# Wavelet Packet Decomposition to Characterize Injection Molding Tool Damage +
## 3 Results and Discussion
### AE Burst Descriptors  
Measured AE bursts are used for derivation of AE descriptors. These descriptors are stacked to form a feature (measurement) vector \(Z\) = (\(z_{1}\),\(z_{2}\),...,\(z_{73}\)), representing acquired AE bursts during different phases of injection molding process. The union of 704 feature vectors, in this paper, composes feature space. AE burst is extracted from acquired AE waveform based on defined rearm time; in our research this was 0.46 ms. An AE burst is over when the rearm time elapses due to the absence of threshold crossings. Calculated time domain descriptors based on acquired AE bursts are: peak amplitude value \(A_{p}\), duration \(d\), rise time \(Rt\), and \(RA\) value. \(RA\) value is the ratio of rise time and amplitude of AE burst [15]. We calculated frequency domain descriptors. The \(P_{L}\) parameter characterizes the lower part of the power spectrum of each AE burst (\(a\) = 90 kHz, \(b\) = 190 kHz). Parameter \(P_{H}\) characterizes the higher part of the power spectrum (\(a\) = 250 kHz, \(b\) = 350 kHz), and parameter \(P_{S}\) covers all power spectrum frequencies (\(a\) = 50 kHz, \(b\) = 550 kHz). The parameters \(P_{L}\), \(P_{H}\), and \(P_{S}\) carry information about the energies at various frequencies in a sensor's frequency spectrum:  
\[g_{i}=\frac{y\overline{yi}}{m} \tag{1}\]  
\[m=\max\left(\mathbf{Y}\cdot\overline{\mathbf{Y}}\right) \tag{2}\]  
\[P_{L}=\sum_{i=90}^{b=190}g_{i} \tag{3}\]  
\(\mathbf{Y}=\left(y_{1}\),\(y_{2}\),...,\(y_{n}\right)\) represents the vector of the discrete Fourier transform using FFT. We introduced the kurtosis and skewness parameters describing the amplitude distribution of recorded AE burst. Kurtosis is a measure of the \"peakedness\" of the probability distribution for a real-valued random variable. In our case we calculated the kurtosis of measured AE burst amplitude values. Kurtosis was used in our research to characterize infrequent extreme deviations of amplitude values in AE burst as:  
\[K=\frac{E(a_{i}-\overline{a})^{4}}{s^{4}} \tag{4}\]  
where \(s\) is a sample standard deviation, \(a_{i}\) is an amplitude value. Skewness \(S\) is defined as a third central moment of amplitude values in AE burst about its mean value. Skewness is in our research was used as a measure of asymmetry about the mean amplitude value of AE burst and is defined as:  
\[S=\frac{E(a_{i}-\overline{a})^{3}}{s^{3}} \tag{5}\]  
Wavelet Packet Analysis Descriptors  
Wavelet analysis overcomes the pitfalls of Fourier methods and is suitable for processing AE signals during injection molding. The wavelet packet analysis is a subtler multiresolution analysis, which decomposes signal into multi-levels in the whole frequency band [21]. The detail of the signal is decomposed as well, so the frequency resolution is improved [22]. The wavelet packets include information on signals in different time windows at different resolutions. Each packet corresponds to some frequency band. Some packets contain important information while others are relatively unimportant. Wavelet packet transform of energy limited signal \(X(t)\) can be computed using the algorithm below:  
\[P_{0}^{1}\left(t\right)=X\left(t\right) \tag{6}\]  
\[P_{j}^{2i-1}\left(t\right)=HP_{j-1}^{i}\left(t\right) \tag{7}\]  
\[P_{j}^{2i}\left(t\right)=GP_{j-1}^{i}\left(t\right) \tag{8}\]  
where \(P_{j}^{i}\left(t\right)\) is the \(i^{\text{th}}\) packet on \(j^{\text{th}}\) resolution, with time parameter \(t=1,2,\ldots,2^{J-j}\), \(i=1,2,\ldots,2^{j},j=1,2,\ldots,J,J=\text{log}_{2}N.\) Operators \(H\) and \(G\) are convolution sum defined as:  
\[H\left\{\cdot\right\}=\sum_{k}h\left(k-2t\right) \tag{9}\]  
\[G\left\{\cdot\right\}=\sum_{k}g\left(k-2t\right) \tag{10}\]  
where \(h(t)\) and \(g(t)\) are a pair of quadrature mirror filters and a time parameter \(t\) is taken as a series of integers \(k\) (\(t\to k=1,\,2,\ldots\) ).