# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 3 Results and Discussion
### Increased Accuracy in the Permittivity
#### 3.1.1 Size of Frequency Step vs. Number of Averages  
Averaging of multiple measurement samples is a routine commonly used to reduce noise in experimental results. According to theory, the noise reduction follows \(N^{\beta}\), where \(N\) is the number of samples and \(\beta\) the exponent which indicates the noise reduction. In this experiment, \(N\) is the number of probing frequencies multiplied by the number of averages and if applicable multiplied by the sample width of the moving average. In the ideal case \(\beta\) = 0.5 [46]. Hereof in this section, the exponent \(\beta\) as a function of the frequency step size and number of averages is investigated.  
In terms of this experiment, the exponent \(\beta\) is expressed as,  
\[\frac{\hat{C}_{\text{v,high}}}{\hat{C}_{\text{v,low}}}=\Big{(}\frac{N_{\text{ high}}}{N_{\text{low}}}\Big{)}^{\beta}, \tag{19}\]  
where \(\hat{C}_{\text{v,low}}\) and \(\hat{C}_{\text{v,high}}\) are the coefficient of variation for the experiment with the least samples and the most samples, respectively, and similarly for the number of samples \(N_{\text{low}}\) and \(N_{\text{high}}\).  
Figure 10 presents a graph in which \(\beta\) is shown as a function of the experimental settings. The emulated experiment with 4096 averages and 1 kHz step size is used as a reference to compare the other settings to.  
For very small frequency step sizes, \(\beta\) is lower than 0.5 and, more importantly, lower than the equivalent \(\beta\) for the scan of the number of averages. Despite that a parameter scan is only performed along the orthogonal axes of the parameter space, the results presented in this graph strongly indicate that it is best to reduce the number of probing frequencies over the width of the peak to \(\sim\)50, i.e., a frequency step size of 200 kHz while the remaining budget in measurement time can be used for probing additional discharges to increase the number of averages.  
The exponent \(\beta\) for the scan of the number of averages fluctuates around 0.5. It can be concluded that the noise is reduced as expected for stationary noise with a mean which is zero [46]. It appears that the resolution in \(Q\) and \(f_{\text{res}}\) scales in the same manner with the sample size. The spread in the exponent \(\beta\) related to the number of averages decreases for the decreasing number of averages. This can be explained by the principle that the noise becomes smaller when the difference in sample size becomes larger.  
The triangular symbols indicate the exponent related to the increase in resolution by the post-processing step in which 20 subsequent samples of \(f_{\text{res}}\) and \(Q\) were averaged. It appears that the reduction of noise by this method is less efficient than that by including additional samples as a result of probing more discharges. However, this moving average is relatively cheap in measurement time as it does not require additional discharges. Note that an increase in the sampling frequency to accommodate the moving average could lead to a longer duration of the experiment as more data needs to be saved. It is also important to mention that a moving average acts as a low-pass filter. Hence, it increases the 'tardiness' of the response of the cavity.