Two-dimensional direction finding of coherent signals with a linear array of vector hydrophones

Abstract

In this paper, we propose a parallel factor (PARAFAC) analysis based two dimensional direction finding algorithm for coherent signals using a uniformly linear array of vector hydrophones. By forming a PARAFAC model using spatial signature of vector hydrophone array, the new algorithm requires neither spatial smoothing nor vector-field smoothing to decorrelate the signal coherency. We also establish that the azimuth-elevation angles of K coherent signals can be uniquely determined by PARAFAC analysis, as long as the number of hydrophones $L \ge 2K - 1$. In addition, because the vector hydrophone array manifold contains no phase information, this new algorithm can offer high azimuth-elevation estimation accuracy by setting vector hydrophones to space much farther apart than a half-wavelength. Simulation results are finally presented to verify the efficacy of the proposed algorithm.

Appendix

In this appendix, we will show that for an vector hydrophone, every two vector hydrophone response vectors with distinct azimuth-elevation angles are linear independent. For the problem under consideration, the values of $\theta _k$ and $\phi _k$ are confined as $\theta _k \in [0, \pi /2)$ and $\phi _k \in [0, 2\pi )$. Since $\mathbf{c}_k = \mathbf{c}_{\ell }$ if $\theta _k = \theta _{\ell } = 0$, regardless of the values of $\phi _k$ and $\phi _{\ell }$. Thus, the azimuth-elevation angles of the $k\hbox {th}$ and the $\ell \hbox {th}$ signals are distinct if

$$\begin{aligned} \left\{ \begin{array}{ll} (\theta _k, \phi _k) \ne (\theta _{\ell }, \phi _{\ell }) &{} \mathrm{if} \ \theta _k \ne 0 \\ \theta _k \ne \theta _{\ell } &{} \mathrm{if} \ \theta _k = 0 \\ \end{array} \right. \end{aligned}$$

Next, consider the following cases

• 1. $\theta _1 = 0$, $\phi _2 \in \{\pi /2, 3\pi /2\}$. In this case, the determinant of the matrix

  $$\begin{aligned} \mathrm{det}\left[ \begin{array}{cc} 1 &{}\quad 1 \\ \sin \theta _1 \sin \phi _1 &{}\quad \sin \theta _2 \sin \phi _2 \\ \end{array} \right] \ne 0 \end{aligned}$$

  Therefore, $\mathbf{c}_1$ and $\mathbf{c}_2$ are linear independent.

• 2. $\theta _1 = 0$, $\phi _2 \in \{0, \pi \}$. In this case, the determinant of the matrix

  $$\begin{aligned} \mathrm{det}\left[ \begin{array}{cc} 1 &{}\quad 1 \\ \sin \theta _1 \cos \phi _1 &{}\quad \sin \theta _2 \cos \phi _2 \\ \end{array} \right] \ne 0 \end{aligned}$$

  Therefore, $\mathbf{c}_1$ and $\mathbf{c}_2$ are linear independent.

• 3. $\theta _1 \ne 0$, $\theta _2 \ne 0$, $\phi _2 \ne m\pi /2, m = 0, 1, 2, 3$, $\phi _2 = \phi _1 + n \pi , n = 0, 1$. In this case, the determinant of the matrix

  $$\begin{aligned} \mathrm{det}\left[ \begin{array}{cc} 1 &{}\quad 1 \\ \sin \theta _1 \cos \phi _1 &{}\quad \sin \theta _2 \cos \phi _2 \\ \end{array} \right] \ne 0 \end{aligned}$$

  Therefore, $\mathbf{c}_1$ and $\mathbf{c}_2$ are linear independent.

• 4. $\theta _1 \ne 0$, $\theta _2 \ne 0$, $\phi _2 \ne m\pi /2, m = 0, 1, 2, 3$, $\phi _2 \ne \phi _1 + \pi$. In this case, the determinant of the matrix

  $$\begin{aligned} \mathrm{det}\left[ \begin{array}{cc} \sin \theta _1 \cos \phi _1 &{}\quad \sin \theta _2 \cos \phi _2 \\ \sin \theta _1 \sin \phi _1 &{}\quad \sin \theta _2 \sin \phi _2 \\ \end{array} \right] = \sin \theta _1 \sin \theta _2 (\cos \phi _1 \sin \phi _2 - \sin \phi _1 \cos \phi _2) \ne 0 \end{aligned}$$

  Therefore, $\mathbf{c}_1$ and $\mathbf{c}_2$ are linear independent.

Finally, we combine all the above results and conclude that for all distinct $(\theta _1, \phi _1)$ and $(\theta _2, \phi _2)$, the vectors $\mathbf{c}_1$ and $\mathbf{c}_2$ are linear independent.