An Attention-Guided Complex-Valued Transformer for Intra-Pulse Retransmission Interference Suppression

Abstract

With the maturation of digital radio frequency memory (DRFM) technology, various intra-pulse retransmission interference methods have emerged. These flexible and changeable retransmission interference methods pose significant challenges to radar detection tasks, particularly in modern battlefields. This paper proposes an attention-guided complex-valued transformer (AGCT) as a solution. First, the encoder maps the received signal contaminated by interference and noise into a high-dimensional space. Then, the dilated convolution block (DCB) group and attention block (AB) group in the mask estimator extract the delicate multi-scale features and large-scale features of the interference, respectively, to obtain a multidimensional space mask. Finally, the decoder restores interference to the time domain and outputs the estimated target echo using residual learning. Considering the characteristics of intra-pulse interference, we introduced the energy attention block (EAB) at the end of the DCBs and the ABs within our network. This addition ensures a heightened focus on extracting interference features. Furthermore, we implemented a curriculum learning strategy during the network training. This approach gradually acclimates the network to fit different types of retransmission interference, starting from simpler to more complex scenarios. Our extensive experiments, conducted under various conditions, have provided compelling evidence of the AGCT’s superior performance. Compared to the comparative network, the AGCT’s advantages are particularly pronounced under more harsh conditions, demonstrating its robustness and effectiveness.

1. Introduction

The strategic advantage of electronic warfare [1] has proven pivotal in conflicts such as the Gulf war and the recent Russo-Ukrainian war. As the most effective electronic reconnaissance equipment today, radar has been widely used in a variety of military equipment. In remote sensing technology, the linear frequency modulation (LFM) signal application is particularly prominent [2,3]. An LFM signal can increase radio frequency (RF) pulse width and average transmit power while maintaining sufficient spectrum width. This configuration does not compromise the range resolution of radar and allows for significant system gains [4].

The widespread adoption of digital radio frequency memory (DRFM) [5,6] has given rise to interference techniques such as smeared spectrum (SMSP) [7,8], interrupted sampling repeater jamming (ISRJ) [9,10], and interrupted non-uniform sampling repeater jamming (INUSRJ) [11,12]. These methods generate numerous false targets near the actual target by retransmitting the intercepted radar signal segments, which poses a significant threat to radar detection. Consequently, suppressing intra-pulse retransmission interference has become a research hotspot in the field of radar signal processing.

Autocorrelation methods and spectral kurtosis maximization are utilized to estimate the periodicity, initial phase, and interference suppression factor of SMSP interference [13]. The parameters of the SMSP signal are estimated from the time–frequency (TF) distribution, followed by interference suppression through signal reconstruction [14]. Li et al. [15] also suppressed SMSP interference in the TF domain and effectively compensated for the target echo information lost during the interference suppression process. A fractional Fourier transform-based algorithm is applied, to suppress SMSP interference [16], capitalizing on the significant difference in chirp rates between the SMSP interference signal and the target echo. To address the edge estimation errors of SMSP interference, an approach combining various box filters and time domain deconvolution curves has been proposed [17].

A novel bandpass filter design uses TF analysis to enhance ISRJ suppression in LFM radar [18], significantly improving signal clarity and detection probability. In [19], an ISRJ cancellation algorithm was proposed that aligns the cancellation signal with the target echo, to enhance signal discrimination under overlapping jamming conditions. Wang et al. [20] proposed a scheme that suppresses ISRJ by jointly designing complementary sequences and receiving filters, which are optimized using a gradient-based solver and the Lagrange multiplier method. In [21], an algorithm was proposed for suppressing ISRJ in intra-pulse frequency modulation slope agile radar, utilizing fractional Fourier filtering. This approach leverages the interference’s distinct energy distribution characteristics to enhance the interference suppression’s effectiveness. Lu et al. [22] introduced a truncated matched filter method to suppress ISRJ in LFM radars by capitalizing on the distinct energy differences between interference and target echoes in the matched filter results.

INUSRJ, as an improved version of ISRJ [23], is often employed to interfere with inverse synthetic-aperture radar [24,25]. Wu et al. [26] proposed an interference suppression method combining the inter-pulse energy function with compressed-sensing technology.

However, the above suppression methods are model-driven, designed for specific types of interference, which may fail in complex real-world scenarios, due to model mismatches. With the advent of deep learning [27], its powerful fitting [28] and feature-extraction [29] capabilities have provided new solutions for different types of interference suppression issues. The classic encoder, mask estimator, and decoder structures in Conv-TasNet [30] have achieved excellent results in single-channel blind source separation of speech signals. Chen et al. [31] proposed an adaptive multi-scale residual auto-encoder to suppress interference in short-range detection systems. In [32], a complex-valued fully convolutional network was proposed to solve the interference mitigation problem of frequency-modulated continuous wave (FMCW) radar. In [33], the weights and activations of anti-interference networks were quantitatively discussed and analyzed.

