# A New Signal Processing Method for Time-of-Flight and Center Frequency Estimation
## 3 Experimental Signal Processing by the Proposed Method
### Estimating the Parameters of the Reflected Signals  
The simplest guided wave propagation experiment was performed on a 0.8 \(\times\) 0.45 m\({}^{2}\) sheet molding compound (SMC) plate with a thickness of \(d\) = 2.3 mm (Figure 4). The ultrasound system \"ULTRALAB\", developed at the Ultrasound Research Institute of Kaunas University of Technology, was used in the experiment. In the SMC plate, the S\({}_{0}\) mode of guided waves was generated by PZT contact transducers excited at a resonant frequency \(f_{\text{ex}}\) = 180 kHz. The received signal was also recorded by a PZT transducer at a distance of L\({}_{1}\) = 600 mm. A 65 V, 3-period sine signal with a Gaussian envelope was used to excite the transmitter.  
The received signal is shown in Figure 5a. In this signal, we can see the directly transmitted signal (direction transmitter-receiver (L\({}_{1}\))) (Figure 5a, (1)), reflections from  
Figure 4: The structural scheme of GW S\({}_{0}\) mode signal generation and recording in an SMC plate.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**ToF** & **SNR** & **0 dB** & **50 dB** & **45 dB** & **40 dB** & **35 dB** & **30 dB** & **25 dB** & **20 dB** & **15 dB** & **10 dB** \\ \hline
60 \(\upmu\)s (100 kHz) & 0 & 0.0007 & 0.0015 & 0.0029 & 0.0061 & 0.0069 & 0.0071 & 0.0161 & 0.0382 & 0.0532 \\
46.7 \(\upmu\)s (300 kHz) & 0 & 0.0008 & 0.0011 & 0.0033 & 0.0033 & 0.0066 & 0.0093 & 0.0150 & - & - \\
44 \(\upmu\)s (500 kHz) & 0 & 0.0013 & 0.0014 & 0.0012 & 0.0015 & 0.0027 & 0.0035 & - & - & - \\
42 \(\upmu\)s (1000 kHz) & 0 & 0.0072 & 0.0065 & 0.0068 & 0.0090 & 0.0150 & - & - & - & - \\ \hline \end{tabular}
\end{table}
Table 2: The relative error of time-of-flight detection (ToF) obtained for different SNRs.  
the edges of the plate (direction transmitter-edge-transmitter-receiver (2L\({}_{2}\) + L\({}_{1}\)) and transmitter-receiver-edge-receiver (L\({}_{1}\) + 2L\({}_{3}\))) (Figure 5a, (2)) and double reflection from the edges (direction transmitter-edge-transmitter-receiver-edge-receiver (2L\({}_{2}\) + L\({}_{1}\) + 2L\({}_{3}\))) (Figure 5a, (3)).  
The proposed algorithm enables the determination of the propagation times and center frequencies of all these signals. In the given case, we obtained the propagation times of the signals \(t_{\rm f1}\) = 209.5 \(\upmu\)s, \(t_{\rm f2}\) = 268.0 \(\upmu\)s, and \(t_{\rm f3}\) = 326.5 \(\upmu\)s. Meanwhile, the corresponding center frequencies of the signals were \(f_{\rm c1}\) = 171.1 kHz, \(f_{\rm c2}\) = 169.5 kHz, and \(f_{\rm c1}\) = 165.6 kHz. The presented measurement results provide information indicating that the central frequency \(f_{\rm c}\) of the guided wave S\({}_{0}\) modes decreases as they propagate in the plate.  
Knowing the propagation time of signals and the distances they travel, it was possible to calculate the propagation velocity of these signals. However, this cannot be done directly for a propagated signal (L1), because the delay time of the exited signal was unknown. The propagation times of two signals (\(t_{\rm f1}\) and \(t_{\rm f2}\)) that have traveled different distances (L\({}_{1}\) and 2L\({}_{2}\) + L\({}_{1}\)) were used to determine the propagation velocity of the signals.  
\[c_{\rm S0}=\frac{(2L_{2}+L_{1})-L_{1}}{t_{\rm f2}-t_{\rm f1}}=\frac{2L_{2}}{t_ {\rm f2}-t_{\rm f1}}=3418.8\;{\rm m/s}, \tag{20}\]  
Figure 5: (**a**) Received GW signal on SMC plate. (**b–d**) Individual extracted signals. (**e–g**) Determination of propagation times of individual signals.  
This plate was also measured by longitudinal scanning. From the obtained B-scan, the S\({}_{0}\) mode propagation velocity was calculated using the 2D FFT method and was determined to be c\({}_{s0}\) = 3388 m/s. The obtained value differs slightly (0.9%) from the value determined by the proposed method.