2.1.1. Cross-Correlation Method and Cross-Spectral Method

First, the cross-correlation method is introduced [16]: the two-hydrophone receiver model is shown in Figure 1, where *d* is the array element spacing and θ is the signal incoming wave direction.

**Figure 1.** Received signal model using two hydrophones.

*x*1(*t*), *x*2(*t*) are the received signals of hydrophone 1 and hydrophone 2, respectively, and their cross-correlation functions can be expressed as:

$$R\_{x\_1x\_2}(\tau) = E[\mathbf{x}\_1(t)\mathbf{x}\_2(t-\tau)] \tag{1}$$

where *E*[•] is a mathematical expectation. When the noise and the signals are independent of each other, and the SNR is high enough, after calculating the delay τ0, corresponding to the correlation peak, the direction of arrival (DOA) estimation can be acquired, according to Equation (2):

$$
\pi\_0 = d \cos \theta / c \tag{2}
$$

where *c* is the speed of sound in water. In addition to the cross-correlation delay estimation algorithm, the commonly used algorithm also has a cross-spectral method [17]. Let the Fourier transform of *x*1(*t*) be *X*1(*f*), and the Fourier transform of *x*2(*t*) can be obtained as *X*1(*f*)*ej*2π*<sup>f</sup>* <sup>τ</sup><sup>0</sup> , according to the delay characteristic of the Fourier transform. Then, the cross-spectrum of hydrophone 1 and the hydrophone 2 can be obtained as follows:

$$Z\_X(f) = X\_1 "\left(f\right) X\_2(f) = \left| X\_1(f) \right|^2 e^{j2\pi f\tau} \tag{3}$$

It can be found, from Equation (3) that the time delay τ<sup>0</sup> is included in the phase information of the cross-spectrum, namely:

$$2\pi f d \cos \theta / c = \arctan \left\{ \frac{\text{Im}[Z(f)]}{\text{Re}[Z(f)]} \right\} \tag{4}$$

The DOA can be estimated according to Equation (4). However, such algorithms first have requirements on SNR concerning the received array signals. Secondly, for wideband signals, when using cross-correlation time delay estimation, the cross-correlation function graph shows many periodic peaks [18], which further increases the difficulty of peak finding. Therefore, implementing DOA estimation based on two hydrophones at a low SNR is very important. We found that neither of these algorithms effectively utilized the phase relationship between the frequencies. The FDA technique in a MIMO radar will be described below, which effectively utilizes the phase relationship between the frequencies.