Deep learning techniques have also been applied to address intra-pulse retransmission interference. Jiang et al. [34] utilized a series of auto-encoders to extract signal and RF features, followed by another auto-encoder to reconstruct radar signals. He et al. [35] proposed a multi-stage multi-domain joint anti-jamming depth network to suppress ISRJ. This network integrates U-Net and transformer architectures to execute interference suppression through a series of steps, including TF filtering, feature reconstruction, and residual reconstruction. In [36], a multi-residual encoder–decoder network (MRDENet) was designed to suppress intra-pulse retransmission interference and to analyze the network’s performance under complete and incomplete datasets.

To sum up, there are currently few studies on applying deep learning methods to solving various types of intra-pulse retransmission interference, and the characteristics of interference signals are ignored in network design. This paper provides a solution: an attention-guided complex-valued transformer (AGCT). The main contributions of this paper are as follows:

(1)

In this paper, a complex-valued network is utilized to align closely with complex-valued signals, incorporating a basic architecture of encoder, mask estimator, and decoder, which demonstrates superior performance in complex electromagnetic environments through extensive validation;

(2)

The design leverages attention blocks (ABs) that integrate both inter-pulse and intra-pulse attention for extracting large-scale interference characteristics, complemented by dilated convolution blocks (DCBs) responsible for the extraction of delicate multi-scale features, providing a robust foundation for interference reconstruction;

(3)

The energy attention blocks (EABs) play a crucial role in enhancing network performance. Integrated into both the ABs and the DCBs, they focus more on interference, thereby improving the network’s ability to handle complex electromagnetic environments. Additionally, the implementation of a curriculum learning strategy accelerates model fitting and enhances the network’s generalization capabilities.

The remainder of this paper is organized as follows: Section 2 describes the signal model; Section 3 proposes the AGCT and provides a detailed description of the system model; Section 4 presents and discusses the experiments and analysis; Section 5 concludes.

Notation: ${\left(·\right)}^T$, ${\left(·\right)}^H$, $\mathbb{E}(·)$, $var(·)$, $sum(·)$, ${∥·∥}_2$, ⊗, ⊙, ∘, ∗, $\left|·\right|$, $\mathrm{vec}(·)$, $\mathrm{unvec}(·)$, $\mathfrak{R}(·)$ and $\mathfrak{I}(·)$ denote the operations of transpose, conjugate transpose, mathematical expectation, variance computation, summation, Euclidean norm, Kronecker product, element-wise product, tensor multiplication, convolution, absolute, vectorization, matrixization, taking real part, and taking imaginary part, respectively; $\mathbf{0}$ and $\mathbf{1}$ refer to the zero vector and one vector, respectively.

2. Signal Model

This paper considers the challenge of single-channel intra-pulse retransmission interference suppression so that the received signal of the ith pulse can be expressed as

$$y_i\left(t\right)=x_i\left(t\right)+J_i\left(t\right)+n_i\left(t\right),$$

(1)

where $x_i\left(t\right)$, $J_i\left(t\right)$, and $n_i\left(t\right)$ are the target echo, retransmission interference signal, and noise of the ith pulse, respectively. In this paper, the target echo is the LFM signal, and the retransmission interference methods include SMSP, ISRJ, and INUSRJ. The target echo is given by

$$\begin{array}{c}{x}_i\left(t\right)=\varphi e^{j2\pi ({\phi }_i+f_{c}t+0.5k{t}^2)},\\ {\phi }_i=\frac{f_d(i-1)}{f_r},f_d=\frac{2v{f}_c}{c},\end{array}$$

(2)

where $\varphi$, $f_d$, k, v, $f_c$, $f_r$, and c denote amplitude, Doppler frequency, chirp rate, target velocity, carrier frequency, pulse repetition frequency, and light velocity, respectively; ${\phi }_i$ is the Doppler phase of the ith pulse.

The multi-pulse received signal $\mathbf{Y}\in {\mathbb{C}}^{P\times T}$ can be expressed in the form of tensor multiplication as

$$\mathbf{Y}=\mathbf{d}\circ ({\mathbf{x}}_1+{\mathbf{J}}_1)+\mathbf{N}=\mathbf{X}+\mathbf{J}+\mathbf{N},$$

(3)

where ${\mathbf{x}}_1\in {\mathbb{C}}^{1\times T}$ and ${\mathbf{J}}_1\in {\mathbb{C}}^{1\times T}$ are the target echo and interference signal of the first pulse; $\mathbf{d}={[1,{\phi }_2,\dots ,{\phi }_P]}^T\in {\mathbb{C}}^{P\times 1}$, P, and T denote the Doppler phase vector, pulse samples, and pulse number, respectively; and $\mathbf{X}$, $\mathbf{J}$, and $\mathbf{N}$ are matrices of multi-pulse target echo, interference signal, and noise, respectively.

