## 3 Results
### Derivation of a Simulation Model from the Experimental Setup  
The simulation model developed for this study is designed to predict the measured ultrasonic echoes from an input coded excitation signal and is divided in two sections: The first section covers the transformation of an ideal coded excitation signal from a user-defined parameter space to the output signal of the signal generator (Figure 1b, Section 1). The second section predicts the resulting echoes taking into account both piezoelectric transducers (transmitter and receiver) as well as the interaction with the object under investigation (Figure 1b, Section 2). The first section can be described numerically and is directly included in the simulation (Figure 3a-e), while the components in the second section are represented by an experimentally determined transfer function (Figure 3f red line) based on impulse-echo measurements (Figure 3f-h). The direct mathematical description of these components would require complex physical modeling. As in this study only the resulting echo is of interest for optimization, the replacement through the transfer function is appropriate.  
To derive the simulation model, the different components of the testing system were subsequently modelled. The process is illustrated for a sine wave with four periods in Figure 3.  
In a first step, a spectral analysis of the LUT-based signal Figure 3a is performed using an FFT (Figure 3b, zoomed in around center frequency in Figure 3c) to adjust the number of data points for each signal to match the center frequency of the piezoelectric transducer (Figure 3d). The adjustment of the number of datapoints is necessary because the transmission of the signal is performed using a system sampling frequency of 122.88 MHz.  
Figure 2: Illustration of the CCF of two ultrasonic echoes resulting from a sine-wave excitation. (**a**) Measurement of two backwall echoes of an ultrasonic wave in 60 mm steel, (**b**) CCF of the two echoes. The * markers highlight the extrema of the CCF.  
The next component, the output stage, is represented through conversion to the pulse-width-modulation (PWM) signal, shown in Figure 3e.  
The remaining components of the testing system are represented through a transfer function [29]. This function is calculated through the Fast-Fourier-Transform (FFT) of a recorded impulse-echo. Multiplication of the FFT of the PWM signal (Figure 3f blue line) with the transfer function (Figure 3f red line) and subsequent reverse FFT results in the predicted echo (Figure 3g) from which the auto-correlation function (ACF) is calculated  
Figure 3: Steps of the simulation model exemplarily shown based on a sine wave: (**a**) signal based on LUT, (**b**) power spectrum of original signal, (**c**) selected region of the power spectrum, (**d**) signal after datapoint reduction, (**e**) PWM signal, (**f**) power spectrum of PWM signal and transfer function, (**g**) calculated echo, (**h**) normalized ACF. The * markers highlight the extrema of the ACF.  
(Figure 3h). The ACF of the calculated echo shape is taken as an approximation for the CCF expected in the experiments.  
To quantify the effect of the signal shapes on the echo, the signals are normalized to the height of the main maximum and relative heights of the positive as well as absolute side maximum are evaluated. In the case of the sine wave, there is a difference of 3.2 percentage points for the absolute side maximum and 10.8 percentage points for the positive side maximum (Figure 3h).