# A Method for Underwater Acoustic Target Recognition Based on the Delay-Doppler Joint Feature
## 2 Theory
### Convolutional Neural Network  
A basic convolutional neural network (CNN) consists of an input layer, a convolutional layer, a pooling layer, a fully connected (FC) layer, and an output layer. The convolutional layer has three main functions: it extracts features from the input data, minimizes the number of deep learning parameters, and maintains the input data space. The complete convolutional neural network structure is shown in Figure 4. Convolution kernel size \(f\), convolution kernel move step \(s\), and convolution kernel number _filter_ are some of the fundamental parameters that make up the convolution layer's basic structure. To determine the output value of the window, the convolution kernel is applied in the \(6\times 6\) window that  
Figure 3: Relationship between Time–Frequency and Delay-Doppler domains.  
Figure 2: The conversion relationship between the signal’s Time–Frequency and Delay-Doppler domains.  
corresponds to the input data. The count of convolution kernel skips in the input matrix is known as the moving step. The convolution kernel window moves \(s\) units to carry out the convolution operation on the corresponding window data again, and the convolution data are obtained after traversing and convolutioning all the input data.  
The convolution process essentially multiplies the elements between the convolution kernel and the corresponding input data and finally sums all multipliers. \(X_{ij}\) and \(K_{ij}\) are the values of the input data and the convolution check, and \(R\) is the convolution data. The precise formula is as follows:  
\[R=\sum_{i=1}^{n}\sum_{j=1}^{n}X_{ij}K_{ij} \tag{3}\]  
The data are processed using convolution and pooling layers, and then, an FC layer is used to minimize the dimensionality of the data and create the feature matrix. Next, the SoftMax layer is used to categorize and recognize the feature matrix. The SoftMax layer is a classifier used for deep learning. It can map the output classification result to the range of zero to one. The likelihood that the sample will belong to the class is represented by the probability of the output-related result. The SoftMax layer is the classification layer of the CNN. Let n be the number of target categories to be classed and identified. The output vector obtained through the FC layer is \(f(x)\). The output result of the SoftMax layer is shown in Formula (4) [25], where \(p_{i}(x)\) represents the output probability of the \(i\) type of target and \(f_{i}(x)\) represents the output vector of the target FC layer of type \(i\). Finally, SoftMax selects the target with the highest probability \(\{p_{i}\}\) as the output value of the signal, which is the label of the target identified by the model.  
\[p_{i}(x)=\frac{e^{f_{i}(x)}}{\sum_{j=1}^{n}e^{f_{i}(x)}},i=1,\ 2,\ldots,n \tag{4}\]  
The process of convolution and pooling is essentially a process of feature extraction and data compression. The selection of a convolution kernel with different attributes can extract different dimensional data information. Therefore, the CNN is a kind of network widely used in classification.