# First SIMO Harmonic Radar Based on the SFCW Concept and the HR Transfer Function
## 2 The HR Equation and HR Transfer Function
This frequency converting reflection parameter can be further divided into distinct system contributions captured by this measurement, when mismatching between individual MF-parameters blocks is ignored and only the superposition of multiple reflection effects to the forward transmission is considered. This first order approximation is applicable and valid here due to the large free-space wave propagation losses. Starting from the interrogator, the first relevant MF-parameter block is the linear forward propagation of the fundamental frequency stimulus signal through the medium to the input of the frequency doubler, which shall be denoted as \(S^{I}_{21,\text{Ref}}\). The nonlinear, input power dependent MF-parameter of the doubler can be described by the parameter \(S^{II,I}_{43}|_{P_{kof}}\), while the linear return path through the medium at the harmonic frequency is captured by \(S^{II}_{12\text{Ref}}\). A graphical representation of the HR signal flow is shown in Figure 2. The complete reference measurement \(S^{II,I}_{11,\text{Ref}}\) can therefore be described using  
\[S^{II,I}_{11,\text{Ref}}=S^{I}_{21,\text{Ref}}\cdot S^{II,I}_{43}|_{P_{Ref}} \cdot S^{II}_{12,\text{Ref}}\quad. \tag{7}\]  
Equation (7) describes the HR transfer function for the reference measurement. This reference result is used further on to normalize the measurement results obtained when a material or object under test is inserted into the pathway, which corresponds to the mixed-frequency equivalent of the reflection normalization procedure used in linear VNAs (see [26]). This measurement implicitly contains the propagation properties of the channel at the two frequencies as well as the specific conversion behavior of the HR tag. Additionally, these data are also used to correct for systematic and repeatable phase offsets in the signal generation and receiver parts of the system (see [19] for details). An object- or material-under-test (MUT) shall have the scattering parameters \(S^{I}_{21,\text{MUT}}\) and \(S^{II}_{21,\text{MUT}}\) at the two frequencies I and II, which describe the superposition of all changes in transmission wave propagation caused by the material. When this material- or object-under-test is inserted into the propagation path, replacing a section of the free space reference measurement, and a measurement \(S^{II,I}_{11,\text{MUT}_{2}}\) is performed, a superposition of diffraction and refraction effects in addition to an observed change in phase velocity \(v_{p}\) and losses caused by the dielectric and magnetic properties of the material influencing the wave transmission through the material will occur. Since the material replaces an equivalent shaped section of free space contained in the reference measurement, it is necessary to de-embed the corresponding free-space section from the material properties, resulting in scattering parameters \(S^{I}_{21,\text{MUT}_{2}}\) and \(S^{II}_{21,\text{MUT}_{2}}\), which only capture the relative change compared to an equivalent section of free-space propagation. The corresponding HR transfer function for the measurement including the MUT is therefore  
\[S^{II,I}_{11,\text{MUT}_{2}}=S^{I}_{21,\text{Ref}}\cdot S^{I}_{21,\text{MUT}_{ 2}}\cdot S^{II,I}_{43}|_{P_{MUT}}\cdot S^{II}_{12,\text{Ref}}\cdot S^{II}_{12, \text{MUT}_{2}}\quad. \tag{8}\]  
The mechanical distance between the HR and the tag is unchanged compared to the reference measurement. The added material/object creates the observed additional attenuation and phase rotation \(S^{I}_{21,\text{MUT}_{2}}\) and \(S^{II}_{12,\text{MUT}_{2}}\).