SMSP is a classic interference method for LFM. Its core content is to compress the intercepted target echo in the time direction and retransmit it R times, and the chirp rate of each sub-pulse is R multiples of the target echo. It can be expressed as

$$\begin{array}{cc}\hfill & {\mathbf{J}}_{\mathrm{smsp}}=\left[{\mathbf{1}}_{1\times R}\otimes {\mathbf{J}}_{sub}\right],\hfill \\ \hfill & {\mathbf{J}}_{\mathrm{sub}}={\varphi }_J{e}^{j2\pi \left(f_c\dot{\mathbf{t}}+0.5kR{\dot{\mathbf{t}}}^2\right)},\dot{\mathbf{t}}\in \left[0,\frac{1}{f_s},\dots ,\frac{T-R}{f_{s}R}\right],\hfill \end{array}$$

(4)

where ${\mathbf{J}}_{\mathrm{sub}}$ and ${\varphi }_J$ are the interference sub-pulse and the interference amplitude.

The essence of ISRJ involves repeatedly retransmitting multiple intercepted segments of the target echo, which can be formulated as

$$\begin{array}{cc}\hfill & {\mathbf{x}}_{{\mathrm{seg}}_1}=\left[x_1\left(t_{i_1}\right),\dots ,x_1\left(t_{e_1}\right)\right],\hfill \\ \hfill & {\mathbf{J}}_{\mathrm{ISRJ}}=\left[\mathbf{0},{\mathbf{1}}_{1\times R}\otimes {\mathbf{x}}_{{\mathrm{seg}}_1},\mathbf{0},\dots ,\mathbf{0},{\mathbf{1}}_{1\times R}\otimes {\mathbf{x}}_{{\mathrm{seg}}_N},\mathbf{0}\right],\hfill \end{array}$$

(5)

where ${\mathbf{x}}_{{\mathrm{seg}}_1}$ and ${\mathbf{x}}_{{\mathrm{seg}}_N}$ denote the first and the Nth intercepted segments, respectively. N, $t_{i_1}$, and $t_{e_1}$ are the number of interceptions, the starting moment of the first segment, and the ending moment of the first segment, respectively.

INUSRJ is an advanced version of ISRJ. The main improvement is that the retransmission number of each intercepted segment is random, thus increasing the complexity of the interference. It can be expressed as

$${\mathbf{J}}_{\mathrm{INUSRJ}}=\left[\mathbf{0},{\mathbf{1}}_{1\times R_1}\otimes {\mathbf{x}}_{{\mathrm{seg}}_1},\mathbf{0},\dots ,\mathbf{0},{\mathbf{1}}_{1\times R_N}\otimes {\mathbf{x}}_{{\mathrm{seg}}_N},\mathbf{0}\right],$$

(6)

where $R_1$ and $R_N$ denote the retransmissions numbers of the first and the Nth segment, respectively.

In Figure 1, the TF diagrams, time domain diagrams, and pulse compression results of the received signal interfered with by three types of retransmission interference (SMSP, ISRJ, INURSJ) are illustrated (jamming-to-signal ratio (JSR) = 20 dB, signal-to-noise ratio (SNR) = 0 dB). In Figure 1a–f, it is evident that under conditions of high-level noise and strong interference, the target echo is nearly drowned out in both the TF and time domains by noise and interference signals. Both SMSP and ISRJ are periodic interference methods, but the latter is more complicated than the former, due to retransmitting different target echo segments. In Figure 1g–i, the dense false targets generated by these three methods obscure the main lobe of the target echo, thereby degrading the detection performance of the radar.

3. System Model

3.1. Complex-Valued Network

Typically, previous research [34,35,36] has dealt with complex-valued signals by separating the real and imaginary parts, either by concatenating or by processing them through two channels. This approach treats the real and imaginary parts as independent, overlooking the inherent phase information of complex signals. Compared to real-valued networks, complex-valued networks offer certain advantages regarding network flexibility, extraction of correlated features between the imaginary and real parts of signals, and the mapping of signal feature transformations [37]. To address these issues, the proposed network adopts a complex-valued structure.

The critical difference between real-valued and complex-valued convolutional and fully connected layers is that the latter involves complex-valued trainable parameters [38,39]. When the input complex-valued feature tensor $\mathbf{H}$ passes through a complex-valued convolutional layer, it undergoes convolution with $C_{out}$ complex-valued weight tensors $\left\{{\mathbf{W}}_1,\cdots ,{\mathbf{W}}_{C_{out}}\right\}$, which can be expressed as

