Research on modulation recognition method of electromagnetic signal based on wavelet transform convolutional neural network

2.1.1. Electromagnetic signal modulation model

Analyzing the electromagnetic signal modulation model [15] is helpful to improve the efficiency of subsequent electromagnetic signal modulation recognition. The modulated electromagnetic signal is usually sampled and stored in the form of IQ (Intelligence Quotient) data, so the received electromagnetic signal can be expressed as:

Among them: ${\gamma }_Q\left[\psi \right]$ and ${\gamma }_{\psi }\left[\psi \right]$ respectively represent the $\psi$ orthogonal component and in-phase component of an electromagnetic signal and a set of sample values, $\mathrm{{\rm Y}}$ represents a constant.

With the complexity of electronic environment and physical environment, electromagnetic signals are seriously distorted in poor communication environment. There are many kinds of interference noises in the process of electromagnetic signal transmission. In this paper, the noise is divided into additive noise $o_1$ and multiplicative noise $o_2$ according to the relationship between signal and noise, whose expression is:

where, $m\left(t\right)$ represents white noise in electromagnetic signals; $o\left(t\right)$ represents electric field intensity. The sampling rate is 8 kHz, the sensitivity is 50 V/m, and the measurement unit is the electric field strength.

Because the modulation recognition accuracy of electromagnetic signals is at a very low level and it is difficult to improve when they are directly recognized under different signal-to-noise ratios. Therefore, empirical mode decomposition and variational mode decomposition methods were not effective in denoising electromagnetic signals in the past [16]. Therefore, wavelet transform will be combined to denoise electromagnetic signals with serious distortion, which will lay the foundation for electromagnetic signal modulation identification of convolutional neural networks and improve the accuracy of electromagnetic signal modulation identification under different signal-to-noise ratios.

2.1.2. Wavelet transform electromagnetic signal denoising processing

Based on the analysis of the electromagnetic signal modulation model, the original electromagnetic signal is decomposed by wavelet transform, and the high-frequency electromagnetic signal with small amplitude and the low-frequency electromagnetic signal with large amplitude can be obtained. Through the mapping of threshold function, a series of wavelet coefficient prediction values are obtained, and these prediction values are used for reconstruction to achieve the purpose of denoising.

The general form of wavelet basis function is as follows:

where: constant $c$ can change the scale of the wavelet basis function and change the time-frequency resolution. Constant $d$ causes the wavelet basis function to translate in time. In this paper, continuous wavelet transform is used, and the complex Morlet function is selected by wavelet basis. The basic steps are as follows:

(1) The original electromagnetic signal $o\left(t\right)$ and wavelet basis function ${\mathrm{\Gamma }}_{c,d}\left(t\right)$ performing inner product operation to obtain wavelet coefficients:

The value of $d$ is changed in a step of $k$ to shift the wavelet base until the coverage of the modulated electromagnetic signal $o\left(t\right)$ ends.

(3) Changing the wavelet scale $c$, repeat steps (1) and (2) to obtain wavelet coefficients with different resolutions.

The main methods of wavelet denoising threshold selection are fixed threshold, adaptive threshold, heuristic threshold, maximum and minimum threshold. The specific process of threshold selection is that the mother wavelet function can decompose the original electromagnetic signal into n details ${\chi }_i$ and $n$ profiles $a_i$, if the threshold of ${\chi }_i$ is $W$, then:

where: ${\chi }_{ij}$ is a wavelet coefficient without threshold processing; ${\stackrel{-}{\chi }}_{ij}$ is the truncated value of ${\chi }_{ij}$. From the threshold processing results and the $n$th layer $a_n$ reconstruct the electromagnetic signal, and the denoising result of the electromagnetic signal can be obtained. Set $\epsilon \left(x\right)$ is a scale function of multi-resolution analysis, and its specific expression is as follows:

where: $\epsilon \left(x\right)$ is a scale function; $g$ is a real number; $2x-g$ is the standard orthogonal basis of multi-resolution analysis; $P_g$ is the scale coefficient. In the construction of wavelet, it is not necessary to know the specific forms of scale function and wavelet function, but only to know $P_g$ we can calculate the wavelet decomposition and recovery.

