# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## 2 Signal-to-Noise and Interference Ratio by a High-Q Resonator as Cooperative Target
### Noise Power  
The thermal environment of the resonator has the temperature \(\delta\) and therefore emits thermal noise. The noise power \(N_{p}\) within the receiver bandwidth \(B\), which is received in addition to the resonator signal, is given by \(k_{B}\delta B\), where \(k_{B}\) is the Boltzmann constant. The low-noise amplifier increases this noise power by the factor \(F_{LNA}\). This results in a total noise power \(N_{p}\) of  
\[N_{p}=F_{LNA}k_{B}\delta B. \tag{15}\]  
The carrier frequency of the Lorentz curve can only be determined correctly if the entire spectrum of the Lorentz curve is contained in the analyzed frequency range. Each windowing of the Lorentz curve changes the decay constant and also shifts the apparent carrier frequency. For example, if the carrier frequency is outside the resolved frequency band, the signal processing cannot determine the correct frequency. The receiver bandwidth \(B\), thus, must be larger than the resonance bandwidth of the resonator to resolve the decaying resonance  
\[B>\frac{f_{0}}{Q}. \tag{16}\]  
A measurement bandwidth of approximately three to four times this minimum bandwidth would be optimal for a maximum signal-to-noise ratio. However, in order to reliably detect the shift in the center frequency, a larger bandwidth than this minimum usually must be sampled, for example 0.5% to 1% \(f_{0}\). In the ISM and SRD frequency bands, the bandwidth is restricted by legal requirements.  
The receiver bandwidth relevant for the noise power results from the sampling frequency \(f_{s}\) or the sampling interval \(T\) of the receiver  
\[B=\frac{1}{2}f_{s}=\frac{1}{2T}. \tag{17}\]  
In order to minimize the noise power in the sampled signal, the received signal is limited to the minimum bandwidth of 0.5\(f_{s}\) before sampling. If the bandwidth of the signal is lowered below the value given in Equation (17), we call the signal over-sampled. The over-sampled content of the signal contains no additional information, and we can reduce in the following analysis the sampling frequency to half of the bandwidth.  
After data collection is complete, usually a new measurement can be started. If neither the measured value, i.e., the desired resonance frequency, nor the transmission channel changes, either the extracted resonance frequencies can be averaged or, in some cases, the individual complex sample values also can be averaged coherently to lower the equivalent noise bandwidth \(B\).  
If the noise can be described by a mean-free Gaussian process, it is completely characterized by the variance \({\sigma_{n}}^{2}\), where  
\[{\sigma_{n}}^{2}=N_{p}. \tag{18}\]  
Since the amplitude of the exponentially decaying resonator signal decreases with time, the signal-to-noise ratio becomes time-dependent. However, we can define the initial signal-to-noise ratio \(\eta\) as the ratio of the initial signal power \(P_{gatal}\) to the noise floor \(N_{p}\), which is given by  
\[\eta=\frac{P_{gatal}}{N_{p}}=\frac{P_{gatal}}{{\sigma_{n}}^{2}}. \tag{19}\]  
Figure 6 shows the initial signal-to-noise ratio \(\eta\) in the reader after the time gate for the same frequency ranges and resonator qualities as in Figure 5. The noise bandwidths used in the left graph were 1.75 MHz for the 433 MHz band, 2 MHz for the 868 MHz band, 26 MHz for the 915 MHz band, 100 MHz for the 2.45 GHz band and 150 MHz for the 5.8 GHz band. However, sampling the same relative bandwidths \(B=0.5\%f_{0}\) results in the graph to the right with significant improvements in the signal-to-noise ratio at the wide bands 915 MHz, 2450 MHz and 5800 MHz.