# First SIMO Harmonic Radar Based on the SFCW Concept and the HR Transfer Function
## 6 Verification of the HR Transfer Function
Four coaxial SMA attenenuators (see Figure 17) with different nominal transmission attenuation values (3 dB, 6 dB, 10 dB and 20 dB), made by three different manufacturers in slightly varying form factors resulting in distinct transmission phases, were used as a well-defined coaxial replacement devices for an MUT. S-parameter measurements were obtained for all attenuators using a Rohde & Schwarz ZVA67 VNA in a frequency step-size of 1 MHz at an IF bandwidth of 100 Hz for all relevant frequency ranges. A Rosenberger precision 3.5 mm male/female calibration kit in combination with the unknown-thru calibration and correction procedure was used for full two-port S-parameter error correction and to account for the insertable mixed SMA connector gender (VNA port 1 male, port 2 female) of the attenuators with a correct position of the S-parameter reference plane. The acquired complex linear S-parameters (\(S_{21}^{I}\) and \(S_{12}^{II}\)) of all devices are shown in Figure 18.  
This setup used here illustrates that an HR can also be used as a vectorial measuring network analyzer, whereby the measured values contain the classical transfer function \(S_{21}\), which in practice also corresponds to the reverse transfer function \(S_{12}\) due to reciprocity, several times and in two frequency ranges. For this coaxial verification measurement scenario, the following assumption is applied to the transfer functions of the reference measurements:  
\[S_{21,\text{Ref}}^{I}=\left[ \begin{array}{cc}0&1\\ 1&0\end{array} \right]\qquad\text{and}\qquad S_{12,\text{Ref}}^{II}=\left[ \begin{array}{cc}0&1\\ 1&0\end{array} \right]\quad, \tag{14}\]  
which leads to the position of the normalization reference plane for \(S_{11}^{II,I}\) as shown in Figures 16 and 17. The nonlinear one-port element is assumed to obey the memory-less square law transfer function as explained during the derivation of Equation (11) and to show an unknown but repeatable harmonic conversion \(S_{43}^{II,I}\) performance. This unknown offset is removed by storing the initial measurement of the nonlinear element without a device inserted into the reference plane and normalizing the subsequent measurements performed with devices-under-test (DUT) to this result (nonlinear reflection normalization). It is important to note that, in contrast to the freespace formulation of Equation (11), these devices here are completely inserted into the position of the reference plane; therefore, we can replace the designation MUTz with the designation MUT in Equation (11), as no part of the setup present under the reference measurement conditions is replaced for the DUT measurements. Therefore, the original Equation (11) can be rewritten in the form of  
\[S_{11,\text{Norm.}}^{II,I}=(S_{21,\text{DUT}}^{I})^{2}\cdot S_{12,\text{DUT}}^ {II}\quad, \tag{15}\]  
which is the equivalent form for the insertable coaxial case of the original free-space reference measurement formulation of this equation and is used in the following to verify the validity of the original assumption. Both equations share the same the rough modeling of the doubler. In order to carry out this verification, all four attenuators were measured with the best accuracy available to the authors with a correct definition of the connector insertion plane using an VNA. The results are shown in Figure 18. An indication for the high measurement accuracy are the very small deviations for the magnitude values, which are less than 0.01 dB in the band up to 2940 MHz and less than 0.02 dB in the band from 5800 MHz. These linear S-parameter values were then used to calculate the expected results for \(S_{11,\text{Norm. }}^{II,I}\) using Equation (15) for all four devices. Then, the HR measurement using the setup shown in Figures 16 and 17 were carried out. The system was allowed to reach thermal equilibrium for a duration of 2 h before the measurements took place. In order to gain some statistical significance of the results, four distinct measurement runs containing all the devices were conducted sequentially.