Saravanan et al. Yu et al. They extracted features from energy content of the clusters and used probabilistic neural network for bearing fault diagnosis. Nikolaou et al. Rafiee et al. Boskoski et al. Li et al. Pan et al. They used energy of reconstructed signals from wavelet packet nodes as feature vector.

Bin et al.

In this method, features were extracted by empirical mode decomposition from signals reconstructed from wavelet packets. In this research the superiority of reconstructed signal from wavelet packet node over wavelet packet coefficients for feature extraction has been demonstrated by one example. This fact is supported by the data acquisition from a real test rig of Yamaha motorcycle gearbox to classify four different health condition of the gearbox.

Wavelet packet is a linear combination of usual wavelet functions, which inherits the attributes of its corresponding wavelet functions such as orthonormality and time-frequency localization [7]. Discrete filters h k and g k are quadrature mirror filters associated with scaling function and mother wavelet function [11].

Wavelet packet coefficients c j , k i of f t signal are computed using Eq. Each wavelet packet can be reconstructed to make a part of the signal.

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Inverse wavelet transform is used to reconstruct signal. Usually, all the wavelet packets are used for signal reconstruction. But, in this research signal reconstruction is done for each wavelet packet independently. Therefore f j i is the reconstructed signal from wavelet packet j , i based on Eq. The original signal can be obtained by summing reconstructed signals from packets of j th level decomposition as follows:. Frequency band in each packet at j th level decomposition is as follows:.

Down-sampling is done to avoid generating redundant data after each decomposition, as shown in Fig. Sorting function S changes Paley order i to frequency order S i by the following recursive equations [12]:. Starting frequency of each packet is S i F j. One advantage of wavelet transform is diversity of mother wavelet. Selection of proper wavelet is very important because it can affect the analysis results [13]. In this research, maximum energy to Shannon entropy criterion was used for mother wavelet selection. This criterion states that the mother wavelet that has produced the maximum energy to Shannon entropy ratio should be chosen as the most appropriate wavelet.

Energy of signal is defined as follows:. In the above equations, E j i is energy of i th wavelet packet node at j th level of the signal. E j is sum of all energy packets in j th level which is equal to the energy of original signal. Shannon entropy S e n t r o p y j is defined as follows:.

## Frequency Analysis Using the Wavelet Packet Transform

Based on maximum energy to Shannon entropy criterion, mother wavelet is selected and this criterion is computed by Eq. Since discrete wavelet packet has been used in this research, mother wavelets like Gaussian, Mexican hat, Meyer and other complex wavelets could not be used. In this research, 75 mother wavelets were considered, as shown in Table 1.

Table 1.

Mother wavelet functions used in this research. MD measures distance between two groups of samples. In this section reconstructed signal and wavelet packet coefficients are compared by one example. Vibration signal of a rotating machine with a specific fault was simulated based on Eqs. In each simulation white Gaussian noise n t with signal to noise ratio 5 was added to signal:. In Eq. Amplitude variation in the fault frequency is considered based on g t function. Values of g t based on fault severity were considered as 0, 0. For each of these four fault severity, 30 signals in time span of [0 0.

Discrete Meyer dmey mother wavelet was selected based on maximum energy to Shannon entropy criterion. Signals were decomposed to level 4.

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## SIAM Review

In order to compare energy feature of wavelet packet coefficients and reconstructed signal of 4, 3 Packet node, they were normalized between 0 and 1. As is shown in Figs. Normalized energy of coefficients in 4, 3 wavelet packet with dmey mother wavelet for four different fault severity. Table 2 shows Mahalanobis distances for four different conditions of the simulated signals. In all cases, Mahalanobis distance of two different conditions of features extracted based on normalized energy of reconstructed signal of 4, 3 Packet node, is greater than normalized energy of coefficients in 4, 3 wavelet packet.

Therefore, compared with the normalized energy of coefficients in 4, 3 , the normalized energy of reconstructed signal of 4, 3 Packet node have higher distinguishability. Normalized energy of reconstructed signal from 4, 3 wavelet packet with dmey for four different fault severity.

Table 2. Mahalanobis distances for four different conditions of the simulated signals. In order to show effectiveness of feature extracted from reconstructed signals, a Yamaha motorcycle gearbox was used for experimental test. The input shaft of a motorcycle's four-speed gearbox that contained gearbox oil while collecting signals was rotated by an electric motor with the nominal speed of RPM.

A load mechanism that was a friction wheel was positioned on its output shaft. The vibration signals were collected by the sampling frequency of Hz using an accelerometer sensor which was installed on the outer surface of the gearbox housing, near its input shaft. The real rotational speed of the motor was measured by a tachometer. There were four shock absorbers under the bases of the test-bed. Data collection of the gearbox was implemented in four states: with faultless gear, slight-worn, medium-worn, and broken tooth gear. The faults were created on the B4 gear at the 4th stage of the gearbox mating.

In this stage, the A4 and B4 gears were mating. The vibration data collected under all of the four gear conditions were divided into 90 pieces indicated by the pulses acquired by the tachometer, with each piece corresponding to one revolution of the input shaft. First, the proper mother wavelet function was chosen by maximum energy to Shannon entropy criterion. As shown in this figure, the 39th wavelet which was db39 had the maximum ratio of energy to Shannon entropy and thus was chosen as the best mother wavelet function.

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I want to estimate dominate frequencies in the original signal. Imagine I have following signal:. So my question, what sampling rate should I select for this case to identify these 3 freq? I've select 10Hz for sampling rate and what level of decomposition should I select to see all of these 3 frequencies?

Here is my little scrip for this signal with wavelet packet.

You can achieve this through separation of the three components you have into different bands. Assume a signal of two sinusoids 0. Wavelet packet transform is not well suited for creating such a spectrogram. The spectrum of higher-order wavelet packets has several strong peaks not just one , which explains why you are getting spurious results.

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