## 3 The Radon Transform as Sensor Shape Design Tool  
The goal of having transducers with the desired relations (3) and (4), is achieved if the sensors are shaped so that the angle-dependent frequency responses \(V_{P2}(\omega,\theta)\) and \(V_{P3}(\omega,\theta)\) (of P2 and P3, respectively) are linked to \(V_{P1}(\omega)\) (i.e., the isotropic one of the reference circular sensor P1), so that:  
\[ \begin{array}{l}V_{P2}(\omega,\theta)=V_{P1}(\omega)\exp(-jk_{0}(\omega) \rho_{2}(\theta))\\ V_{P3}(\omega,\theta)=V_{P1}(\omega)\exp(-jk_{0}(\omega)\rho_{3}(\theta))\end{array}  \tag{7}\]  
i.e., the frequency response of the three piezo patches should be equal but for a phase shift which is proportional to the \(\rho_{2}\) and \(\rho_{3}\) quantities which we have introduced in the previous section. The term \(k_{0}(\omega)\) is the wave vector-frequency relationship of a guide wave mode, i.e., the so called dispersion curve of the Lamb wave that is captured by the sensor. Replacing \(k_{0}(\omega)\) with \(\omega/v(\omega)\) where \(v(\omega)\) is the phase wave velocity, under the hypothesis of narrow-band waves, the variations of \(v(\omega)\) can be neglected and, given \(\omega_{0}\) the center frequency, a constant velocity \(v(\omega_{0})\) can be considered. The relationships (7) then create the relationships (3) and (4) over time. To impose the relationships (7) three key elements can be exploited:  
1. The model of the frequency response of a sensor in the presence of a Lamb wave mode. 2. The Radon Transform (RT) and its inverse (IRT). 3. The Projection Slice Theorem. For the element (a), the model that was proposed by [PERSON] and [PERSON] [21] can be adopted:  
\[V_{P}(\omega)=jU(\omega)k_{0}(\omega)H_{P}(\theta)D_{P}(\omega,\theta) \tag{8}\]  
where \(U(\omega)\) denotes the amplitude and the polarization of the wave component relevant to the piezo properties of the patch at the considered frequency, \(k_{0}(\omega)\) is the wave vector that characterizes the propagation, \(H_{P}(\theta)\) is a quantity that is related to the material properties of the piezo-structure system, and finally \(D_{P}(\omega,\theta)\) is the sensor Directivity function, which is the only function that is dependent on the shape of the sensor. It can be computed by the following integral:  
\[D_{P}(\omega,\theta)=\int\limits_{\Omega_{P}}e^{-jk_{0}(\omega)(x\cos\theta+ y\sin\theta)}\phi_{P}(x,y)dxdy \tag{9}\]  
where \(\phi_{P}(x,y)\) is referred to as shape function and it describes the shape of sensor as a step function of form:  
\[\phi_{P}(x,y)=\left\{ \begin{array}{ll}1&(x,y)\in\Omega_{P}\\ 0&(x,y)\
otin\Omega_{P}\end{array} \right. \tag{10}\]  
with \(\Omega_{P}\) the area of the piezoelectric path. Without lack of generality, here we consider the case of a single piezoelectric patch, with a single polarization and costant piezoelectric properties. From Equation (9), the Directivity function can be computed as the bidimensional Fourier Transform (FT) at the angle \(\theta\) of the shape function \(\phi_{P}(x,y)\). Furthermore, the output variable is evaluated in \(k_{0}(\omega)\), the wave vector of the Lamb wave mode considered (e.g., A\({}_{0}\), S\({}_{0}\), etc. modes). The element (b), which is the Radon Transform [22], is defined, as follows:  
\[R_{\theta}(\rho)[\varphi(x,y)]=\int\int_{\Omega_{P}}\phi(x,y)\delta(\rho-x\cos \theta-y\sin\theta)dxdy \tag{11}\]  
and it consists of multiple line-integrals. It admits inverse. Finally (Point (c)), the Projection Slice Theorem [23] states that the bidimensional FT at the angle \(\theta\) of a given function, is equal to the mono-dimensional FT of the Radon Transform of that function. This implies that the Directivity function, and therefore the frequency response, can simplybe calculated as the monodimensional FT of the RT of the shape function and by evaluating into \(k_{0}(\omega)\).