# 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
### Interference Due to Environmental Echoes  
After switching off the read signal, the reader receives the signals not only from the decaying oscillation of the resonators but also from a variety of other echoes from the environment. A wireless measurement of the natural frequency of a resonant passive transponder is only possible if the decaying signal by the resonator lasts significantly longer than all ambient echoes picked up by the reader. Transducers with high directivity help to suppress ambient echoes, but they require precise alignment and limit the read area. The signals of the decaying oscillation of the resonator can be separated in the reader if the quality factor of the resonator is significantly higher than the equivalent quality factor of the power delay profile of the transmission channel. Figure 4 shows schematically an input signal in the reader after the end of the stimulation signal as a function of time. The left diagram shows the real part of the sampled values and the diagram to the right shows both the real and imaginary part. For signal processing, only sampled values according to the sampling theorem are relevant, i.e., two samples per oscillation. For a better overview, 10 samples per oscillation are shown here and they are also connected. The first samples are dominated by stochastic ambient echoes, which, however, soon fade away. As the process progresses, the signals from the decaying resonator stand out from the noise. After about 10 oscillations, these usable signals disappear into the noise. The quality factor of the oscillator in this schematic drawing is 10 and the signal-to-noise ratio at t = 0 is 20 dB. Ambient echoes would cause a systematic error in the evaluation of the resonance frequency, which cannot be eliminated later. Also, averaging would not lower this systematic error. Therefore, the ambient echoes must be eliminated by a time gate \(t_{gate}\) before the resonator signal is analyzed by the reader unit. Choosing this time gate is relevant. If the time gate is set too short, systematic measurement errors result. If it is set too long, only an unnecessarily decayed signal is available for evaluation and the measurement accuracy suffers. In some applications, the  
Figure 4: Schematic representation of the complex input signals of the reader, real part on the left and real and imaginary parts on the right, both as a function of time. Frequency and time are scaled as in Figure 2. Initially, the ambient echoes are larger than the resonator’s decaying signal, but they have a smaller quality factor and soon disappear. In the next phase, the resonator signals dominate, which later also disappear into the noise. The resonator in this schematic has a quality factor of 10.  
power delay profile of the transmission channel can be measured before deploying the passive transponder. A method for adjusting the time window is suggested in Section 4.4. Due to the exponentially decaying oscillation, the time gate results in a further attenuation of the initial power \(P_{in}\) that can be analyzed in the reading device to \(P_{gated}\) with  
\[P_{gated}=\frac{1}{2}\cdot P_{t}\cdot{g_{t}}^{2}\cdot{g_{r}}^{2}\cdot e^{-2 \alpha t_{gate}}\cdot e^{-2\alpha_{m}R}\cdot\left(\frac{\lambda}{4\pi R}\right) ^{4}\cdot{H_{R}}^{2}(\omega)\cdot\left(1-e^{(i\omega_{d}-i\omega-\alpha)t_{ load}}\right)^{2}. \tag{13}\]  
It describes the link budget when reading a cooperative resonant passive target by a radar system. The term \(\left|1-e^{(i\omega_{d}-i\omega-\alpha)t_{load}}\right|^{2}\) describes the state of the loading of the resonator after the loading time \(t_{load}\); \({H_{R}}^{2}\) represents the results of both the matching of the readout frequency \(\omega\) to the natural frequency \(\omega_{d}\) and the electrical matching of the impedance of the transducer to the resonator.