2.1.2. FDA Technique

As shown in the Figure 2, the frequency of the waveform radiated from each sensor was incremented by Δ*f* from element to element.

**Figure 2.** Schematic diagram of an FDA space structure.

By means of quadrature modulation or matched filtering, only the corresponding frequency signal is received. It is easy to obtain the phase difference between the adjacent elements (*m*th and *m*+1th), which can be written as:

$$
\Delta\!\!\!\/p = 2\pi f\_0 d \sin\theta \,/\,\text{c} - 2\pi R\_m \Delta f / \,\text{c} + 2\pi \Delta f d \sin\theta \,/\,\text{c}.\tag{5}
$$

where *Rm* is the distance from the sound source to the *m*th sensor. When the far field condition is met, *Rm* can be recorded as:

$$R\_m = R\_1 - d\sin\theta.\tag{6}$$

According to the beamforming principle [19], it can be calculated that the phase shift of Equation (5) cause the beam at some apparent angle θ [20]:

$$\theta' = \arcsin\left\{\sin\theta - \frac{R\_1\Delta f}{df\_0} + \frac{\Delta f \sin\theta}{f\_0}\right\}.\tag{7}$$

Equation (7) associates the scan angle θ , the target azimuth θ and the target distance. Therefore, FDA can estimate the DOA and distance and can also suppress clutter interference.

It should be noted that the DOA estimation of the MIMO radar and two passive hydrophones have two important differences: (1) When FDA is applied in the MIMO radar, it is used for multiple array elements, and the frequency of the transmitted signal varies with the number of elements. In the algorithm of this paper, only two array elements are used, and the received signal is sampled in the frequency domain. (2) In the MIMO radar, the waveform of each transmitted signal is known. Therefore, the initial phase of the received signal for each element at each frequency is controllable. In the algorithm of this paper, since it is applied to a passive sonar, the initial phase of each frequency is unknown. Therefore, the application of the idea of FDA to the DOA estimation of two passive hydrophones has to be greatly changed.
