## 2 Problem Description
### Array Signal Model  
The uniform linear vector hydrophone model studied in this paper consists of a series of vector hydrophones arranged in a straight line at equal intervals. Each vector hydrophone is capable of simultaneously measuring both the pressure component and the particle velocity components of the acoustic field, thus providing rich information about the acoustic environment. This model boasts advantages such as a simple structure, ease of deployment and maintenance. Through signal processing techniques such as beamforming and noise suppression, it enables precise detection, localization, and tracking of underwater targets. Furthermore, the uniform linear vector hydrophone model has broad application prospects in marine environment monitoring, seabed terrain exploration, and marine ecological research, providing powerful technical support for research and applications in the field of underwater acoustics. In this paper, we use the uniform linear array (ULA) as shown in Figure 2 to simulate hydrophone array signal reception under far-field narrowband conditions. There are \(M\) elements in a ULA and each element corresponds to an individual channel. The array element spacing \(d\) is half of the signal wavelength. Assuming that the signal source and array are in the same plane, the incident signal can be simulated as a plane wave. This assumption is grounded in the principles of wave propagation. When the distance between the signal source and the receiver array is considerably larger than the physical dimensions of the array, the curvature of the wavefront that emanates from the source becomes negligible as it approaches the array. Under these conditions, the wavefront can be approximated as a plane, or more specifically, as a plane wave. Simulating the incident signal as a plane wave in the context of signal processing and array signal analysis is a valid and widely adopted assumption. This assumption simplifies the mathematical modeling and enables the extraction of critical information from the received signals. Figure 1: Four disrupted signals (**left**) and their corresponding noise spectrum (**right**). We take the first array element as the reference, so the signals reaching different antennas will have wave path difference \(D_{m}\); it can be formulated as follows:  
\[D_{m}=(m-1)d\sin\theta,m=1,2,\ldots,M \tag{1}\]  
The difference of the wave path leads to the time difference of the received signal. Assuming the speed of signal propagation is \(v\), the time difference of the signal received by the m-th array element can be expressed as \(\tau_{m}=\frac{(m-1)d\sin\theta}{v}\). The frequency domain phase difference of the signal received by the array element can be expressed as \(\beta=e^{-\
icefrac{{\mu}}{{2\pi}}\frac{{\pi(m-1)d\sin\theta}}{\lambda}}\) with \(\lambda=\frac{v}{j}\), where \(\lambda\) is the wavelength and \(f\) is the frequency of signal. So, we obtain the flow pattern matrix of the array \(A\); it can be formulated as follows:  
\[A(\theta)= \begin{bmatrix}1&1&\cdots&1\\ e^{-\
icefrac{{\mu}}{{2\pi}}\frac{{d\sin\theta_{1}}}{\lambda}}&e^{-\
icefrac{{ \mu}}{{2\pi}}\frac{{d\sin\theta_{2}}}{\lambda}}&\cdots&e^{-\
icefrac{{\mu}}{ {2\pi}}\frac{{d\sin\theta_{1}}}{\lambda}}\\ \vdots&\vdots&\vdots&\vdots\\ e^{-\
icefrac{{\mu}}{{2\pi}}\frac{{(M-1)d\sin\theta_{1}}}{\lambda}}&e^{- \
icefrac{{\mu}}{{2\pi}}\frac{{(M-1)d\sin\theta_{1}}}{\lambda}}&\cdots&e^{- \
icefrac{{\mu}}{{2\pi}}\frac{{(M-1)d\sin\theta_{1}}}{\lambda}}\end{bmatrix}  \tag{2}\]  
where \(L\) is the number of narrowband far-field sources and \(j\) is \(\sqrt{-1}\). In this paper, colored noise is used to simulate the actual noise environment. The original signal vector and noise vector at the \(t\)-th snapshot can be expressed as:  
\[S(t)= \begin{bmatrix}s_{1}(t)\\ s_{2}(t)\\ \vdots\\ s_{L}(t)\end{bmatrix} ,N(t)= \begin{bmatrix}n_{1}(t)\\ n_{2}(t)\\ \vdots\\ n_{M}(t)\end{bmatrix}