# Performance Evaluation of Adaptive Tracking Techniques with Direct-State Kalman Filter +
## 4 Experimental Setup
### Performance Metric  
The metric to evaluate the PLL tracking and system performance is the same as in previous studies [7]. The tracking performance \(\text{P}_{\text{Tracking}}\) evaluates the tracking of a single satellite vehicle (SV), whereas the system performance \(\text{P}_{\text{System}}\) considers all the visible SVs. \(\text{P}_{\text{Tracking}}\) in meters is characterized as:  
\[\text{P}_{\text{Tracking}}=(\overline{\sigma}_{\theta}^{u}-\sigma_{\text{LB}} ^{u})\cdot\lambda_{\text{L1}} \tag{49}\]  
where \(\lambda_{\text{L1}}\) is the wavelength of the GPS L1 C/A signal, \(\overline{\sigma}_{\theta}^{u}\) is the average of the last ten minutes un-smoothed carrier phase error's standard deviation, and \(\sigma_{\text{LB}}^{u}\) is the square root of the CRB [7]. \(\overline{\sigma}_{\theta}^{u}\) in cycles is defined as:  
\[\overline{\sigma}_{\theta}^{u}=\frac{1}{K_{\text{sim}}}\sum_{k=1}^{K_{\text{ sim}}}\sigma_{\theta}^{u}[k] \tag{50}\]  
where \(K_{\text{sim}}\) is the number of evaluation epochs in samples. The total evaluation time \(T_{\text{sim}}\) in seconds is:  
\[T_{\text{sim}}=\frac{K_{\text{sim}}}{f_{\text{log}}} \tag{51}\]  
where \(f_{\text{log}}\) is the data-logging rate in Hertz. \(\sigma_{\text{LB}}^{u}\) in cycles is represented as:  
\[\sigma_{\text{LB}}^{u}=\left(\frac{1}{2\pi}\right)\sqrt{\frac{1}{2\tau_{\text {int}}C/N_{0}}\left(1+\frac{1}{2\tau_{\text{int}}C/N_{0}}\right)} \tag{52}\]  
where the \(\left(1+\frac{1}{2\tau_{\text{int}}C/N_{0}}\right)\) term is the squaring loss [1]. The three-sigma rule-of-thumb conservative upper threshold \(\sigma_{\theta}^{th}\) is considered in the tracking performance evaluation. For a two-quadrant phase discriminator, \(\sigma_{\theta}^{th}\) in cycles is expressed as:  
\[\sigma_{\theta}^{th}=\frac{1}{24} \tag{53}\]  
A low value of \(\text{P}_{\text{Tracking}}\) denotes good tracking performance. If the measured \(\text{P}_{\text{Tracking}}\) from \(\overline{\sigma}_{\theta}^{u}\) is less than \(\text{P}_{\text{Tracking}}\) considering \(\sigma_{\theta}^{th}\), one can ensure stable tracking and no cycle slips [1]. On the contrary, if the measured \(\text{P}_{\text{Tracking}}\) is bigger than the conservative threshold, the probability of losing the lock increases. The metric used for the system performance \(\text{P}_{\text{System}}\) is expressed as:  
\[\text{P}_{\text{System}}=\overline{\text{PLI}}\times\overline{\text{N}}_{ \text{sat}} \tag{54}\]  
where \(\overline{\text{PLI}}\) is the average of the phase-lock indicator (PLI) with respect \(K_{\text{sim}}\) and the accumulated number of tracked SVs \(N_{\text{sat}}^{\text{Acc}}\), and \(\overline{\text{N}}_{\text{sat}}\) is the normalized average number of visible satellites being tracked. The expression of \(\overline{\text{PLI}}\) is:  
\[\overline{\text{PLI}}=\frac{1}{K_{\text{sim}}\,N_{\text{sat}}^{\text{Acc}}} \sum_{k=1}^{K_{\text{sim}}}\sum_{l=1}^{N_{\text{sat}}^{\text{Acc}}}\text{PLI} ^{l}[k] \tag{55}\]and \(\mathrm{PLI}^{I}[n]\) is calculated based on the prompt in-phase \(I_{p}^{I}\) and prompt quadrature \(Q_{p}^{I}\) correlation samples of the corresponding \(I^{th}\) SV:  
\[\mathrm{PLI}^{I}=\frac{(I_{p}^{I})^{2}-(Q_{p}^{I})^{2}}{(I_{p}^{I})^{2}+(Q_{p}^{ I})^{2}} \tag{56}\]  
The second term of Equation (54), \(\overline{\mathrm{N}}_{\mathrm{sat}}\), is defined as:  
\[\overline{\mathrm{N}}_{\mathrm{sat}}=\frac{1}{K_{\mathrm{sim}}N_{\mathrm{sat }}^{\mathrm{total}}}\sum_{k=1}^{K_{\mathrm{sim}}}N_{\mathrm{sat}}[k] \tag{57}\]  
where \(N_{\mathrm{sat}}^{\mathrm{total}}\) is the total number of SVs during the simulation. \(\overline{\mathrm{PLI}}\) and \(\overline{\mathrm{N}}_{\mathrm{sat}}\) are normalized. Consequently, \(\mathrm{P}_{\mathrm{System}}\) is also normalized. A value near to one of \(\mathrm{P}_{\mathrm{System}}\) refers to good system performance, whereas a value near zero indicates poor system performance and the possibility of not achieving a PVT solution.