# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 2 Signal-to-Noise and Interference Ratio by a High-Q Resonator as Cooperative Target
### Signal Power
The vibration characteristics of the resonator is described by the natural angular frequency of the undamped oscillation \(\omega_{0}=2\pi f_{0}\) and the loaded quality factor \(Q\), which gives the damping constant \(\alpha\) of free oscillation  
\[\alpha=\frac{\omega_{0}}{2Q}=\frac{\pi f_{0}}{Q}. \tag{4}\]  
The natural angular frequency of the damped oscillation \(\omega_{d}=2\pi f_{d}\) is given by  
\[\omega_{d}=\sqrt{{\omega_{0}}^{2}-\alpha^{2}}. \tag{5}\]  
After the load time \(t_{load}\), the stimulation signal is switched off, and the oscillation will decay at its damped natural angular frequency \(\omega_{d}\) with the signal \(s_{decay}(t)\), which will start at the power \(P_{decay}\)[49]  
\[s_{decay}(t)=\sqrt{P_{decay}}\cdot e^{(-\alpha+i\omega_{d})(t-t_{load})+i \varphi_{1}}. \tag{6}\]  
The accumulated phase shift \(\varphi_{1}\) and the power \(P_{decay}\) at the start are given by  
\[\varphi_{1} = \varphi_{0}+\omega t_{load} \tag{7}\] \[P_{decay} = P_{r}{H_{R}}^{2}(\omega)\Big{(}1-e^{(i\omega_{d}-i\omega-\alpha )t_{load}}\Big{)}^{2}. \tag{8}\]  
In Figure 2, the dashed lines in blue show the function of time during which the stimulation signal is provided by the antenna due to the picked-up read signal and in a solid black line the oscillation which is built up in the resonator. The amplitude of the excitation signal is normalized to one in the image. The frequency of the excitation and the decaying resonator is plotted in \(f/f_{0}\) and the time in \(t\cdot f_{0}\). The stimulation signal stops at \(t_{load}\) and the oscillation decays. The signal of the decaying resonator is divided between the source impedance of the antenna and the internal resistance of the resonator. Figure 2: Schematic representation of the stimulation signal \(s_{load}(t)\) delivered from the antenna to the resonator and the stored oscillation \(s_{decay}\) in the resonator. The amplitude and frequency of the excitation signal were normalized to one, which is the time to \(t\cdot f_{0}\). The resonator in this example has a quality factor of 10. Shown is the real part of \(s_{load}(t)\) and \(s_{decay}\). The stimulation signal stops at \(t_{load}\), and the oscillation decays. Half of the signal from the decaying resonator is applied to the source impedance of the antenna, the other half to the internal resistance of the resonator. The figure is taken from [49]. Near resonance, i.e., \(\omega\approx\omega_{d}\), the term \(\left(1-e^{(i\omega_{d}-i\omega-\alpha)t_{load}}\right)\) increases for \(\alpha t_{load}<1\) linear with \(t_{load}\) and approaches 1 for \(\alpha t_{load}>\pi\). The decaying oscillation in Equation (6) supplies the response signal, whereby, if matched (\(H_{R}(\omega)=1\)), half of the power is dissipated in the internal resistor of the resonator and half is coupled to the transducer and sent out as a backscattered wave with power \(P_{backscattered}\) to the reading device:  
\[P_{backscattered}=\frac{1}{2}\cdot P_{decay} \tag{9}\]  
This signal again propagates through the media back to the reader. The reader, at the distance \(R\), picks up a fraction \(s_{in}(t)\) of the backscattered signal, which will start at the power \(P_{in}\)  
\[P_{r}=P_{t}\cdot g_{t}\cdot g_{r}\cdot e^{-\alpha_{m}R}\bigg{(}\frac{\lambda}{ 4\pi R}\bigg{)}^{2}. \tag{2}\]0 \[P_{r}=P_{t}\cdot g_{t}\cdot g_{r}\cdot e^{-\alpha_{m}R}\bigg{(}\frac{\lambda}{ 4\pi R}\bigg{)}^{2}. \tag{2}\]1 (11) \[P_{r}=P_{t}\cdot g_{t}\cdot g_{r}\cdot e^{-\alpha_{m}R}\bigg{(}\frac{\lambda}{ 4\pi R}\bigg{)}^{2}. \tag{2}\]2 \[P_{r}=P_{t}\cdot g_{t}\cdot g_{r}\cdot e^{-\alpha_{m}R}\bigg{(}\frac{\lambda}{ 4\pi R}\bigg{)}^{2}. \tag{2}\]3  
Figure 3 shows the initial received power \(P_{in}\) at the end of stimulation in the reader for ISM [41] and SRD [42] bands that are often used to read passive resonant transponders. The initial received powers are independent of the quality factor \(Q\) of the resonator.