# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 2 Definition of the Signal Error and Its Properties
We assume that \(x(t)\) can be described as the sum of the successive harmonics \(x(t,\omega)\) of this signal and \(\hat{x}(t)\) is described by the equation:  
\[\dot{x}(t)=\sum_{i=0}^{\infty}\dot{x}(t,\omega_{x,i}), \tag{3}\] \[\dot{x}(t,\omega)=E_{x,\omega}(\omega)\sin(\omega t+\varphi_{x, \rho}(\omega)), \tag{4}\]  
where \(\omega_{x,i}\) is pulsation of \(i\)-th harmonic of this signal, \(E_{x,\rho}(\omega)\) is the amplitude and \(\varphi_{x,\rho}(\omega)\) is the phase shift in the harmonic corresponding to pulsation \(\omega\). In the real case, however,it is assumed that the signal \(x(t)\) disturbed by the resultant error signal \(e_{x,\Sigma}(t)\) can be presented as:  
\[\bar{x}(t)=\dot{x}(t)+e_{x,\Sigma}(t)=e_{x,r}(t)+\sum_{i=0}^{\infty}\bar{s}(t, \omega_{x,i}), \tag{5}\]  
\[\bar{x}(t,\omega)=\dot{x}(t,\omega)+E_{x,e}(\omega)\sin(\omega t+\varphi_{x,e} (\omega)). \tag{6}\]  
According to Equation (5), the resultant error signal \(e_{x,\Sigma}(t)\) related to the signal \(x(t)\) can be divided into components related to the share of random errors \(e_{x,r}(t)\) (non-deterministic) and a component related to deterministic errors which, in turn, can be divided into static \(e_{x,s}(t)\) and dynamic \(e_{x,d}(t)\). Due to the above assumptions, describing the single \(n\)-th harmonic of the error signal \(e_{x,d}(t)\) with the equation:  
\[e_{x,d,n}(t)=E_{x,e,n}\sin(\omega_{x,e,n}t+\varphi_{x,e,n}), \tag{7}\]  
the variance of the analyzed error component in the case when \(\omega_{x,e,n}>0\) rad/s can be determined according to the relation [18,19]:  
\[\sigma_{x,d,n}^{2}=\frac{E_{x,e,n}^{2}}{2}. \tag{8}\]  
In the absence of a correlation between the successive components of the error signal, the resulting error variance is equal to the sum of the error variances [13]. However, under certain conditions these waveforms are correlated with each other, and therefore these correlations should be taken into account, which is discussed in the next section, presenting the method of determining the parameters of the resultant error, which is the result of all partial error waveforms. Based on Equation (7), it is possible to introduce an additional division due to the nature of the error waveform. For a pulsation value close to zero, successive realizations of the error within a single measurement window will not change. Therefore, this error can be classified into the group of static errors described as:  
\[e_{x,s}(t)=e_{x,d,0}(t)=E_{x,e,0}, \tag{9}\]  
assuming that \(\varphi_{x,e,0}=\pi\). In cases where subsequent implementations will change significantly, these waveforms should be classified as dynamic errors. For errors of a static nature, attention should be paid to the fact that their variance within a single measurement window will be zero, while for subsequent measurement implementations within multiple measurement windows, it will be nonzero when these errors occur. Therefore, this group of errors can be analyzed separately. In further considerations, it is assumed that, in the case of a dynamic error signal \(e_{x,d}(t)\), all harmonics of this signal with non-zero pulsation will be analyzed, while the role of the constant component \(E_{x,e,0}\) in this signal, the value of which may change for subsequent window numbers measurement, will be analyzed within the static error signal \(e_{x,s}(t)\). According to the properties of WT algorithms, described earlier in [17], in the case of random error signals, the only important information is the value of the variance \(\sigma_{x,r}^{2}\) in the resulting random error signal \(e_{x,r}(t)\) contained in the input quantities \(x(t)\) of the algorithm used. This value can be determined by knowing the variance values of the subsequent components of the random error signal and the values of the correlation coefficients of these signals based on equation [13]:  
\[\sigma_{\Sigma}^{2}=\left[ \begin{array}{c}\sigma_{0}\\ \sigma_{1}\\ \vdots\\ \sigma_{N-1}\end{array} \right]^{T}\left[