# A Planer Moving Microphone Array for Sound Source Localization
## 2 Introduction to the FPMMS and Building Procedure
The steering vector can be written as  
\[e(f,\theta_{k})=\alpha e^{-j\omega_{f}\tau_{1},\,\theta_{k}}\times\left[1,e^{- j\omega_{f}(\tau_{2},\,\theta_{k}-\tau_{1},\,\theta_{k})},\ldots,e^{-j\omega_{f}( \tau_{M,\,\theta_{k}}-\tau_{1},\,\theta_{k})}\right]^{T} \tag{4}\]  
For all linear microphone arrays, including the FPMMS, 1 \(\times\) 4 microphone array, 4 \(\times\) 4 microphone array, and the circular microphone array, the relative time delay can be calculated as follows:  
\[\tau_{m,\,\theta_{k}}-\tau_{1,\,\theta_{k}}=\frac{d_{m,\theta_{k}}-d_{1,\theta _{k}}}{c} \tag{5}\]  
\(d_{m,\theta_{k}}\) is the distance from the \(k_{th}\) source to the \(m_{th}\) microphone, and \(d_{1,\theta_{k}}\) is the same for the \(1_{st}\) reference microphone. \(c\) is the sound speed. The only special adjustment for the circular microphone array is as follows:  
\[x_{m}(t)=\sum_{k=1}^{K}\alpha_{m,k}s_{k}(t-\tau_{m,\;\theta_{k}})+v_{m}(t) \tag{1}\]0  
\(l_{m}\), as shown in Figure 4, is the distance in the circular microphone array between the \(m_{th}\) microphone and the \(1_{st}\) reference microphone. \(\gamma_{m}\) denotes the angle of the \(m_{th}\) microphone to the \(1_{st}\) reference microphone with respect to the center of the circle. For practical applications, \(\gamma_{m}\) and \(l_{m}\) can be calculated by the following geometrical relationships:  
\[x_{m}(t)=\sum_{k=1}^{K}\alpha_{m,k}s_{k}(t-\tau_{m,\;\theta_{k}})+v_{m}(t) \tag{1}\]1  
\[x_{m}(t)=\sum_{k=1}^{K}\alpha_{m,k}s_{k}(t-\tau_{m,\;\theta_{k}})+v_{m}(t) \tag{1}\]2  
Here, \(r\) is the radius of the circular microphone array. As shown in the flow chart, the four-time-collected data form together as a whole vector \(\mathbf{x}(n,f)\) and is sent to a TF-WISE algorithm [36] for multiple-sound source angular estimation. For planar distance estimation, since the distance from the sound source to microphone \(L\) is much smaller than \(\frac{2d^{2}}{\lambda}\), \(d\) is the inter-mic distance, and \(\lambda\) is the wavelength. Thus, the near-field model is chosen [39; 40]. Figure 5 depicts the procedure for the planar distance estimation model:  
\[x_{m}(t)=\sum_{k=1}^{K}\alpha_{m,k}s_{k}(t-\tau_{m,\;\theta_{k}})+v_{m}(t) \tag{1}\]3 Interpretation of circular microphone array geometrical relationships. Here, \(s\) is the sound source. The TF-WISE algorithm [41] gives time lag information, supposing that \(\tau_{1,\ \theta_{2}},\tau_{1,\ \theta_{3}}\) represent the time lag between the first microphone (reference microphone), second microphone, and third microphone. The following relationships are formed:  
\[x_{m}(t)=\sum_{k=1}^{K}\alpha_{m,k}s_{k}(t-\tau_{m,\;\theta_{k}})+v_{m}(t) \tag{1}\]4  
\[x_{m}(t)=\sum_{k=1}^{K}\alpha_{m,k}s_{k}(t-\tau_{m,\;\theta_{k}})+v_{m}(t) \tag{1}\]5  
where \(r_{1},\ r_{2},\ r_{3}\) are the distances from the sound source to the microphones, and \(c\) represents sound speed. Utilizing the law of cosines,  
\[x_{m}(t)=\sum_{k=1}^{K}\alpha_{m,k}s_{k}(t-\tau_{m,\;\theta_{k}})+v_{m}(t) \tag{1}\]6  
\[x_{m}(t)=\sum_{k=1}^{K}\alpha_{m,k}s_{k}(t-\tau_{m,\;\theta_{k}})+v_{m}(t) \tag{1}\]7  
\(\tau_{1,\ \theta_{2}},\ \tau_{1,\ \theta_{3}},\ d\), and \(c\) are estimated or known values. Using Equations (1)-(4), \(r_{1},\ r_{2},\ r_{3}\) and \(\theta_{1}\) can be calculated. It should be clear that for angular estimation based on TF-WISE, the angle/geometry is for the center of the microphone array, while the \(\theta_{1}\) here is the angle estimation for the reference microphone. As for height estimation, we still utilize a near-field model. Apart from the first pattern, which will be discussed later, the remaining two patterns only have two microphones for time lag estimation, while distance estimation needs at least three microphones. The situation is simplified to the far field here, meaning that only the angle in the vertical surface can be estimated. Thus, we utilize the angle and distance estimated previously, fixing the sound source on the vertical axis.