## 2 Automatic Tracking Method of Weak Acoustic Targets within Jamming Environment
### Target Motion Analyzing (TMA)  
This paper's technique initially collects some snapshots of trajectory information of interested target, then obtains the TMA parameters like slope and intercept by analyzing the prior trajectory. Normally, the underwater acoustic target varies slowly in a short time; if the bearing of the target at time \(t_{i}\) is \(\theta_{i}\), it can be thought of as linear \(\Delta t\) over a specific short period of time. The trajectory of the interested target can be described as \(t_{i}=k\theta_{i}+b\), where \(k\) is the slope and \(b\) is the intercept to be estimated. Let \(\mathbf{\alpha}=(k,b)^{T}\) represent the parameter vector, so we have \(\mathbf{T}=\mathbf{H}\mathbf{\alpha}+\mathbf{N}\), where \(\mathbf{T}=[t_{1},t_{2},\cdots,t_{n}]^{T}\), \(\mathbf{H}=\left[ \begin{array}{cccc}\theta_{1}&\theta_{2}&\ldots&\theta_{n} \\ 1&1&\ldots&1\end{array} \right]^{T}\),and **N** is the noise vector with Gaussian distribution. Then calculate its Cramer-Rao Lower Band (CRLB) to obtain the minimum error. According to the knowledge of probability theory, it is known that  
\[\int_{-\infty}^{+\infty}f(t,\mathbf{\alpha})dt=\textbf{I} \tag{9}\]  
where \(f(t,\mathbf{\alpha})\) is the probability density function including two variables \(t\) and \(\mathbf{\alpha}\). Therefore, we have  
\[\frac{\partial}{\partial\mathbf{\alpha}}\int_{-\infty}^{+\infty}f(t,\mathbf{\alpha}) dt=\int_{-\infty}^{+\infty}\frac{\partial}{\partial\mathbf{\alpha}}f(t,\mathbf{\alpha}) dt=\textbf{0} \tag{10}\]  
Let \(\mathbf{\hat{\mathbf{\alpha}}}\) be the unbiased estimation of \(\mathbf{\alpha}\):  
\[\int_{-\infty}^{+\infty}\mathbf{\hat{\mathbf{\alpha}}}f(t,\mathbf{\alpha})dt=E(\mathbf{\hat{ \mathbf{\alpha}}})=\mathbf{\alpha} \tag{11}\]  
where \(E(*)\) returns the expectations of the input variables. From Equation (11), the following equation can be derived:  
\[\frac{\partial}{\partial\mathbf{\alpha}}\int_{-\infty}^{+\infty}(\mathbf{\alpha}f(t, \mathbf{\alpha}))dt=\frac{\partial}{\partial\mathbf{\alpha}}(E(\mathbf{\hat{\mathbf{\alpha}}} ))=\frac{\partial}{\partial\mathbf{\alpha}}(\mathbf{\alpha})=\textbf{I} \tag{12}\]  
Equation (12) can be also expressed as  
\[ \begin{split}\frac{\partial}{\partial\mathbf{\alpha}}\int_{-\infty }^{+\infty}(\mathbf{\alpha}f(t,\mathbf{\alpha}))dt&=\int_{-\infty}^{+ \infty}\frac{\partial}{\partial\mathbf{\alpha}}(\mathbf{\alpha}f(t,\mathbf{\alpha}))dt\\ =\int_{-\infty}^{+\infty}(\textbf{I}*f(t,\mathbf{\alpha})+\mathbf{\alpha }\frac{\partial}{\partial\mathbf{\alpha}}f(t,\mathbf{\alpha}))dt&= \textbf{I}+\int_{-\infty}^{+\infty}\mathbf{\alpha}\frac{\partial}{\partial\mathbf{ \alpha}}f(t,\mathbf{\alpha})dt\end{split}  \tag{13}\]  
Therefore, we have  
\[\int_{-\infty}^{+\infty}\mathbf{\alpha}\frac{\partial}{\partial\mathbf{\alpha}}f(t, \mathbf{\alpha})dt=\textbf{0} \tag{14}\]  
Computing the difference of Equations (11) and (14), we obtain  
\[\int_{-\infty}^{+\infty}(\mathbf{\hat{\mathbf{\alpha}}}-\mathbf{\alpha})\frac{\partial} {\partial\mathbf{\alpha}}f(t,\mathbf{\alpha})dt=\textbf{I} \tag{15}\]  
Because \(\frac{\partial}{\partial\mathbf{\alpha}}f(t,\mathbf{\alpha})=\frac{\partial}{\partial \mathbf{\alpha}}(\ln f(t,\mathbf{\alpha}))f(t,\mathbf{\alpha})\), Equation (15) can be rewritten as  
\[ \begin{split}\int_{-\infty}^{+\infty}(\mathbf{\hat{\mathbf{\alpha}}}- \mathbf{\alpha})\frac{\partial}{\partial\mathbf{\alpha}}(\ln f(t,\mathbf{\alpha}))f(t, \mathbf{\alpha})dt\\ =\int_{-\infty}^{+\infty}(\mathbf{\hat{\mathbf{\alpha}}}-\mathbf{\alpha}) \sqrt{f(t,\mathbf{\alpha})}\frac{\partial}{\partial\mathbf{\alpha}}(\ln f(t,\mathbf{ \alpha}))\sqrt{f(t,\mathbf{\alpha})}dt=\textbf{I}\end{split}  \tag{16}\]  
According to the Cauchy-Bunyakovsky-Schwarz inequality, the following can be derived  
\[