# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 1 Introduction  
Sensors are essential parts of any control system. Wireless sensors, called transponders, are used at moving or rotating control points, on or in living beings, and in situations where wiring would be expensive. Passive transponders are used in applications where traditional battery, RFID or energy harvesting-based radio technology cannot be operated or only at high operating costs. Chipless passive transponders require neither a battery nor an integrated circuit. Such a transponder system consists of two parts, a reading unit and passive transponders that act as cooperative targets. Both parts are connected by transducers via a wireless connection, for example a radio, inductive, capacitive or ultrasonic connection, as shown in Figure 1. To stimulate the passive transponders, the reader sends a read signal to the transponder nodes via the communication channel. Three types of chipless passive transponders are described in the literature: delay lines, resonators and mixer types [1]. This contribution focuses on resonators with a high quality factor \(Q\), where the quantity being measured changes the resonant frequency of the resonator. The read signal stimulates an oscillation in the resonator, which decays exponentially after the read signal is switched off. A part of the energy stored in the oscillation is sent back to the reader as a backscatter signal, where it is received, sampled and evaluated. The resonant frequency can be extracted if the decay time of the oscillation is long enough to separate it in time domain from the read signal and some ambient echoes in the wireless transmission channel. Mixer types sometimes mix the read frequency down to an intermediate frequency, which then also excites a resonator [2; 3]. So far, ceramic dielectric resonators [2; 4], bulk [5; 6] and surface acoustic wave resonators (SAWs) [1; 7; 8; 9; 10; 11], HF cavity resonators [12; 13], LC resonators [14; 15; 16], coplanar resonators [4; 17], spiral resonators [18], and air-filled substrate-integrated waveguide resonators [19] are presented in the scientific literature for wireless temperature [6; 11], pressure [1], blood pressure [20], torque [7; 21], strain [4; 10; 12; 19], transient load [9], vibration [22], mass flow [4], corrosion of steel [23], pH value [16], and food quality [24] measurements together with other physical parameters. The review [25] provides a good overview of the current state of research, development and applications of wireless resonance sensors and [26; 27] of the subset of passive LC wireless sensors. Due to the low complexity of chipless passive transponders, which enable operation without batteries or electronic circuits, the instrumentation technology is assessed as maintenance-free, robust and suitable for use in harsh environments. Systems using chipless passive transponders usually surpass the read distance of RFID-based transponder systems by far, since they are linear time-invariant systems where the read distance is limited only by the receiver noise, which blocks detection of the response signal from distances beyond the maximum read distance. There are three types of error sources: systematic errors, statistical errors and unknown systematic errors [28]. This manuscript addresses the evaluation of noisy exponentially decaying complex exponential functions, where the noise is assumed to be Gaussian distributed and distortion-free. In addition, linear, time-invariant transmission channels without interference from other transmitters as well as linear and time-invariant transponders with constant natural frequencies are assumed. Due to the linear, time-invariant characteristics of the transducer and the resonator, a capacitive or inductive load on the transducer can pull the natural frequency of the resonator and thus distort the measurement result [27].