## 2 The Proposed Method
### GSC Method with Sidelobe Neutralization
#### 2.3.2 Beamforming Sidelobe Neutralization  
When the configuration of the microphone array was determined, the sidelobe of the beamforming pattern could be calculated as in Equation (13). If there are two sound sources, \(s_{1}\) and \(s_{2}\), deployed in the direction vector \(\
u_{1}\) and \(\
u_{2}\), respectively, in the sound field, the sidelobe neutralization method could be derived using the following equations:  
\[S=[s_{1},s_{2}],\
u=[\
u_{1},\
u_{2}]; \tag{14}\]  
the received signal of microphones \(x_{m}\) could be written as:  
\[x_{m}=Sv=s_{1}\
u_{(1,m)}+s_{2}\
u_{(2,m)} \tag{15}\]  
The output signal of the beamforming function to the corresponding direction is:  
\[y_{v1}=\sum_{m=1}^{M}\omega_{(s1,m)}^{*}x_{m}=\sum_{m=1}^{M}\
u_{(1,m)}^{*}s_ {1}\
u_{(1,m)}+\
u_{(1,m)}^{*}s_{2}\
u_{(2,m)}=s_{1}+\sum_{m=1}^{M}\
u_{(1,m)} ^{*}s_{2}\
u_{(2,m)} \tag{16}\]  
\[y_{v2}=\sum_{m=1}^{M}\omega_{(s2,m)}^{*}x_{m}=\sum_{m=1}^{M}\
u_{(2,m)}^{*}s_ {1}\
u_{(1,m)}+\
u_{(2,m)}^{*}s_{2}\
u_{(2,m)}=s_{2}+\sum_{m=1}^{M}\
u_{(2,m)} ^{*}s_{1}\
u_{(1,m)} \tag{17}\]  
\[\text{Let }u_{ab}=\sum_{m=1}^{M}\
u_{(a,m)}^{*}\
u_{(b,m)}=B(\phi_{a},\phi_{b}), \tag{18}\]  
\[y_{v1s2}=y_{v1}-u_{12}y_{v2}=s_{1}+u_{12}s_{2}-u_{12}s_{2}-u_{12}u_{21}s_{1}=s _{1}(1-u_{12}u_{21})=s_{1}\Big{(}1-|u_{12}|^{2}\Big{)} \tag{19}\]  
where \(u_{ab}\) represents the attenuation coefficient of the sidelobe when the sound source is placed at direction b and beamformed to direction \(a\). Moreover, the attenuation coefficient will be the conjugate of \(u_{ab}\) when the cast direction is inverse. Hence, Equation (19) could be used for direction \(v_{1}\) to reduce the noise received from direction \(v_{2}\). This equation proves that the noise signal from the interference direction can theoretically be removed entirely from the beamforming output signal of the look direction. When the attenuation coefficient is not equal to one, the signal amplitude from the look direction could be maintained by multiplying a proper scale factor. On the other hand, if the attenuation coefficient of the frequency is equal to one, this frequency signal would be lost completely and could not be compensated. Therefore, the sidelobe neutralization method could be modified as in Equations (20) and (21). The desired signal component in the noise direction was calculated and subtracted from the beamforming output of the noise direction as in Equation (20). Hence, the estimated noise signal becomes the pure noise without the desired signal. The output of the sidelobe neutralization method was calculated using Equation (21). This equation proves that the sidelobe neutralization method would not miss the frequency signal with an attenuation coefficient of even one. Because most of the time, \(u_{ab}\) is less than 1, the cube of \(u_{ab}\) is less than itself. Combining Equations (16) and (21), it means that \((u_{12})^{3}\) should be less than \(u_{12}\) most of the time. Thus, the beamforming output with less noise is obtained. \[y_{v2s1}=y_{v2}-u_{21}y_{v1}=s_{2}+u_{21}s_{1}-u_{21}s_{1}-u_{21}u_{12}s_{2}=s _{2}(1-u_{21}u_{12})=s_{2}\Big{(}1-|u_{21}|^{2}\Big{)} \tag{20}\]  
\[y_{v1s2s1}=y_{v1}-u_{12}y_{v2s1}=s_{1}+u_{12}s_{2}-u_{12}s_{2}\Big{(}1-|u_{21} |^{2}\Big{)}=s_{1}+s_{2}(u_{12})^{3} \tag{21}\]  
However, the estimated noise direction is often inaccurate in practice. The error of the estimated noise direction would affect the performance of the sidelobe neutralization method.