# Investigations into the Recognisability of Gear Damage
## 4 Linear Prediction Spectra
### Linear Prediction Theory  
Linear prediction describes a mathematical method of time series analysis in which future values of a signal are estimated based on a linear combination of past values. For this purpose, linear prediction coefficients \(\alpha_{m}\) are calculated according to [24]. These coefficients can be considered as filter coefficients of an all-pole filter. In linear prediction, it is assumed  
Figure 11: Averaged frequency spectra of the measurement series of the gearbox with damage to gear \(z_{1}\).  
Figure 12: Averaged frequency spectra of the measurement series of the gearbox with damage to gear \(z_{3}\).  
that a sufficiently stationary signal section \(x[n-m]\) is given by a linear prediction \(\hat{x}[n]\) according to  
\[\hat{x}[n]=-\sum_{m=1}^{p}\alpha_{m}x[n-m] \tag{17}\]  
with the order \(p\) of the prediction coefficients. For the calculation of the coefficients, an error term \(\varepsilon_{\text{LP},n}\) is further calculated from the input signal and the prediction according to  
\[\varepsilon_{\text{LP},n}=x[n]-\hat{x}[n]=x[n]+\sum_{m=1}^{p}\alpha_{m}x[n-m]. \tag{18}\]  
To determine the prediction coefficients, the accumulated squared error \(e_{\text{LP}}\) is defined to be  
\[e_{\text{LP}}=\sum_{n=-\infty}^{\infty}\varepsilon_{\text{LP},n}^{2}=\sum_{n=- \infty}^{\infty}\Big{(}x[n]+\sum_{m=1}^{p}\alpha_{m}x[n-m]\Big{)}^{2}. \tag{19}\]  
In order to minimise the quadratic error, the following step involves the partial derivation of the error term  
\[\frac{\partial e_{\text{LP}}}{\partial\alpha_{m}}=0,\quad\text{for }m=1,\ldots,p. \tag{20}\]  
This results in the system of linear equations  
\[\sum_{m=1}^{p}\alpha_{m}\sum_{n=-\infty}^{\infty}x[n-i]x[n-m]=-\sum_{n=-\infty }^{\infty}x[n-i]x[n] \tag{21}\]  
with \(i=1,\ldots,p\). By this, \(p\) linear equations are given for \(p\) unknown parameters \(\alpha_{1},\ldots,\alpha_{p}\). The established system of linear equations can be rearranged by the terms of the discrete autocorrelation function \(R_{\text{xx}}\) (see Equation (11)). Using the ACF definition and the property that ACFs are even functions (\(R_{\text{xx},i}=R_{\text{xx},-i}\)), the system of linear equations from Equation (21) can be rearranged to the Yule-Walker equations [25]  
\[\sum_{m=1}^{p}R_{\text{xx},|i-m|}\alpha_{m}=-R_{\text{xx},i},\quad\text{for }i=1,\ldots,p. \tag{22}\]  
These \(p\) linear equations can be described in matrix form as Equation (23).  
\[\begin{split}\left[\begin{array}{cccc}R_{\text{xx},0}&R_{ \text{xx},1}&\cdots&R_{\text{xx},p-1}\\ R_{\text{xx},1}&R_{\text{xx},0}&\cdots&R_{\text{xx},p-2}\\ \vdots&\vdots&\ddots&\vdots\\ R_{\text{xx},p-1}&R_{\text{xx},p-2}&\cdots&R_{\text{xx},0}\end{array}\right] \left[\begin{array}{c}\alpha_{1}\\ \alpha_{2}\\ \vdots\\ \alpha_{p}\end{array}\right]=\\ -\left[\begin{array}{c}R_{\text{xx},1}\\ R_{\text{xx},2}\\ \vdots\\ R_{\text{xx},p}\end{array}\right]\end{split} \tag{23}\]  
The autocorrelation matrix represents a Toeplitz matrix in which the values of the main and secondary diagonals are equal. In digital signal processing, the system of equations is usually solved efficiently via a Levinson-Durbin recursion [25]. As stated before, the estimation of the prediction coefficients \(\alpha_{m}\) of the signal can be interpreted as coefficients of an all-pole filter. This is shown in Figure 13. A frequency spectrum via an FFT together with the spectra via the prediction coefficients is shown. The number of coefficients \(p\) was chosen to be 5, 25, 50, 100 and 1000. For the calculation of the signals, a vibration signal with 8192 samples at a sampling rate of 44 kHz was considered. For better comparability, a frequency range up to 10 kHz is considered.  
The effect of the prediction coefficient's order on the spectrum can be seen in the figure. For a very low number of coefficients, the spectrum is strongly smoothed, but hardly resembles the FFT spectrum. If, on the other hand, a very high ordinal number is selected, the spectrum approaches the FFT spectrum.