2.2.2. Performance Analysis

According to the above analysis, it is easy to obtain the directivity function of the passive two-hydrophone algorithm based on FDA technology:

$$R(\theta) = \left| \frac{\sin(\pi M \Delta f \frac{d \sin \theta}{c})}{M \sin(\pi \Delta f \frac{d \sin \theta}{c})} \right|. \tag{14}$$

The azimuth resolution, based on half the width of the main lobe, is defined as in Equation (14):

$$\theta\_{\mathcal{I}} = \arcsin(\frac{c}{N\Delta f d}).\tag{15}$$

In order not to obtain a grating lobe, the scanning angle <sup>θ</sup> needs to satisfy sin(θ) <sup>≤</sup> *<sup>c</sup>* <sup>2</sup>Δ*f d* . In general, the scanning angle is −90◦ to 90◦, so when frequency domain sampling is performed on the signal, the frequency interval <sup>Δ</sup>*<sup>f</sup>* <sup>≤</sup> *<sup>c</sup>* <sup>2</sup>*<sup>d</sup>* should be satisfied.

According to the beamforming of Equation (13), the output SNR is:

$$SNR\_{out} = 10\log\frac{\sum\_{m=1}^{M} \left| X\_1(f\_m) \right|^2}{\sum\_{m=1}^{M} A\_{f\_m} Z\_N(f\_m)}.\tag{16}$$

It can be seen from Equation (16) that the less related the ambient noise between the frequencies, the higher the output SNR.
