# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 3 Cramer-Rao Lower Bounds for Exponentially Decaying Phasors
### Limit Value for the Frequency Resolution \(\sigma_{\dot{\omega}}\)  
Sampled signals with a finite spectrum that satisfy the Nyquist criterion are completely determined, and the parameters of the signal, such as the carrier frequency, can therefore be estimated without errors. If a noise component is added to the signal, the signal can no longer be separated from the noise even if the Nyquist criterion is met and the carrier frequency can only be estimated with a variance that depends on the signal-to-noise ratio. If the noise is Gaussian distributed and bias-free, the minimum variance approaches zero as the number of sample points approaches infinity. In this case, the signal can be completely separated from the noise component. However, with sampled, exponentially decaying phasors, the measurement accuracy approaches a finite limit value even with an infinitenumber of sampling points, since the useful signals disappear into the noise after a few time constants of the decay.  
\[\lim_{N\rightarrow\infty}r(\alpha NT) = \frac{e^{-2\alpha T}}{e^{-2\alpha T}}=1 \tag{40}\] \[\lim_{N\rightarrow\infty}{\sigma_{\omega_{d}}}^{2}=\lim_{N \rightarrow\infty}{\sigma_{\omega}}^{2} \geq \frac{\left(1-e^{-2\alpha T}\right)^{3}}{2T^{2}e^{-2\alpha T}\eta}\] (41) \[\lim_{N\rightarrow\infty}\frac{{\sigma_{b}}^{2}}{\tilde{b}^{2}}= \lim_{N\rightarrow\infty}{\sigma_{\tilde{\varphi}}}^{2} \geq \frac{1-e^{-2\alpha T}}{2\eta}\bigg{[}1+\frac{e^{-4\alpha T}}{e^{ -2\alpha T}}\bigg{]}=\frac{1-e^{-4\alpha T}}{2\eta} \tag{42}\]  
If the measurement cycles of excitation of the resonator, scanning and evaluation of the decaying resonator signal are repeated several times, a measurement accuracy below any specified lower limit can still be achieved by averaging the results.  
The residual \(r(\alpha NT)\) very quickly approaches the limit value 1, as Figure 7 shows. As a result, the variance \(\sigma_{\omega_{d}}^{2}\) reaches its final value when the received signal is sampled along \(Q\) oscillations. In this case, \(\alpha NT=\pi\) and \(r(\pi)\approx 1\).  
Figure 7 shows that the limit value specified in Equation (40) is nearly reached when \(\alpha NT=\pi\), i.e., after sampling of Q oscillations. \(\sigma_{\omega_{d}}^{2}\) increases significantly, or the resolution decreases significantly, when fewer than Q oscillations are sampled. However, a longer sampling does not increase the resolution but rather costs time that could possibly be used for another measurement cycle and subsequent averaging. In addition, it increases the noise energy in the signal, while the energy of the useful signal remains almost constant. There are two ways to solve this problem: either the signal is truncated after the time \(0.85\ldots 1.0\cdot Q/f_{d}\) or the signal is weighted with truncation as a special case of weighting.  
An analysis in the Appendix C shows that an excitation and sampling time of \(0.85\ldots 0.9\cdot Q/f_{d}\) with a subsequent restart of the measurement and an averaging of the measurement results promises the lowest variance \(\sigma_{\omega_{d}}^{2}\). The literature gives also \(0.9\cdot Q/f_{d}\) as the optimal length between improving resolution and reducing the impact of noise for a single signal to be analyzed [39]. However, the minimum toward longer stimulation and sampling times is very flat; the reduction in variance for a sampling time of \(0.85\cdot Q/f_{d}\) is less than 1% compared to stimulation and sampling of \(1.0\cdot Q/f_{d}\). After complete loading and sampling, the oscillation can reduce the ripple in the spectrum that would be caused by a hard on/off cycle.