# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 2 Definition of the Signal Error and Its Properties  
Errors contained in the processed measurement signal can be divided in many ways, but in the most general case, random and deterministic errors can be distinguished [11]. Random errors result from the implementation of a stochastic process, the operation of which cannot be described in a deterministic way. Such errors can, therefore, only be described using a probabilistic description, since the course of the error signal is unknown. The second group of errors is deterministic errors, the origin of which usually results from the nonideal transmittance of a fragment of the measurement chain located in front of the wavelet transformation algorithm. These errors can be described by an appropriate equation, which can be derived by knowing the model of the measurement chain and the estimated spectrum of the processed signal. A slightly more detailed division in the literature includes groups of static, dynamic and random errors [12]. This division refers to the nature of the error from the size of the point of view of the measurement window. The realizations of the static error within the measurement window do not change (or change slightly), and therefore the transmittance of subsequent fragments of the measurement chain does not affect the propagation of these errors. In the case of dynamic and random errors, their implementation changes within the measurement window, while, unlike random errors, dynamic errors can be described in a deterministic way. In the general case, the error signal \(e_{x,\Sigma}(t)\) for the quantity \(x(t)\) can be defined as the difference between the ideal \(\hat{x}(t)\) and the actual \(\hat{x}(t)\) course of this quantity:  
\[e_{x,\Sigma}(t)=\hat{x}(t)-\hat{x}(t). \tag{2}\]  
In addition to the time course of the presented error signal, which it may not always be possible to indicate, this signal will also be associated with parameters such as variance, expected value, the probability density distribution of obtaining the indicated value of the realization of this signal and expanded uncertainty, depending on the adopted confidence level [13]. Each fragment of the measurement chain, by processing the signal in question, can modify the indicated parameters in accordance with its properties. Additionally, the analyzed signals may be mutually correlated, which should also be taken into account during the analysis. From the point of view of the process of creating the uncertainty budget, the most important information will be the signal variance \(\sigma_{x,\Sigma}^{2}\), the associated standard uncertainty \(\sigma_{x,\Sigma}\) and the expanded uncertainty \(U_{x,\Sigma}\), depending on the value of the standard uncertainty \(\sigma_{x,\Sigma}\) and the expansion coefficient \(c_{x,\Sigma}\), depending on the shape of the density function of the probability of the occurrence of selected error signal values \(e_{x,\Sigma}(t)\)[13]. Therefore, it can be noted that the course of the \(e_{x,\Sigma}(t)\) signal itself does not have to be known to be able to develop an uncertainty budget. As wavelet transform algorithms constitute a set of filters with a specific transmittance resulting from the adopted algorithm parameters, the parameters of signal errors should be analyzed in the frequency domain [14; 15; 16]. In the case of non-deterministic signals, the relationship between the variance values of these signals as a function of pulsation should be determined, as presented, among others, in [17], while in the case of deterministic signals, the resultant value of the variance in the selected pulsation of the resulting deterministic error signal should be indicated.