# 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
### Sampling  
The results of the Cramer-Rao lower bound calculations depend on an agreement on the starting time of the samples, as this changes the phases. There are two special time frames: either the samples are taken symmetrical to \(t=0\), or sampling starts at \(t=0\). The second choice seems to best fit for an exponentially decaying signal. Therefore, we shift the time frame to start sampling at \(t=0\). The analogue mixed down signal \(s_{md}(t)\) is then given by  
\[s_{md}(t)=\sqrt{P_{gatal}}\cdot e^{(-\alpha+\beta u_{d}-\mu_{i})t+\beta v_{3}}, \tag{20}\]  
with \(P_{gatal}\) given in Equation (13). Downconversion reduces the number of oscillations in the signal, but \(\alpha\) is preserved.  
If we neglect the phase shift due to the interference between the forced oscillation and the natural oscillation in the resonator, the phase shift \(\varphi_{3}\) of the mixed down analog signal compared to the local oscillator of the reader is given by  
\[\varphi_{3}=(\omega+\omega_{d})\frac{R}{c}+(\omega-\omega_{d})t_{gate}. \tag{21}\]  
Since \(R\) and \(\omega_{d}\) are unknown, \(\varphi_{3}\) is also unknown. In order to simplify the following formulas a bit, we introduce the following abbreviations  
\[\tilde{b}=\sqrt{P_{gatal}},\qquad\qquad\tilde{\omega}=(\omega_{d}-\omega), \qquad\qquad\tilde{\varphi}=\varphi_{3}. \tag{22}\]  
Figure 6: Remaining signal-to-noise ratio \(\eta\) in the reader after the time gate at the start of the sampling. The same frequency ranges and resonator qualities were used as in Figure 5. The noise bandwidths in the left graph were selected according to the regulations for the ISM [41] and SRD [42] bands, to 1.75 MHz for the 433 MHz band (red line), 2 MHz for the 868 MHz band (green line), 26 MHz for the 915 MHz band (blue line), 100 MHz for the 2.45 GHz band (magenta line) and 150 MHz for the 5.8 GHz band (black line). The resonator quality factors used were 1000 (solid line), 100 (dashed line) and 10 (dotted line). In the right graph, constant relative bandwidths of 0.5% of the carrier frequency are assumed. All other parameters correspond to those in Figure 3.  
The exponentially decaying signal then can be written as  
\[s_{md}(t)=\tilde{b}e^{(-\alpha+\hat{\omega})t+\hat{\varphi}\hat{\varphi}}. \tag{23}\]  
This signal is completely described by four variables that can be combined into a state vector \(\chi\), which is given by  
\[\chi=\left(\begin{array}{c}\hat{\omega}\\ \tilde{\varphi}\\ \alpha\\ \tilde{b}\end{array}\right). \tag{24}\]  
Two variables, \(\hat{\omega}\) and \(\tilde{\varphi}\), describe the phase and two, \(\alpha\) and \(\tilde{b}\), the amplitude of the decaying phasor, with one variable each characterizing the change over time (\(\hat{\omega}\) and \(\alpha\)) and one variable (\(\tilde{\varphi}\) and \(\tilde{b}\)) the initial state. The sampling delivers the discrete-time signal \(s_{n}\), where \(n\) is the index of an element.  
\[s_{n}=s_{md}(t_{n})=\tilde{b}e^{(-\alpha+\hat{\omega})t_{n}+\hat{\varphi}\hat{ \varphi}}=\tilde{b}e^{(-\alpha+\hat{\omega})t_{n}T+\hat{\varphi}\hat{\varphi}}, \ \ \text{for}\ \ n=0,1\ldots N-1. \tag{25}\]  
In addition to \(s_{n}\), the sampled complex signal \(z_{n}\) contains noise \(n_{n}\). The average noise power is given by Equation (15). The sampled common signal \(z_{n}\) is given by  
\[z_{n}=s_{n}+n_{n}. \tag{26}\]