 36 ].

It has been proved by Daubechies that arbitrary wavelet transforms can be implemented by using a lifting scheme [ 37 ]. The second generation wavelet package transform belongs to the lifting scheme based on wavelet packet transform, which contains the forward transform decomposition and inverse transform reconstruction. The inverse transform can be realized by running the forward transform backwards.

The detailed procedure is explained as follows:. The forward transform of a second generation wavelet package for signal decomposition contains three steps: split, prediction and update.

The reason for splitting an original signal into two series is that adjacent samples are much more correlated than those far from each other. Therefore, the odd and even series are highly correlated. Prediction and update : Several samples of even series can be used to predict a certain sample in the odd series, and the prediction difference is defined as a detail signal. The even series can be updated using the obtained detail signal and the modified even series is defined as an approximation signal:. The forward transform of second generation wavelet package transform is illustrated in Figure 1.

The inverse transform for signal reconstruction can be derived from the forward transform by running the lifting scheme as illustrated in Figure 1 backwards. The signal in one frequency band after decomposition is set to be reconstructed, and the others are set as zero. The signal reconstruction of second generation wavelet package transform for an appointed frequency band is carried out as follows:.

Since the second generation wavelet package transform obeys the energy conservation principle due to its bi-orthogonal basis, each of the attained 2 l frequency bands has the same bandwidth and end to end after the l th decomposition and reconstruction. A comprehensive vibration monitoring study was conducted on a 50 MW turbo generator unit shown in Figure 3 to ensure its normal start-up and operation. Therefore, the condition monitoring emphasis is focused on the vertical vibration of the 4 bearing.

In the start-up process with empty load, the peak to peak vibration in the vertical direction of 4 bearing is Moreover, the peak to peak vibration in the vertical direction is increased to The acquired vibration waveform is shown in Figure 5 , which shows disorder and dissymmetry in the top and bottom of the vibration signal. The sampling frequency is 2 KHz. The FFT spectrum of the vibration signal is shown in Figure 6. The amplitude of the running frequency is the largest in the whole frequency range.

In addition, there are some harmonic frequency components from two times the running frequency to ten times the running frequency, whose amplitudes are also large. In order to further analyze the sensor-dependent vibration signal, a second generation wavelet package is adopted to decompose the original signal to the extent of level 2, level 3 and level 4, respectively.

Each frequency band has the same bandwidth end to end. Afterwards, the relative energy of the corresponding frequency band analyzed by the second generation wavelet package is computed according to Equation The sensor-dependent signals analyzed by the second generation wavelet package in level 2. The relative energy of the first frequency band is the largest, while the relative energy of the second frequency band is larger than that of the last two frequency bands.

On the basis of the level 2 study, the original signal is further decomposed to the extent of level 3 as shown in Figure 9. The relative energy is distributed in the eight frequency bands as shown in Figure The energy of the first four frequency bands is much larger than that of the rest. The sensor-dependent signals analyzed by second generation wavelet package in level 3. On the basis of the level 3 study, the original signal is further decomposed to the extent of level 4 as shown in Figure The relative energy of the first band is much larger than that of the other frequency bands.

Except for the first largest frequency bands, the energy of the fourth to the ninth frequency band is larger than that of the rest. It is inferred that there are some fault information contained in the fourth to the ninth frequency band. Since mechanical faults and unhealthy parameters can make machinery operational condition become uncertain, the probability distribution of the monitored vibration signal become uncertain and thus induces a much lower operational safety. The operational safety degree before maintenance from sensor-dependent vibration signals. As the above analysis, the amplitude of the running frequency is the largest among the whole frequency range and some harmonic frequency components from two times the running frequency to ten times the running frequency are also large in Figure 6.

