## 3 Noise Source Localization Method Using GIB with 'p+v' Joint Processing
### GIB Based on the Joint Processing of 'p+v'
According to Equations (15) and (16), the combined form of 'p+v' is  
\[p_{v}+v=\left(\frac{G(s,r_{1})+G(s,r_{2})}{2}+\frac{1}{j\omega\rho}\frac{G(s,r_{1 })-G(s,r_{2})}{r_{1}-r_{2}}\right)Q=G_{pv}Q \tag{17}\]  
It can be seen from the above formula that the sound field transfer function after the joint treatment of 'p+v' is  
\[v=\frac{1}{j\omega\rho}\
abla p=\frac{1}{j\omega\rho}\frac{jkr_{v}-1}{4\pi r_{v} ^{2}}e^{jkr_{v}}Q=\frac{jkr_{v}-1}{j\omega\rho r_{v}}p_{v} \tag{11}\]0  
Substitute the transfer function \(G_{pv}\) of the sound pressure velocity joint processing method into the GIB algorithm, there can obtain the output results of the GIB based on the 'p+v' joint processing:  
\[v=\frac{1}{j\omega\rho}\
abla p=\frac{1}{j\omega\rho}\frac{jkr_{v}-1}{4\pi r_{v} ^{2}}e^{jkr_{v}}Q=\frac{jkr_{v}-1}{j\omega\rho r_{v}}p_{v} \tag{11}\]1  
Because the virtual concentric dual-radius open SMA in this paper is obtained using a virtual SMA extrapolation method, the joint processing transfer function obtained by the finite difference method needs to be substituted into the GIB algorithm to achieve noise source localization and identification. After measuring velocity by using the dual-radius open SMA element with the same azimuth for differential processing, the sound pressure information of the velocity point location is obtained. In previous studies, the mean value method (\((p_{1}+p_{2})/2\)) was used to obtain the sound pressure. However, in this paper, the virtual SMA extrapolation method can also be used to solve the approximate sound pressure at this point. According to the process of the proposed method, we next provide an error comparison of the two calculation methods. In this paper, the sound field on a virtual sphere (the sphere is between the actual SMA and the sound source) is reconstructed by the virtual SMA extrapolation method, and \(M\) points (virtual SMA elements) are taken out on the sphere (the azimuth and elevation of these points are the same as elements of the actual SMA). In order to evaluate the data accuracy of virtual SMA elements, the normalized least squares reconstruction error \(\varepsilon\) of these points is [16]:  
\[v=\frac{1}{j\omega\rho}\
abla p=\frac{1}{j\omega\rho}\frac{jkr_{v}-1}{4\pi r_{v} ^{2}}e^{jkr_{v}}Q=\frac{jkr_{v}-1}{j\omega\rho r_{v}}p_{v} \tag{11}\]2  
where \(p_{the}\) and \(p_{rec}\) are the theoretical and approximate values of sound pressure on the virtual sphere, respectively. Figure 3 shows the sound pressure at the velocity point calculated by the SMA extrapolation method and mean value method, respectively. Figure 3a shows the sound pressure results and theoretical values calculated by the above two methods, respectively, when the sound source frequency is 500 Hz. Figure 3b shows the calculation error of the two calculation methods when the sound source frequency is 100-1000 Hz. (Simulation conditions: sound source distance is 1 m, simulated spherical array radius is 0.3 m, virtual spherical array radius is 0.4 m, SNR = 20 dB, and sound velocity in airc = 340 m/s.) Figure 3a shows that the two methods can effectively calculate the sound pressure at the velocity point, and the calculation results are accurate with the exception of a few points. Figure 3b shows that the sound pressure error at this point increases with the increase in sound source frequency, causing a decrease in the error values of the results calculated by the virtual array extrapolation method.