## 4 Characterization and Performance Analysis
### Other Practical Issues  
* The SQM variation envelopes show the maximum level of SQM sensitivity under specific multipath delay and power. However, in practice due to satellite motion and other effects, there are ever-present phase variations between LOS and multipath signals which cause SQM outputs to fluctuate between in-phase and out-of-phase envelopes. This will reduce the practical sensitivity of SQM metrics during multipath detection. The same argument applies to multipath range error envelopes and SQM effectiveness.
* In the extracted SQM profiles and multipath error envelopes, the power of reflected signals is assumed to be constant for different multipath delays. In practice, the reflected power should preferably be defined as a function of the multipath relative delay and a model of practical observations.
* In the signal monitoring approach, the practical statistics of SQM metrics may deviate from the nominal values based on the actual shape of the correlation function which depends on receiver parameters such as pre-correlation bandwidth and filter characteristics. In addition, the nominal statistics provided in Table 1 are calculated based on theoretical analysis with simplified approximations and assumptions. Therefore, for a reliable detection, the metric statistics and consequently detection thresholds should be calibrated based on practical observations.
* The SQM metrics variations are affected either by C/N\({}_{0}\) value fluctuations or correlation peak distortions. Although the C/N\({}_{0}\) can be also considered a monitoring metric itself, its effect on SQM-based detection performance is filtered through the tuning detection threshold based on the effective C/N\({}_{0}\) at each epoch. If the effective C/N\({}_{0}\) is smoothed, the effect of instant changes in C/N\({}_{0}\) values will be projected on the SQM metric.
* The multipath detection process can be considered as a general maximizing procedure of the likelihood ratio by setting an appropriate detection threshold for each PRN. By setting this threshold, the SQM threshold excess can be monitored as a primary real-time indication of anomaly detection. However, to optimize detection performance and have more reliable detection (in terms of false alarm and detection probability), a filtering process should be applied at the expense of delayed decision-making. Herein, the detection procedure is defined based on a fixed interval detector called the _M of N_ detection strategy usually used in GNSS acquisition procedures [32]. The _M of N_ search detector takes a window of \(N\) samples and compares them to a predefined threshold. If \(M\) or more samples exceed the threshold, then the detection output will be 1 and otherwise 0. This procedure is repeated for the next window in the search pattern. With this detection strategy, the overall probability of false alarm in \(N\) trials is [32]: \[P_{FA}=\sum_{n=M}^{N}\left(\begin{array}{c}N\\ n\end{array}\right)P_{fa}^{n}\left(1-P_{fa}\right)^{N-n}=1-\sum_{n=0}^{M-1} \left(\begin{array}{c}N\\ n\end{array}\right)P_{fa}^{n}\left(1-P_{fa}\right)^{N-n}\] (7) where \(\left(\begin{array}{c}N\\ n\end{array}\right)\) is the binomial coefficient of \(n\) and \(N\) integers referring to the number of combinations of \(N\) items taken \(n\) at a time. \(P_{fa}\) is the false alarm probability in each trial equals to 0.27% under a normal distribution assumption for the SQM metric and 3 times the standard deviation (\(\pm 3\sigma\)) as the detection threshold.