# Rail Diagnostics Based on Ultrasonic Guided Waves: An Overview
## 5 Implementation of On-Board Systems
### Passive Approach
#### 5.2.2 Defect Detection Principle  
In this case, defect detection requires the estimation of the time domain _Green's function_\(G_{AB}(t)\) between the two receivers \(A\) and \(B\), without prior knowledge of the source excitation spectrum. This function represents the response of the test object measured a \(B\) from an impulse excitation at \(A\). If the frequency domain [PERSON]'s transfer function \(G_{AB}(\omega)\) can be estimated, discontinuities in the rail (such as defects) can be detected by means of a procedure similar to that illustrated for the pitch-catch ultrasonic guided-wave approach. In fact, discontinuities can be perceived as a change in the structural impulse response, \(G_{AB}(t)\), obtained from \(G_{AB}(\omega)\) through an inverse Fourier transform. In terms of defect detection, as long as the reconstructed [PERSON]'s function is stable during a test run, successful passive structural inspection becomes possible.  
The estimation of the Green's function is based on the computation of the ensemble averaged _cross-power spectrum_ between the output and the excitation, normalized to a modified ensemble averaged _auto-power spectrum_ of the excitation. The following assumptions are made: (a) the system is linear and piecewise stationary, so that the statistics of the excitation \(W(\omega)\) do not change in the observation interval; (b) the noise terms are uncorrelated, have a zero DC component, and tend to zero on average, thus enabling for separating the useful signal component from noise. The function \(G_{AB}(\omega)\) is estimated by computing the deconvolution of the cross and auto power spectra. In fact, the spectra of the responses at \(A\) and \(B\) can be expressed as  
Figure 13: Representation of the linear wheel-rail interaction in the frequency domain [49].  
\[V_{A}(\omega)=W(\omega)\cdot WA(\omega)\cdot A(\omega)+N_{A}(\omega) \tag{2}\]  
and  
\[V_{B}(\omega)=W(\omega)\cdot WA(\omega)\cdot G_{AB}(\omega)\cdot B(\omega)+N_{B}( \omega), \tag{3}\]  
respectively; here, \(V_{A}(\omega)\) (\(V_{B}(\omega)\)) is the response measured at receiver \(A\) (\(B\)), \(W(\omega)\) is the wheel excitation spectrum, \(WA(\omega)\) is the transfer function of the rail between the wheel and the transducer at \(A\), \(A(\omega)\) (\(B(\omega)\)) is the frequency response of the receiving sensor \(A\) (\(B\)), \(N_{A}(\omega)\) (\(N_{B}(\omega)\)) is the uncorrelated noise originating from the environment at \(A\) (\(B\)) (see Figure 13).  
Based on Equations (2) and (3), it can be shown that [49]  
\[\left\langle\text{Cross Power}\right\rangle=\left|W(\omega)\right|^{2}\left| WA(\omega)\right|^{2}G_{AB}(\omega) \tag{4}\]  
and  
\[\left\langle\text{Auto Power}\right\rangle=\left|W(\omega)\right|^{2}\left| WA(\omega)\right|^{2}, \tag{5}\]  
in which the symbol \(\left\langle\right\rangle\) represents the _ensemble average_ operator.  
By using Equations (4) and (5), the frequency Green's Function between the two receivers is computed as  
\[\frac{\left\langle\text{Cross Power}\right\rangle}{\left\langle\text{Auto Power}\right\rangle}=\frac{\left|W(\omega)\right|^{2}\left|WA(\omega) \right|^{2}G_{AB}(\omega)}{\left|W(\omega)\right|^{2}\left|WA(\omega)\right|^ {2}}=G_{AB}(\omega). \tag{6}\]  
Then, the time domain Green's function can be evaluated by taking the inverse Fourier transform from the left-hand side of the last equation, i.e., as  
\[G_{AB}(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}G_{AB}(\omega)\,e^{i\omega t} \,d\omega. \tag{7}\]  
The last integral is computed in an approximate fashion through an _inverse fast Fourier transform_ (IFFT).