# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 3 Measuring Chain Error Model
### Analog Part of the Measurement Chain  
The analog part of the measurement circuit, which converts the signal \(s(t)\) into its voltage representation, marked as \(y(t)\), can be described by a model using the transmittance \(G_{y}(j\omega)\) and the processing function \(f_{y}(x)\) in accordance with the diagram shown in Figure 2. Based on the proposed scheme, the output quantity \(y(t)\) of the object can be described as:  
\[\dot{y}(t)=\dot{f}_{y}(\dot{u}(t)), \tag{23}\] \[\ddot{y}(t)=\dot{y}(t)+e_{y,\Sigma}(t), \tag{24}\]  
where \(e_{y,\Sigma}(t)\) is the resultant error signal included in the output quantity \(y(t)\) which will be discussed in more detail later in this paper. Knowing the transmittance \(G_{y}(j\omega)\) makes it possible to determine the gain \(K_{y}(\omega)\) and the phase shift \(\varphi_{y}(\omega)\) as a function of pulsation:  
\[K_{y}(\omega)=\big{|}G_{y}(j\omega)\big{|} =\sqrt{\left(\Re\big{(}G_{y}(j\omega)\big{)}\right)^{2}+\left( \Im\big{(}G_{y}(j\omega)\big{)}\right)^{2}}, \tag{25}\] \[\varphi_{y}(\omega) =\arctan\Biggl{(}\frac{\Im\big{(}G_{y}(j\omega)\big{)}}{\Re\big{(} G_{y}(j\omega)\big{)}}\Biggr{)}, \tag{26}\]  
Figure 2: Model of a fragment of the analog part of the measurement chain. propagated random error signals \(e_{u,rp}(t)\), it is impossible to determine the deterministic form of these signals, and therefore only their variances should be determined based on Equation (27). In order to determine the average value of the variance \(\sigma_{u,rp}^{2}\) in the signals of propagated random errors in the pulsation interval \(\omega\in[b;a]\), it is possible to use the following equation [18]:  
\[\sigma_{u,rp}^{2}=\frac{1}{a-b}\int_{b}^{a}\tilde{K}_{y}^{2}(\omega)\sigma_{s, r}^{2}(\omega)d\omega; \tag{36}\]  
however, this activity should be carried out only at the last stage of the analysis, where subsequent fragments of the measurement chain will not affect the spectrum of the processed random error signals. The error signals contained in the quantity \(u(t)\) described so far constitute the resultant error signal \(e_{u,\Sigma}(t)\) associated with this quantity, with  
\[e_{u,\Sigma}(t)=e_{u,sw}(t)+e_{u,sp}(t)+e_{u,dw}(t)+e_{u,dp}(t)+e_{u,rw}(t)+e_{ u,rp}(t). \tag{37}\]  
The symbol \(e_{u,rw}(t)\) denotes an additional component constituting a random error introduced by the object, the variance of which is marked with the symbol \(\sigma_{u,rw}^{2}\). Next, the signal \(u(t)\) is processed according to the object processing function \(f_{y}(x)\); therefore  
\[\dot{y}(t)=\dot{f}_{y}(\dot{u}(t)). \tag{38}\]  
It should be noted that the real form \(\tilde{f}_{y}(x)\) of the discussed function may actually differ from the ideal form \(\dot{f}_{y}(x)\) assumed by the designer of the measurement chain. In the real case we can write  
\[\ddot{y}(t)=\tilde{f}_{y}(\dot{u}(t)+e_{u,\Sigma}(t))+f_{z}(\mathbf{z}(t))= \dot{y}(t)+e_{y,\Sigma}(t), \tag{39}\]  
where \(f_{z}(\mathbf{z})\) is a function that takes into account selected quantities disturbing the process of determining the value of quantity \(y(t)\) (e.g., environmental parameters). Based on Equation (39), one can distinguish the own error signal \(e_{y,fw}(t)\), which results directly from the imperfection of the object processing function, and the own error signal \(e_{y,zw}(t)\), which results from the participation of disturbing factors. These signals can be described as:  
\[\dot{y}(t)=\dot{f}_{y}(\dot{u}(t)), \tag{23}\]0 \[\dot{y}(t)=\dot{f}_{y}(\dot{u}(t)), \tag{23}\]1  
The error signal \(e_{y,fw}(t)\) will be deterministic if the form of the function \(\tilde{f}_{y}(x)\) is known. Otherwise, it is possible to describe its parameters in a probabilistic category. The nature of the error signal \(e_{y,zw}(t)\) will depend on the properties of the analyzed object. In the case of the influence of environmental parameters on the measurement process, it can be assumed that these values will be slowly changing, hence this signal will usually be included in the group of static error signals.