$$\begin{array}{cc}\hfill \mathbf{O}[:,i,:]& ={\mathbf{W}}_i\ast \mathbf{H}+{\mathbf{B}}_i=[\mathfrak{R}\left({\mathbf{W}}_i\right)+j\mathfrak{I}\left({\mathbf{W}}_i\right)]\ast [\mathfrak{R}\left(\mathbf{H}\right)+j\mathfrak{I}\left(\mathbf{H}\right)]+[{\mathfrak{R}}_i\left({\mathbf{B}}_i\right)+j\mathfrak{I}\left({\mathbf{B}}_i\right)]\hfill \\ \hfill & =[\mathfrak{R}\left({\mathbf{W}}_i\right)\ast \mathfrak{R}\left(\mathbf{H}\right)-\mathfrak{I}\left({\mathbf{W}}_i\right)\ast \mathfrak{I}\left(\mathbf{H}\right)+\mathfrak{R}\left({\mathbf{B}}_i\right)]\hfill \\ \hfill & +j[\mathfrak{I}\left({\mathbf{W}}_i\right)\ast \mathfrak{R}\left(\mathbf{H}\right)+\mathfrak{R}\left({\mathbf{W}}_i\right)\ast \mathfrak{I}\left(\mathbf{H}\right)+\mathfrak{J}\left({\mathbf{B}}_i\right)],\hfill \end{array}$$

(7)

where $\mathbf{O}$ and $\mathbf{B}$ denote the output complex-valued feature tensor and complex-valued bias tensor. The operation details are shown in Figure 2. Similarly, the operation of a complex-valued feature tensor passing through a complex-valued fully connected layer can be expressed as

$$\begin{array}{cc}\hfill \mathbf{O}& =\mathbf{W}\circ \mathbf{H}+\mathbf{B}=\left[\mathfrak{R}\right(\mathbf{W})+j\mathfrak{I}(\mathbf{W}\left)\right]\left[\mathfrak{R}\right(\mathbf{H})+j\mathfrak{I}(\mathbf{H}\left)\right]+\left[\mathfrak{R}\right(\mathbf{B})+j\mathfrak{I}(\mathbf{B}\left)\right],\hfill \\ \hfill & =\left[\mathfrak{R}\right(\mathbf{W})\circ \mathfrak{R}(\mathbf{H})-\mathfrak{I}(\mathbf{W})\circ \mathfrak{I}(\mathbf{H})+\mathfrak{R}(\mathbf{B}\left)\right]+j\left[\mathfrak{I}\right(\mathbf{W})\circ \mathfrak{R}(\mathbf{H})+\mathfrak{R}(\mathbf{W})\circ \mathfrak{I}(\mathbf{H})+\mathfrak{J}(\mathbf{B}\left)\right].\hfill \end{array}$$

(8)

The operation of various common real-valued activation functions and complex-valued activation functions can be expressed as

$$\begin{array}{c}\hfill ReLU\left(h\right)=max(0,h),PReLU\left(h\right)=max(\kappa h,h),\\ \hfill Sigmoid\left(h\right)=\frac{1}{1+e^h},Tanh\left(h\right)=\frac{e^h-e^{-h}}{e^h+e^{-h}},Softmax\left(h\right)=\frac{e^h}{sum\left(e^{\mathbf{H}}\right)},\end{array}$$

(9)

$$\begin{array}{cc}\hfill \mathbb{C}ReLU\left(\mathbf{H}\right)& =ReLU\left(\mathfrak{R}\right(\mathbf{H}\left)\right)+jReLU\left(\mathfrak{I}\right(\mathbf{H}\left)\right),\hfill \\ \hfill \mathbb{C}PReLU\left(\mathbf{H}\right)& =PReLU\left(\mathfrak{R}\right(\mathbf{H}\left)\right)+jPReLU\left(\mathfrak{I}\right(\mathbf{H}\left)\right),\hfill \\ \hfill \mathbb{C}Sigmoid\left(\mathbf{H}\right)& =Sigmoid\left(\mathfrak{R}\right(\mathbf{H}\left)\right)+jSigmoid\left(\mathfrak{I}\right(\mathbf{H}\left)\right),\hfill \\ \hfill \mathbb{C}Tanh\left(\mathbf{H}\right)& =Tanh\left(\mathfrak{R}\right(\mathbf{H}\left)\right)+jTanh\left(\mathfrak{I}\right(\mathbf{H}\left)\right),\hfill \end{array}$$

(10)

where h and $\kappa$ denote an element of $\mathbf{H}$ and a learnable parameter, respectively. These complex-valued activation functions involve separately applying nonlinear mappings to the real and imaginary parts of the complex-valued feature tensor. Layer normalization in transformers is a common normalization technique applied across the feature dimension. Standardizing each feature to have a mean of 0 and a variance of 1 enhances the model’s training efficiency and generalization ability, facilitating more stable and faster convergence during the training process. The complex-valued layer normalization procedure can be expressed as

$$\begin{array}{cc}\hfill \mathbb{C}LayerNorm\left(\mathbf{H}\right)& =LayerNorm\left(\mathfrak{R}\right(\mathbf{H}\left)\right)+jLayerNorm\left(\mathfrak{I}\right(\mathbf{H}\left)\right).\hfill \end{array}$$

(11)

For simplicity in the following discussions, all components mentioned previously in the proposed network are in their complex-valued form.

