3.2.1. TDOA Measurement

The signals observed by a pair of sensors can be mathematically described as:

$$z\_1\left(t\right) = u\_1s\left(t\right) + n\_1\left(t\right) \tag{16}$$

$$z\_2\left(t\right) = \mathfrak{a}\_2\mathfrak{s}\left(t - \mathfrak{r}\_{1.2}\right) + \mathfrak{n}\_2\left(t\right),\tag{17}$$

where *z*<sup>1</sup> (*t*) and *z*<sup>2</sup> (*t*) are the signals received by the pair of sensor array, *s*(*t*) is the true signal, *n*<sup>1</sup> (*t*) and *n*<sup>2</sup> (*t*) are noise signals, *τ*1:2 is the time difference between two sensors detecting the signal, *α*<sup>1</sup> and *α*<sup>2</sup> are signal amplitudes [8].

The time difference can be estimated by the generalized cross correlation (GCC) method [4]:

$$R\_{\rm GCC}\left(\pi\_{\emptyset}\right) = \int\_{-\infty}^{\infty} \psi\_{12}\left(\omega\right) Z\_1\left(\omega\right) Z\_2^\*\left(\omega\right) e^{-j\omega\pi\_{\emptyset}} d\omega \tag{18}$$

$$\text{fit}\_q = \arg\max R\_{\text{GCC}} \left( \text{tr}\_q \right) \tag{19}$$

$$\psi\_{1,2}\left(\omega\right) = \frac{1}{\left|G\_{x\_1x\_2}\left(\omega\right)\right|} = \frac{1}{\left|Z\_1\left(\omega\right)Z\_2^\*\left(\omega\right)\right|}.\tag{20}$$

Here, *RGCC* ' *τq* ( is the GCC, where *Z*<sup>1</sup> (*ω*) is the Fourier transforms of *z*<sup>1</sup> (*t*) and *Z*<sup>∗</sup> <sup>2</sup> (*ω*) is the Fourier transform conjugate of *z*<sup>2</sup> (*t*). *ψ*1,2 (*ω*) is the weight function of GCC. To reduce environmental noise and reverberation interference, we choose the phase transform (PHAT) as our weight function, the formula is given by *ψ*1,2 (*ω*) = <sup>1</sup> |*Gx*1*x*<sup>2</sup> (*ω*)| .