When processing wavelet packet coefficients, it is easy to produce pseudo-Gibbs phenomenon by using hard threshold function to reduce noise; Soft threshold function is used to reduce noise, which is easy to cause distortion such as edge blur. Therefore, in view of the defects of soft and hard threshold methods, combined with the energy distribution characteristics of wavelet packet coefficients in each frequency band after decomposition, a new threshold function is constructed as follows:

where: $\varpi$ is the wavelet coefficient value; ${\varpi }_{\tau }$ is the wavelet coefficient value after the threshold is applied; $e$ is any normal number; $\tau$ is the set threshold. The threshold function can be changed by changing $\tau$ value to get any function between soft and hard threshold functions.

In order to improve the denoising effect of electromagnetic signals, wavelet transform needs to construct a two-scale equation of Daubechies wavelet [17], which is specifically expressed as:

Among them: $\eta \left(x\right)$ is a wavelet function; $Z$ is an integer set; $P_g$ is the scale coefficient.

The process of constructing the two-scale equation of Daubechies wavelet is as follows:

Step 1: Select a positive integer that $n\ge 2$.

Step 2: Select a polynomial with the following expression:

Among them: $H$, $P_n\left(s\right)$ is algebraic polynomials with real coefficients; $s$ is a real number between [0, 1].

Step 3: Select algebraic polynomials with real coefficients.$Q\left(Z\right)$, make$P\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^2\frac{\varpi }{2}\right)={\left|Q\left(e^{\varpi }\right)\right|}^2$, in which:$\varpi =0~2\pi$.

Step 4: Take$\tilde{V}\left(o\right)=Q\left(o\right)\left(\frac{1+o}{2}\right)$, then $\tilde{V}\left(o\right)=\frac{\sqrt{2}}{2}\sum _{g=0}^L{P}_g{o}^g$, in which: $L$ is the maximum value of non-zero data in a two-scale sequence; $o$ is Gaussian white noise; $\tilde{V}\left(o\right)$ is a low-pass filter with linear phase. The smoothness of the Daubechies wavelet increases with the number of decomposition layers $n$.

The specific process of denoising the original electromagnetic signal by wavelet transform is shown in Fig. 1.

Fig. 1Data processing flow of electromagnetic signal denoising based on wavelet transform

As can be seen from Fig. 1, the data processing flow of electromagnetic signal denoising based on wavelet transform mainly includes three parts: wavelet decomposition, threshold quantization of decomposed high-frequency coefficients and wavelet reconstruction. The analog signals of all channels are collected, and the decomposition levels are determined. At the same time, the scale equation and wavelet threshold are selected as the mother wavelet of electromagnetic signal decomposition. The collected original electromagnetic signals are processed according to the decomposition algorithm, and the high-frequency electromagnetic noise signal coefficients are filtered to obtain useful electromagnetic signals, and these useful electromagnetic signals are reconstructed to obtain the denoised electromagnetic signals. The adaptive threshold selection method can dynamically adjust the threshold according to the local characteristics of the signal to distinguish the noise from the signal more effectively. Sparse representation theory holds that real signals can be represented with fewer non-zero coefficients under certain wavelet bases. Therefore, the signal can be filtered by using the sparsity property of the signal combined with the wavelet transform. For example, an optimization algorithm based on sparse constraints, such as compressed sensing, can be used to improve the filtering effect. The multi-scale decomposition and reconstruction methods are tried to better preserve the information of signals in different frequency ranges.

2.2.1. Convolution extends through the network

Convolution stretching network is a kind of feedforward neural network [18], and the basic network model is generally composed of multiple convolution layers, pooling layers and fully connected layers. The convolution layer acts as a filter, extracting features from input data, and extracting new features by weighted summation of neurons in the upper layer. The calculation formula is as follows:

Among them: ${\alpha }_j$ represents the input of the convolutional layer of layer $j$, and this paper refers to the eigenvectors carried by electromagnetic signals; $b_j$ represents the offset value of the layer $j$; ${\alpha }_j$ represents the weight of the layer $j$.

