# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 4 Application of the Proposed Analysis Method
### Model Application for Polharmonic Input Signal  
In this section we assume that \(s(t)\) is a triangle signal. According to the assumptions, the processed signal \(s(t)\) can be described in the form:  
\[\dot{s}(t)=D_{s,\rho}+E_{s,\rho}\frac{\pi}{8}\sum_{i=1}^{\infty}(-1)^{i-1}(2i-1) ^{-2}\sin(\omega_{s,\rho}t(2i-1)+\varphi_{s,\rho}), \tag{65}\]  
where \(D_{s,\rho}\) is a DC component of the signal, \(E_{s,\rho}\) is the amplitude of the fundamental harmonic, \(\varphi_{s,\rho}=0\) rad is signal phase, \(\omega_{s,\rho}\) is fundamental harmonic pulsation. In the experiment it was assumed that \(\dot{s}(t)\cong\dot{s}(t)\).  
According to the previous assumptions, the parameters of the random error signal \(e_{x,r}(i)\) remain identical to those of the monoharmonic signal. However, a significant difference is in the parameters of the dynamic error signal \(e_{x,d}(i)\), for which, according to Equation (34), the subsequent harmonics taken into account in Equation (65) should be analyzed. Therefore, for each harmonic \(e_{x,d,i}(i)\) of the error signal \(e_{x,d}(i)\), its variance should be determined, similarly to the case of the monoharmonic signal. In the discussed case, for the \(i\)-th harmonic it occurs that:  
\[\sigma_{x,d,i}^{2}=\frac{\left(E_{s,\rho,i}\cos\left(\bar{\varphi}_{y}(\omega_ {s,\rho,i})\right)-E_{s,\rho,i}\right)^{2}+\left(E_{s,\rho,i}\sin\left(\bar{ \varphi}_{y}(\omega_{s,\rho,i})\right)\right)^{2}}{2}, \tag{66}\]  
wherein the amplitude for the \(i\)-th harmonic is determined according to the relationship:  
\[E_{s,\rho,i}=\frac{\pi}{8}(2i-1)^{-2}E_{s,\rho}. \tag{67}\]  
The value of variance for the resultant error signal \(e_{x,\Sigma}(i)\) in the analyzed case can be determined according to the relationship:  
\[\sigma_{x,\Sigma}^{2}=\sigma_{x,r}^{2}+\sum_{i=1}^{\infty}\sigma_{x,d,i}^{2}; \tag{68}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Frequency \(f_{s,\rho}\), Hz** & **Measured \(\sigma_{m}^{2}\) \(\mu\)V** & **Calculated \(\sigma_{c}^{2}\), \(\mu\)V** & **Relative Error \(\delta\), \%** \\ \hline
1000 & 1.90 & 1.64 & \(-13.93\) \\
2000 & 6.46 & 6.24 & \(-3.35\) \\
3000 & 13.16 & 14.08 & 7.01 \\
4000 & 21.86 & 25.26 & 15.58 \\
5000 & 39.07 & 39.93 & 2.21 \\
6000 & 54.  
however, we propose analyzing only those harmonics for which it occurs that \(kf_{s,o}\leq\frac{1}{2}f_{s}\) for \(k=(2i-1)\) (all harmonic components for which the frequency is less or equal to the Nyquist frequency).  
In order to verify the presented considerations, a measurement experiment using the Monte-Carlo method was carried out. In this case \(s(t)\) was a triangle signal with a given frequency \(f_{s,o}\) in the range \(\hat{f}_{s,o}\in[1;5]\) kHz with parameters \(D_{s,o}=0.5\) V, \(E_{s,o}=0.475\) V and a random initial phase. For the selected signal frequency values, 30,000 samples of \(x(i)\) were taken in order to determine the value of the error signal \(e_{x,\Sigma}(i)\) based on Equation (54). Based on the obtained measurement results, the actual value of the variance \(\sigma_{m}^{2}\) in the analyzed error signal was determined and compared with the value \(\sigma_{c}^{2}\) estimated on the basis of Equation (68). The obtained results are summarized in Table 2. The source of the \(s(t)\) signal was an RIGOL DG1011 arbitrary waveform generator [30].