3.2. Attention-Guided Complex-Valued Transformer

The network proposed in this paper utilizes the Doppler-shift information contained between different pulses by stacking multi-pulse received signals into one-dimensional tensors for network input as ${\mathbf{Y}}^{\prime }\in {\mathbb{C}}^{B\times 1\times PT}$, and B denotes batch size. The advantage of the transformer over convolutional neural networks (CNN) lies in its ability to extract the relationship of information in long sequences [40,41], which is crucial for suppressing intra-pulse retransmission interference. The schematic of the AGCT architecture is shown in Figure 3. The network consists of an encoder, a mask estimator, and a decoder. First, the encoder maps the long received sequence to a high-dimensional sparse space for representation. Then, the mask of intra-pulse interference in the sparse space is obtained through the mask estimator. Finally, the decoder restores the estimated intra-pulse interference to the time domain, and a residual connection is applied to achieve interference suppression.

(1)

Encoder: The encoder is a one-dimensional convolutional layer and a PReLU layer, which performs convolution on a received sequence ${\mathbf{Y}}^{\prime }$ using C filters of length $k_1$ with stride of $k_1/2$ and nonlinear activation. This process generates a high-dimensional sparse representation $\mathbf{D}\in {\mathbb{C}}^{B\times C\times L}$ as

$$\mathbf{D}=PReLU(EncodingConv\left({\mathbf{Y}}^{\prime }\right)).$$

(12)

This approach effectively captures the essential features of the signal for further processing and analysis. In contrast to short-time Fourier transform with fixed basis functions, a convolutional encoder can learn more adaptive representations for feature extraction through training [30].

(2)

Mask estimator: At the beginning of the mask estimator, similar to many transformer models, the normalized $\mathbf{D}$ undergoes position encoding to imbue the sparse representation with position information. A pointwise convolution, serving as a bottleneck layer, adjusts the number of channels for transmitting features in the mask estimator and enhances the model’s performance.

The core of the mask estimator consists of $u_1$ DCBs and $u_2$ ABs. The DCBs focus on capturing finer multi-scale details, while the ABs are designed to extract large-scale information from interference.

The feature tensor is repeated r times through the DCB group with gradually increasing dilation factors $\left\{2^0,2^1,\dots ,2^{u_1}\right\}$. Each element of the final output feature tensor integrates extensive sequence information from the input feature tensor, ensuring comprehensive representation across the network. The detailed structure of the DCBs is illustrated in Figure 4a. The dilated depthwise convolution layer significantly increases the network’s receptive field, while the pointwise convolution layer enhances the network’s depth, collectively improving the overall functionality of the network. In the DCBs, the number of channels is consistently maintained at C, with a depthwise convolution kernel length of $k_2$. The same padding is employed to preserve the length of the sequence.

To effectively mask target signals, interference often has significantly more vital energy than the target echo, even in high-dimensional feature space where the interference regions exhibit higher energy levels. However, traditional anti-interference networks often overlook this energy distribution information. Therefore, we have incorporated an EAB at the end of the DCB, enabling the network to capture interference information better. This design prioritizes salient features across the sequence and channel dimensions, as depicted in Figure 4b. The structure of both branches is fundamentally the same. However, the feature tensor needs to be transposed before entering the channel branch. Adaptive average pooling is employed to amalgamate global information across another dimension, to produce an initial attention vector. This vector then passes through two depthwise convolution layers, generating a weight vector via a sigmoid layer. The process can be formulated as

$$\begin{array}{cc}\hfill & \mathbf{S}\mathbf{W}=Sigmoid\left({\sigma }_{\mathrm{s}2}\left(PReLU\left({\sigma }_{\mathrm{s}1}\left(\frac{1}{C}\sum _{i=1}^C\left|\mathbf{I}\mathbf{F}\mathbf{T}\right|\right)\right)\right)\right),\hfill \\ \hfill & \mathbf{C}\mathbf{W}=Sigmoid\left({\sigma }_{\mathrm{c}2}\left(PReLU\left({\sigma }_{\mathrm{c}1}\left(\frac{1}{L}\sum _{i=1}^L\left|\mathbf{I}\mathbf{F}{\mathbf{T}}^T\right|\right)\right)\right)\right),\hfill \end{array}$$

(13)

where $\mathbf{IFT}\in {\mathbb{C}}^{B\times C\times L}$, $\mathbf{SW}\in {\mathbb{C}}^{B\times 1\times L}$, $\mathbf{CW}\in {\mathbb{C}}^{B\times C\times 1}$, and $\sigma \left(·\right)$ denote the input feature tensor, sequence weight vector, channel weight vector, and depthwise convolution operation. The convolution kernel lengths of the above four depthwise convolutional layers are all $k_3$. These two weight vectors are multiplied to generate an energy weight tensor $\mathbf{EW}\in {\mathbb{C}}^{B\times C\times L}$. This tensor is then element-wise multiplied with $\mathbf{IFT}$ to produce the output feature tensor $\mathbf{OFT}\in {\mathbb{C}}^{B\times C\times L}$ as

$$\mathbf{OFT}=\mathbf{IFT}\odot \left(\mathbf{CW}\circ \mathbf{SW}\right)=\mathbf{IFT}\odot \mathbf{EW}.$$

(14)

The EAB assigns higher weights to high-energy elements, enabling the network to clearly identify which sequence positions and channels to focus on. The output of the DCB branches into two paths: one employs a skip connection to transport gradients and transmit features to the next DCB efficiently. The other path accumulates feature tensors from the EAB across the DCBs with varying dilation factors, continuously integrating features of different scales.

