## 2 Localization of Microphones Based on Sound Source Array  
### Time Delay of Arrival (TDOA) Estimation  
The GCC algorithm is currently the most widely used time estimation method. Aiming at the existence of a large amount of noise and reverberation in the actual environment, and thus causing the maximum peak value of the cross-correlation function to be inconspicuous or to have multiple peaks, GCC is better than the basic cross-correlation algorithm for its ability to suppress noise and reverberation in terms of peak detection difficulty. It is assumed here that the sound signal model received by the microphone is an ideal model; that means only ambient noise is considered. The ambient noise is approximately replaced by white Gaussian noise [42; 43]. It is also assumed that there is no correlation between noise and noise, and noise and sound source. The mathematical model of the signal received by the microphone is shown as  
\[x_{i}(t)=\alpha_{i}s(t-\tau_{i})+n_{i}(t) \tag{1}\]  
where considering the sound signal \(s(t)\), the signal received by the \(i\)-th microphone \(x_{i}(t)\), \(\alpha_{i}\), \(\tau_{i}\), and \(n_{i}(t)\) represent the amplitude attenuation coefficient, time delay, and ambient noise of the sound signal reaching the \(i\)-th microphone, respectively. The cross-correlation function of sound signals reached to the \(i\)-th and \(j\)-th microphone is shown as  
\[R_{x_{i}x_{j}}(\tau)=E\big{[}x_{i}(t)x_{j}(t-\tau)\big{]} \tag{2}\]  
Since noise and noise, and noise and sound source are not related to each other, Equation (3) can be derived from Equations (1) and (2):  
\[R_{x_{i}x_{j}}(\tau)=\alpha_{i}\alpha_{j}R_{ss}\big{(}\tau-\tau_{ij}\big{)} \tag{3}\]  
Figure 1: Reference frame of the system and the main components. where \(R_{ss}(\tau)\) is the autocorrelation function of \(s(t)\), \(\tau_{ij}\) is the relative time delay in which the \(i\)-th and \(j\)-th microphones receive the sound signal. According to the nature of the autocorrelation function, it is known that \(R_{x_{i}x_{j}}(\tau)\) reaches the maximum value when \(\tau-\tau_{ij}=0\). Then the time delay estimate \(\tau_{ij}\) can be calculated according to the cross-correlation function of the signals received by the two microphones and the point where the maximum value is located. Equation (4) is obtained by Fourier transform [44; 45] of Equation (3)  
\[\varphi_{xi;x_{j}}(\omega)=\alpha_{i}\alpha_{j}\varphi_{ss}(\omega)e^{-j\omega \tau_{ij}}+\varphi_{n_{i}n_{j}}(\omega) \tag{4}\]  
where \(\varphi_{ss}(\omega)\), \(\varphi_{x_{i}x_{j}}(\omega)\), and \(\varphi_{n_{i}n_{j}}(\omega)\) represent the corresponding power spectrum of \(R_{ss}(\tau)\), \(R_{x_{i}x_{j}}(\tau)\), and \(R_{n_{i}n_{j}}(\tau)\), respectively. \(R_{n_{i}n_{j}}(\tau)\) is the cross-correlation function of \(n_{i}(t)\) and \(n_{j}(t)\). Furthermore, by weighting Equation (4) and then performing the inverse transform, the generalized cross-correlation function can be obtained, as is shown  
\[R_{ij}(\tau)=\int\limits_{-\infty}^{\infty}\varphi_{x_{i}x_{j}}(\omega)\theta_ {ij}(\omega)e^{j\omega\tau}d\omega \tag{5}\]  
where \(\theta_{ij}(\omega)\) is a generalized weighting function [46]. There are weighted functions such as Roth, SCOT (smooth coherent transformation), PHAT (phase transformation) [47; 48; 49], etc. As the PHAT weighting function has less volatility in the case of constantly changing signal-to-noise ratios, and the anti-noise performance is also better, the PHAT weighting function was selected for analysis in this paper. As shown in Figure 2, there was a part of a sound signal waveform received by two microphones (m0, m1). The frequency of the sound wave was 2 kHz, and the sampling frequency was 16 kHz. Analyzing Figure 2, the time difference of arrival between m0 and m1 was approximately 0.008 s. In this paper, the GCC method was used to estimate the time difference of sound. As shown in Figure 3, according the values determined for the peak of the cross-correlation function, the TDOA can be estimated.