## 2 Theoretical Analysis
### Decoupling the Cladding-Induced Interference  
As shown in Equation (1), the solution of the PECT signal is mainly depended on three parameters: the sensor spatial frequency spectra, \(S(\alpha)\); generalized reflection coefficient of the four-layered structure, \(R^{\prime}{}_{5,4}(\alpha)\); and lift-off coefficient, \(e^{-2\alpha I}\). According to Equations (1)-(3), \(R^{\prime}{}_{5,4}(\alpha)\) is determined by the ferromagnetic plate, the insulation and the cladding. \(e^{-2\alpha I}\) is related to the sensor lift-off. Then through analyzing \(R^{\prime}{}_{5,4}(\alpha)\) and \(e^{-2\alpha I}\), the interference factors could be discussed. As \(R^{\prime}{}_{5,4}(\alpha)\) indicates that the interference caused by the cladding and the insulation are coupled with the ferromagnetic plate, thus, \(R^{\prime}{}_{5,4}(\alpha)\) is discussed firstly. \(R^{\prime}{}_{5,4}\) (\(\alpha\)) is the result of the ratio between the amplitudes of the reflected wave and the incident wave [22]. It essentially indicates the electromagnetic wave propagation in the four-layered structure. As shown in Figure 2, the amplitude of the incident wave in layer 5 is set to 1, and then \(R_{5,4}\) and \(T_{5,4}\) denote the single reflection and transmission from the interface between layers 5 and 4, respectively. In addition, as the thickness of layer 4 is finite, there are multiple reflections and transmissions in layer 4, for example \(T_{5,4}R^{\prime}{}_{4,3}\eta\), \(T_{5,4}R^{\prime}{}_{4,3}R_{4,5}\eta^{2}\), \(T_{5,4}R^{\prime}{}_{4,3}^{2}R_  
Wherein \(T_{5,4}=1+R_{5,4}\), \(R_{4,5}=-R_{5,4}\), \(\eta=e^{\beta 4(d3-d4)}\). To explain this more clearly, \(R^{\prime}{}_{5,4}\) (\(\alpha\)) in (3) could be rewritten as follows:  
\[R^{\prime}{}_{5,4}(\alpha)=R_{5,4}(\alpha)+R^{\prime}{}_{4,3}(\alpha)e^{-2\beta _{4}(d_{3}-d_{4})}\frac{1-R_{5,4}(\alpha)R_{5,4}(\alpha)}{1+R_{5,4}(\alpha)R^{ \prime}{}_{4,3}(\alpha)e^{-2\beta_{4}(d_{3}-d_{4})}} \tag{7}\]  
where the first term is the single reflection coefficient, and the second term is the sum of the other \(n\)-th reflections. According to Equation (1), \(R_{5,4}(\alpha)=(\alpha\mu_{r4}-\beta_{4})/(\alpha\mu_{r4}+\beta_{4})\), wherein \(\beta_{4}=(\alpha^{2}+j\omega\mu_{0}\mu_{r4}\sigma_{4})^{1/2}\). It indicates that \(R_{5,4}\) (\(\alpha\)) is only related to the cladding material. Therefore, \(R_{5,4}(\alpha)\) could be regarded as part of the cladding-induced interference. In addition, [PERSON] demonstrated that the amplitude of the first reflection was much larger than that of the other reflections [8]. Therefore, the interference caused by \(R_{5,4}(\alpha)\) is significant. To further decouple the interference caused by the cladding, the electromagnetic wave propagation in layers 2 and 3 is studied with the same method, and the generalized reflection coefficient \(R^{\prime}{}_{4,3}\) is rewritten as follows:  
\[R^{\prime}{}_{4,3}(\alpha)=R_{4,3}(\alpha)+R^{\prime}{}_{3,2}(\alpha)e^{-2\beta _{3}(d_{2}-d_{3})}\frac{1-R_{4,3}(\alpha)R_{4,3}(\alpha)}{1+R_{4,3}(\alpha)R^{ \prime}{}_{3,2}(\alpha)e^{-2\beta_{3}(d_{2}-d_{3})}} \tag{8}\]  
where the first term, \(R_{4,3}(\alpha)\), could be expressed as \(R_{4,3}(\alpha)=(\mu_{r3}\beta_{4}-\mu_{r4}\beta_{3})/(\mu_{r3}\beta_{4}+\mu_{ r4}\beta_{3})\). As the insulation is always composed of non-conducting material, such as rock wool or foamed glass, then \(\mu_{r3}=1\), \(\beta_{3}=\alpha\). Thus, \(R_{4,3}(\alpha)\) could be rewritten as \(R_{4,3}(\alpha)=(\beta_{4}-\mu_{r4}\alpha)/(\beta_{4}+\mu_{r4}\alpha)=-R_{5,4 }(\alpha)\). Therefore, \(R_{4,3}(\alpha)\) is also related only to the cladding material. If Equation (8) is completely substituted into Equation (7), the expressions of \(R^{\prime}{}_{5,4}(\alpha)\) will become very complicated, which is not conducive to decoupling.