## 2 Dynamic Error Model  
Dynamic error signals are signals whose realized values change for subsequent samples of the algorithm's input quantities within the analyzed measurement window, and the nature of these changes is deterministic. An example of such an error signal is a signal resulting from the introduction of a phase shift in the output quantities of the analog part of the measurement chain. Let us assume that a fragment of the measurement chain converts a time-varying physical quantity \(s(t)\) into a voltage signal representing it, \(y(t)\), in accordance with the diagram in Figure 2.  
As shown in [16; 18], the most important source of the dynamic error signal in this case is the introduction of a phase shift, \(\phi_{y,j}\), by the analyzed fragment of the measurement chain, which differs from the expected value, \(\phi_{y,j}\). An additional source of the signal in question may also be the introduction by the object of a gain \(\hat{R}_{y,j}\) other than the desired one \(\hat{K}_{y,j}\). It can be observed that the analyzed fragment of the measurement chain acts as a filter with parameters determined by its physical properties, while ideally the transmittance of this object should have the following form:  
\[\hat{G}_{y}(j\omega)=s_{y}, \tag{8}\]  
where \(s_{y}\) is the sensitivity of the analyzed object. According to the adopted assumption, the ideal gain and phase characteristics are given by the following:  
\[\hat{K}_{y}(\omega)=s_{y}, \tag{9}\] \[\phi_{y}(\omega)=0\text{ rad.} \tag{10}\]  
Assuming the measurement process is performed multiple times, the variance of the subsequent harmonics of the dynamic error signal [24] can be determined as follows:  
\[\sigma_{y,d,j}^{2}=\frac{1}{2}E_{y,\epsilon,j}^{2}, \tag{11}\]  
and the expanded uncertainty can be calculated as follows:  
\[U_{y,d,j}=\frac{1}{\sqrt{2}}c_{d}E_{y,\epsilon,j}, \tag{12}\]  
where \(c_{d}\) is the expansion factor for the sine function distribution at the assumed confidence level \(\gamma=1-\alpha\)[17; 26]. If the resulting dynamic error signal \(e_{y,d}(t)\) consists of many harmonics with different angular frequencies, the resultant variance of this signal is given by Equation [24]:  
\[\sigma_{y,d}^{2}=\sum_{j=1}^{\infty}\sigma_{y,d,j}^{2}=\frac{1}{2}\sum_{j=1}^ {\infty}E_{y,\epsilon,j}^{2}. \tag{13}\]  
However, determining the expanded uncertainty for a given confidence level in this case is complex and can be achieved using methods described in [27; 28; 29; 30; 31; 32; 33; 34]. If the error signal \(e_{y,d}(t)\) contains harmonics with identical angular frequencies, the resultant values of these harmonics should be determined according to the method presented in [16; 24; 34]. Assuming that the signal \(y(t)\) is the input quantity of the A/D converter, this converter transfers the dynamic error signal \(e_{y,d}(t)\) present in this signal to the output. Additionally, the converter introduces its own errors [14; 35; 36] to the output quantity \(x(i)\). These error signals result from the finite resolution of the converter, imperfections of the sample-and-hold (S/H) system, inhomogeneity of the internal structure of the standard, or imperfections of the reference voltage source [35; 37]. Throughout the remainder of the paper, it is assumed that the metrological properties of the S/H system are included in the error model of the analog part of the measurement chain. Hence, the value of the converter output quantity, expressed in the unit of the input quantity \(y(t)\), is given by Equation [14]:  
\[\tilde{u}_{AC}(x)=q\left|\frac{x}{q}+0.5\right|, \tag{14}\]where \(q=\frac{b-a}{N_{q}}\) is the value of a single quantum of the transducer that processes voltage in the range \(x\in[a;b]\), where \(a,b\in\mathbb{R}\), with resolution \(N_{q}\in\mathbb{N}\).