# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 2 Definition of the Signal Error and Its Properties
\begin{array}{cccc}1&r_{0,1}&\cdots &r_{0,N-1}\\ r_{1,0}&1&&r_{1,N-1}\\ \vdots&&\ddots&\vdots\\ r_{N-1,0}&\cdots&\cdots&1\end{array} \right]\left[ \begin{array}{c}\sigma_{0} \\ \sigma_{1}\\ \vdots\\ \sigma_{N-1}\end{array} \right], \tag{10}\]where \(\sigma_{i}^{2}\) is the variance of the \(i\)-th random error signal \(e_{i}(t)\) and \(r_{i,j}=r_{j,i}\) is the Pearson correlation coefficient of the pair of signals \(e_{i}(t)\) and \(e_{j}(t)\), equal to:  
\[r_{i,j}=r_{j,i}=\frac{\sigma_{i,j}^{2}-\sigma_{i}^{2}-\sigma_{j}^{2}}{2\sigma_{ i}\sigma_{j}}\,, \tag{11}\]  
where \(\sigma_{i,j}^{2}\) is variance in signal \(e_{i,j}(t)=e_{i}(t)+e_{j}(t)\). This indicated property of WT algorithms results from the assumptions of the central limit theorem and the fact that these algorithms process many samples of the same input quantity [3; 12; 13; 17; 20]. In the case of deterministic error signals, each component of these signals should be analyzed separately, but in the case of dynamic error signals, it is possible to determine the resulting variance value for the selected pulsation value. The first case considered is a situation where there are several error components with the same pulsation, but with different phase shifts and amplitudes. The papers [14; 20] propose a way to determine the variance of the resultant error in this case, while the presented method does not allow the resultant phase shift to be determined, the knowledge of which is sometimes necessary in the case of analyzing subsequent fragments of the measurement chain. Therefore, the paper proposes the use of the description of successive \(N\) harmonics of the error signal with the same pulsation in the vector form:  
\[\mathbf{e}_{i}= \begin{bmatrix}e_{i,a}&e_{i,b}\end{bmatrix} = \begin{bmatrix}E_{i} \cos(\varphi_{i})&E_{i}\sin(\varphi_{i})\end{bmatrix} , \tag{12}\]  
on the basis of which the resultant vector for the error signal can be determined in the form  
\[\mathbf{e}_{\Sigma}= \begin{bmatrix}e_{\Sigma,a}&e_{\Sigma,b}\end{bmatrix} = \begin{bmatrix}\sum_{i=0}^{N-1}e_{i,a}&\sum_{i=0}^{N-1}e_{i,b}\end{bmatrix} . \tag{13}\]  
Then, the resultant amplitude and phase shift of the error signal with the analyzed pulsation can be determined in the form  
\[E_{\Sigma} =\sqrt{\epsilon_{\Sigma,a}^{2}+\epsilon_{\Sigma,b}^{2}}, \tag{14}\] \[\varphi_{\Sigma} =\arctan\left(\frac{e_{\Sigma,b}}{e_{\Sigma,a}}\right). \tag{15}\]  
The correlation coefficient of successive error signals composed in this way can be determined on the basis of their phase shift, which is described by the equation [19]:  
\[r_{i,j}=\cos\left(\varphi_{j}-\varphi_{i}\right), \tag{16}\]  
where \(i\) and \(j\) are the harmonic indexes of the error signal for which the correlation coefficient \(r_{i,j}\) is determined, assuming that \(\omega_{i}=\omega_{j}\). It is also necessary to consider the combination of the received error signals with different pulsations, where \(\omega_{i}\
eq\omega_{j}\) occurs. In this case, there is no need to consider the correlation of the successive components of the error signal, because they are linearly independent of each other and therefore uncorrelated [14] unless they are caused by the same phenomena. A universal way to determine the variance in the resultant error, which proves the above thesis, is the method described by the following equation [14; 18]:  
\[\sigma_{\Sigma}^{2}=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}\left(\sum_{i=0 }^{N-1}E_{e,i}\sin(\omega_{e,i}t+\varphi_{e,i})\right)^{2}dt, \tag{17}\]where \(N\) is the number of components in the error signal. To determine the variance of the resultant error \(e_{x,d}(t)\) of all its harmonics, it is also possible to apply equation (10) in which the correlation coefficients are equal to  
\[r_{i,j}= \begin{cases}\cos\bigl{(}\varphi_{j}-\varphi_{i}\bigr{)}&\text{ where }\omega_{i}=\omega_{j}\\ 0&\text{ where }\omega_{i}\
eq\omega_{j}\end{cases} .