A High-Efficient Method for Synthesizing Multiple Antenna Array Radiation Patterns Simultaneously Based on Convolutional Neural Network

Zhang, Shiyuan; Shi, Chuan; Bai, Ming

doi:https://doi.org/10.1155/2023/6666997

International Journal of Antennas and Propagation

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Abstract Introduction Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 6666997 | https://doi.org/10.1155/2023/6666997

A High-Efficient Method for Synthesizing Multiple Antenna Array Radiation Patterns Simultaneously Based on Convolutional Neural Network

Shiyuan Zhang,¹Chuan Shi,¹and Ming Bai¹

Academic Editor: Arpan Desai

Received19 Jul 2023

Revised18 Sept 2023

Accepted20 Sept 2023

Published17 Oct 2023

Abstract

This paper proposes a high-efficient method that utilizes deep learning technology for synthesizing multiple antenna array radiation patterns simultaneously. More in details, the mathematical feasibility of using neural networks to optimize and synthesize radiation patterns of antenna arrays is demonstrated. Boundary functions are designed to reshape the important characteristics of target radiation patterns and transform them into a two-channel mask matrix, allowing for the simultaneous input of multiple target radiation patterns into the neural network without sacrificing computational efficiency. During training, the cost function is designed to represent the difference between each synthesized radiation pattern and the corresponding target radiation pattern, guiding self-learning. The main framework of the method is a convolutional neural network, where the convolutional layer is used to reduce the expansion of input parameters due to the simultaneous input of multiple mask matrices. Simulation experiments have been conducted to synthesize multiple incoherent target radiation patterns simultaneously on a patch antenna array layout, and the computation time is compared with the combined time required to compute each one individually. The results demonstrate that this method offers the advantage of computational efficiency for simultaneous synthesis of multiple incoherent radiation patterns.

1. Introduction

Antenna arrays are an effective means for radar and communication electronic systems to obtain antenna beams with strong directionality, low sidelobe, and easy scanning and beamforming capabilities [1–4]. The radiation pattern of an antenna array with excellent performance can be obtained by solving the excitations of each antenna element. Based on this, many radiation pattern synthesis methods are proposed and developed, including mathematical optimization synthesis methods [5–7], global optimization algorithms [8–10], and convex optimization algorithms [11–13]. The rapid development of deep learning technology [14–16] has drawn the attention of numerous antenna researchers, resulting in the creation of several neural network frameworks aimed at addressing challenges in the field of antenna [1, 17]. When it comes to radiation pattern synthesis, a variety of neural network methods have been proposed and developed [18–20], which can be broadly classified into two main types. The first category is the numerous training samples driven method. The neural network is pretrained on a large dataset to learn the relationship between the target radiation patterns and the element excitations. As a result, the trained neural network gains the ability to generalize and solve problems related to radiation pattern synthesis. The second category is the optimal approximation method for radiation patterns. In this method, the neural network constantly adjusts its own parameters during training to ultimately approximate the mapping between the target radiation patterns and the element excitations.

For synthesizing multiple radiation patterns, almost all optimization methods and synthesis techniques require each pattern to be synthesized individually, and the computational efficiency and results will be affected by the initial evolutionary sample. Inspired by the fact that neural networks can effectively approximate almost any mapping, a high-efficient method based on convolutional neural networks is proposed for synthesizing multiple antenna array radiation patterns simultaneously. The universal approximation theorem of nonlinear input-output mapping is utilized to demonstrate the mathematical feasibility and establish a link between deep learning networks and solving the problem of radiation pattern synthesis, which is rare in similar research. Moreover, a multiparameter control expression is proposed to restrict the radiation characteristics of the target radiation pattern, so that multiple targets can be input simultaneously without compromising the computational efficiency. Through simulation experiments, we have successfully synthesized multiple incoherent radiation patterns on a patch antenna array layout and compared the calculation time with the combination time required to calculate each one individually.

2. Feasibility Proof of Solving Radiation Pattern Synthesis by Neural Network

Assuming that an antenna array consisting of n radiation elements is arranged in the XOY plane, the far-field pattern FF(θ, φ) in any direction (θ, φ) is as follows:where the excitation weight vector is denoted as and a_i(θ, φ) represents the element pattern of the i-th antenna. The array factor AF_i(θ, φ) is determined by the free-space wave number β and the position (x_i, y_i) in the coordinate system.

