# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 4 Calculating the Natural Frequency of a Decaying Oscillator
### Detection of the Ambient Echoes of the Transmission Channel  
After the natural angular frequency \(\tilde{\omega}\) and the decay constant \(\alpha\) are determined using Equations (51) and (52), the phase angle \(\tilde{\varphi}\) and the amplitude \(\tilde{b}\) at \(t=0\) can be calculated by Equations (31) and (32). For this purpose, the natural frequency in the individual sampling points \(z_{n}\) is eliminated by multiplying with \(e^{-\tilde{\omega}nT}\), and the influence of noise is reduced by weighting with \(e^{-\alpha nT}\). All data are then added up. The phase of this sum gives the phase angle \(\tilde{\varphi}\) at \(t=0\). The amplitude \(\tilde{b}\) at \(t=0\) is obtained by normalizing the amplitude of this sum with the amplitude of a corresponding geometric series with a starting amplitude of 1. Figure 20 shows on the left the phase angles of the individual sampling points \(z_{n}\) with the natural frequency eliminated and weighted with \(e^{-\alpha nT}\). In addition, the phase angle \(\tilde{\varphi}\) is displayed. The middle graph shows the associated amplitudes of the individual sample points \(z_{n}\) on a logarithmic scale along with \(\tilde{b}\).  
After all four variables of the state vector \(\chi\) have been determined, and the resulting estimates \(\mu_{n}\) for the sampled signals \(z_{n}\) at times \(n\cdot T\) can be calculated according to Equation (27) and with these, using Equation (33), the noise components together with the echoes in the transmission channel. Figure 20 shows on the right in red the noisy signal of the decaying resonator with a signal-to-noise ratio of 20 dB together with the echoes in the transmission channel. In order to eliminate these echoes, the signal was only evaluated from \(t=0\). For the curve shown in blue, the estimated values \(\mu_{n}\) were subtracted from the measurement values \(z_{n}\). The ambient echoes and noises remain.  
Figure 19: Cumulative density functions from 5000 simulation runs using the weighting function Equation (56) together with the cumulative density functions of the corresponding Gaussian functions. All other parameters correspond to those in Figure 14. The curves are so close together that they can hardly be distinguished. The signal-to-noise ratios are again 20 dB (**left**), 10 dB (**middle**) and 0 dB (**right**).  
Figure 20: The left graph shows the remaining phase angles of the individual sampling points \(z_{n}\), where the natural frequency was eliminated by multiplying with \(e^{-\tilde{\omega}nT}\) and where the impact of noise was reduced by weighting with \(e^{-\alpha nT}\). In addition, the phase angle \(\tilde{\varphi}\) at \(t=0\) is plotted according to Equation (31). For the center graph, the amplitudes of the sample points weighted with \(e^{-\alpha nT}\) were plotted together with the amplitude \(\tilde{b}\) at \(t=0\) calculated according to Equation (32). The noisy signal from the decaying resonator with a signal-to-noise ratio of 20 dB is shown in red on the right along with the echoes in the transmission channel. The blue curve shows the remaining signal when the estimated values \(\mu_{n}\) have been subtracted from the measured values \(z_{n}\).  
Since the echoes in the transmission channel are systematic and do not change between individual readings, they add up coherently. If several measurement signals are summed up, in which the estimated values of the decaying resonator have been subtracted, the echoes of the transmission channel stand out clearly from the noise. Figure 21 shows the coherent sum of 50 signals with subtracted estimated values, on the left in linear scale and on the right in logarithmic scale. The systematic interference signals disappear into the noise at \(t=-15\), and the signal evaluation could be started earlier by this interval, resulting in a higher resolution.