## 2 The Principle of Leak Location Method Based on the Generalized Cross-Correlation Function  
### The Modified Generalized Cross-Correlation Location Method  
Figure 1 presents a pressurized piping system: \(l\) is the distance between the two sensors, and \(l_{1}\) and \(l_{2}\) are the distances between the leak point and sensors 1 and 2, respectively. If there is a leak between the two AE sensors, the two measured signals \(x_{1}(t)\) and \(x_{2}(t)\) can be mathematically modeled [21] as  
\[x_{1}(t)=h_{1}(t)\otimes s(t)+n_{1}(t),x_{2}(t)=h_{2}(t)\otimes s(t+d)+n_{2}(t), \tag{1}\]  
where \(s(t)\) is the leakage-induced signal measured at the leak location, the parameter \(t\) is the discrete-time counter, and \(d\) denotes the time delay between two measured signals. \(h_{1}(t)\) and \(h_{2}(t)\) are the discrete-time impulse response functions between the leak point and each sensor, respectively, and they provide the relationships between an impulse signal and each sensor signal. \(n_{1}(t)\) and \(n_{2}(t)\) are the background noises picked up by the two sensors, \(\otimes\) is the discrete-time convolution operator. \(s(t)\), \(n_{1}(t)\) and \(n_{2}(t)\) are assumed as stationary random signals. The noises measured by the two sensors are assumed to be mutually unrelated and unrelated with the leakage-induced signal, so the CCF between two measured signals with noise and without noise are equivalent, and can be expressed as follow:  
\[R_{s1s2}(\tau)=R_{x1x2}(\tau), \tag{2}\]  
where \(R_{s1s2}(\tau)\) and \(R_{x1x2}(\tau)\) are the CCF without noise and with noise, respectively, in terms of the time delay \(\tau\). Since the CCF is sensitive to the non-leak vibration noises [21], the time delay is commonly estimated by the GCC function [22], which can be written as:  
\[R_{x1x2}(\tau)=F^{-1}\Big{\{}\Psi_{g}(\omega)S_{x1x2}(\omega)\Big{\}}, \tag{3}\]  
where \(F^{-1}\{\cdot\}\) denotes the inverse discrete Fourier transform, \(\omega\) denotes the frequency, and \(\Psi_{g}(\omega)\) is the frequency weighting function of the GCC function. \(S_{x1x2}(\omega)\) is the cross-spectral density between two measured signals \(x_{1}(t)\) and \(x_{2}(t)\)[23], and can be written as:  
\[S_{x1x2}(\omega)=\frac{1}{2\pi}\lim_{T\rightarrow\infty}E\Bigg{[}\frac{X_{1} ^{*}(\omega)X_{2}(\omega)}{T}\Bigg{]}=S_{ss}(\omega)\Big{|}H_{1}^{*}(\omega)H _{2}(\omega)\Big{|}e^{i\omega d}, \tag{4}\]  
where \(E[\cdot]\) denotes the expectation operator, \(X_{1}(\omega)\) and \(X_{2}(\omega)\) are the N-point discrete Fourier transforms (DFT) of \(x_{1}(t)\) and \(x_{2}(t)\), respectively, \(H_{1}(\omega)\) and \(H_{2}(\omega)\) are the DFT of \(h_{1}(t)\) and \(h_{2}(t)\), respectively, \(S_{ss}(\omega)\) is the auto-spectral density of \(s(t)\), and \((\cdot)^{*}\) means complex conjugation. Since multiplication in the frequency domain corresponds to convolution in the time domain [24], the GCC function between  
Figure 1: Schematic of the pressurized piping system. two measured signals in terms of the time delay \(\tau\) can be derived by substituting Equation (4) into Equation (3), and expressed as:  
\[R_{x1x2}(\tau)=F^{-1}\Big{\{}S_{ss}(\omega)\Psi_{g}(\omega)\Big{|}H_{1}^{*}( \omega)H_{2}(\omega)\Big{|}e^{i\omega d}\Big{\}}=R_{ll}(\tau)\otimes_{g}(\tau) \otimes\delta(\tau+d), \tag{5}\]  
where \(R_{ll}(\tau)=F^{-1}\{S_{ss}(\omega)\}\) is the autocorrelation function of \(s(t)\), and \(\Psi_{g}(\tau)=F^{-1}\{\Psi_{g}(\omega)|H_{1}{}^{*}(\omega)H_{2}(\omega)|\}\), \(\delta\left(\tau+d\right)=F^{-1}\{e^{i\omega d}\}\). The time delay between two measured signals is equal to the quotient of the offset of the GCC maximum peak and the sampling frequency. \(\delta\left(\tau+d\right)\) is a function equal to zero everywhere except for \(\tau=-d\), and the Dirac Delta function's integral over the entire real line is equal to one. If \(\Psi_{g}(\omega)=1\), the Dirac delta function \(\delta\left(\tau+d\right)\) delayed by \(d\) will be broadened by \(S_{ss}(\omega)\), \(H_{1}(\omega)\) and \(H_{2}(\omega)\)[25].