# 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  
Various radar techniques for readers for passive chipless transponders have been presented in the literature such as time domain sampling, frequency domain sampling and hybrids [45; 46; 47; 48]. The highest signal-to-noise ratio and thus the highest measurement accuracy of the frequency is obtained by the reading device sending out a CW signal with a carrier frequency \(\omega=2\pi f\) very close to the natural frequency of the resonator as the read signal and sampling the decaying resonator signal after switching off the stimulation. The reader transmits this signal with a power \(P_{t}(t)\) which is sent via a transducer, e.g., an antenna with the gain \(g_{t}\) over a wireless channel. Electromagnetic or acoustic channels have been used as wireless channels for the far-field transmission of the read and the backscattered decaying signal [5; 6]. The transponder receives a fraction \(P_{r}(t)\) of the emitted signal at a distance \(R>\lambda\), where \(\lambda\) is the wavelength in the transmission channel. If the transponder's transducer gain is given by \(g_{r}\), then \(P_{r}\) is ideally given by the Friis equation [44]  
\[P_{r}=P_{t}\cdot g_{t}\cdot g_{r}\cdot\bigg{(}\frac{\lambda}{4\pi R}\bigg{)}^{ 2}. \tag{1}\]  
An additional propagation loss \(L_{a}\) might be caused by absorption of the wave in the media [9]. The absorption per unit distance is given by an absorption coefficient of the media \(\alpha_{m}\). The received power \(P_{r}\) is then given by  
\[P_{r}=P_{t}\cdot g_{t}\cdot g_{r}\cdot e^{-\alpha_{m}R}\bigg{(}\frac{\lambda}{ 4\pi R}\bigg{)}^{2}. \tag{2}\]  
The transmission of signals between the reader and the passive transponder does not correspond to free space propagation if the signals at the receiver are changed by superimposing signal components reflected in the transmission channel. If the quality of the resonator is significantly higher than the quality of the transmission channel and neither the mutual arrangement nor the distance between reader and transponder changes during communication, a constant effective distance can be specified with which the received power can be determined according to Equations (1) or (2). The effective distance sometimes differs from the geometric distance. The signal \(s_{load}(t)\) at the feed point of the transponder antenna can be written through  
\[s_{load}(t)=\sqrt{P_{r}}\cdot e^{i(\alpha t+\varphi_{0})}, \tag{3}\]  
where the phase shift \(\varphi_{0}\) is given by the delay caused by the transmission between the reader and transponder. \(H_{R}(\omega)\) describes the response of the resonator loaded with the impedance of the transducer by a signal at the frequency \(\omega\). The total received power \(P_{r}\) is absorbed by the resonator when the source resistance of the transducer is matched to the impedance of the resonator at resonance and when the carrier frequency of the read signal is close to the natural frequency of the resonator. The resonator then begins to oscillate in a forced oscillation. Incomplete matching of the antenna to the resonator reduces the power transfer twice, during stimulation from the antenna into the resonator and during decay from the resonator to the antenna, \(H_{R}(\omega)\) thereby is reduced by the mismatch loss. The damping constant \(\alpha\) of free oscillation lengthens slightly in this case because the resonator is operated somewhere between open and loaded.