# A New Signal Processing Method for Time-of-Flight and Center Frequency Estimation
## 2 Methodology for Estimation of Time-of-Flight and Center Frequency of Signals
### Signal Processing of Ultrasound Signals with Noises
\[U_{1}(f)=\mathrm{FT}[u_{1}(t)],\ U_{2}(f)=\mathrm{FT}[u_{2}(t)]. \tag{7}\]  
The upper (\(f_{\mathrm{H1}}\) and \(f_{\mathrm{H2}}\)) and lower (\(f_{\mathrm{L1}}\) and \(f_{\mathrm{L2}}\)) frequency limits at the 6 dB level (0.5 level) are determined by frequency spectra \(U_{1}(f)\) and \(U_{2}(f)\)  
\[f_{\mathrm{H1}}=\max(\arg\{U_{1}(f)>0.5\}),\ f_{\mathrm{L1}}=\min(\arg\{U_{1} (f)>0.5\}), \tag{8}\]  
\[f_{\mathrm{H2}}=\max(\arg\{U_{2}(f)>0.5\}),\ f_{\mathrm{L2}}=\min(\arg\{U_{2} (f)>0.5\}), \tag{9}\]  
then the corresponding frequency bands (Figure 1, (5) and (6))  
\[\Delta f_{1}=f_{H1}-f_{\mathrm{L1}},\ \Delta f_{2}=f_{H2}-f_{\mathrm{L2}}. \tag{10}\]  
Based on research results [23], we selected a package of five Gaussian filters (Figure 1, (5) and (6), red line) to filter each separated signal \(u_{1}(t)\) and \(u_{2}(t)\)  
\[U(f)=\mathrm{FT}[u(t)], \tag{2}\]0  
\[U(f)=\mathrm{FT}[u(t)], \tag{2}\]1  
\[U(f)=\mathrm{FT}[u(t)], \tag{2}\]2  
where \(B_{in}(f)\) is the frequency response of \(i\)-th bandpass filter, \(i=1\)-5, \(n\)--number of separated signal, \(df_{n}=\frac{f_{\mathrm{H}n}-f_{\mathrm{L}n}}{\Delta a_{\mathrm{H}}}\) are the steps between central frequencies of two neighboring filters,and \(\Delta B_{n}\) are the bandwidths of the filters. The bandwidths of the filters were chosen from the \(\Delta f_{n}/\Delta B_{n}=2.5\) condition [23]. Next, in each filtered signal, the zero-crossings of the signals are identified (Figure 2):  
\[U(f)=\mathrm{FT}[u(t)], \tag{2}\]3  
\[U(f)=\mathrm{FT}[u(t)], \tag{2}\]4  
where \(t_{kn}^{0}\) are the zero-crossings of \(i\)-th filter, \(k=1,2,\ldots\) K is the number of zero-crossings, K is the total number of zero-crossings, \(f_{in}\) is the central frequency of \(i\)-th filter. Finally, the ToF of the separated signals \(t_{f1}\) (Figure 1, (7)) and \(t_{f2}\) (Figure 1, (8)) are determined by averaging the times of the concentrated zero-crossings instances \(t_{iMn}^{0}\)  
\[U(f)=\mathrm{FT}[u(t)], \tag{2}\]5  
\[U(f)=\mathrm{FT}[u(t)], \tag{2}\]6  
where M is the number of zero-crossings in the \(i\)-th filter. Figure 2 shows the original signal (black line) and the signals filtered by a five-filter package (No. 1-5, color line). The zero-crossings are marked in these signals. The concentration of zero-crossings is clearly visible. It corresponds to the ToF (\(t_{f}\)) of the signal. The five-filter package was chosen based on the studies conducted in [23] because, as the number of filters increases, the uncertainties in determining the zero-crossing concentrations begin to increase. Nearby zero-crossing instances (\(t_{p1}\) and \(t_{p2}\)) were assigned to both the original signal and the median filtered signal. We used these instances to determine the center frequency of the original signal. \[U(f)=\mathrm{FT}[u(t)], \tag{2}\]7  
A simulated signal with two central frequencies was processed using the algorithm described above. The following values were obtained from the following calculations: the ToF of the separated signals \(t_{f1}=46.7\)\(\upmu\)s and \(t_{f2}=69.0\)\(\upmu\)s; and the center frequency of the signals \(f_{c1}=302\) kHz and \(f_{c2}=492\) kHz. \[U(f)=\mathrm{FT}[u(t)], \tag{2}\]8 The original signal and the signals filtered by a five-filter package.