## 4 A Comparative Performance of the Model
### Comparison on Network Scattering Parameters Data \(S_{11}\)  
The first comparative example is concerned with fitting network scattering parameter data. Network scattering parameters are powerful tools for the analysis and design of high-frequency and microwave networks, and they can be tested by users in a guided example with Matlab Radio Frequency toolbox [21; 22]. As stated in Matlab R2021b release documentation, rationalfit approximates data using stable rational function object and is based on vector fitting method [1], along with its improvement [23]. The original paper [1] gained popularity in the electrical engineering community, with high number of paper citations and patent. The data in this example are the same used by the author to illustrate its robust performance in fitting S-parameters. We compared the model functions obtained by both schemes to fit the real and imaginary parts of the responses \(S_{11}\) along with their corresponding errors, Figures 6-9.  
A more in-depth comparison is presented below in Figure 7 with a pointwise error plot of each method and a difference plot between the errors (error with rationalfit minus error with scaled null space).  
Next, we show a similar profile when we fit the imaginary part of the responses in terms of frequencies. These comparisons are reported in Figures 8 and 9. First, we report the fitting of both methods with scaled null space and with Matlab rationfit method in Figure 8.  
In Figure 9, we show a pointwise comparison that illustrates the error plot and the difference between the magnitude of the errors (error with rationalfit minus error with scaled null space).  
Figure 6: (**a**) Fitting the real part of \(s_{11}\) parameter with scaled null space method on the left and with Matlab rationalfit on the right. Number of measurement data (\(s\), Re(\(s_{11}\))) is m = 202. We observe higher accuracy in fitting with scaled null space method. (**b**) We magnified (**a**) to show differences between two methods.  
Figure 7: A pointwise comparison of fitting errors in the example above. Left-hand side graph shows absolute value of errors of our method scaled null space versus Matlab rationalfit. Right-hand side  
graph represents difference between errors in rationallift and our method, where positive y values indicate how much rationallift is less accurate, and negative y values show how much our method is less accurate. Range of errors in rationallift is [1.5492 \(\times 10^{-5}\), 0.0212], while with scaled null space range is [2.7089 \(\times 10^{-14}\), 0.0198], which clearly exhibits a better accuracy, especially over most of frequencies.  
Figure 8: (a) Fitting imaginary part of \(s_{11}\) parameter with scaled null space method on left and with Matlab rationallift on right. Number of measurement data (\(s\), Im(\(s_{11}\))) is \(m\) = 202. Again, a higher accuracy is achieved when fitting with scaled null space method. (b) We have zoomed up (**a**) to show the differences between the two methods.  
Figure 11: Approximation of sine function over interval \([0,\,5\pi]\) using 201 uniformly selected sample points. Both functions are almost overlapping over whole interval \([0,\,5\pi]\) as shown in top plot. Bottom graph is a magnified window to manifest a little contrast and visualize with more certainty high recovery accuracy. This example shows that fitting function a solid continuous representation of given discrete distribution.  
Figure 12: Fitting same discrete sample of 201 uniformly selected datapoints from sine function over \([0,\,5\pi]\) with MATLAB rationalfit. Residual errors are significant over most of points in \([0,\,5\pi]\).  
Figure 13: Left graphs show fitting functions via null space method without scaling frequencies \(s\) to interval \([0,\,1]\). Error plots on right side suggest failure to obtain an accurate curve fitting without scaling process.