Research on the Anti-Jamming Performance of Particle Swarm Optimization in Deep Neural Networks for Navigation Satellite Systems

Zhiyong Wei^1,a*, Feng Yang ^1,b and Yilong Zhai^1,c

¹The Twenty-ninth Research Institute of China Electronics Technology Group Corporation, Chengdu, 610000 China

^aEmail: w13881899174@163.com

^bEmail: eyangf@sina.com

^cEmail: 1278399622@qq.com

^*Corresponding Author

Abstract—As the external electromagnetic environment becomes increasingly complex and satellite navigation systems rapidly develop, the demand for anti-jamming performance and response speeds increases. This study proposes a method based on Particle Swarm Optimization of a Deep Neural Network for processing navigation signals. The direction vector of the received signal serves as the sample input, while the weight vector optimized by the Particle Swarm Optimization algorithm acts as the sample label. This approach facilitates training deep neural networks. The simulation results demonstrate that, compared to the conventional PI algorithm, the proposed Particle Swarm Optimization of Deep Neural Network processing algorithm exhibits superior anti-jamming capabilities and achieves a fourfold increase in computational speed. When multiple jamming signals traverse space to reach the receiver, the covariance matrix may show partial linear correlation. Contemporary processing algorithms struggle with matrix inversion, which significantly undermines anti-jamming performance. PSO-DNN processing enhances anti-jamming capabilities, ensuring the effective suppression of jamming signals.

Keywords: Particle Swarm Optimization; Deep Neural Network; Power Inversion; Signal Processing; Satellite Navigation;

INTRODUCTION

With the accelerated progress of navigation systems, satellite navigation receiving terminals have been widely implemented in military and civilian applications [1]. Due to the significant distance between navigation satellites and the Earth’s surface, satellite signals propagating through space incur space loss, which reduces the power of the signals reaching the receiver, typically 20–30 dB lower than the noise [2]. Furthermore, navigation signals are susceptible to jamming, especially human jamming; it is crucial to respond quickly to jamming and suppress it when processing the navigation of the received signal [3]. The traditional spatial anti-jamming processing algorithm requires performing matrix inverse operations on the received signal. When the incidence angle of either the jamming signal or the desired signal changes, the weight vector must be updated. This process involves considerable computational load, making it difficult for the traditional algorithm to update the optimal weights in real-time due to higher requirements. With advancements in artificial intelligence and neural network technology, deep neural networks (DNN) have been employed in anti-jamming navigation processing, with received data pre-trained by neural networks. Particle Swarm Optimization (PSO) [4] is a classical swarm intelligence algorithm inspired by the social behaviors of birds during flight and foraging. Flocks of birds search for a global optimal point through information exchange among individuals. The PSO algorithm boasts several advantages: it has a simple principle, requires fewer parameter settings, is easy to implement and comprehend, and follows clear and straightforward procedures; additionally, it considers individual characteristics and overall traits, allowing for faster convergence [5]. The optimal weight vectors were obtained as sample labels using the particle swarm algorithm, and the trained network model was utilized to output the weight vectors adaptively during the generalization phase. Compared to the conventional matrix inverse airspace processing algorithm, which demands extensive computations, the neural network exhibited a significantly reduced response time during the generalization stage. As a result, the navigation array antenna can leverage a neural network to enhance jamming suppression.

Anti-jamming techniques using neural networks and particle swarms have numerous applications in radar, communication, and sonar systems. PSO algorithms are widely employed in power systems [6], system reliability design [7], image processing [8], and other domains. Studies [9]-[11] have developed beamformers based on network architectures like DNN, CNN, and Generative Adversarial Network (GAN) to generate high-quality ultrasonic images quickly. Bo Mo-yu and colleagues designed a three-layer deep neural network. Rapid beam formation is achievable with extensive echo data; however, this method imposes similarity constraints on the combined weight vectors of the same incident signal. Moreover, the process is constrained by the sample setup of six fixed-angle jamming scenarios [12]. The improved PSO algorithm proposed in [13] is utilized for beam formation, which decreases the side lobes and deepens the zero trap; nevertheless, issues such as premature convergence, tendency to fall into local optima, and a limited search range persist. The authors of [14] achieved enhanced fault coverage in circuit detection using fewer test vectors via a DNN-driven PSO approach. Bibhuprasad Sahu et al. [15] implemented a CPSO-DNN algorithm to identify feature genes from high-dimensional data, enhancing disease detection accuracy. Some researchers have selected the PSO algorithm and its hybrid forms as a more predictive algorithm for DNN optimization [16].