Signals analyzed by the second generation wavelet package and wavelet package energy distributions in level 2, level 3 and level 4 exhibit non-stationary, nonlinear and colored noise characteristics. Considering the start-up process with no load and loading operation conditions, the vertical vibrations of the 3 and 5 bearings which are adjacent to the 4 bearing are not high under 20 m. Different from the 3 and 5 bearings, the vibration of the 4 bearing increases with increased speed and load. It is concluded that the vibration is not caused by imbalance and misalignment for the reason that vibrations would be out of limits in multiple bearing position if an imbalance or misalignment fault occurs.

Therefore, the problem is focused on the 4 bearing itself. It is inferred that the monitored non-stationary and nonlinear components in the vibration signal of the 4 bearing may be caused by mechanical looseness and local friction, so the bearing force and support status of the sizing block and bearing lodgement must be checked. With the above analysis, the turbo generator unit is stopped and overhauled.

The preload of the 4 bearing bushing is about 0. The gaps of the left and right sizing block are checked by a filler gauge. The 0. The gap in the bottom of the 4 bearing bushing is also far away from the obligate gap of 0. Therefore, the gaps of the 4 bearing bush are re-corrected and the preload is added to the requirement of 0. After maintenance, the turbo generator unit is operated again. The sensor-dependent vibration signal is decreased in the start-up process with increasing speed.

The FFT spectrum of the vibration signal is shown in Figure 14 , which is different from the spectrum before maintenance shown in Figure 6. The amplitudes of the harmonic frequency components from two times the running frequency to ten times the running frequency are decreased. The second generation wavelet package is adopted to analyze the acquired vibration signal on level 2, level 3 and level 4, respectively.

Afterwards, the relative energy of the corresponding frequency band analyzed by the second generation wavelet package is computed according to the Equation The waveform of the sensor-dependent vibration signal in time domain after maintenance. The four obtained signals analyzed by the second generation wavelet package in level 2 are illustrated in Figure It is shown that the relative energy in the second frequency band of Figure 8 before maintenance is generated by the machinery fault information of the gaps in the 4 bearing bushing and the whole energy is decentralized in frequency bands.

Signal analysis by the second generation wavelet package in level 2 after maintenance.

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## Wim Sweldens - Google Scholar Citations

The sensor-dependent vibration signal is processed to the extent of level 3, which is shown in Figure Signal analysis by the second generation wavelet package in level 3 after maintenance. The vibration signal acquired from the test rig is utilized to analyzing according to the method proposed in Section 2. In order to validate the effectiveness of method proposed in this paper, a method based on the second generation wavelet denoising and the maximum Lyapunov exponent is utilized.

It can be seen from Fig. So, the effectiveness of the method can be validated.

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The prediction results of two forecasting methods. In the proposed method, comparing with the first generation wavelet denoising, the SGWT is more effective in denoising. Finally, a bearing test rig is established and the effectiveness of the proposed method and the necessity of SGWT are validated by experimental data.

This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Views 9. Reads 3. Downloads Introduction Bearings are the basic components of many machines, and the bearing abnormities have significant effects on the operation of machines. Framework of the proposed method 2. The second generation wavelet denoising The second generation wavelet transform, proposed by Wim Sweldens, is a new wavelet construction method using lifting scheme.

### Short-term trend prediction of bearing based on SGWT and MLE method

In the split step, an approximate signal a l at level l is split into even samples and odd samples: 1. The comparison between the first and the second generation wavelet denoising a b c d. The improved chaotic time series prediction model based on the maximum Lyapunov exponent As the chaotic time series conceal the dynamic information of chaotic system and the phase space reconstruction method can reveal the dynamic information, we can reconstruct the chaotic oscillator which depicts the trajectory of the chaotic system .

The comparison between the real data and the forecasting result a b c d. Case study 3. Experimental setup and data acquisition A mechanical test bed in the RCM laboratory of Mechanical Engineering College is used in this research to validate the effectiveness of the proposed method in this paper. The test rig. Results analysis and discussion The vibration signal acquired from the test rig is utilized to analyzing according to the method proposed in Section 2. The original signal without SGWT a b.

Time Frequency Analysis & Wavelets

The original signal with SGWT.