# Distinction of Scrambled Linear Block Codes Based on Extraction of Correlation Features
## 4 Scrambled Linear Block Code Identification Based on Correlation Features Extraction Network
### Network Training Process  
The convolution kernel of convolutional layer 1 is \(2\times 1\). Since the input sample is a sequence of the code elements of the receiver signal and autocorrelation features, which form the real and imaginary parts of the input sample, respectively, the input sample is not the received sequence \(z=[z(0),z(1),\ldots,z(k-1)]\). Therefore, it is necessary to first combine the real and imaginary parts of the scrambled sequence and to then extract the features based on the correlation.  
As can be seen in Figure 3, when the \(1\times 2\)-dimensional convolution kernel is used to extract the signal features of the scrambled sequences, the deep features obtained from their convolution must show different patterns due to the different correlations of the linear block code, the self-synchronization of the linear block code, and the synchronization of the linear block code. In fact, the process of the convolution of the received sequence using a \(1\times 2\)-dimensional convolution kernel is similar to the calculation of its correlation features under time delay. However, for the self-synchronization of linear block code and the synchronization of linear block code, the correlation features of code elements separated by more than two code elements are not as obvious as those of two adjacent code elements, so convolutional kernels with lengths longer than \(1\times 2\) dimensions do not have this differentiation advantage, and they increase the parameters of network training and increase network complexity. Therefore, the \(1\times 2\)-dimensional convolution kernel is used to convolve the layer 1 output features in order to obtain correlation features that are more consistent with the nature of scrambled signals.  
The weight and bias training update of the correlation feature extraction network is mainly divided into two processes: forward propagation and back propagation. Among them, the forward propagation process is to use the input training samples in order to calculate the neuron activation value of the network, and the back propagation process is to perform a reverse calculation in order to obtain the gradient corresponding to the weight and bias of each error; finally, the gradient-descent algorithm is used to calculate the weight and to offset update adjustment.  
Consider the training data \(\left\{\left(y^{(1)},q^{(1)}\right)\right\}\), \(\left(y^{(2)},q^{(2)}\right)\), \(\cdots\), \(\left(y^{(m)},q^{(m)}\right)\right\}\) with a sample size of \(m\), where \(y^{(i)}(i=1,2,\ldots,m)\) is the input scrambled data, and \(q^{(i)}(i=1,2,\cdots,m)\) is the type label corresponding to the scramble code. The \(W\) and \(b\) update formulas are, respectively, expressed as follows:  
\[W^{t+1}=W^{t}-\alpha\frac{\partial J(W,b;y,q)}{\partial W^{t}}, \tag{18}\]  
\[b^{t+1}=b^{t}-\alpha\frac{\partial J(W,b;y,q)}{\partial b^{t}}. \tag{19}\]  
where \(\alpha\) is the learning rate, and \(J(\cdot)\) is the loss function. The loss function used is categorical_crossentropy, which is a loss function for mul-ticlass classification tasks and is suitable for the multiclass problem.  
The expression of \(J(\cdot)\) is  
\[J(W,b;y,q)=-\frac{1}{m}\sum_{i=1}^{m}q^{(i)}\cdot\log\Bigl{(}f\Bigl{(}W,b;y^ {(i)}\Bigr{)}\Bigr{)}+\lambda{\sum{\left\|W\right\|}^{2}}. \tag{20}\]  
where \(\lambda\) is the regularization coefficient used to prevent the network from overfitting.  
The input \(x_{m}^{(l)}(j)\) and output \(y_{m}^{(l)}(j)\) of each layer of neurons in the network have the following relationship:  
\[y_{m}^{(l)}(j)=f\Bigl{(}x_{m}^{(l)}(j)+b_{m}^{(l)}(j)\Bigr{)}. \tag{21}\]  
where \(l\) is the layer number of the network, \(j\) is the \(j\)-th neuron of the feature map, \(m\) is the \(m\)-th feature map in the network layer number, \(f(\cdot)\) is the activation function used by this layer, and \(b\) is the bias.