This study proposes a PSO-DNN-based navigation anti-jamming processing algorithm, building on existing research. Vectors composed of random expected and jamming signal incidence angles serve as inputs to the DNN. The optimal weight vector obtained through the PSO algorithm is used as the sample label. The DNN structure includes four hidden layers activated by the Leaky ReLU function, while the tanh function activates the output layer. Simulation results demonstrate that as the angle of arrival of the incident signal changes, zero-trapping forms adaptively based on the angle of the jamming signal, enabling anti-jamming processing in the airspace. The response to disturbances is four times faster than that of conventional power inversion (PI) algorithms. In cases of two incidents, when the jamming signals arrive at the receiving end, the covariance matrix exhibits partial linear correlation. Traditional airspace processing algorithms, such as the PI algorithm, cannot invert the covariance matrix, resulting in ineffective jamming suppression. Algorithmic processing based on PSO-DNN facilitates accurate weight prediction and the establishment of zero traps in the direction of incoming jamming, allowing for adaptation to more complex jamming scenarios.

SIGNAL MODE

This study examines a uniform circular array consisting of M array elements with a spacing of d between them. In the x-y plane, M array elements are evenly distributed on a circle with radius r. Since the M array elements are uniformly arranged along the circle, their numbering coincides with their positioning. This distribution establishes that the angle between the m^th array element and the first array element is given by φ_m = 2πm/M, where m = 1, 2, …, M. The array model is depicted in Fig. 1.

Fig. 1. Uniform circular array

If the origin is used as the phase reference point, then the phase of the received signal at the m^th element of the array is as follows:

(1)

Where θ represents the pitch angle, φ denotes the azimuth angle, λ indicates the signal wavelength, and k describes the direction of the incoming signal.

(2)

Where k represents the position vector of the m^th array element, according to the equation above, the orientation matrix of the uniform circular array is expressed as follows:

(3)

If the incident signal is represented as S(t) and the noise signal is defined as n(t), then the received array signal can be expressed as follows:

(4)

The power inversion algorithm [17] obtains the optimal weights of the array by minimizing the output power value of the array. It applies simple constraints on the target signal to prevent obtaining the relevant solution during weighting, thus achieving the complete output of the processed signal. Its cost function is constructed as follows:

(5)

In (5), s = [1, 0, …, 0] ^T is an M×1-dimensional vector. The Lagrange multiplier method was applied to the equation above, and the expression for the weight vector of the PI algorithm is as follows:

(6)

The output of the array is as follows:

(7)

The normalized gain can be expressed as follows.

(8)

PSO-DNN ALGORITHM

This study optimizes network hyperparameters for DNN processing by utilizing PSO to achieve improved parameter settings. Additionally, PSO is used to identify optimal solutions for weight vectors, which serve as sample labels for DNNs, enhancing their anti-jamming performance.

A. PSO Algorithm

As a nonlinear optimization algorithm, the PSO algorithm draws inspiration from the social behavior of birds in group movement, incorporating evolutionary computation and group intelligence. Compared to genetic algorithms such as GA and SA [18], PSO algorithms are less prone to premature convergence or getting trapped in local extremes. They are developing rapidly and are characterized by having few parameters, fast convergence, and simple implementation. The PSO algorithm employs a population of M individuals (particles). It repeatedly searches for the optimal solution in a D-dimensional space, where particles move in space exhibiting two characteristics: velocity and position. The population utilizes the fitness function as a criterion to evaluate the search performance of the particles, continuously adjusting their current positions to locate the particle position that optimizes the fitness function [19]. The study [20] outlines the specific ideas, development history, and current research status of PSO algorithms.

Assuming that the position of the i^th particle in the particle swarm is denoted as x_i, the position vector in the search for the best fitness is the individual extreme, referred to as pbest. The position vector when all particles of the population search for the best fitness is called the global extreme value, denoted by gbest. During each iteration, the velocity and position of particle i are updated, as shown in (9).

(9)

Where t represents the number of current iterations (i = 1, 2, … M), w is the inertia weight that balances the global and local search capabilities of the algorithm, c₁ and c₂ are the acceleration coefficients, and r₁ and r₂ are random constants within the range [0,1]. The primary steps of the PSO algorithm are illustrated in Fig. 2.

Fig. 2. PSO algorithm.

The core of the PSO algorithm is updating the particle’s velocity and position in each iteration. Based on (9), the particle iteratively searches through the space, continuously learns the individual optimum and population optimum, and adjusts the search speed and particle position until an optimal solution is found or the maximum number of iterations is reached. By designing a suitable fitness function, the PSO algorithm can effectively control the directional map of a navigation signal array. The fitness function is expressed as follows:

(10)

Where σ_s², σ_i², and σ_n² represent the powers of the navigation signal, jamming signal, and noise, respectively. α(θ_s) and α(θ_i) indicate the array guidance vectors for navigation and jamming signals, respectively. In this study, we optimized the network hyperparameters for DNN processing using the PSO algorithm to identify improved parameter settings and optimal solutions for weight vectors as sample labels for DNNs, thereby accelerating DNN processing.

