# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 4 Application of the Proposed Analysis Method
### Random Errors Sources  
In the case of random error signals, there are many more sources related to them and they occur in the whole measurement chain. It must be noticed that all phenomena for which a deterministic description is not possible must be considered as random error sources. These phenomena are, e.g., noise present in the input signal, the reference voltage and supply voltage, quantization errors, internal structure heterogeneity, integral errors and differential errors. As the analysis of all these phenomena separately is not possible without detailed knowledge of the parameters and the internal structure of the components used, we propose estimating the parameters of the resulting random error signal by performing the experiment described next. In the experiment, a nominally constant voltage \(s(t)\) was applied to the input of the measurement chain from a voltage source. Then, for the selected input voltage value, many implementation values of the quantity \(c(i)\) were taken. Based on the average obtained values of the realization of the quantity \(c(i)\), it is possible to determine the static characteristics of the part of the measurement circuit which includes the amplifier and the measuring transducer, as well as to determine the inverse function for it, which will enable the reconstruction of the quantity \(s(t)\) in the form of the quantity \(x(i)\). To allow this, it is necessary to repeat the experiment for subsequent input voltage values. The described experiment was carried out in each \(k\)-th measurement series of 30,000 values for the realization of the quantity \(c(i)\) of the input voltage values in the range \(\hat{s}(k)\in[25;975]\) mV, where this value changed by \(\Delta\hat{s}(k)=25\) mV for each measurement series. Based on values obtained for the realization of the quantity \(c(i)\), the average value of these realizations \(\overline{c}(k)\) was determined for the \(k\)-th measurement series. Then, on the basis of the obtained results, a linear approximation of the characteristics of the processing of the quantity \(s(t)\) to the quantity \(c(i)\) was performed:  
\[\overline{c}(k)=f_{c}\big{(}f_{y}(\hat{s}(k))\big{)}\approx 4093.1\cdot\hat{s}(k) +0.4519, \tag{57}\]  
where \(\hat{s}(k)\) is the set voltage value for the analyzed series of measurements. Assuming that the sensitivity of quantity \(x(i)\) with respect to quantity \(s(t)\) is 1 V/V, the estimated average value of \(\overline{x}(k)\) as a function of the value of \(\hat{s}(k)\) can be described as:  
\[\overline{x}(k)=f_{x}(\overline{c}(k))=f_{x}\big{(}f_{c}\big{(}f_{y}(\hat{s}( k))\big{)}\big{)}=\frac{\overline{c}(k)-0.4519}{4093.1}\approx\hat{s}(k). \tag{58}\]  
The voltage source for the \(s(t)\) signal used in the experiment was the RIGOL DG1011 arbitrary waveform generator [30]. It was assumed that the errors introduced by the generator are negligible in relation to the errors arising in the analyzed section of the measurement chain. It can be noticed that the static characteristic described by Equation (58) is a processing function for the part of the measurement chain that converts the quantity \(s(t)\) into the quantity \(x(i)\). By determining the standard deviation of the differences between the measured realization values \(\overline{c}(k)\) and the values determined in accordance with Equation (58), the standard uncertainty value related to the nonlinearity of the discussed characteristic is obtained. Additionally, on the basis of all the realization values of the quantity \(c(i)\), it is possible to determine a histogram of the realization of the random error signal of this quantity, obtained under the experimental conditions.