The activation function layer is usually located after the convolution layer. The extracted linear features are mapped into nonlinear features by activating functions. Among them, the activation function ReLU is widely used in convolutional neural networks. The calculation formula is as follows:

Among them: ${\alpha }_{j+1}$ represents the input of the layer $j+1$, $f\left({\alpha }_{j=1}\right)$ represents the output obtained by nonlinear mapping of the input by the activation function.

Pool layer is mainly responsible for down sampling. By calculating the nodes in different regions of the feature, the dimension of the nonlinear feature map is reduced. The advantage of this operation method is that it can reduce parameters, reduce computational complexity and prevent over-fitting. The calculation formula is as follows:

Among them: ${\beta }_v$ represents the output obtained by maximizing the pool of nonlinear features.

The fully connected layer is to map features to the sample label space and integrate features together. Classify by softmax, express the output as probability, select the maximum probability as the prediction label and output it to complete the classification. The calculation formula is as follows:

Among them: ${\widehat{\beta }}_v$ represents a prediction tag, ${\beta }_{\theta }$ represents the class $\theta$ sample; $K$ represents the number of categories.

The training process of convolutional neural network is as follows:

Step 1: Parameter initialization. Before the convolutional neural network starts training, the parameters of each node need to be initialized. The method adopted is to initialize each parameter to a random value close to 0.

Step 2: Parameter updating. Convolutional neural network uses backward propagation gradient descent to make the parameters of each node gradually approach the optimal solution [19]. After the parameters of each node are initialized, the model prediction results are obtained by forward calculation, and the residual of each node in the last layer of the network is calculated according to the predicted values, and then the residual of each node in each layer is calculated reversely from the last layer, so as to obtain the partial derivative of the loss function to the parameters, and then the parameters are updated.

The concrete process of calculating the residual error and updating the parameters of CNN network: The labeled sample data set used for training convolutional neural network is actually a collection of several input-output pairs, and the loss function is used to measure the deviation between the predicted output value and the actual output value of convolutional neural network during training. The smaller the loss function, the better the model performance. Commonly used loss functions include 0-1 loss function, absolute loss function and mean square error loss function. The definition of residual is the loss function to the$j$floor level$i$partial derivatives of the nodes are used to calculate the partial derivatives of the loss function to the parameters. The formula of residual is as follows:

Among them:$R$represents loss function; $\partial {\upsilon }_i^{\left(j\right)}$ represents the $j$ floor level $i$ nodes; ${\sigma }_i^{\left(j\right)}$ represents the residual of the node.

The expression of the residual of each node in the last layer is as follows:

Among them: ${\sigma }_i^{\left(j\right)}$ represents the actual value; $f\text{'}$ represents that derivative of the activation function; ${\sigma }_i^{\left(j\right)}$ represents the $U$ floor level $i$ nodes; ${\delta }_i^{\left(U\right)}$ represents the activation value of the last node.

The calculation formula of node residual in other layers is as follows:

Among them: $w_{\varsigma i}^{\left(j-1\right)}$ represents the weight of nodes $j-1$ floor level $i$ node to node $j$ floor level $\varsigma$ node.

Using the residual of each node, the partial derivative of the loss function to the parameter can be calculated, and the specific solution formula is:

Among them: $b_i^{\left(j\right)}$ represents the $j$ floor level $i$ node offset.

Finally, the update is implemented according to the parameter update, and the parameter update is as follows:

Among them: represent $\rho$ learning rate.

2.2.2. Electromagnetic signal modulation identification process

The electromagnetic signal modulation identification process based on wavelet transform convolutional neural network is as follows:

Step 1: The electromagnetic signal obtained by wavelet transform is used as CNN input.