B. Deep Neural Network

Deep neural networks, which are the fundamental architecture of neural networks, draw on biological processes to mimic the information processing mechanisms observed in the human brain [21]. A DNN explores complex data patterns and applies them for regression modeling, classification, and unsupervised data clustering across various domains [22]. A deep neural network consists of an input, an output, and a continuous stream of data [23]. The main characteristic of deep learning is its nonlinearity and the potential to include several hidden layers. This study set the number of hidden layers to four according to [12].

Fig. 3. DNN.

C. Activation Function

The sigmoid function is the most commonly used activation function in deep learning due to its differentiable properties that are well-suited for backpropagation training algorithms. However, it is prone to gradient vanishing and slow model convergence, making the forward propagation of the computation more time-consuming [24]. The tanh function, another prevalent activation function, offers faster model convergence but is more computationally intensive because of the exponential operations involved [25]. The ReLU activation function [26] addresses the vanishing gradient problem, is computationally efficient, and converges much faster than the sigmoid and tanh functions. If the input is x, the ReLU activation function can be expressed as follows:

(11)

However, if x < 0, the gradient is zero. Gradients less than zero are set to zero in the ReLU function, which may cause the neuron never to be activated by any data again, leading to the neuron becoming necrotic. The Leaky ReLU activation function [27] can address the issue of neuronal necrosis. When x < 0, the Leaky ReLU maintains a slight gradient (α), allowing the neuron to update the weights even in negative intervals. The Leaky ReLU activation function can be expressed as shown in (11), where the slope α of the negative part is a minimal positive number (usually set to 0.01), referred to as the ‘leakage factor.’

(1)

In the design of neural networks, Leaky ReLU can serve as an activation function in the hidden layer. The output layer usually employs a linear activation for regression tasks, allowing the network to generate an arbitrary range of values. Given that the real and imaginary parts of the weight coefficients are constrained to the range [-1,1], the study selects the tanh function as the activation method for the output layer to ensure that the network’s output values remain within the interval [-1,1]. The mathematical expression for this function is as follows:

(12)

D. Hyperparameter Optimization

PSO was also used to automate the DNN’s hyperparameter tuning. This chapter presents an enhanced PSO framework to optimize architectural parameters (neuron counts) collaboratively, regularize parameters (L2 coefficient and dropout rate), and train parameters (learning rate and batch size) in fully connected neural networks.

An 8-dimensional search space is defined as follows:

(13)

Where N₁ ∈ [1024, 2048], N₂ ∈ [512, 1024], N₃ ∈ [256, 512], and N₄ ∈ [24, 512] represent the neurons in each layer, “η” ∈ [10^-6, 10^-2] represents the initial learning rate, “λ” ∈ [10^-7, 10^-3] represents the L2 regularization coefficient, “p” ∈ [0.1, 0.5] represents the dropout probability, and B ∈ [64, 256] indicates the batch size. The discrete parameters were converted using the nearest-integer rounding.

The mean squared error (MSE) from five-fold cross-validation served as the evaluation metric:

(14)

Converting the negative sign aligns PSO’s default maximization process with the objectives of MSE minimization.

E. PSO-DNN Processing Flow

The PSO algorithm is used to obtain the corresponding weight vectors, and the imaginary and real parts of the weight vectors are transposed separately to obtain w_re+im, which yields L training samples: w_re+im∈R^{L× 2M}. The PSO algorithm accelerated DNN processing, and the estimated value of the weight vector output by the network was denoted as w^*_re+im. The steps of the PSO-DNN algorithm proposed in this study are as follows.

Step 1: Randomly generate L angle vectors θ for the input data of the DNN training samples, where θ_train∈R^L×(K+1).

Step 2: The PSO algorithm was utilized to determine the weight vector. The result is w_re+im.

Step 3: Train the DNN with the obtained training samples {θ_train, w_re+im}.

Step 4: Enhance the hyperparameters after fine-tuning the MSE using 5-fold cross-validation with PSO.

Step 5: Test with the trained DNN. The θ_test network adaptively outputs the weight vector by inputting new test data.

Step 6: Convert the network output into a complex weight vector, denoted by w^*_re+im, and utilize w^*_re+im for anti-jamming processing.

SIMULATION ANALYSIS

This study employed a uniform circular array to conduct the simulation using an Intel 17-14650 CPU, 32GB RAM, MATLAB 2023, and TensorFlow 2.0. The simulation parameters were set as follows:

Table 1. Navigation simulation configuration.

Parameter	Setting
Number of antenna elements	4
Element spacing	λ/2
SNR	-20
INR	40
Snapshots	1024
Azimuth range	[0,180]
Pitch range	[0,90]
Angle step	0.001

Table 2. DNN configuration after optimization.