If the array factor and element pattern of each antenna are predefined, varying the excitation weight vector can generate different far-field patterns FF(θ, φ). Therefore, the problem of synthesizing the antenna array radiation pattern can be rewritten as finding the excitation weight vector that maps to the target radiation pattern FF(θ, φ), which can be achieved through the inverse function = F⁻¹ (FF(θ, φ)). It is evident that there exists a general mapping between the excitation weight vector and the desired radiation pattern FF(θ, φ). Interestingly, a trainable neural network structure can be used as a tool to implement this general mapping [21]. Hence, a neural network method is proposed to solve the inverse problem mentioned above. According to the universal approximation theorem of nonlinear input-output mapping, let σ(•) be a nonconstant, bounded, and monotone-increasing continuous functions. Let I_m0 denote the m0-dimensional unit hypercube [0, 1]^m0. The space of continuous functions on I_m0 is denoted by C(I_m0). Then, given any function f ∋ C(I_m0) and ε > 0, there exists an integer m1 and sets of real constants a_k, b_k, and ω_kq, where k = 1, …, m1 and q = 1, …, m0 such that we may define the following:as an approximate realization of the function f(•); that is,for all that lies in the input space s₁,…, s_m0.

The linear calculation process in (3) is analogous to the forward superposition calculation principle used for synthesizing radiation patterns. Universal approximation theorem is directly applicable to neural network [21]. In the neural network, we note σ(•) as activation function and note that (3) represents the output of a neural network as follows: (1) the network has m0 input nodes and a single hidden layer consisting of m1 neurons; the inputs are denoted by s₁,…, s_m0; (2) hidden neuron k has synaptic weights ω_k1,…, ω_m0 and bias b_k; and (3) the network output is a linear combination of the outputs of the hidden neurons, with a₁,…, a_m1 defining the synaptic weights of the output layer.

The depth of the neural network can be enhanced by increasing the number of hidden layers, and (2) can be expressed in the following form of nested activation functions:

In this paper, the target pattern FF(θ, φ) is treated as the function f(•) and the excitation weight vector is taken as the output of the neural network. Then, by substituting (4) with (1), the output radiation pattern FF_R(θ, φ) of the neural network during training is obtained:

Based on the universal approximation theorem and (3), the radiation pattern FF_R(θ, φ) output by the neural network after training will be the approximate realization of the target radiation pattern FF(θ, φ).

3. Design of the Neural Network

3.1. Design of the Neural Network Input and Cost Function

In the target radiation patterns, the main beam direction, sidelobe level, and main beam width are typically the most crucial radiation characteristics. Therefore, it is not necessary to restrict all scanning angles, but only the scanning angle region where the radiation properties of interest are present to limit. In this paper, we express the target radiation pattern FF(θ, φ) as a function and relax it to a space bounded by the upper boundary UB(θ, φ) and lower boundary LB(θ, φ). The upper boundary UB(θ, φ) confines the desirable upper radiation features, such as the sidelobe level SLL and -3 dB width of the main beam 2θ_w, while the lower boundary LB(θ, φ) limits the desirable lower radiation features, including the main beam direction (θ₀, φ₀) and, if necessary, the lowest level value V, which are defined by various parameters. To facilitate parallel data processing and improve computational efficiency, the two boundaries are transformed into a two-channel mask matrix with θ and φ as coordinate axes, as shown in Figure 1. Under these conditions, the neural network’s output radiation pattern FF_R(θ, φ) will satisfy

If there are S desired radiation patterns that need to be synthesized simultaneously, the s-th cost function l_s() is computed as the difference between the radiation pattern FF^s_R(θ, φ) and its corresponding mask matrix:where D and G are the total number of sampling directions in the angular space (θ, φ). Different attention weight coefficients ν₁ and ν₂ are assigned to Uloss and Vloss for effective self-learning.

The pooling layer provides spatial and translational invariance to convolutions, allowing features to be detected regardless of their location in the input. However, for radiation pattern synthesis, the main beam direction is the critical information that the neural network must learn. As such, the output must be sensitive to the main beam direction information, which is described in the input. Thus, pooling layers must be removed between convolutional layers. After multiple convolutional layers, the critical information of the input is extracted and output. Finally, a few dense layers are constructed to integrate the global information output of the convolutional structure and map them to the sampling space. The detailed structure of the neural network is presented in Figure 1.

The number of convolution layers and dense layers can be chosen based on the following factors:(1)The characteristics of the antenna array are the number of array elements, array spacing, and so on. For complex antenna arrays, more convolution layers may be necessary to extract additional information, and more dense layers may be required for information mapping.(2)Design objectives of the radiation pattern: For the design objectives with high accuracy, such as the beam direction and peak radiation intensity of the antenna array, more convolution layers may be necessary to extract more precise information.(3)Computing resources: The number of convolutional and dense layers must also consider the limitations of available computing resources. More convolutional layers and dense layers may require more computing resources, such as GPUs and longer training times.

In conclusion, the selection of the number of convolutional and dense layers should be based on the specific antenna design problem. Multiple experiments and adjustments may be required to determine the optimal number of convolutional and dense layers and other hyperparameters.

In this work, the structural parameters of the convolutional layer and dense layer are based on those of the AlexNet neural network, but all pooling layers are removed. The number of neurons in the output layer is related to the number of antenna units. Additionally, a batch normalization layer is added after each convolutional layer to enhance learning. To accelerate convergence and prevent gradient vanishing, the activation functions of all layers are set to “relu” except for the output layer which uses the “sigmoid” activation function to control the excitation weight vector range.

It is noteworthy that in the training process, the function of backpropagation technique is to reduce the difference between the output radiation pattern FF_R(θ, φ) and the target radiation pattern FF(θ, φ). During the calculation, the output includes the amplitude and phase excitations of all antenna elements in the s-th group (s = {1, …, S}), as shown in Figure 1. Forward propagation entails computing all the cost function l_s. The neural network continuously updates the network’s parameters by identifying the direction of gradient reduction of all S-cost functions. Ultimately, all S synthesized radiation patterns meet their respective predefined requirements. For the backpropagation process, we used the “Adam” optimization algorithm to update the parameters of the neural network and seamlessly integrate it into the TensorFlow framework.

4. Numerical Examples

In this section, we demonstrate the successful simultaneous synthesis of five radiation patterns, each with a different main beam direction, using our proposed neural network framework. The numerical examples were implemented by the TensorFlow framework, which is supported by GPU-based acceleration, resulting in efficient and effective computation.

In the example, the radiation patterns are synthesized in the upper half space of the array for directions (θ, φ) and the layout of the antenna array is shown in Figure 2. The constructed mask matrices have a resolution of 1° for both θ and φ and have resolution 1° in the upper half space, resulting in D = 360 and G = 90. To optimize all radiation patterns, weight coefficients ν₁ = 1 and ν₂ = 500 are applied with a learning rate of μ = 5 ∗ 10⁻⁶.

The requirements for the five target radiation patterns outlined in this paper are as follows: each main beam direction needs to be pointed precisely in a different direction, and lower sidelobe levels can be achieved by setting the sidelobe level value to SLL = −18. The amplitude (normalized) and phase results of the radiation pattern with different main beam directions are shown in Figures 3 and 4, respectively, and the main radiation properties of the different results are shown in Table 1.

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

(d)

(e)

All the aforementioned results were synthesized simultaneously, taking only 87 seconds, which is a half of the combined time required to compute each one individually (185 seconds). This demonstrates the high efficiency of the proposed neural network framework in synthesizing multiple incoherent radiation patterns simultaneously.

5. Discussion

We believe that the reason behind this high efficiency is the unique computational approach in neural networks: First of all, backpropagation technology is one of the reasons to ensure the high efficiency of neural networks. And another point is that we add the convolutional layer structure as an important improvement over our previous work. The utilization of weight sharing and local connections in the convolutional layers serves to significantly reduce network parameters. This, in turn, allows for the deepening of the network while preserving computational efficiency. Furthermore, for synthesizing multiple antenna array radiation patterns simultaneously, adding the convolutional layer structure can also reduce the expansion of input parameters due to the simultaneous input of multiple mask matrices and further ensure computational efficiency. It is also worth noting that the appropriate selection of the learning rate, determined through several experiments, not only accelerates the learning efficiency but also leads to satisfactory results. Moreover, in the target radiation pattern presented in this paper, the direction of the main beams is the most important radiation characteristic, and thus the weight ν₂ associated with controlling the direction of the main beam is given a higher value. By increasing the weight ν₁ of the Uloss, which is related to controlling the sidelobe level, the sidelobe level value in the result can be further reduced. The proposed neural network framework has the flexibility to synthesize more than five incoherent radiation patterns. However, increasing the number of radiation patterns synthesized will result in a larger input and output for the neural network framework. The ability of a neural network to process data concurrently is related to the number of neurons at each layer, and thus the number of neurons in each layer will also increase with the number of radiation patterns synthesized simultaneously. Furthermore, if this method is used for a larger antenna array, the number of neurons in the output layer must match the number of antenna units. Under the above circumstances, there is a greater demand for computing equipment and sufficient computing resources.

6. Conclusions

In this paper, we present a method for synthesizing multiple antenna array radiation patterns simultaneously, using CNN. Numerical examples demonstrate that the proposed method is highly efficient. Multiple parameters control the upper and lower boundary functions, allowing flexibility to reshape the various requirements of the target radiation patterns. Additionally, the transformation of the mask matrix not only facilitates parallel computation but also offers a method for the neural network to handle multiobjective optimization problems. The authors believe that utilizing the proposed neural network to simultaneously synthesize a large number of radiation patterns will offer higher computational efficiency advantages and can be extended to address many other electromagnetic field problems.

Data Availability

The data, models, or code used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Microwave Engineering Laboratory, Beihang University.

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Copyright © 2023 Shiyuan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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