## 2 Automatic Tracking Method of Weak Acoustic Targets within Jamming Environment
### Target Motion Analyzing (TMA)
\begin{split}\int_{-\infty}^{+\infty}(\mathbf{\hat{\mathbf{\alpha}}}- \mathbf{\alpha})\sqrt{f(t,\mathbf{\alpha})}\frac{\partial}{\partial\mathbf{\alpha}}(\ln f (t,\mathbf{\alpha}))\sqrt{f(t,\mathbf{\alpha})}dt\\ \cdot\int_{-\infty}^{+\infty}(\mathbf{\hat{\mathbf{\alpha}}}-\mathbf{\alpha}) \sqrt{f(t,\mathbf{\alpha})}\frac{\partial}{\partial\mathbf{\alpha}}(\ln f(t,\mathbf{ \alpha}))\sqrt{f(t,\mathbf{\alpha})}dt\\ \leq var(\mathbf{\hat{\mathbf{\alpha}}})\cdot E[(\frac{\partial}{\partial \mathbf{\alpha}}(\ln f(t,\mathbf{\alpha})))^{2}]\end{split}  \tag{17}\]  
where \(var(\mathbf{\hat{\mathbf{\alpha}}})=\int_{-\infty}^{+\infty}(\mathbf{\hat{\mathbf{\alpha}}}- \mathbf{\alpha})^{2}f(t,\mathbf{\alpha})dt\), \(E[(\frac{\partial}{\partial\mathbf{\alpha}}(\ln f(t,\mathbf{\alpha})))^{2}]=\int_{- \infty}^{+\infty}(\frac{\partial}{\partial\mathbf{\alpha}}(\ln f(t,\mathbf{\alpha})))^ {2}f(t,\mathbf{\alpha})dt\). Hence, it has  
\[var(\mathbf{\hat{\mathbf{\alpha}}})\cdot E[(\frac{\partial}{\partial\mathbf{\alpha}}(\ln f (t,\mathbf{\alpha})))^{2}]\geq\textbf{I} \tag{18}\]Now the CRLB \(Fisher^{-1}(\mathbf{\alpha})\) of the motion analysis can be obtained:  
\[var(\mathbf{\hat{x}})\geq Fisher^{-1}(\mathbf{\alpha}) \tag{19}\]  
where \(Fisher(\mathbf{\alpha})=E[\frac{\partial}{\partial\mathbf{\alpha}}(\ln f(t,\mathbf{\alpha})) ^{2}]\) is the Fisher information. The optimal is obtained when \(\mathbf{\hat{x}}-\mathbf{\alpha}=C\frac{\partial}{\partial\mathbf{\alpha}}(\ln f(t,\mathbf{ \alpha}))\), where \(C\) is a constant. Now we can start to discretize the CRLB of motion analysis. Considering that the error vector \(\mathbf{N}\) follows the Gaussian distributions, we have \(\mathbf{N}=\mathbf{T}-\mathbf{H}\mathbf{\alpha}\sim N(0,\sigma^{2}*\mathbf{I})\), and its joint probability distribution \(F(\mathbf{T},\mathbf{\alpha})\) is  
\[ \begin{split} F(\mathbf{T},\mathbf{\alpha})&=\Pi_{k=1} ^{n}\frac{1}{\sqrt{2\pi\sigma}}exp(-\frac{(t_{k}-\theta_{k}\mathbf{\alpha})^{T}*(t _{k}-\theta_{k}\mathbf{\alpha})}{2\sigma^{2}})\\ &=(\frac{1}{\sqrt{2\pi\sigma}})^{n}exp(-\frac{(\mathbf{T}- \mathbf{H}\mathbf{\alpha})^{T}*(\mathbf{T}-\mathbf{H}\mathbf{\alpha})}{2\sigma^{2}}) \end{split}  \tag{20}\]  
Take the natural logarithm of Equation (20) and then compute the partial derivatives:  
\[\frac{\partial}{\partial\mathbf{\alpha}}\ln F(\mathbf{T},\mathbf{\alpha})=(\frac{ \mathbf{H}^{T}\mathbf{H}}{\sigma^{2}})((\mathbf{H}^{T}\mathbf{H})^{-1}\mathbf{ H}^{T}\mathbf{T}-\mathbf{\alpha}) \tag{21}\]  
According to the optimal constraint of the Cauchy-Schwarz inequality, the best estimation of \(\mathbf{\alpha}\) is obtained with  
\[\mathbf{\hat{x}}-\mathbf{\alpha}=C*\frac{\partial}{\partial\mathbf{\alpha}}(\ln F( \mathbf{T},\mathbf{\alpha})) \tag{22}\]  
Let \(C\) be \((\frac{\mathbf{H}^{T}\mathbf{H}}{\sigma^{2}})^{-1}\); then, the minimum variance unbiased estimate (MVUE) is obtained:  
\[\mathbf{\hat{x}}=(\mathbf{H}^{T}\mathbf{H})^{-1}\mathbf{H}^{T}\mathbf{T} \tag{23}\]  
So we obtain  
\[\left( \begin{array}{cc}\sum_{i=1}^{n}\theta_{i}^{2}&\sum_{i=1}^{n}\theta_{ i}\\ \sum_{i=1}^{n}\theta_{i}&n\end{array} \right)\left( \begin{array}{c}k\\ b\end{array} \right)=\left( \begin{array}{c}\sum_{i=1}^{n}\theta_{i}t_{i}\\ \sum_{i=1}^{n}t_{i}\end{array} \right) \tag{24}\]  
The parameter \(k\) therefore can be estimated via  
\[k=\frac{\sum_{i=1}^{n}\theta_{i}\sum_{i=1}^{n}t_{i}-n\times\sum_{i=1}^{n} \theta_{i}t_{i}}{\sum_{i=1}^{n}\theta_{i}\sum_{i=1}^{n}\theta_{i}-n\sum_{i=1} ^{n}\theta_{i}^{2}} \tag{25}\]  
where \(n\) is the count of snapshots including prior target information. Then, we need to make \(\theta=\arctan(k)\), which is the tilt angle of the target trajectory.