# First SIMO Harmonic Radar Based on the SFCW Concept and the HR Transfer Function
## 2 The HR Equation and HR Transfer Function
\(P_{RXT}\) can be calculated by combining the normal one-way path loss formula to calculate the illumination field strength at the slant range \(R\) and the effective antenna area of the fundamental frequency receiving antenna of the tag in the form of the HR radar equation  
\[P_{RXT}=\underbrace{\frac{P_{TX}\cdot G_{TX}}{4\pi\cdot R^{2}}}_{\text{Illumination power}}\quad\underbrace{\frac{G_{RXT}\cdot\lambda_{1}^{2}}{4\pi}}_{\text{area of the tag}\,\text{of}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  
The phase component \(\angle(P_{RX,2})\) of the measured harmonic return signal phasor can be determined by following the progression of the signal phase through all system components, which is shown extensively in [19; 24]. In this context, it is only important to know that the received harmonic signal phase can be calculated using  
\[\angle(P_{RX,2})=\frac{e^{2j(\omega^{I}t+\varphi_{0})}}{\text{Freq. doubled}}\cdot\underbrace{e^{-j\hbar\omega^{I}v_{p}^{-1}\cdot R}}_{ \text{Slant range}}\, \tag{5}\]  
with \(\omega^{I}\) denoting the illumination signal's angular frequency, \(\varphi_{0}\) describing an arbitrary but repeatable phase offset, the phase propagation velocity \(v_{p}=c_{0}(\sqrt{\varepsilon_{I}\mu_{r}})^{-1}\) through the dispersion-less medium, and \(R\) describing the mechanical slant range covered during propagation, when a memory-less power series model for the frequency doubler is assumed. It is important to note for this application that the frequency doubling process results in an increased sensitivity of the propagation phase with regard to the propagation medium or material when compared with fundamental frequency SFCW radar. A similar observation can be made for the loss contributions of the material under test due to the square-law dependence of the return magnitude upon the received illumination signal power by the tag. Both observations combined lead to a greater sensitivity of such a system for small dielectric and loss variations in the material or object under test. In reality, nonlinear frequency doublers are, however, far from ideal components and must be designed and optimized for a specific set of conditions relating to the measurement setup at hand. A longer discussion about the design of these components in the context of HR can be found in [19]. For the scope of this article, it is sufficient to note that the use of HR leads to additional boundary conditions for the system, which are imposed by the transmission through a lossy medium between the HR interrogator and the tag. These conditions must be met for the validity of Equations (1) and (5). Ideally, the HR tag is operated in its well-defined square law transfer function region without saturation by the illumination signal, while still providing enough harmonic output power to obtain a good SNR at the receiver. In order to describe the actual frequency converting nonlinear measurement in the HR setup, it is useful to describe the transfer functions using mixed frequency scattering parameters (MF-parameters), which are much easier to employ and understand than X-parameters in this context and are described in-depth in the text book [25]. In keeping in-line with the definition of normal linear S-parameters, the measured frequency converting reflection parameter \(S_{11,M}\) measured by the system shown in Figure 1 from the stimulus frequency \(I\) to the harmonic response frequency \(II\) can be written as \(S_{11,M}^{II,I}\), which relates the received harmonic complex power wave \(b_{1}^{II}\) at point D in Figure 1 to the transmitted fundamental frequency power wave \(a_{1}^{I}\) at point A as measured by a VNA, formally described as  
\[S_{11,M}^{II,I}=\frac{b_{1}^{II}}{a_{1}^{I}}. \tag{6}\]  
When this measurement is performed without a material or object under test, as shown in Figure 1, the obtained result is the so-called reference measurement \(S_{11,\text{Ref}}^{II,I}\) and must be performed and stored for each HR tag element.