## 5 Extended Kalman Filter Tracking
### Second Order Models  
In this section, we assume that the relative propagation delay \(\tau\) and time offset \(T\) instead follow the second-order Markov model defined in Equations (17) and (18). The second-order transition matrix has the structure \(\mathbf{F}(L_{A})=\text{diag}(\,\mathbf{f}(L_{A})\,\,,\,\mathbf{f}(L_{A})\,)\), where the state space variables \(\mathbf{x}\) and transition matrix \(\mathbf{f}(L_{A})\) are defined as:  
\[\mathbf{x}= \begin{bmatrix}\tau,\,\,\hat{\tau},\,\,T,\,\,\hat{T},\,\,\hat{T} \end{bmatrix} ^{\top}\,\,\,,\,\,\,\,\mathbf{f}(l)= \begin{bmatrix}1&l&\frac{1}{ 2}l^{2}\\ 0&1&l\\ 0&0&1\end{bmatrix} . \tag{46}\]  
The delay covariance \(\mathbf{Q}_{\tau}=\text{cov}[\,\tau,\hat{\tau},\hat{\tau}\,]\) under a Gaussian acceleration perturbation model is derived in [44] and is given by:  
\[\mathbf{Q}_{\tau}(l)= \begin{bmatrix}\frac{1}{4}\,\sigma_{\tau}^{2}\,l^{4}& \frac{1}{2}\,\sigma_{\tau}^{2}\,l^{3}&\frac{1}{2}\,\sigma_{\tau}^{2}\,l^{2} \\ \frac{1}{2}\,\sigma_{\tau}^{2}\,l^{3}&\frac{1}{2}\,\sigma_{\tau}^{2}\,l^{2}& \sigma_{\tau}^{2}\,l\\ \frac{1}{2}\,\sigma_{\tau}^{2}\,l^{2}&\sigma_{\tau}^{2}\,l&\sigma_{\tau}^{2} \end{bmatrix} \,, \tag{47}\]  
where \(\sigma_{\tau}^{2}\) is the variance in relative acceleration. The offset covariance \(\mathbf{Q}_{T}=\text{cov}[\,T,\hat{T},\hat{T}\,]\) is a reflection of the clock errors expressed as functions of their Allan variance characteristics. This offset covariance is derived in [41] and is given by:  
\[\mathbf{Q}_{T}(l)=\sigma_{T}^{2}\,q_{1}(l)+\sigma_{\tau}^{2}\,q_{2}(l)+\sigma_ {\tau}^{2}\,q_{3}(l)\,, \tag{48}\]  
\[q_{1}(l)= \begin{bmatrix}l&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix} \,\,\,,\,\,q_{2}(l)= \begin{bmatrix}\frac{1}{3}l^{3}&\frac{ 1}{2}l^{2}&0\\ \frac{1}{2}l^{2}&l&0\\ 0&0&0\end{bmatrix} \,\,\,,\,\,\,q_{3}(l)= \begin{bmatrix}\frac{1}{20}l^{5}& \frac{1}{8}l^{4}&\frac{1}{6}l^{3}\\ \frac{1}{8}l^{4}&\frac{1}{3}l^{3}&\frac{1}{2}l^{2}\\ \frac{1}{6}l^{3}&\frac{1}{2}l^{2}&l\end{bmatrix} . \tag{49}\]  
The model parameters \(\sigma_{\tau}^{2}\,,\,\sigma_{\tau}^{2}\), and \(\sigma_{\tau}^{2}\) may be rigorously defined as functions of the Allan variances of the oscillators that drive these radios, but this examination is beyond the scope of this report. We refer the reader to [41] for more detailed information. The measurement prediction model is given in Equation (27), where the measurements \(\mathbf{z}\) and controls \(\mathbf{u}\) remain unchanged from Equation (38), but the transition function is \(\mathbf{h}(\cdot)\) now written as  
\[\mathbf{h}\Big{(}\hat{\mathbf{x}}_{(-)}^{(n-1)}\Big{)}= \begin{bmatrix}\tau^{(n-1 )}-T^{(n-1)}\\ \tau^{(n-1)}+\tau^{(n-1)}\,l_{A}^{(n-1)}+\tau^{(n-1)}\,{l_{A}^{(n-1)}}^{2}+\\ T^{(n-1)}+\hat{T}^{(n-1)}\,l_{A}^{(n-1)}+\hat{T}^{(n-1)}\,l_{A}^{(n-1)}{}^{2} \end{bmatrix} , \tag{50}\]  
where the second-order frame length estimate [9] is given by Equation (24). Following the definition presented in Equation (41), the measurement Jacobian is now given by Equation (51). We compute and simplify these six partial derivatives using Equation (50) and summarize the results in Equations (52)-(57). These derivatives themselves include partial derivatives of \(l_{A}\) with respect to \(T\), \(\hat{T}\), and \(\hat{T}\), which we can compute using the definition provided in Equation (23). We include the simplified results in Equations (58)-(60) for convenience. \[\left.\frac{\partial\mathbf{h}(\hat{\mathbf{x}})}{\partial\hat{\mathbf{x}}} \right|_{\hat{\mathbf{x}}_{(-)}^{(n-1)}} = \begin{bmatrix}\frac{\partial\mathbf{h}(\hat{\mathbf{x}})}{ \partial\tau}&\frac{\partial\mathbf{h}(\hat{\mathbf{x}})}{\partial\hat{\tau} }&\frac{\partial\mathbf{h}(\hat{\mathbf{x}})}{\partial\hat{\tau}}&\frac{ \partial\mathbf{h}(\hat{\mathbf{x}})}{\partial T}&\frac{\partial\mathbf{h}( \hat{\mathbf{x}})}{\partial\hat{T}}&\frac{\partial\mathbf{h}(\hat{\mathbf{x} })}{\partial\hat{T}}\end{bmatrix} \right|_{\hat{\mathbf{x}}_{(-)}^{(n-1)}},\] (51) \[\left.\frac{\partial\mathbf{h}(\hat{\mathbf{x}})}{\partial\tau} \right|_{\hat{\mathbf{x}}_{(-)}^{(n-1)}} =