# Performance Evaluation of Adaptive Tracking Techniques with Direct-State Kalman Filter +
## 2 Tracking Scheme of Direct-State Kalman Filter
### Process and Measurement Noise Covariance Matrix  
The Kalman gains \(\mathbf{K}\) are calculated for each loop iteration based on Equations (17), (20), (21) and (23). \(\mathbf{K}\) depends on the process noise covariance \(\mathbf{Q}\) and measurement noise covariance \(\mathbf{R}\). This paper calculates these matrices considering a third-order DSKF for carrier phase tracking. In addition, a constant-acceleration model is assumed [31; 32] for the \(\mathbf{Q}\) calculation. The BET is used to discretize the noise that is added in the acceleration state:  
\[\mathbf{w}_{\mathrm{k}}=\mathbf{A}\:\mathbf{w}\tau_{\mathrm{int}}=\mathbf{A} \begin{bmatrix}0\\ 0\\ w_{\mathrm{a}}\end{bmatrix}\tau_{\mathrm{int}}=\begin{bmatrix}w_{\mathrm{a}} \tau_{\mathrm{int}}^{3}\\ w_{\mathrm{a}}\tau_{\mathrm{int}}^{2}\\ w_{\mathrm{a}}\tau_{\mathrm{int}}\end{bmatrix} \tag{28}\]  
where \(\mathbf{w}\) is the continuous noise vector, \(\mathbf{w}_{\mathrm{k}}\) is the discretized noise vector, and \(w_{\mathrm{a}}\) is the zero-mean Gaussian distributed perturbation that suffers the acceleration in cycles/\(s^{3}\).  
\(\mathbf{Q}\) is obtained performing the variance of the discretized noise:  
\[\mathbf{Q}=\mathrm{E}\Big{(}\mathbf{A}\:\mathbf{w}\:\mathbf{w}^{\mathrm{T}} \:\mathbf{A}^{\mathrm{T}}\Big{)}\tau_{\mathrm{int}}^{2}=\begin{bmatrix}\tau_{ \mathrm{int}}^{6}&\tau_{\mathrm{int}}^{5}&\tau_{\mathrm{int}}^{4}\\ \tau_{\mathrm{int}}^{2}&\tau_{\mathrm{int}}^{4}&\tau_{\mathrm{int}}^{4}\\ \tau_{\mathrm{int}}^{4}&\tau_{\mathrm{int}}^{2}&\tau_{\mathrm{int}}.\end{bmatrix} \:q \tag{29}\]  
where \(q\) is the variance of \(w_{\mathrm{a}}\) in cycles\({}^{2}\)/\(s^{6}\) and determines the uncertainty of the states. A higher \(q\) implies higher state uncertainty, leading to higher confidence in the incoming measurements. On the contrary, a lower the \(q\) leads to higher confidence of the states and less dependence on the measurement.  
The measurement noise covariance \(\mathbf{R}\) determines the validity of the incoming measurements. A high value indicates an increased uncertainty, whereas a low value defines high confidence. An adequate model for \(\mathbf{R}\) is the Cramer-Rao bound (CRB) of the STL since it represents the minimum error variance of a time of arrival (ToA) unbiased estimator [33; 34]. In this case, the measurement residual of the DSKF is the discriminator's output \(\epsilon_{\mathrm{u}}\) of an STL. The CRB of \(\epsilon_{\mathrm{u}}\) is achieved when the error estimation does not feedback on additional noise. Only the thermal noise of the incoming error parameter \(\epsilon\) is present. If the carrier phase offset parameter (\(\epsilon=\theta\)) is taken as a measurement, and under the assumption of a two-quadrant discriminator, \(R\) in cycles\({}^{2}\) is represented as [28; 34; 35]:  
\[R=\mathrm{VAR}(\theta)=\frac{1}{2\tau_{\mathrm{int}}(C/N_{0})_{1}}\Big{(}1+ \frac{1}{2\tau_{\mathrm{int}}(C/N_{0})_{1}}\Big{)} \tag{30}\]  
where \((C/N_{0})_{1}\) is the linear \(C/N_{0}\) in Hertz. This relation is commonly used in \(C/N_{0}\)-based DSKF [24; 25].