Parameter	Setting
N₁	1768
N₂	984
N₃	392
N₄	96
η	3.2e-4
λ	8.7e-6
B	192
p	0.25

The neural network optimized with the PSO algorithm significantly enhances convergence speed and predictive ability.

Fig. 4. Loss curve comparison.

Fig. 5. Loss curve comparison.

Fig. 4 illustrates that, after optimizing the hyperparameters with PSO, the neural network achieves a faster convergence speed, facilitating rapid training. Fig. 5 demonstrates that the optimized network exhibits fewer minor errors on both the test and cross-validation sets, making it more suitable for predicting the weight vectors.

A. Single Jamming Signal Simulation Analysis

A desired navigation signal and a jamming incident signal exist. Upon reaching the receiver, the signal incident angle assumes various values and is assigned randomly. The DNN training structure shown in Fig. 3 is utilized for 100 sets of epochs, and the number of nodes in the four hidden layers is divided into 1768, 984, 392, and 96, respectively, with two and eight nodes in the input and output layers, respectively. The incidence angles of jamming are (40^°, 50^°).

Fig. 6. Single jamming 2D radiation pattern.

Fig. 7. Single jamming 3D beam pattern

As illustrated in Fig. 6 and Fig. 7, both the trained neural network and the conventional PI algorithm can form a deeper zero-sag at the jamming location. The PI algorithm generates a zero-sag in the incoming directions (40^°, 50^°) with a zero-sag depth of 67.3587 dB, while the DNN predicted the actual location of the zero-sag (40.031^°, 50.061^°) with a depth of 75.7215 dB. The normalized gain of the PSO-DNN algorithm forms accurate zero traps at jamming locations, minimizes redundant jamming suppression at non-jamming locations, and achieves a greater trap depth compared to the PI algorithm for improved performance.

Table 3. Computational efficiency: PI vs. PSO-DNN.

Weight vector acquisition method	Time(s)
PI	0.041073
PSO-DNN	0.010016

The processing speed of the proposed algorithm (PSO-DNN) was compared to that of the PI algorithm. The computation time required for the PI algorithm to calculate the weight vector and the time needed for the PSO-DNN to predict the weight vector were statistically analyzed based on the input angles of the test samples. The average values were obtained from 100 simulation runs for each method. As shown in Table 3, the trained network structure has a faster response speed than the PI algorithm, with a speed increase of approximately four times, making it suitable for rapid response requirements in the context of satellite navigation anti-jamming.

B. Analysis of Dual-Correlated Jamming Signals

The desired navigation signal and two jamming signal incidents exist. Upon reaching the receiver, the signal incidence angles were considered different values and randomly assigned. The jamming incidence angles are (40^°, 40^°) and (40^°, 125^°). When two incoming jamming signals propagate through space to reach the receiver, the covariance matrix becomes partially linearly correlated. Under these conditions, the traditional algorithm was compared and analyzed using the algorithm described in this study.

Fig. 8. Dual-correlated jamming 2D radiation pattern.

Fig. 9. Dual-correlated jamming PSO-DNN radiation pattern.

Fig. 10. Dual-correlated jamming PI radiation pattern.

Fig. 8, Fig. 9, and Fig. 10 show that the PSO-DNN processing can accurately establish zero trapping. In contrast, the PI algorithm finds it difficult to invert the covariance matrix, resulting in a significant drop in the performance of anti-jamming processing and making it challenging to create precise zero trapping in the direction of jamming. The actual positions of the zero traps formed by the PSO-DNN predicted weights are (40.024^°, 40.130^°) and (40.024^°, 124.963^°). The error was less than 0.04, and the zero trapping of the jamming depths is 72.9375 and 59.8279.

CONCLUSION

With the rapid advancement of satellite navigation, there are growing demands for the robustness and accuracy of positioning in complex electromagnetic environments. This study proposes a PSO-DNN navigation anti-jamming processing method to achieve optimal weight labels using the PSO algorithm for DNN training. The simulation results indicate that under single jamming conditions, the algorithm presented in this study demonstrates superior anti-jamming performance compared to the traditional PI algorithm, resulting in a deeper zero-trapping depth at the location of the jamming source. The processing speed of the algorithm utilized in this study is approximately four times faster than that of the PI algorithm. When two jamming signals arrive at the receiver, the covariance matrix shows partial linear correlation between the jamming signals. Conventional algorithms necessitate matrix inversion operations; therefore, the covariance matrix’s linear correlation significantly decreases anti-jamming performance, making jamming suppression challenging. Through PSO-DNN processing, rapid jamming response and suppression can still be achieved, which is crucial for enhancing the navigation receiver’s processing speed and anti-jamming performance.

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