# Error Model of a Measurement Chain Containing the Discrete Wavelet Transform Algorithm
## 1 Introduction  
Wavelet transform (WT) algorithms find their practical applications in many areas. They are used in finance, medicine, power engineering, geology and computer science computations, among others [1; 2; 3; 4]. The continuous wavelet transform (CWT) algorithm, described in more detail in the textbook in [5], in the case of processing a time-varying signal \(x(t)\), the implementations of which belong to the domain of real numbers, can be presented using the following equation:  
\[w_{a,b}=\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}x(t)\psi\left(\frac{t-b}{a} \right)dt, \tag{1}\]  
where \(w_{a,b}\) is a wavelet transform coefficient calculated for a scale parameter \(a\) and a time shift parameter \(b\), where \(a\in(0;\infty)\), \(b\in(-\infty;\infty)\) and \(\psi(t)\) is the mother wavelet function. In the case of a discrete wavelet transform (DWT) there is an additional limitation on the available scale and shift parameters values as \(a=2^{m}\), \(b=n2^{m}\) for \(m,n\in\mathbb{N}\) where \(m\) is scale number and \(n\) is shift number (window number) [5]. The contemporary literature describes many types of wavelets, but the equation for this function is not always defined explicitly for each mother wavelet function \(\psi(t)\) (usually the description of the family includes assumptions regarding the mutual relations of the scaling factors \(c_{n}\) related to the selected wavelet [6; 7]). The wavelet transform algorithm is, therefore, a tool that allows signal analysis to be carried out simultaneously in the frequency and time domains. The possibility of using various wavelets enables the properties of this algorithm to be adjusted to the specificity of the analyzed phenomenon or process. Unfortunately, the above-mentioned features mean that an analysis of the properties of metrological measurement chains that use the discussed algorithms in their structure is not easy, which is why it is often omitted, as wasthe case, for example, in [8; 9; 10]. It should be noted that the person using these algorithms is usually an expert in the field of the phenomenon under study and does not have expert knowledge in the field of wavelet transform algorithms. Omitting the analysis of the metrological properties of the algorithm used, which constitutes a very important part of the measurement chain, makes it impossible to estimate the uncertainty budget of this chain, which means that decisions made on the basis of the value of the output values may be inappropriate. Currently, no proposals for a unified error model have been presented in the literature that would adequately describe the metrological properties of measurement chains using the wavelet transform algorithm. The authors' previous works assumed the presentation of the wavelet transformation algorithm in a matrix form and indicated how to identify the values of the transformation matrix in the case of an existing implementation of any algorithm. However, the proposed method requires the indication of the uncertainty budget for the input quantities of the algorithm used, and therefore it is necessary to present an error model for part of the measurement chain, which is the source of the input quantities of the wavelet transform algorithm. This algorithm is usually the last part of the measurement chain. Taking into account the comments indicated above and the current state of knowledge, the article proposes a general error model that is suitable for measurement chains containing wavelet transform algorithms in their structure. This model is based on the definition of the error signal and assumes the division of error signals due to their properties. The purpose of using the proposed model is to enable the determination of the uncertainty budget for the input values of the WT algorithm in such a way that it is possible to apply the previously proposed description of the impact of this algorithm on the errors contained in the signal it processes and ultimately determine the uncertainty budget of the entire measurement chain.