The data flows in the proposed network are all three-dimensional. The first dimension is the batch size, and the last two are the number of channels and the sequence length. Therefore, the transpose operation mentioned in this paper transposes the second and third dimensions.

In this paper, the structure of the ABs is consistent, as depicted in Figure 4c. Each AB consists of four ECBs, fusion attention, and an EAB. The detailed structure of the ECB is shown in Figure 4d. To align with its configuration, the transpose operation needs to be performed before the feature tensor is input to the ABs. The fully connected layer is employed to adjust the output feature channels followed by dropout to prevent overfitting. Subsequently, the feature tensor passes through a depthwise convolution layer with a kernel length of $k_4$, to enhance the receptive field and preserve fine features. The process concludes with a skip connection for output.

Upon entering the AB, the feature tensor $\mathbf{Z}\in {\mathbb{C}}^{B\times L\times C}$ is processed through ${\mathrm{ECB}}_{\mathrm{qk}}$ to generate $\mathbf{U}\in {\mathbb{C}}^{B\times L\times \theta }$, which then undergoes offset and scaling operations to produce the query $\mathbf{Q}\in {\mathbb{C}}^{B\times L\times \theta }$ and key $\mathbf{K}\in {\mathbb{C}}^{B\times L\times \theta }$ components essential for the attention mechanism. Parallel to this process, $\mathbf{Z}$ is also mapped through ${\mathrm{ECB}}_{\mathrm{v}}$ to obtain value $\mathbf{V}\in {\mathbb{C}}^{B\times L\times E}$. $\theta$ and E, which denote the output channel numbers from the fully connected layers of ${\mathrm{ECB}}_{\mathrm{qk}}$ and ${\mathrm{ECB}}_{\mathrm{v}}$, respectively. In this paper, $L^{\prime }$ is significantly greater than $\theta$ and E.

The self-attention mechanism [40] can be formulated as

$${\mathbf{A}}_s=Softmax\left(\frac{\mathbf{Q}\circ \mathbf{K}}{\sqrt{\theta }}\right)\circ \mathbf{V}.$$

(15)

The entire process requires $L^2\theta +L^{2}E$ multiplication operations. When dealing with feature tensors of long sequence lengths where L is large, the computational cost becomes significantly expensive.

Inspired by the low-complexity mixed chunk attention in [42,43], integrating inter-pulse attention with intra-pulse attention has been implemented to enhance feature extraction from interference. To ensure the feature sequence length is evenly divisible into P parts, padding operations are applied to $\mathbf{Q}$, $\mathbf{K}$, and $\mathbf{V}$ to extend their sequence lengths to $L^{\prime }$. Figure 5 illustrates the schematic of the fusion attention. Inter-pulse attention ${\mathbf{A}}^{\prime }$ can be calculated as

$${\mathbf{A}}^{\prime }=\mathbf{Q}\circ \left(\frac{1}{L^{\prime }}{\mathbf{K}}^T\circ \mathbf{V}\right).$$

(16)

${\mathbf{A}}^{\prime }\in {\mathbb{C}}^{B\times L\times E}$ considers the long dependency relationships between pulses from a global perspective. $\mathbf{Q}$, $\mathbf{K}$, and $\mathbf{V}$ are each divided into P sub-tensors with sequence length M, which are used to calculate the intra-pulse attention as

$$\begin{array}{cc}\hfill {\mathbf{A}}_p^{″}& ={ReLU}^2\left(\frac{1}{M}{\mathbf{Q}}_p\circ {\mathbf{K}}_p^T\right)\circ {\mathbf{V}}_p,\hfill \\ \hfill {\mathbf{A}}^{″}& =\left[{\mathbf{A}}_1^{″},{\mathbf{A}}_2^{″},\dots ,{\mathbf{A}}_P^{″}\right],\hfill \end{array}$$

(17)

where ${\mathbf{Q}}_p\in {\mathbb{C}}^{B\times M\times \theta }$, ${\mathbf{K}}_p\in {\mathbb{C}}^{B\times M\times \theta }$, ${\mathbf{V}}_p\in {\mathbb{C}}^{B\times M\times E}$, and ${\mathbf{A}}_p^{″}\in {\mathbb{C}}^{B\times M\times E}$ represent the query, key, value, and attention tensor corresponding to the pth pulse, respectively, and where ${ReLU}^2(·)$ denotes the squared complex-valued ReLU operation. ${\mathbf{A}}^{″}\in {\mathbb{C}}^{B\times L^{\prime }\times E}$ considers the fine dependencies within each pulse from a local perspective. The integration of these two attention tensors can be expressed as

$$\mathbf{A}={\mathbf{A}}^{\prime }[:,1:L,:]+{\mathbf{A}}^{″}[:,1:L,:].$$

(18)

The multiplication operation count of the fusion attention is $2L^{\prime }\theta E+M^2\theta P+M^{2}EP$, which is cheaper than (15).

