# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 3 Measuring Chain Error Model
### Properties of the Analog-to-Digital Converter  
The analog-to-digital converter (ADC) has a discrete set of possible output values, and therefore introduces an error \(e_{AC,q}(x)\) related to quantization into the output signal \(x(i)\) (where \(i=nT_{s}\), \(n\in\mathbb{N}\) and \(T_{s}=\frac{1}{J_{s}}\)). This error can be described in the form [21]:  
\[e_{AC,q}(x)=x-\tilde{u}_{AC}(x)=x-q\left[\frac{x}{q}+0.5\right], \tag{45}\]  
where \(q\) is the value of a single quantum and \(\tilde{u}_{AC}(x)\) is the value of the ADC output for the input voltage value equal to \(x\), expressed in the unit of the input quantity. It can be noticed that the realization values of the discussed error signal depend on the realization values of the ADC input quantities. According to the research results presented in [12; 21; 22; 23], this correlation can be neglected. Therefore, we propose describing the signal \(e_{x,q}(i)\) associated with the \(i\)-th realization of the quantization error using an uncorrelated additive noise model with the same probability of obtaining each of the possible realization values in the interval:  
\[-\frac{q}{2}\leq\hat{e}_{x,q}(i)\leq\frac{q}{2}, \tag{46}\]  
therefore, the variance in this signal can be determined according to the relationship [13]:  
\[\sigma_{x,q}^{2}=\frac{q^{2}}{12}. \tag{47}\]  
Given the above assumptions, in the ideal case the output value \(x(i)\) of the ADC can be given by the equation:  
\[\dot{x}(i)=\dot{y}(t_{i}), \tag{48}\]  
where \(t_{i}\) is the selected time at which the input signal \(y(t)\) was sampled. In the real case, the quantity \(x(i)\) can be described as:  
\[\dot{x}(i)=\dot{x}(i)+e_{x,\Sigma}(i). \tag{49}\]  
The resulting error signal \(e_{x,\Sigma}(i)\) at the output of the ADC can therefore be described as:  
\[e_{x,\Sigma}(i)\cong e_{y,\Sigma}(t_{i})+e_{x,q}(i)+e_{x,w}(i), \tag{50}\]  
where \(e_{x,q}(i)\) is the error signal related to the rounding operations introduced in accordance with Equation (45), the variances of which are described by Equation (47), while \(e_{x,w}(i)\) is an additional self-error signal. Moreover, for an ideal quantizer system \(e_{x,w}(i)=0\). Therefore,the variance \(\sigma_{x}^{2}\) in the signals at the output of the A/D converter can be described by the equation:  
\[\sigma_{x}^{2}\cong\sigma_{y}^{2}, \tag{51}\]  
where \(\sigma_{y}^{2}\) is the variance in the signal at the object input. These assumptions allow us to assume that the analyzed object has no influence on the form of processed error signals:  
\[e_{x}(i)\cong e_{y}(t_{i}), \tag{52}\]  
where \(e_{x}(i)\) is the error signal at the output of the ADC related to the error signal \(e_{y}(t)\) at its input. The validity of the adopted assumptions was demonstrated, among others, in [17; 21].  
In the case of a real ADC, it is necessary to determine the parameters of the self-error signal, and this error most often results from ADC integral nonlinearity, differential nonlinearity, gain error, zero error, the heterogeneity of the internal structure of the converter or the imperfection of the reference voltage source [24; 25]. Additionally, parameters related to the sample-and-hold circuit (S/H) should be considered separately. In this case, the model proposed for the analog part of the measurement chain, the parameters of which will be appropriate for the S/H circuit, will be used.