# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 3 Cramer-Rao Lower Bounds for Exponentially Decaying Phasors
### Relative Frequency Accuracy by a Wireless Readout of a High-Q Resonator  
If we substitute the signal-to-noise ratio in Equations (19) and (13) into Equation (36) or (38), we obtain the Cramer-Rao lower bound for the natural frequency.  
Figure 7: **Left chart**: reduction in the residual \(r(\alpha NT)\) according to Equation (39), **right chart**: increase in the resulting measurement resolution, both graphs as a function of \(\alpha NT\). A resonator signal with a quality factor of 3 and a sampling of 2 samples per oscillation was used.  
\[\sigma_{\omega_{d}}{}^{2}=\sigma_{\alpha}{}^{2} \geq \frac{\left(1-e^{-2\alpha T}\right)^{3}}{2T^{2}e^{-2\alpha T}}\cdot \frac{1}{\eta}\cdot r(\alpha NT),\text{ with} \tag{43}\] \[\frac{1}{\eta} = \frac{F_{LNA}k_{\text{B}}8f_{s}}{P_{t}\cdot g_{t}{}^{2}\cdot g_{t} {}^{2}\cdot e^{-2\alpha t_{gate}}\cdot e^{-2\alpha_{m}R}{H_{R}}^{2}(\omega) \cdot\left(1-e^{(j\omega_{d}-j\omega-\alpha)t_{\text{load}}}\right)^{2}}\cdot \left(\frac{4\pi R}{\lambda}\right)^{4} \tag{44}\]  
and \(r(\alpha NT)\) given in Equation (39). The relative frequency error \(\sigma_{f_{0}}/f_{0}\) results from the above equations to  
\[\frac{\sigma_{f_{0}}}{f_{0}}=\frac{1}{4\pi^{2}f_{0}}\cdot\sigma_{\omega_{d}} \tag{45}\]  
Figure 8 shows the Cramer-Rao lower bound for the relative frequency error \(\sigma_{f_{0}}/f_{0}\) that can be achieved in a single measurement for some selected ISM and SRD bands. For the left diagram, the maximum bandwidth in the frequency bands was used; in the right picture, a measurement bandwidth of 0.5%/\(f_{0}\) was used. Resonators with quality factors of 10, 100 and 1000 were used in each frequency range.  
The bandwidths of the resonators with a quality factor of 10 were wider than the permitted frequency bands and, therefore, these resonators were not taken into account. The frequency bands 433 MHz and 868 MHz were also too narrow for the resonators with a quality factor of 100. With a measurement bandwidth of 0.5%/\(f_{0}\), only the resonators with quality factor of 1000 could be read.  
To analyze the influence of the quality factor on the lower Cramer-Rao bound for the relative frequency error \(\sigma_{f_{0}}/f_{0}\), a sweep on \(Q\) in Equation (45) was performed. Figure 9 shows the reduction in relative frequency error for one single readout cycle with increasing resonator quality factor. A readout distance \(R=1\lambda\) was assumed for the illustration. The displayed \(\sigma_{f_{0}}/f_{0}\) must be multiplied by \(\left(R/\lambda\right)^{2}\) in order to scale to a different read distance \(R\). The diagrams start with the quality factor at which the resonance bandwidth fits into the ISM or SRD band. For a quality factor where the resonator bandwidth became smaller than one fifth of the assigned bandwidth, the sampled bandwidth was fixed at five times the resonator bandwidth.  
Figure 8: Cramer–Rao lower bound for the relative frequency error \(\frac{\sigma_{f_{0}}}{f_{0}}\) as a function of the readout distance. The same frequency ranges, resonator quality factors and noise bandwidth as in Figure 6 were used in the left diagram, in the right graph, constant relative bandwidths of 0.5% of the carrier frequency are assumed. All other parameters correspond to those in Figure 3.  
Equations (38) and (39) can be further simplified if \(Q\) oscillations of the RF signal or \(Q\cdot B/f_{0}\) oscillations in the base band are sampled. In this case, we obtain \(\alpha NT=\pi\) and \(r(\alpha NT)\approx 1\):  
\[{\sigma_{\omega_{d}}}^{2}\geq\frac{\left(1-e^{-2\alpha T}\right)^{3}}{2T^{2}e^{ -2\alpha T}}\cdot\frac{1}{\eta} \tag{46}\]  
Above approximation is shown by dotted lines in Figure 9.