## 2 Theoretical Approach  
To investigate the feasibility of estimating material density inside a cardboard box for the purposes of automobile safety, we use crumpled paper, whose density can easily be controlled by using different numbers of sheets. In addition, we can simulate light materials to heavy materials depending on the quantity of paper. We assume that randomly accumulated papers are uniformly distributed in the box. It is necessary to analyze the radar returns theoretically to later verify measurement results by calculating the magnitude of reflections from the sealed box. When the electromagnetic wave from the radar impinges the box located on the road, it is expected that the major reflections will occur in two boundaries, (i) from air to box and (ii) from box to air, as noted in Figure 1a. These reflections produce two corresponding peaks in the range profile, as demonstrated in Figure 1b. Even though there are multiple peaks from multiple reflections, two peaks are dominantly observed in the range profile because other echoes from multiple reflections experience severe attenuation due to a long pathway. To calculate the magnitudes of the peaks, \(A_{1}\) and \(A_{2}\), that are related to the material characteristics inside, we used a five-layer model consisting of air, cardboard box, enclosed material, cardboard box, and air. The problem setup is demonstrated in Figure 2.  
For the theoretic estimation of the magnitudes of \(A_{1}\) and \(A_{2}\), the reflection coefficient and the transmission coefficient should be calculated at each boundary. The reflection coefficients are found using the intrinsic impedance values that are the respective values for permeability and permittivity. \[\Gamma_{NM}=\frac{\eta_{M}-\eta_{N}}{\eta_{M}+\eta_{N}}\,,\,T_{NM}=1-\Gamma_{NM} \tag{1}\]  
where \(\eta\) is intrinsic impedance, \(T_{NM}\) is the reflection coefficient between the \(M\)th and \(N\)th layers, and \(T_{NM}\) is the transmission coefficient between the \(M\)th and \(N\)th layers. The electromagnetic characteristics of the material, the permittivity, and the permeability of five layers are required to calculate wave impedance, the propagation constant, and the attenuation constant. Because the materials considered are nonmagnetic, the relative permeability is set to 1 for all layers. The relative permittivity of the cardboard box was found to be 1.78 [15]. The dielectric constant of paper can range from 1.7 to 4.0 depending on the manufacturing characteristics, humidity, and paper composition [16]. In this study, the permittivity values were unique in each case because of the different densities of paper at each stage. The dielectric constant for the last stage, the densest, was found by calculating a value that provided high attenuation, resulting in a negligible magnitude for the second reflection. The last stage, which had the most paper, dissipated the most energy from the incident wave. The \(\varepsilon\) values for the first four stages were calculated under the assumption that they are linearly proportional to the paper quantity per stage, referring to stage five. Besides, the cardboard and paper exhibited conductivity, producing complex permittivity:  
\[\varepsilon_{c}=\epsilon^{\prime}-je^{\prime\prime}=\epsilon^{\prime}-j\frac{ \sigma}{\omega} \tag{2}\]  
The complex component \(\epsilon^{\prime\prime}\) can be found using the operating frequency of the radar and the electric conductivity of the material. The cardboard and printer paper are both made of wood, whose conductivity varies from \(10^{-4}\) to \(10^{-3}\) (S/m) depending on humidity and the manufacturing of the material [17]. The electric conductivity value of \(10^{-3}\) was used for the cardboard box. The conductivity value of \(\sigma\) = 0.000468 (S/m) was chosen for the last stage. For other stages, it was assumed that the number of papers inside the box per stage was linearly proportional to conductivity. Table 1 shows paper quantity, dielectric constant, and conductivity values per stage used in the simulation.