## 4 Characterization and Performance Analysis
### SQM Sensitivity and Effectiveness in Multipath Detection  
To evaluate the performance and effectiveness of SQM approaches for multipath detection, the monitoring variation profiles should be compared with the multipath error envelopes. To this end, SQM \"sensitivity\" and \"effectiveness\" are now defined to make the subsequent discussion clearer. The _sensitivity_ of the SQM approach is defined based on the magnitude of the SQM variation profiles. If the SQM outputs exceed a given threshold, the multipath is detected and the SQM method is sensitive. Otherwise, the multipath signal is not detectable and the SQM metric is therefore not sensitive. The _effectiveness_ of the SQM approach is defined based on the magnitude of the multipath error. A SQM approach is effective when the SQM outputs are above the detection threshold (multipath is detected) and at the same time multipath has induced a significant error on range measurements. The effectiveness of a SQM approach is important since ideally SQM based measurement weighting should be based on its effectiveness and not sensitivity. Consider the Double-Delta (DD) metric for BPSK(1) signaling. To evaluate the effect of correlator spacing on the SQM effectiveness, two sets of correlator spacings are considered according to Table 2. In addition to the previously defined DD metric (Test 1), Test 2 is defined based on 0.1 and 0.4 chips tracking and monitoring the correlator spacing, respectively. The nominal statistics of Test 1 and Test 2 are summarized in Table 2. Figure 6 shows the SQM profiles of the Double-Delta Test 1 and Test 2 for SMR = 3 dB and BPSK(1) signaling (blue solid and dashed lines). For a better comparison, the corresponding ranging error envelopes are also plotted on the right-hand vertical axis in brown. For a C/N\({}_{0}\) value of 45 dB-Hz, detection thresholds have been set as three times the corresponding nominal SD to satisfy a probability of false alarm of 0.0027. Figure 5: BPSK(1) correlation peak and its distortion under a single-ray multipath. According to Figure 6, for both NC and HRC discriminators, multipath with relative delays less than 10 m is not detectable. For the NC discriminator (Figure 6a), under medium and long-delay multipath, the deviations of the SQM metrics from the detection threshold (SQM sensitivity) coincide with the non-zero tracking range error envelopes (SQM effectiveness). For the HRC discriminator (Figure 6b), the Double-Delta SQM metric will not be effective for most medium-delay multipath scenarios when the resulting ranging errors decrease to zero. In order to evaluate the effect of correlator spacing, narrowing the tracking correlator mitigates the overall effect of multipath on tracking range errors (compare the solid and dotted brown lines in Figure 6) as is well known. In terms of SQM sensitivity, narrower monitoring correlators lower nominal SD and thus detection thresholds. This improves the sensitivity of the SQM metric in the case of short-delay multipath scenarios in which case narrower monitoring correlators results in higher SQM variations (compare the solid and dashed blue lines). In the case of the medium and long-delay multipath scenarios, both detection thresholds and SQM variation envelopes are decreased by narrowing the monitoring correlators. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**SQM Metric** & **Metric Definition** & **Nominal Mean** & **Nominal SD** & **Nominal SD for C/N0 = 45 (dB-Hz)** \\ \hline Double Delta Test 1 & \(m_{3}=\frac{\big{(}I_{-0.5}-I_{-i,0.5}\big{)}-\big{(}I_{-0.1}-I_{+0.1}\big{)}}{I _{0}}\) & 0 & \(\sqrt{\frac{1.6}{2(C/N_{0})T_{t}}}\) & 0.0356 \\ \hline Double Delta Test 2 & \(m_{4}=\frac{\big{(}I_{-0.2}-I_{+0.2}\big{)}-\big{(}I_{-0.05}-I_{+0.05}\big{)}}{I _{0}}\) & 0 & \(\sqrt{\frac{0.6}{2(C/N_{0})T_{t}}}\) & 0.0218 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Double-Delta tests for BPSK(1) signaling. Figure 6: _Cont._  
With similar arguments applied to Figures 2-4, the following conclusions can be made.