## 2 Automatic Tracking Method of Weak Acoustic Targets within Jamming Environment
### De-Nosing  
The PCA method is used in this study to denoise BTR diagrams. There exists a transforming matrix \(P\) in a sample matrix \(X=[x_{1},x_{2},\ldots,x_{m}]^{T}\) in \(n\) dimensions that can lower its dimension number from \(n\) to \(l\) by transforming \(Y=PX\), where \(Y=[y_{1},y_{2},\ldots,y_{m}]^{T}\). PCA enables the reduction of information loss by dimensionality reduction. It can reduce the correlation between samples in order to keep the data diverse in each dimension; this is  
Figure 2: Frame of the proposed method after denoising.  
accomplished by maximizing variance across all dimensions. The covariance matrix of the reduced dimension sample set \(Y\) is  
\[C_{Y}^{l\times l}=\frac{1}{m}YY^{T}=\left(\begin{array}{ccc}\lambda(y_{1},y_{1 })&\cdots&\lambda(y_{1},y_{l})\\ \vdots&\ddots&\vdots\\ \lambda(y_{l},y_{1})&\ldots&\lambda(y_{l},y_{l})\end{array}\right) \tag{3}\]  
where the diagonal elements are the variances of the random variables, and the non-diagonal elements are the covariances between the random variables. According to the principle of PCA, the covariance should be zero, whereas the variance should be non-zero, which means the covariance matrix is a diagonal matrix.  
Substitute \(Y=PX\) into the covariance matrix \(C_{Y}^{l\times l}\)  
\[\begin{split}& C_{Y}^{l\times l}=\frac{1}{m}YY^{T}=\frac{1}{m}PX( PX)^{T}\\ &=P\frac{1}{m}XX^{T}P^{T}=P_{l\times n}C_{X}^{l\times l}P_{l\times n }^{T}\end{split} \tag{4}\]  
where \(C_{X}^{l\times l}\) is the covariance matrix of \(X\). Compute the eigenvalue decomposition of \(C_{X}^{l\times l}\)  
\[\Sigma_{n\times n}=U_{n\times m}C_{X}^{l\times l}U_{n\times n} \tag{5}\]  
Equations (4) and (5) demonstrate that the transformation matrix \(P_{l\times n}\) is composed of the first \(l\) eigenvectors of matrix \(C_{X}^{l\times l}\) in rows. The principal component analysis therefore can be performed by computing the eigenvalue decomposition of \(C_{X}^{l\times l}=\frac{1}{m}XX^{T}\).  
The singular decomposition (SVD) of rectangle matrix \(A\in R^{m\times n}\) is  
\[A_{m\times n}=U_{m\times m}\Sigma_{m\times n}V_{n\times n}^{T} \tag{6}\]  
where \(U_{m\times m}\) and \(V_{n\times n}\) are orthogonal matrices, \(\Sigma_{m\times n}=diag(\sigma_{1},\ldots,\sigma_{n})\) is a diagonal matrix and \(\sigma_{i}\) is the singular value of \(A_{m\times n}\). Then,  
\[\begin{split}& A_{m\times n}^{T}A_{m\times n}=(U_{m\times m} \Sigma_{m\times n}V_{n\times n}^{T})^{T}U_{m\times m}\Sigma_{m\times n}V_{n \times n}^{T}\\ &=V_{n\times n}^{T})\Sigma_{m\times n}^{T}U_{m\times m}^{T}U_{m \times m}\Sigma_{m\times n}V_{n\times n}^{T}=V_{n\times n}(\Sigma_{m\times n} )^{2}V_{n\times n}^{T}\end{split} \tag{7}\]  
where \(\Sigma_{m\times n}^{2}\) is a diagonal matrix composed of \(A_{m\times n}^{T}A_{m\times n}\)'s eigenvalues \(\lambda_{i}\). It can also be seen that the singular value \(\sigma_{i}\) of \(A_{m\times n}\) is the square root of \(A_{m\times n}^{T}A_{m\times n}\)'s eigenvalue \(\lambda_{i}\). Therefore, either PCA or SVD can be effected by computing the eigenvalue decomposition of a real symmetric matrix. Let \(A\) be \(\frac{X^{T}}{\sqrt{m}}\); it yields  
\[AA^{T}=\frac{X^{T}}{\sqrt{m}}^{T}\ \frac{X^{T}}{\sqrt{m}}=\frac{1}{m}XX^{T} \tag{8}\]  
Due to this equivalence relation, the PCA problem can be solved via SVD, which is more efficient because it avoids the calculation of \(AA^{T}\).