# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 5 Summary and Outlook  
The accuracy of a remote readout of a passive resonator depends sensitively on the initial signal-to-noise and interference ratios as well as the decay constant of the signal. This manuscript is limited to linear, time-invariant transmission channels without interference from other transmitters, linear, time-invariant transponders with constant resonance frequency, and Gaussian distributed, bias-free measurement uncertainties. For such a system, we showed the dependence of the initial signal-to-noise and interference ratio on the radar parameters' transmission power, gain of the transducer to the transmission medium, reading distance and stimulation duration as well as on the passive transponder parameters' resonance frequency, quality factor and impedance matching of the resonator to the transducer. The equations for the maximum likelihood estimation of the parameters of the exponentially decaying discrete-time signal and their Cramer-Rao lower bounds were presented. A stimulation and sampling time of approximately \(Q/f_{d}\) for a single remote reading of the passive resonator promises the highest measurement resolution when subsequently averaged over multiple measurements. An iterative numerical algorithm enables an accuracy in determining the resonator parameters' natural frequency and decay constant down to the Cramer-Rao limit. With these parameters, the systematic interference caused by echoes in the transmission medium could be determined. The algorithm described in this manuscript allows to minimize the statistical errors in a wireless reading of a passive resonant transponder. Systematic errors, such as a pull in the resonance frequency due to external capacitive or inductive loads on the transducer, interference caused by stochastic jamming, shadowing or crosstalk in the reader, and time-varying transmission channels due to movements or rotations must be evaluated using more advanced algorithms [27; 29; 30]. The presented analysis allows a comparison between the theoretically possible measurement accuracy and an actual one of any experimental or commercial system based solely on the physical and electrical parameters of the measurement system. This comparison allows the qualitative and quantitative assessment of the performance of the system and  
Figure 21: Sum of the complex values of 50 measurement signals where the estimated values of the decaying resonator have been subtracted; on the left in linear scale and on the right in logarithmic scale. The systematic interference signals disappear into the noise at \(t=-15\), and the signal evaluation could be started earlier by this interval, resulting in a higher resolution. shows any potential for optimization. It also allows the prediction of the measurement accuracy which can be expected at a given distance or what minimum resonator quality is required for a given measurement distance and a given measurement accuracy. Conceptualization, L.M.R., T.A., T.S., T.O., W.L. and S.J.R. ; Methodology, L.M.R., T.A., T.O. and S.J.R. ; Software, L.M.R. ; Validation, L.M.R. ; Formal analysis, L.M.R. and T.A. ; Investigation, L.M.R., T.A. and T.O. ; Resources, L.M.R., W.L. and S.J.R. ; Data curation, L.M.R. ; Writing--original draft, L.M.R. ; Writing--review & editing, L.M.R., T.S., T.O. and W.L. ; Visualization, L.M.R., T.A. and T.S. ; Supervision, L.M.R. ; Project administration, L.M.R. ; Funding acquisition, L.M.R. and S.J.R. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data presented in this study and the corresponding MATLAB codes are available on request from the corresponding author. The authors thank the re-viewers and staff of MDPI for their valuable advice on improving the manuscript. Author [PERSON] is employed by the RSSI GmbH and offers various commercial products for wireless interrogation of SAW sensors. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships.