Through-the-Wall Radar under Electromagnetic Complex Wall: A Deep Learning Approach

In this paper, we considered five distinct wall models: homogeneous wall, air gap wall, inhomogeneous wall, anisotropic wall, and inhomogeneous-anisotropic wall. To approximate an inhomogeneous wall, we use multiple homogeneous layers, each with a specified value of $\varepsilon$. When the homogeneous layers are combined, an inhomogeneous wall is created.

If the wall is a perfect nonmagnetic dielectric with only one direction of inhomogeneity, the permittivity is given by $\varepsilon(x,\omega)$, $\mu=1$, and $\sigma=0$. The inhomogeneous wall is approximated by a series of homogeneous layers adjacent to one another; for example, a ten-centimeter wall is approximated by ten layers with varying permittivity, the permittivity values of which were $\varepsilon_{r}=[3,6,4,5,6,3,6,3,5,2]$. The wall view is depicted in Figure 2.

In isotropic environments, the orientation of the environment’s electric or magnetic polarization is precisely determined by the direction of the external electric or magnetic field. As a result, the electrical and magnetic permeability coefficients of such environments, $\varepsilon$ and $\mu$, are expressed as numerical coefficients. On the other hand, there are numerous materials whose electrical or magnetic polarization is not oriented in the direction of electric or magnetic fields. As a result, the electrical and magnetic permeability coefficients of these environments, also referred to as anisotropic environments, must be expressed using matrices and tensors. In mathematics, anisotropy is defined as the lack of symmetry with respect to a collection of spatial rotation transformations. An anisotropic structure is one that appears differently from different axes. An anisotropic structure is one that has at least one non-scalar structural parameter in electromagnetism. In other words, in anisotropic environments, field $D(\bar{r,\omega})$ is not in the $E(\bar{r,\omega})$ direction, or field $B(\bar{r,\omega})$ is not in the $H(\bar{r,\omega})$ direction.

Where :

We utilize a uni-axial anisotropic non-magnetic perfect dielectric wall in this work, where the non-main diagonal of the $\varepsilon$ matrix is equal to zero, and only the main diagonal has a value of $\mu=1$, $\sigma=0$, and permittivity of the wall is equal to the following tensor.

Another model for the wall is inhomogeneous-anisotropic. We represent the wall as multiple homogeneous layers, with each layer having a distinct tensor $\varepsilon$ for inhomogeneity and anisotropy. Indeed, we have a wall composed of many homogeneous-anisotropic layers stacked on top of one another to form an inhomogeneous-anisotropic wall. The following is the problem’s assumed permittivity:

For the sake of simplicity, let $\varepsilon=[\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{zz}]$ in Equation 5. In this case, we assumed is ten centimeter thick and approximated it with ten distinct layers. The permittivity values of the layers have been assumed to be those in Table \Romannum1. The overview of the inhomogeneous-anisotropic wall is shown in Figure 3.

We modeled the wall with airgap using three layers, placing two layers of homogeneous wall with $\varepsilon_{r}=6$ on either side and in the center of an air layer, which is a practical example of an electromagnetic complex wall.

Artificial neural networks, which are inspired by brain neural cells, emerged in the last two decades and have a wide range of applications, including optimization, artificial intelligence, and many others. By modeling the structure of a cell and the neural network of the brain, an artificial cell and neural network can be constructed. An overview of an artificial neuron and neural cell are depicted in Figure 4. The inputs (input neurons) are as follows: $X_{1}$, $X_{2}$, … . In a neural network, each $X_{i}$ has a weight, denoted by $W_{i}$. Indeed, each input is multiplied by the weight associated with it. The sum function (sigma) in the neural network then adds the product of the X’s and W’s, and an activation function then calculates the output value based on this computation. If $f(u)$ represents the activation function and $b$ represents a bias value, the output of neurons can be written as follows:

A neural network is composed of multiple layers of neurons. In general, a neural network consists of three layers: input, hidden, and output. As the number of layers and neurons in each hidden layer increases, the model becomes increasingly complex. As the number of hidden layers and neurons in our network increases, it becomes a deep neural network, and learning is referred to as deep learning. The activation of Relu and Linear functions was used to design the deep neural network, as shown in Equations 8 and 9.

The electromagnetic solver in the finite difference time domain (FDTD) is one technique for solving Maxwell equations. Ampere’s and Faraday’s Laws can be expressed as the following equations:

$\varepsilon$ and $\mu$ denote the relative permittivity and permeability, respectively. They can be scalar, tensor, or location dependent, as explained previously. We employ $TM_{z}$ polarization in this work in order to rewrite Ampere’s and Faraday’s Laws as follows[29]:

The scalar equations for $TM_{z}$ are obtained from (11) and (12):

Equations (13)–(15) can be expressed in finite-differences form due to the discretization in space-time; future fields can be written in terms of past fields. Assume that the indexes $m$ and $n$ denote the spatial step in the $x$ and $y$ direction, and $q$ denotes the temporal step. Additionally, the spatial step sizes are $\Delta x$ and $\Delta y$ in the $x$ and $y$ directions, respectively. The finite difference approximation of (13) expanded about the space-time point (m$\Delta x$, (n + 1/2)$\Delta y$, q$\Delta t$).The resulting equation is:

For future value in terms of past value, the equation can be rewritten as follows:

We can also write for Equation (14) and (15) expanded about the space-time point ((m + 1/2)$\Delta x$, n$\Delta y$, q$\Delta t$) and (m$\Delta x$, n$\Delta y$, (q + 1/2)$\Delta t$), respectively:

In the simulation, we consider a two-dimensional environment on the X-Y plane with dimensions of 2 x 2 meters, surrounded by perfectly matched layers. The spatial step sizes are set to be $0.1\lambda$ in the X and Y directions . Given that we have assumed a frequency of 3 GHz, $\lambda=c/f=0.1$m. This means that the size of the meshes is 0.01m. Given a space of 2 x 2 meters, the number of meshes in the X and Y directions is 200.

We used a Deep Neural Network (DNN) in this work to estimate the two-dimensional position of targets hidden behind a wall. The DNN algorithm is implemented in Python, and the model is constructed using the Tensorflow and Keras [31] frameworks.We used dense and dropout layers in the proposed deep neural network model. The fully connected (dense) layer contains all of the neurons from the previous layer. The fully connected layer’s primary function is to connect the lower layer’s local features to the upper layer’s local features. Several neurons in the dropout layer are inadvertently ignored during the learning process. Dropout is a regularization technique for neural networks that assists in avoiding the learning process from being manipulated, as well as increasing learning speed and reducing the risk of overfitting. During the training process, dropout should be applied to the connections between layers, but not during network evaluation and testing. We propose a distinct deep learning model for each of the single-target and two-target locating modes that performs well regardless of the assumed wall models. Additionally, we investigated two distinct modes: one without noise, which corresponds to an infinite SNR, and one with noise at various SNRs.

We employed a 9-layer deep neural network to locate a sngle target. In this case, the DNN input’s size is 285, which is equal to the length of the $E_{z}$, $H_{x}$, and $H_{y}$ field vectors, which are composed of three vectors with a length of 95. The DNN’s output has a size of 2, estimating the two numbers that represent the target’s x and y coordinates. We chose 0.0001 as the learning rate and 30 as the batch size. Additionally, we used the Adam optimizer and the Mean Squared Logarithmic Error (MSLE) loss function, which are defined below:

y is the actual value, $\hat{y}$ is estimated value, and $N$ is the total number of data points. In fact, MSLE is the mean of the squared differences between the actual and estimated values following log transformation. The details of the designed DNN for this mode are summarized in Table \Romannum2.

We combined the Dense and Dropout layers in the presented model through trial and error and experience to achieve high accuracy and low error. We attempted to achieve the highest accuracy possible while using the smallest model possible. The accuracy and error graphs for single-target locating over 5000 epochs are shown in Figures 6.(a) and 6.(b). Additionally, we used a 12-layer deep neural network to locate two targets. The accuracy and error graphs for two targets over 1000 epochs are shown in Figure 6.(c), 6.(d), 6.(e), and 6.(f). In these two modes, we used the Mean Squared Logarithmic Error (MSLE) loss function and the Adam optimizer, with the learning rate set to 0.001 and the batch size set to 30. The details of the designed networks are summarized in Table \Romannum3, \Romannum4.

Because the received signal has not changed, the DNN’s input in the two-targets mode is identical to the single-target mode. However, in two-target mode, we must estimate the two targets’ two-dimensional coordinates, resulting in a network output of four neurons. When we compare the network parameters, also known as trainable parameters, for single- and two-target modes, as shown in Tables \Romannum2, \Romannum3, and \Romannum4, we find that single-target mode has 347,642 parameters and two-target mode has 529,714 parameters, indicating that the more targets we have, the more parameters we have. This results in a more dense network, which can be achieved by increasing the number of neural network layers or by increasing the number of neurons contained within the layers.

The accuracy, validation-accuracy, loss, and validation-loss values for the five models considered for the wall, as well as a model incorporating all data in single and double locating, are shown in Table \Romannum5. The proposed deep learning model for location finding has been fine-tuned in such a way that it is insensitive to changing the wall model and does not change the amount of accuracy and error associated with changing the wall model. As can be seen, the problem becomes more complicated as the number of targets increases, and accuracy decreases as the network becomes more dense.

Additionally, adding noise to the signal can help bring the situation closer to reality. In this section, we added noise to the received signal and assumed the wall is a complex electromagnetic wall, increasing the difficulty of the task. To evaluate the model’s performance in the presence of noise, we introduced an Additive White Gaussian Noise (AWGN) with varying SNR values. The accuracy value for the validation dataset in all data mode is shown in Figure 7.

We made no changes to the structure of the neural network models presented in this scenario for single-target and two-target modes. As illustrated in Figure 8, accuracy increases as SNR increases, but the point is that when single-target and two-target diagrams are compared. Single-target mode is less accurate at low SNRs, whereas it is more accurate at high SNRs. To clarify this point, one could argue that because single-target mode has fewer datasets than two-target mode, the neural network would overfit after several epochs, which is why we use early stopping to avoid overfitting. The training procedure is terminated before the neural network becomes overfit. Due to a lack of data in the single-target mode, we have lower accuracy in low SNRs compared to the two-target mode. However, as the SNR increases, the neural network overcomes this situation and achieves greater precision than the two-target mode.