# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 1 Introduction
In the case of non-time-invariant transmission channels, for example if the distance between the reader and transponder varies, in the case of temporary shadowing,  
Figure 1: Schematic of the operation of a transponder system using passive resonant sensors: A reader sends a read signal (blue arrow to the right) to the sensor node over a wireless channel. This signal is received there and then stored in an oscillation in the resonator. When the read signal is switched off, the oscillation resonates with an exponentially decreasing amplitude at a frequency, which is modified by the quantity to be measured. A part of the stored energy is sent back to the reader as a backscatter signal (blue arrow to the left). There, it is received, sampled and evaluated. or if stochastic interference occurs, more complex methods, as described in [29; 30], must be combined with the algorithms presented here. [PERSON] and [PERSON] were the first to calculate the theoretical lower bounds, so-called Cramer-Rao lower bounds, for the parameters of a sampled exponential decaying complex phasor in noise based on a maximum likelihood estimation [31]. Based on their results, this manuscript calculates a theoretical lower limit of the measurement accuracy of a wireless readout of a passive transponder system using resonant sensor nodes. There is an increasing number of publications on extracting the resonance frequency and quality factor of a noisy complex exponential function with exponentially decreasing amplitude. Examples are a numerical fit of Lorentz curves to the measurement data [9; 32], a frequency estimation based on linear prediction and singular value decomposition [33; 34], or super-resolution based methods [35]. An overview and performance comparison of some concepts can be found in [27; 36; 37]. Using \(\sin(x)/x\) interpolation, analytic solutions for some frequency estimators were presented [38]. What is still missing, however, is an analysis that allows to predict the theoretically possible measurement accuracy based solely on the physical and electrical parameters of the measuring system. By comparing this theoretical measurement accuracy with the actual one, the performance of the measurement system can be assessed qualitatively and quantitatively. In addition, it can be predicted which measurement accuracy can be expected at which distance. [PERSON] presented an iterative, high-performance algorithm which approximates the points around the maximum of the measured spectrum by a Lorentz curve, thereby reaching an accuracy of the Cramer-Rao lower bounds [39; 40]. Using this algorithm, several examples of the achievable accuracy are shown for wireless readout of resonators of different quality factors in typical ISM (Industrial, Scientific, Medical, ISM) [41] and SRD (Short-Range Device, SRD) [42] bands as a function of read distance or signal-to-noise ratio. The manuscript begins by presenting a mathematical model of a noisy discrete-time complex signal with exponentially decaying amplitude. When reading a passive resonator remotely, the initial signal-to-noise and interference ratio results from the radar parameters' transmission power, gain of the transducer to the transmission medium, reading distance and stimulation duration as well as from the 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 are then formulated. An iterative numerical algorithm for determining these parameters is also presented, and its performance is compared with the theoretical lower bounds. All visualizations of the formulas and the numerical simulations were carried out using MATLAB, Version 9.13.0 [43].