# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 4 Application of the Proposed Analysis Method
### Model Application for a Monoharmonic Input Signal  
Analyzing Equations (34) and (35) in the case of the non-ideal dynamic properties of the measurement chain, it can be noticed that the parameters of the described signals will depend on the spectrum of the signal \(s(t)\) processed by the measurement chain. The first example of the application of the proposed model concerns a case in which this circuit processes a sinusoidal input signal with a given frequency \(f_{s,o}=\frac{\omega_{s,o}}{2\pi}\), amplitude \(E_{s,o}\) and direct-current (DC) component \(D_{s,o}\). This signal can be described by the equation:  
\[\bar{s}(t)=E_{s,o}\sin(\omega_{s,o}t+\varphi_{s,o}), \tag{61}\]  
whereas in the experiment it is assumed that \(\varphi_{s,o}=0\) rad. According to the assumptions that the environmental conditions do not change during subsequent measurement series and that the errors introduced by the arbitrary waveform generator used are negligibly small in relation to those introduced by the measurement chain, it can be assumed that the \(s(t)\) signal is not disturbed by any error signals, therefore \(\bar{s}(t)\cong\bar{s}(t)\). Based on Equation (55), it can be noted that, in the frequency range \(\hat{f}\in[0;\frac{1}{2}f_{s}]\), the value of the amplification introduced by the measuring amplifier is constant and is compensated by the static playback algorithm, in accordance with Equation (58). It can therefore be concluded that the only important parameter from the point of view of the error model will be the value of the phase shift introduced by the measuring amplifier,  
Figure 4: (**a**) Approximation of mean \(\bar{c}(k)\) of the ADC output quantity \(c(i)\) values obtained for \(k\)-th measures series (calculated according to Equation (57)); (**b**) a histogram of the DWT input quantities random error \(e_{x,r}(i)\) values obtained during the experiment (calculated according to Equation (59)). which is the cause of the dynamic self-error signal, depending on the pulsation of the input signal. The signal in question can be defined based on Equation (34):  
\[ \begin{split} e_{x,d}(i)=&\ E_{s,\rho}(\omega_{s,\rho}) \sin\bigl{(}\omega_{s,\rho}iT_{s}+\varphi_{s,\rho}+\tilde{\varphi}_{y}(\omega_{s,\rho})\bigr{)}-\\ &\ E_{s,\rho}(\omega_{s,\rho})\sin\bigl{(}\omega_{s,\rho}iT_{s}+ \varphi_{s,\rho}+\tilde{\varphi}_{y}(\omega_{s,\rho})\bigr{)}\end{split} , \tag{62}\]  
where \(\varphi_{s,\rho}=0\) rad, \(\phi_{y}(\omega)=0\) rad and the value of \(\tilde{\varphi}_{y}(\omega)\) is estimated according to polynomial (56). The form of the error signal described in Equation (62) consists of two factors with the same frequency, the resultant parameters of which can be be estimated in accordance with the content of equations from (12) to (18). By transforming the indicated equations and taking into account the components of Equation (62), the variance in the dynamic error signal \(e_{x,d}(i)\) can be determined in accordance with the relationship:  
\[\sigma_{x,d}^{2}=\frac{\bigl{(}E_{s,\rho}\cos\bigl{(}\tilde{\varphi}_{y}( \omega_{s,\rho})\bigr{)}-E_{s,\rho}\bigr{)}^{2}+\bigl{(}E_{s,\rho}\sin\bigl{(} \tilde{\varphi}_{y}(\omega_{s,\rho})\bigr{)}\bigr{)}^{2}}{2}. \tag{63}\]  
It can, therefore, be seen that the variance in the described signal depends on the signal frequency \(s(t)\) and its amplitude. To summarize the considerations, the resultant error signal \(e_{x,\Sigma}(i)\) of the input quantity of the DWT algorithm in the discussed case will consist of the random error signal \(e_{x,r}(i)\), the parameters of which were estimated previously, and the dynamic error signal \(e_{x,d}(i)\), whose parameters are estimated according to Equation (63) for the given signal parameters \(s(t)\).