Step 2: The convolution layer in CNN is used to extract the feature vectors of electromagnetic signals, and the specific process is as follows:

The convolutional neural network used in this paper includes three convolution layers, one fully connected layer and one output layer. The first layer of convolutional neural network adopts one-dimensional convolution layer, which performs convolution operation on one-dimensional feature vector of input denoised electromagnetic signal. After parameter optimization, the convolution kernel with the size of is adopted, the number of filters is set to 64, the step size is set to 1, and 1, Padding is set to same. The convolution kernel gradually slides over the original input data, and the value of the convolution kernel is multiplied by the value of the corresponding position in the input feature vector. Then, a nonlinear activation function ReLU is used to improve the representation ability of the network model. The specific formula is as follows:

Among them: ${\alpha }_{\gamma }$ represents the electromagnetic signal to be identified, which carries 100 feature vectors. $w_1$ and ${\beta }_1=R\times \left(b_1+w_1{\alpha }_{\gamma }\right)$ respectively represent the weight and offset value of the convolution layer of the first layer, ${\beta }_1$ represents the output of the first layer.

The second layer of convolutional neural network adopts one-dimensional convolution layer, and performs convolution operation on the output electromagnetic signal feature vector of the first layer again. The convolution kernel is adopted, and the number of filters is set to 256, the step size is set to 2, and the 2, Padding is set to same, which gradually slides over the input data. Similarly, ${\beta }_1$ its value is multiplied by the value of the corresponding position, and then the output of the second layer is obtained through a nonlinear activation function ReLU [20]. The specific formula is as follows:

Among them: ${\beta }_1$ represents the input of the second layer, $w_2$ and $b_2$ respectively represents the weight and offset value of the convolution layer of the second layer, ${\beta }_2$ represents the output of the second layer.

The third layer of convolutional neural network adopts one-dimensional convolution layer, and the output electromagnetic signal feature vector of the second layer is analyzed ${\beta }_2$ perform another convolution operation. Using a convolution kernel of 13, the number of filters is set to 512, the step size is set to 2, padding is set to the same. Gradually slide over the input, multiply its value with the corresponding position, and then get the output of the third layer through a nonlinear activation function ReLU. The specific formula is as follows:

Among them: ${\beta }_2$ represents the input of the third layer, $w_3$ and $b_3$ respectively represent the weight and offset value of the convolution layer of the third layer, ${\beta }_3$ represents the output of the third layer. So far, the electromagnetic signal to be identified has undergone three one-dimensional convolution operations to get 512 features after mapping.

Thirdly, the modulation mode of electromagnetic signal is accurately identified by combining the full connection layer softmax classifier of convolutional neural network.

The fourth layer of convolutional neural network adopts a fully connected layer. First, all the feature maps output by the third layer of convolutional neural network are integrated into a vector, which is used as the input of fully connected layer. After the full connection operation of 128 neurons, the features extracted from the convolution layer are integrated to obtain higher-level features, which lays the foundation for subsequent recognition. The specific formula is as follows:

Among them: ${\beta }_3$ represents the input of the fourth layer, $w_4$ and $b_4$ respectively represent the weight and offset value of the fourth convolution layer, ${\beta }_4$ represents the output of the fourth layer. So far, the electromagnetic signal to be identified has been mapped to 256 features through the full connection layer.

The fifth layer of convolutional neural network is the output layer, and the final classification is realized by full connection operation. The softmax function is selected here, which can map the output of neurons to the probability space and obtain the probability of the modulation type of the electromagnetic signal to be measured. The formula is as follows:

Among them: $w_5$ and $b_5$ respectively represent the weight and offset value of the fourth convolution layer, ${\widehat{\beta }}_v$ represents a prediction tag, ${\beta }_v$ represents the output of the fully connected layer, $y_k$ represents the actual electromagnetic signal type, $y_v$ represents the predicted electromagnetic signal type.

Through the above steps, the modulation recognition of electromagnetic signals is realized.