# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 4 Application of the Proposed Analysis Method
### Deterministic Errors Sources  
The first step of analysis is to determine static and dynamic error signals parameters. In the case of constant environment parameters (temperature, pressure, humidity) it can be assumed that static errors do not exists in the presented system. In the case of any possible change in the described parameters it is necessary to describe how the analyses' parameters impact the measurement chain. In the case of a dynamic error, the origin of this error will be the real transmittance \(\tilde{G}_{y}(j\omega)\) of the measuring amplifier, which is different from the transmittance \(\tilde{G}_{y}(j\omega)\). Therefore, this transmittance must be determined or the parameters marked with Equations (25) and (26) should be determined by experiments. As the exact model of the operational amplifier used was unknown, an experiment was carried out in which a sinusoidal signal was applied to the input of the amplifier its gain and the phase shift was measured. The source of the \(s(t)\) signal was an RIGOL DG1011 arbitrary waveform generator [30], while the \(s(t)\) and \(y(t)\) signals' parameters were measured with an RIGOL DS5062 MA DSO [31] using mean value of 256 subsequent samples. For the experiment described here, the frequency of input signal \(s(t)\) was within range \(\tilde{f}_{s,\rho}\in[0.1;48]\) kHz. The measurements enable the determination of the relationship between the gain and the phase shift introduced by the applied measuring amplifier as a function of pulsation.  
As none of the typical filter models were applicable in the described situation, on the basis of the obtained values, approximations of the tested characteristics were carried out using polynomials, using the least squares method, whereby:  
\[\tilde{K}_{y}(\omega)\approx 3.29, \tag{55}\] \[\tilde{\varphi}_{y}(\omega)\approx-6.26\cdot 10^{-13}\omega^{2}-5. 73\cdot 10^{-7}\omega, \tag{56}\]  
where \(\tilde{K}_{y}(\omega)\) is the gain, while \(\tilde{\varphi}_{y}(\omega)\) is the phase shift in the selected harmonic of the processed signal. The relationships presented will enable the description of the waveform of the output signal of the measuring amplifier and, as a result, the determination of the waveform of the error signal introduced by this element.  
Another dynamic error source is the S/H circuit at the input of the ADC. According to the manufacturer's data, the S/H circuit can be described as a two-stage low-pass passive RC filter [25; 26]. The first filter stage is formed by the internal impedance of the  
Figure 3: Schematic representation of the prototype of the measuring chain, being the object of the experiment.  
input voltage source and circuit capacitance. As the output impedance of the amplifier is relatively low, the influence of the first filter stage can be omitted [13; 24; 25]. The second filter is formed by the ON resistance of the internal S/H switch and a capacitance of typical values 5 k\(\Omega\) and 6 pF, respectively [25; 26]. The resulting cutoff frequency of this filter is relatively high in comparison to the bandwidth of the amplifier so influence of the second filter can be neglected as well.