## 3 Experiments and Validation
### Design of Experiments  
We choose the method of the so-called Latin Hypercube Design (LHD), a well-established procedure in the field of Design of Experiments (DoE). Using the Latin Hypercube Design, a test field can be created in such a way that the variance of the global mean of the variables under consideration is minimized. Compared with other methods, such as the Monte Carlo method, it provides better results in terms of minimizing variance with the same number of test points. Thereby, a \(q\times r\) matrix called \(\mathbf{L}^{q\times r}\), whose columns consist of an arbitrary permutation of the numbers \(\{1,2,3,\ldots,q\}\), is formed. In the framework of the so-called Latin Hypercube Sampling (LHS), a random number from \([0,1)\) is subtracted from each value of the LHD and then divided by \(q\), thus normalizing the entire test field ([18], p. 155). In our investigation, we use the function lhsdesign() from Matlab(r). Furthermore, it is necessary to limit the number of possible combinations and thus the dimensions of the test space to a feasible level. In this connection, [PERSON] et al. propose, in [19], to estimate the number of experiments (\(n_{\text{t}}\)) considering a given number of dimensions \(n_{\text{D}}\) using the following formula:  
\[n_{\text{t}}\approx 10\cdot n_{\text{D}}. \tag{5}\]  
Usually, the variables considered in the LHD are continuously distributed. Nevertheless, we also want to consider different types of signals and thus include several functions as variables that are added randomly. Hence, to define the parameters to be varied, we first specify four functions in total that intersect a plane with offset \(d\) parallel to the \(xy\)-plane; see Table 3. Here, Parabola is a common two-dimensional function and is depicted in Figure 5. On the other hand, the Peaks function depicted in Figure 6 is a standard function of Matlab(r) that is characterized by prominent minima and maxima. Furthermore, we investigate the Rosenbrock function [20] as well as [PERSON]'s function [21] (see Figures 7 and 8). Note that these functions are generally applied as test functions for evaluating algorithms for solving nonlinear optimization problems. As the next parameter to be varied, we choose variable \(d\), which determines the offset of the plane. We want to underline at this point that the RooTri() function is also able to consider arbitrarily oriented planes; however, we limit the analysis to parameter \(d\), since the contour() function can only determine intersections with parallel planes. Thus, we are able to compare the two functions with each other. The last parameter refers to length \(2l\) of the interval on which the functions are to be examined. We define this in such a way that it is divided into a certain number of equidistant increments and extends over a range from \(-l\) to \(l\) in both the \(x\)- and \(y\)-directions using the function linspace(). This gives us a total of three parameters that are used to generate the test field while we vary (half) length \(l\) of the interval in a range from 1 to 10 and offset parameter \(d\) in a range from \(-5\) to 5. According to Equation (5), 30 experiments must, therefore, be carried out and are summarized in Table 4. In this context, we additionally introduce another quantity, denoted by \(\alpha\), that relates the length of the interval (\(2l\)) to the number of increments (\(n_{\alpha}\)). \begin{table}
\begin{tabular}{l} \hline \hline Lenovo T480s \\ Matlab® R2022b \\ Microsoft Windows 10 Enterprise LTSC \\ Intel(R) Core (TM) i7-8550U CPU @ 1.80 GHz, 1992 MHz, 4 cores, 8 logical processors \\ Installed physical memory 16.0 GB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System specifications.