## 2 Methods
### Extraction of Multidimensional Features for Underwater Targets
#### 2.1.3 Distribution of Doppler Frequency Shift  
When a single-frequency acoustic wave interacts with a moving underwater volume target (typically streamlined), variations in factors such as the target's dimensions, structural geometry, radiation angle, and time-varying velocity result in differing frequency shifts produced by each volume element. This results in the formation of a Doppler frequency shift distribution. The Doppler components contained in the echo of the volume target at time \(t\) can be represented as follows:  
\[D_{T}(t,f_{d})=\int_{V_{T}}H_{D}(f,\textbf{r}_{T})dV_{T} \tag{8}\]  
where \(f_{d}\) represents the Doppler frequency shift, and the target Doppler response function \(H_{D}(f,\textbf{r}_{T})\) is a function of the signal frequency \(f\) and the position vector \(\textbf{r}_{T}\) of the volume element. Additionally, \(D_{T}(t,f_{d})\) denotes a non-stationary random process. The Doppler components in the echoes of a transmitted pulse with duration \(T_{b}\) can be expressed as follows:  
\[\begin{split} D_{T}(f_{d})&=\int_{T_{b}}\int_{V_{T }}H_{D}(f,\textbf{r}_{T},t)dVdt\\ &=\tilde{D}(f_{d})+\tilde{D}(f_{d})\end{split} \tag{9}\]  
In Equation (9), \(\tilde{D}(f_{d})\) represents the mean of the Doppler frequency shift, and \(\tilde{D}(f_{d})\) is the random component of the Doppler frequency shift. The Doppler components contained in the echoes of \(N\) source interferences can be expressed as follows:  
\[\begin{split} D_{I}(f_{d})&=\sum_{k=1}^{N}H_{D}(f, \textbf{r}_{T})\\ &=\sum_{k=1}^{N}f_{d}^{k}\end{split} \tag{10}\]  
Therefore, the Doppler components of the echoes from volume targets and multi-source interference exhibit distinct compositional differences. To extract the Doppler feature, two-dimensional correlation analyses involving time delay and frequency shifts are applied to the echo signal, as presented in Equation (11):  
\[\begin{split}\mathcal{F}_{d}(\tau,f_{d})&=\int_{- \infty}^{\infty}r_{x}(t,\tau)e^{-j2\pi f_{d}\tau}dt\\ &=\int_{-\infty}^{\infty}\hat{s}\big{(}t+\frac{\tau}{2}\big{)} \hat{s}^{*}\big{(}t-\frac{\tau}{2}\big{)}e^{-j2\pi f_{d}\tau}dt\end{split} \tag{11}\]  
Figure 5: The ETFF spectrum (RHOA = 30\({}^{\circ}\)): (**a**) multi-source interference; (**b**) volume targets.  
where \(s(t)\) represents the estimated effective echo signal. By calculating the energy distribution of \(\mathcal{F}_{d}(\tau,f_{d})\) over \((\tau,f_{d})\), the Doppler frequency shift distribution feature (DFSDF) is derived, as presented in Equation (12):  
\[\begin{split} DFSDF(\tau,f_{d})&=|\mathcal{F}_{d}( \tau,f_{d})|^{2}\\ &=\mathcal{F}_{d}(\tau,f_{d})\mathcal{F}_{d}^{*}(\tau,f_{d}) \end{split} \tag{12}\]  
The form of \(DFSDF(\tau,f_{d})\) resembles the ambiguity function \(\Psi_{x}(\tau,\varphi)\) used in the analysis of active sonar transmission waveforms [25], where \(\varphi\) denotes the frequency shift. The ambiguity function is employed to evaluate the time-frequency resolution capability of a given waveform. For pseudo-random signals, the ambiguity function exhibits a pin-shaped profile, which contrasts significantly with those of single-frequency and linear frequency modulation signals. As the Doppler components in the echo become more complex, the ambiguity function tends to approach a pinboard or pin-like shape. It can be observed that the Doppler frequency shift distribution feature of the echoes represents an alternative application of the ambiguity function. The DFSDFs for multi-source interference and volume targets are illustrated in Figure 6a and Figure 6b, respectively.