# Rail Diagnostics Based on Ultrasonic Guided Waves: An Overview
## 7 Land-Based Systems Implementation-Evolution: Early Rail Defect Detection Capability
### Signal Pre-Processing
#### 7.3.2 Phased Array Processing  
An array of transducers allows for exciting a specific mode in a desired direction of propagation, even if the array is composed of two transducers only. This result is obtained by feeding both transducers with excitation signals characterized by a small difference in delay or phase. The phase difference depends on the wavelength of the propagation mode selected at the transmission frequency and on the axial distance between the two transducers. The same idea can be exploited at the receive side; in fact, if a phase shift is applied to the signal acquired by a transducer, the energy associated with the desired mode and originating from the desired direction can be captured. It is worth noting that a perfect cancellation is impossible, since other modes characterized by different wavelengths are transmitted and received in both directions. This problem can be mitigated by increasing the overall number of transducers forming the array. Phased array processing can be applied to the full matrix of signals acquired from an array of transducers, as explained by [PERSON] in Ref. [PERSON] (1966). The acquired time signals are converted to the frequency domain and, after processing, are converted back to the time domain. When the first (second) transducer is excited, the responses acquired by both transducers are stored in the first (second) column of the _measured displacement matrix_  
\[\mathbf{V}(\omega)=\left[ \begin{array}{cc}V_{11}(\omega)&V_{12}(\omega)\\ V_{21}(\omega)&V_{22}(\omega)\end{array} \right]; \tag{8}\]  
Here, \(V_{ij}(\omega)\) is the spectrum of the response of the \(i\)-th transducer in the array (with \(i=1\) and \(2\), if the array consists of two transducers only) in response to the excitation injected through the \(j\)-th transducer (with \(j=1\) and \(2\), if the array consists of two transducers only). Figure 17: Pre-processing steps of the acquired signals [65]. The analysis of different combinations of transmitted and received propagation modes requires the computation of _mode shape matrices_. Each column of the mode-shape matrix refers to the displacement of a mode at each of the transducer locations. The mode shape matrices for an array of transducers in a 1D waveguide (such as a rail) contains both mode shape and phase information according to the axial position of the associated transducer, for each mode in every possible direction [71]. Since it may happen that the modes selected for receive processing are different from those chosen for transmit processing, two mode shape matrices are required, one for reception and the other for transmission (denoted \(\mathbf{R}(\omega)\) and \(\mathbf{T}(\omega)\), respectively). If the same modes are selected for reception and transmission, then \(\mathbf{R}(\omega)=\mathbf{T}(\omega)\). The coefficients of these matrices are computed through a SAFE analysis of the rail. If the mode shape matrices \(\mathbf{R}(\omega)\) and \(\mathbf{T}(\omega)\), and the measured displacement matrix \(\mathbf{V}(\omega)\) (8) are known, the contributions of the different modes to the received signal can be computed. The spectrum \(\boldsymbol{\alpha(\omega)}\) of the overall response is evaluated as  
\[\boldsymbol{\alpha(\omega)}=\mathbf{R}(\omega)^{-1}\,\mathbf{V}(\omega)\, \mathbf{T}(\omega)^{*^{-1}}=\left[ \begin{array}{cc}\boldsymbol{\alpha_{r1}t_ {1}}(\omega)&\boldsymbol{\alpha_{r1}t_{2}}(\omega)\\ \boldsymbol{\alpha_{r2}t_{1}}(\omega)&\boldsymbol{\alpha_{r2}t_{2}}(\omega) \end{array} \right]; \tag{9}\]  
Here, the spectrum \(\boldsymbol{\alpha_{r_{1}t_{1}}}(\omega)\) (\(\boldsymbol{\alpha_{r_{2}t_{2}}}(\omega)\)) describes the transmission and reception in the forward (backward) direction, whereas \(\boldsymbol{\alpha_{r_{2}t_{1}}}(\omega)\) (\(\boldsymbol{\alpha_{r_{1}t_{2}}}(\omega)\)) is the transmission in the forward (backward) direction and the reception in the backward (forward) direction.