## 2 Principle of TDOA Method and Beamforming Method
### Beamforming Method  
The beamforming method theoretically can identify the location of any AE source in the near-field region. When the array is focused to a point source at limited distance, the incident AE waves are spherical, as shown in Figure 2. Array output is calculated by [20]  
\[b(\overrightarrow{r},t)=\frac{1}{M}\sum_{m=1}^{M}w_{m}x_{m}(t-\Delta_{m}( \overrightarrow{r})) \tag{7}\]  
Figure 1: Linear array of four sensors. where \(b(\overrightarrow{r},t)\) is the output of the array; \(\overrightarrow{r}\) is the direction vector from the reference sensor to the focused point. The reference point may be arbitrary and it is the first sensor point on the left side in the Figure 2; \(M\) is the number of sensors; \(w_{m}\) the weighting coefficient for the channel of sensor \(m\) (usually no modification, in this article, \(w_{m}\equiv 1\)) and \(x_{m}(t)\) represents the signal acquired from the No. \(m\) sensor; \(\Delta_{m}(\overrightarrow{r})\) indicates the individual time delay of No. \(m\) sensor to the reference point. By adjusting time delay \(\Delta_{m}(\overrightarrow{r})\), the signals associated with the spherical waves, emitting from AE source focus, will be aligned in time before they are summed. As shown in Figure 2, \(\Delta_{m}(\overrightarrow{r})\) can be obtained by  
\[\Delta_{m}(\overrightarrow{r})=\frac{\left|\overrightarrow{r}\right|-\left| \overrightarrow{r}-\overrightarrow{r}_{m}\right|}{c} \tag{8}\]  
where \(\overrightarrow{r}_{m}\) the distance between reference point and No. \(m\) sensor; \(\left|\overrightarrow{r}\right|-\left|\overrightarrow{r}-\overrightarrow{r}_ {m}\right|\) represents the difference between the distance from reference point to focus point and the distance from No. \(m\) sensor to the focus point; \(c\) is the propagation velocity of AE wave. The output of the beamforming method is the integral of \(b(\overrightarrow{r},t)\) over time, which has the meaning of energy. If the focused point is the real source, the signals are aligned at the same wave front and the output of the beamforming is maximum. However, the signals cannot be aligned at the same wave front when the array of sensors is focused on other locations, and the output of the beamforming is not the maximum. The location of the maximum output energy represents the location of the AE source. When the beamforming method is used, every point on the structure is focused, and the output of the beamforming array at every point needs to be calculated. Once the location of the real AE source is focused on, the output of the sensor array reaches the maximum value. Thus, the location of the real AE source can be judged by comparing those outputs. Figure 3 shows the operational principle of the beamforming method. Figure 2: Schematic diagram of the principle of beamforming method. In operation, the localization accuracy can be determined according to actual needs. For example, on a 600 mm by 600 mm plate, if locating is performed with an accuracy of 10 mm, 3721 (61 \(\times\) 61) points need to be scanned. If locating is performed with an accuracy of 1 mm, 361,201 (601 \(\times\) 601) points need to be scanned. Obviously, the higher the localization accuracy, the more points need to be scanned. This means that the longer it takes to locate. Then the output of each point can be calculated. The point with the largest output value is the location of the acoustic emission source. Figure 4 shows the beamforming array output in a typical acoustic emission source location. At this time, normalization is performed according to the maximum output value. It can be seen from the Figure 4 that the location where the output is one is the acoustic emission source. The beamforming method has high localization accuracy. However, the localization speed is very low since great quantities of points need to be scanned.