# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 4 Calculating the Natural Frequency of a Decaying Oscillator
### Weighting for Improving the Signal-to-Noise Ratio  
The algorithm works correctly even with low-quality factors, as long as the maximum in the frequency domain is dominated by the signal of the decaying resonator. If noise, interference, and aliasing signals become too strong, a noise peak might overpower the resonant signal and, therefore, lead to incorrect results. One way to maximize the signal to noise energy is to cut off the sampled signal at \(Q/f_{d}\) or to stop sampling there. The amplitude of the spectrum exactly at resonance frequency is (see Equation (A52))  
\[S(f_{0})=\sum_{n=0}^{N-1}s_{n}=\bar{b}e^{i\varphi}\frac{1-e^{-aNT}}{1-e^{-aT}}. \tag{54}\]  
The exponential function in the numerator is very small and so is the exponent in the denominator. The amplitude at resonance frequency can therefore be approximated as  
\[|S(f_{0})|\approx\frac{\bar{b}}{aT}=\frac{\bar{b}Q}{\pi f_{0}T}=\frac{\bar{b}Q }{2\pi}. \tag{55}\]  
Figure 15 shows the detection probability of the algorithm as a function of the signal-to-noise ratio with 5000 simulations each. A correct detection was assumed if the calculated natural frequency was within an interval of \(\pm 6\sigma\) around the actual natural frequency. The left chart is calculated for a resonator quality factor of 10, the middle for 100 and the right for 1000. Figure 14: Cumulative density functions in blue of the 5000 simulation runs shown in Figure 13 together with the cumulative density functions of the corresponding Gaussians in red. The signal-to-noise ratios again are 20 dB (**left**), 10 dB (**middle**) and 0 dB (**right**). Figure 15: Probability that the algorithms in Equation (50) to Equation (53) result within an interval of \(\pm 6\sigma\) at the actual resonant frequency depicted as a function of the signal-to-noise ratio for a resonator with quality factors of 10 (red line), 100 (blue line) and 1000 (black line). The curves do not scale, as suggested in Equation (55), with \(Q^{2}\cdot\eta\), i.e., the same detection probabilities arise for the same \(Q^{2}\cdot\eta\) product but with \(Q^{0.75}\cdot\eta\). This effect is caused by the usage of different sample lengths chosen to \(Q/f_{d}\). Thus, the noise energy scales from one graph to another by a factor of 10. If we take this into account, the curves scale with \(Q^{1.75}\cdot\eta\), which is quite similar to Equation (55). The curves do not go to zero even at a very low signal-to-noise ratio, especially on the graph with a quality factor of 10, as there remains a finite probability that a maximum noise peak occurs within \(\pm 6\sigma\) around the actual resonance frequency. For \(Q^{0.75}\cdot\eta\approx 6\), the detection probability reaches 50%, and for \(Q^{0.75}\cdot\eta\geq 50\), it approaches to 100%. Figure 16 shows the convergence of 10 runs at signal-to-noise ratios where the detection probabilities are only 80%. In the left diagram, the signal-to-noise ratio is 3.9 dB for signals from resonators with a quality factor of 10, in the middle \(-4.2\) dB for a resonator with a quality factor of 100 and on the right \(-12.85\) dB for a resonator with a quality factor of 1000. Typically, eight trajectories can be seen. In dashed lines, \(\pm 1\sigma\) of the Cramer-Rao lower bound is shown. The convergence of the algorithm decreases as the length of the sampling length increases above \(Q/f_{d}\). Figure 17 shows on the left the convergence as a function of the sample length for every 5000 runs. The quality factor of the resonator is 100 and the initial signal-to-noise ratio is 0 dB. For this combination, the middle graph in Figure 15 shows a detection probability of 100% for a sampling length of \(1\cdot Q/f_{d}\). The sample length here is increased from \(1\cdot Q/f_{d}\) to \(20\cdot Q/f_{d}\), which reduces the probability of success from 100% to 20%. Figure 16: Convergence trajectories for each of the 10 runs at signal-to-noise powers where the detection probabilities are only 80% for a resonator with quality factors of 10 (**left**), 100 (**middle**) and 300 (**right**).