## 3 Performance Analysis of TDOA Method and Beamforming Method Based on the Simulated Signals
### Performance Analysis of TDOA Method  
AE simulation is carried out on steel plates with dimensions of 800 mm by 1000 mm by 5 mm and the location of the 15 sensors is shown in Figure 6. The spacing of each sensor is 50 mm. The coordinates of the first sensor on the left are (50 mm, 100 mm). The location of AE source is set to #1 (300 mm, 600 mm) and #2 (500 mm, 800 mm). Material properties used in the finite element model are tabulated as Table 1.  
Figure 5: Excitation information: (**a**) The loading location; (**b**) The mechanical representation. Figure 6: Steel plate simulation model. The simulated signals received by the 15 sensors are extracted for locating. Based on the first sensor on the left, the TDOA locating of four sensors is carried out according to the sensor spacing of 50 mm, 100 mm, 150 mm, and 200 mm, respectively. The localization results are shown in Table 2.  
It can be concluded from Table 2 that the x-coordinates error located by the TDOA method of the four sensors for the linear array is small. When the x-coordinate of AE source is within the sensor array, the x-coordinate of the localization point is very accurate, but the y-coordinate is generally not accurate. The relationship between the localization of four sensors and the TDOA error is studied by using the TDOA of ideal wave. Set AE wave propagation speed in the plate to 5400 m/s. AE source locations are selected as (350 mm, 600 mm), (550 mm, 600 mm), and (750 mm, 600 mm). Sensor spacing is 200 mm. The coordinates of number 1-4 sensors are (100 mm, 0 mm), (300 mm, 0 mm), (500 mm, 0 mm), and (700 mm, 0 mm). Firstly, it is assumed that the TDOA between the times of AE wave arriving at number 2 and 3 sensors to that of number 1 sensor remains unchanged. The TDOA between number 4 sensor and number 1 sensor is changed to deviate from within 10% of the correct value and observe the localization result. Then, it is assumed that the TDOA between the times of AE wave arriving at number 2 and 4 sensors to that of number 1 sensor remain unchanged. The TDOA between number 3 sensor and number 1 sensor is changed to deviate from within 10% of the correct value. Finally, it is assumed that the TDOA between the times of AE wave arriving at number 3 and 4 sensors to that of number 1 sensor are unchanged. The TDOA between the number 2 sensor and number 1 sensor is changed, and the deviation is also within 10% of the correct value. The localization results are shown in Table 3. N.A. indicates that the localization result has an imaginary number. It can be concluded from Table 3 that when the TDOA fluctuates within a certain range, the localization result of the x-coordinate does not change much, but the range of the y-coordinate of the localization varies greatly. Sometimes the localization result may be an imaginary number. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & \(E\) & \(\rho\) & \(V\) \\ \hline Value & 209 & 7800 & 0.3 \\ Units & GPa & Kg/m\({}^{3}\) & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sensor Spacing** & **Actual Location of \#1** & **Localization** & **Actual Location of \#2** & **Localization** \\
**(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline
50 & (300, 600) & (286, 281) & (500, 800) & (650, 762) \\
100 & (300, 600) & (300, 334) & (500, 800) & (534, 1010) \\
150 & (300, 600) & (301, 381) & (500, 800) & (495, 757) \\
200 & (300, 600) & (300, 586) & (500, 800) & (503, 834) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time difference of arrival (TDOA) locating of different sensor spacing.