The gating module consists of ${\mathrm{ECB}}_1$ and ${\mathrm{ECB}}_2$, which restore the channel numbers of $\mathbf{A}\in {\mathbb{C}}^{B\times L\times E}$ and $\mathbf{V}$ back to C, respectively. ${\mathbf{V}}^{\prime }\in {\mathbb{C}}^{B\times L\times C}$ is activated through a sigmoid function and then element-wise multiplied with ${\mathbf{V}}^{″}\in {\mathbb{C}}^{B\times L\times C}$ to produce ${\mathbf{V}}^{‴}\in {\mathbb{C}}^{B\times L\times C}$. Similarly to the DCB, the EAB and skip connection are added at the end of the AB to improve the performance of the network in extracting interference features. This process can be formulated as

$$\begin{array}{cc}\hfill {\mathbf{Z}}^{\prime }& =EAB\left(Sigmoid\left({ECB}_1\left(\mathbf{A}\right)\right)\odot {\mathrm{ECB}}_2\left(\mathbf{V}\right)\right)+\mathbf{Z}\hfill \\ \hfill & =EAB\left({\mathbf{V}}^{\prime }\odot {\mathbf{V}}^{″}\right)+\mathbf{Z}=EAB\left({\mathbf{V}}^{‴}\right)+\mathbf{Z}.\hfill \end{array}$$

(19)

In the AB group, a serial structure is adopted, where each AB’s input is the output from the previous AB. The output of the last AB is transposed and merged with the multi-scale features from the DCB group through addition. The merged features incorporate the fine details of the interference at smaller scales and the extensive information at larger scales. The merged features pass through a gating unit composed of four pointwise convolution layers and activation functions to generate a high-dimensional interference feature mask $\mathbf{M}\in {\mathbb{C}}^{B\times C\times L}$. This mask is then element-wise multiplied with $\mathbf{D}$ to estimate the representation of interference in the high-dimensional feature space $\mathbf{G}\in {\mathbb{C}}^{B\times C\times L}$, which can be expressed as

$$\mathbf{G}=\mathbf{D}\odot \mathbf{M}.$$

(20)

(3)

Decoder: The decoder is a one-dimensional transposed convolutional. It is the inverse transformation of the encoder, which restores high-dimensional representation to a one-dimensional time sequence as

$${\widehat{\mathbf{X}}}^{\prime }={\mathbf{Y}}^{\prime }-DecodingConv\left(\mathbf{G}\right)={\mathbf{Y}}^{\prime }-{\widehat{\mathbf{J}}}^{\prime },$$

(21)

where ${\widehat{\mathbf{X}}}^{\prime }\in {\mathbb{C}}^{B\times 1\times PT}$ and ${\mathbf{J}}^{\prime }\in {\mathbb{C}}^{B\times 1\times PT}$ denote the output of the network and the estimated intra-pulse interference, respectively. Therefore, the encoder and decoder parameters are consistent, to ensure consistency in the encoding and decoding processes. Then, ${\widehat{\mathbf{X}}}^{\prime }$ is a performed matrixization operation to obtain the multi-pulse estimated target echo $\widehat{\mathbf{X}}\in {\mathbb{C}}^{B\times P\times T}=\mathrm{unvec}({\widehat{\mathbf{X}}}^{\prime })$.

4. Experiments and Analysis

The experimental setup includes a Windows 10 operating system, an AMD Ryzen 7 5800X 8-Core Processor, an NVIDIA RTX 3090 GPU, and 32 GB of RAM. The AGCT used in these experiments is implemented using PyTorch 2.1.2.

4.1. Dataset

The critical parameters of each signal in the system are listed in

Table 1. System configuration.

