2.2.1. Theory

According to the model in Figure 1, the frequency domain expressions of the received signals of the two hydrophones are:

$$\begin{array}{l} S\_1(f) = \begin{vmatrix} X\_1(f) \end{vmatrix} e^{j[2\pi f r/c + \phi(f)]} + N\_1(f) \\ S\_2(f) = \begin{bmatrix} X\_1(f) \end{bmatrix} e^{j[2\pi f d \cos 0/c + 2\pi f r/c + \phi(f)]} + N\_2(f) \end{array} \tag{8}$$

where *N*1(*f*) and *N*2(*f*) are the ambient noise received by hydrophone 1 and hydrophone 2, respectively. φ(*f*) is the random, frequency-dependent phase of the source. It can be observed, from the above equation, that in the phase information of *S*2(*f*), the first item contains the azimuth information of the target, the second term relates to the propagation distance, and the third term is the initial phase of the frequency. Since, in the application condition of the passive sonar, the target distance and the initial phase of the sound source signal are unknown, we first calculate the cross-spectrum of the two sensor signals to remove the phase in the second and third terms:

$$Z(f) = \mathbb{S}\_1 \, ^\*(f)\mathbb{S}\_2(f) = Z\_X(f) + Z\_N(f) \tag{9}$$

where *ZX*(*f*) is the cross-spectrum of the signal, and *ZN*(*f*) is the component related to environmental noise. *ZX*(*f*) and *ZN*(*f*) are denoted as:

$$\begin{aligned} Z\_X(f) &= \left| X\_1(f) \right|^2 e^{j2\pi f d \cos 0/\varepsilon} \\ Z\_N(f) &= X\_1^\*(f) N\_2(f) + N\_1^\*(f) X\_2(f) + N\_1^\*(f) N\_2(f) \end{aligned} \tag{10}$$

After obtaining the cross-spectrum *Z*(*f*), *Z*(*fm*) is obtained by sampling *Z*(*f*) in the frequency domain. According to the idea of the frequency diversity technique, the frequency of the sampling point is *fm*, and the frequency increment is Δ*f*. Vector [*Z*(*f*1),*Z*(*f*2), ... ,*Z*(*fM*)] can be generated as:

$$Z(f\_m) = \left| X\_1(f\_m) \right|^2 e^{j2\pi f\_m d \cos \theta / \varepsilon} + Z\_N(f), \\ f\_m = f\_1 + (m - 1)\Delta f. \tag{11}$$

It can be found, from Equation (11), that the phase difference of *ZX*(*fm*) between the adjacent sampling points is *j*2πΔ*f d* cos θ/*c*. There is no phase relationship between the various frequencies of ambient noise. Array manifolds are generated by the phase relationship in Equation (12):

$$A\_{f\_m} = \mathfrak{e}^{-j2\pi m\Delta f d \cos 0/c}.\tag{12}$$

The beamforming output can be obtained according to the principle of in-phase superposition:

$$\text{Beam}(\theta) = \sum\_{m=1}^{M} Z(f\_m) A\_{f\_m} \tag{13}$$
