## 5 Apparent Power Signature Processing
### Wavelet Packet Decomposition (WPD)  
WPD is an extension of the DWT by generalizing the link between multi-resolution approximation and the Mallat algorithm. In this algorithm, WPD splits the approximation and detail space simultaneously. It has been shown that better frequency resolution is obtained so that more features can be extracted from the original signal [20]. The recursive process of WPD is represented by the type of binary tree in Figure 5. The wavelet packet coefficients are represented by \(P_{j}^{m}\). Each node of the wavelet packet tree is denoted with a pair of integers \((j,m)\), where \(j\) represents the decomposition level and \(m\) represents the order position at the corresponding level. Similar to the DWT, the original signal \(S\) is decomposed into the approximation part \(P_{1}^{1}\) and detail part \(P_{1}^{2}\) at the first level. In the next step, the detail part \(P_{1}^{2}\) is split into the high-frequency part \(P_{2}^{4}\) and the low-frequency part \(P_{2}^{3}\). This process is repeated in the following step. According to the coefficient \(P_{j}^{m}\), a reconstruction signal \(S_{j}^{m}\) of data length \(N\) can be generated, by setting all the other coefficients to be zero and implementing the wavelet packet tree in the inverse procedure [21].  
To distinguish the characteristic harmonic component from the apparent-power signature, the type of mother wavelet and the decomposition level must be well selected. The WPD of the original signal \(S\) can be regarded as projecting the signal onto the wavelet packet basis, in order to obtain a series of the coefficient \(P_{j}^{m}\). For the purpose of extracting the fault signal, the wavelet coefficient should be larger than those of other series. Besides that, the appropriate selection of decomposition level can help to extract the characteristic signal and save the computation time.  
In this paper, the db44 mother wavelet is used for simulation results. By considering the computation time using FEA, a sampling frequency is chosen at 1 kHz and the time acquisition is selected at one second with the frequency resolution of 1 Hz. For the simulation case, the characteristic component frequency appears at \(kf_{r}\) in the apparent power. Therefore, the original apparent power signal is decomposed into 8 levels. The node (8, 10) and the node (8, 21) which contain harmonic components \(f_{r}\) and \(2f_{r}\) are selected in order to reconstruct the signal.  
Figure 6 gives the decomposition result based on the simple use of WPD. The characteristic faulty component, which is approximately 48.34 Hz, is chosen as an analytical object. Moreover, the motor is under 33% of rated load with 2% of purely DE. The FFT spectrum shows that the dominant spectral components are located at a frequency of 48.34 Hz. However, it is noticeable that there exist other  
Figure 5: Wavelet packet decomposition tree for decomposition of the original signal.  
components of 44.18 Hz and 25.39 Hz in the red circle shown in Figure 6. These two parts are not related to the eccentricity fault, which is inevitable for the WPD due to the overlap problem between adjacent bands. Therefore, it is essential to filter these two noisy components in order to improve the SNR. After using the WPD algorithm, the filtered signal needs to be post-processed in order to eliminate or suppress the corresponding noisy components.