2.2.3. Algorithm Extension: Three Hydrophones

From the above theoretical derivation, we can find that the passive two-hydrophone algorithm based on FDA technology is similar to the single-frequency signal processing algorithm of a conventional towed-line array. Similarly, the algorithm can be extended to higher dimensions, for instance, using a wideband signal to obtain a performance similar to the single-frequency processing of a circular array. The specific process is as follows:

Assume that the three hydrophones, a, b, and c, have a radius of *R*. The angle with the reference abscissa are θ*a*, θ*<sup>b</sup>* and θ*c*, respectively. The incident angle of the far-field sound source is θ0. According to the spatial structure of the three hydrophones, as shown in Figure 3, the frequency domain model of the received signals can be obtained as follows:

$$\begin{aligned} S\_a(f) &= e^{j2\pi f(-R\cos(\theta\_a - \theta\_0) + r)/c + \varphi(f)} \\ S\_b(f) &= e^{j2\pi f(-R\cos(\theta\_b - \theta\_0) + r)/c + \varphi(f)} \\ S\_c(f) &= e^{j2\pi f(-R\cos(\theta\_c - \theta\_0) + r)/c + \varphi(f)} \end{aligned} \tag{17}$$

where *r* is the propagation distance. Similarly, we also calculate the cross-spectrum to remove the initial phase:

$$\begin{cases} Z\_{\rm ab}(f\_1) = S\_{\rm a}(f\_1) \cdot S\_{\rm b}"\left(f\_1\right) \\ Z\_{\rm ac}(f\_m) = S\_{\rm a}(f\_m) \cdot S\_{\rm c}"\left(f\_m\right), \ m = 1, \ 2, \ 3, \ldots, M \end{cases} \tag{18}$$

where *f* <sup>1</sup> is the starting frequency, and *fm* is the frequency of the sampling point. Bring Equation (17) into Equation (18) and expand the cosine term to obtain:

$$\begin{aligned} Z\_{\text{ab}}(f\_1) &= e^{j2\pi f\_1 \mathbb{R} \left( (\cos(\theta\_b) - \cos(\theta\_t))\cos(\theta\_0) + (\sin(\theta\_b) - \cos(\theta\_t))\sin(\theta\_0) \right)/c} \\ Z\_{\text{ac}}(f\_m) &= e^{j2\pi f\_m \mathbb{R} \left( (\cos(\theta\_t) - \cos(\theta\_t))\cos(\theta\_0) + (\sin(\theta\_t) - \cos(\theta\_t))\sin(\theta\_0) \right)/c} \end{aligned} \tag{19}$$

Let *A*<sup>1</sup> = cos(θ*b*) − cos(θ*a*) and *B*<sup>1</sup> = sin(θ*b*) − sin(θ*a*), and then take the conjugate of two cross-spectra and multiply:

$$Y\_{m} = Z\_{\text{ab}}(f\_1) \cdot Z\_{\text{ac}}\, ^\ast(f\_m) \tag{20}$$

Let <sup>γ</sup> <sup>=</sup> arccos <sup>√</sup> *<sup>A</sup>*<sup>3</sup> *A*3 <sup>2</sup>+*B*<sup>3</sup> 2 , *A*<sup>3</sup> = *f*1*A*<sup>1</sup> − *fmA*2, *B*<sup>3</sup> = *f*1*B*<sup>1</sup> − *fmB*2, and Equation (20) be transformed into:

$$\mathcal{Y}\_I = e^{j2\pi R} \sqrt{A\_3^{\cdot 2} + B\_3^{\cdot 2}} \cos(\gamma - \theta\_0) / c \tag{21}$$

In order to construct a circular array manifold, i.e., *ej*2π*R*cos(*m*θ*s*−θ0))/*<sup>c</sup>* , where θ*<sup>s</sup>* = 2π/*M*. Let γ = *m*θ*s*, Equation (21) can be obtained:

$$\cos m\theta\_s = \frac{A\_3}{\sqrt{A\_3^2 + B\_3^2}}\tag{22}$$

According to Equation (23), *fm* can be solved:

$$f\_{\rm m} = \frac{B\_1 \cos(m\theta\_s) - A\_1 \sqrt{1 - \cos^2(m\theta\_s)}}{B\_2 \cos(m\theta\_s) - A\_2 \sqrt{1 - \cos^2(m\theta\_s)}} f\_1 \tag{23}$$

Therefore, it is only necessary to know the coordinates of the three array elements, and it is easy to obtain *R*, θ*a*, θ*b*, θ*<sup>c</sup>* according to the geometric relationship. The target position can be estimated. Beamforming is shown in Equation (24):

$$\text{Beam}(\theta) = \sum\_{m=1}^{M} Z\_{\text{ab}}(f\_1) Z\_{\text{ac}} ^\ast(f\_m) e^{j2\pi R} \sqrt{A y^2 + B y^2} \cos(m \theta\_\circ - \theta\_0) / \varepsilon \tag{24}$$

In this way, we have implemented a wideband signal based on the azimuth estimates of the three hydrophones. Because the algorithm is designed with reference to the circular array, there is no problem with port and starboard ambiguity.

**Figure 3.** Schematic diagram of the spatial structure of the three hydrophone.