	Parameter	Value
Target echo $\mathbf{X}$	Sampling frequency $f_s$ (MHz)	5
	Carrier frequency $f_c$ (GHz)	4
	Pulse duration ($\mu$s)	160
	Pulse repetition frequency (Hz) $f_r$	2000
	Number of samples in each pulse T	800
	Number of pulses P	64
	Target velocity v (m/s)	[5:20]
	Chirp rate $\left|k\right|$ (GHz/s)	[3:30]
	JSR (dB)	[−10:45]
SMSP interference ${\mathbf{J}}_{\mathrm{smsp}}$	Number of retransmissions R	$\left\{4,5\right\}$
ISRJ ${\mathbf{J}}_{\mathrm{isrj}}$	Number of retransmissions R	$\left\{4,5\right\}$
	Number of interceptions N	$\left\{4,5\right\}$
INUSRJ ${\mathbf{J}}_{\mathrm{inusrj}}$	Number of retransmissions R	[1:1:8]
	Number of interception N	$\left\{2,4,5,8\right\}$
Noise	SNR (dB)	[−60:0]

. Initially, 20,000 target echo samples were randomly generated within the specified chirp rate and target velocity ranges. Gaussian white noise was then randomly added to these samples within the range of −60 dB to 0 dB. Three types of retransmission interference—SMSP, ISRJ, and INURSJ—were added to each noisy sample, and the level of JSR ranged from −10 dB to 40 dB. The dataset was meticulously divided into a training set and a validation set at a ratio of 5:1, ensuring a comprehensive representation of the data.

Next, 1000 chirp rates were evenly selected within the range of [−30:3] GHz/s and [3:30] GHz/s, to generate target echo samples. These samples were then added to noise and three types of interference signals under each SNR [−60:5:0 dB] and JSR [−10:5:45 dB], to create the test set for this paper.

The training, validation, and test sets had 50,000, 10,000, and 156,000 samples, respectively.

The noisy interfered-with samples were vectorized and normalized to accelerate the training process and enhance the network’s ability to fit the model. The normalization process can be expressed as

$${\mathbf{Y}}^{\prime }=\frac{vec\left(\mathbf{Y}\right)-\mathbb{E}(vec(\mathbf{Y}\left)\right)}{\mathrm{var}\left(\mathrm{vec}\right(\mathbf{Y}\left)\right)}.$$

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4.2. Evaluation Metrics and Loss Function

In this paper, the root mean of the square error (RMSE) $\zeta$, the correlation coefficient $\psi$, and the detection probability $\chi$ are used as evaluation metrics, which can be expressed as

$$\begin{array}{cc}\hfill & \zeta =\sqrt{\mathbb{E}\left({\left|vec\left(\mathbf{X}\right)-{\widehat{\mathbf{X}}}^{\prime }\right|}^2\right)}\hfill \\ \hfill & \psi =\frac{|vec\left(\mathbf{X}\right){\widehat{\mathbf{X}}}^{\prime H}|}{\sqrt{{\parallel vec\left(\mathbf{X}\right)\parallel }_2^2{∥{\widehat{\mathbf{X}}}^{\prime }∥}_2^2}},\hfill \\ \hfill & \chi =\frac{{\varpi }_s}{{\varpi }_t},\hfill \end{array}$$

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where ${\varpi }_s$ and ${\varpi }_t$ denote the number of successful and total detections, respectively. The closer the value of $\zeta$ is to 0, the less error during the recovery process of the target echo, suggesting a more accurate restoration. Similarly, the value of $\psi$ closer to 1 indicates a higher similarity between the estimated target echo and the actual target echo. This research aims to enhance target-detection performance by suppressing retransmission interference, with detection probability [44,45] serving as a direct metric for assessing target-detection performance.

We propose a weighted correlation coefficient as the loss function, which continuously optimizes the network through training, to achieve effective interference suppression. It can be expressed as

$$\begin{array}{cc}\hfill {\psi }_i& =\frac{-vec(\mathbf{Y}-\mathbf{X}){\left[vec\left(\mathbf{Y}\right)-{\widehat{\mathbf{X}}}^{\prime }\right]}^H}{\sqrt{{\parallel vec(\mathbf{Y}-\mathbf{X})\parallel }_2^2{∥vec\left(\mathbf{Y}\right)-{\widehat{\mathbf{X}}}^{\prime }∥}_2^2}},\hfill \\ \hfill \rho & =\frac{{\parallel vec\left(\mathbf{X}\right)\parallel }_2^2}{{\parallel vec(\mathbf{Y}-\mathbf{X})\parallel }_2^2+{\parallel vec\left(\mathbf{X}\right)\parallel }_2^2},\hfill \\ \hfill loss& =-\rho \psi +(1-\rho ){\psi }_i,\hfill \end{array}$$

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where ${\psi }_i$ is the correlation coefficient of the irrelevant components, and where $\rho$ is the weight assigned to $\psi$; $\psi$ provides phase information guidance for the recovery of the target echo. After supplementing ${\psi }_2$, the amplitude of the target echo can be effectively estimated. By assigning weights, the contribution of each component in the loss function can be meticulously balanced.

4.3. Training Setup

The hyperparameter settings of the network are detailed in

Table 2. Network configuration.

	Parameter	Value
Decoder	Length of convolution kernel $k_1$	32
	Number of output channels C	256
DCB group	Number of repetitions r	3
	Number of DCBs $u_1$	6
	Length of dilated convolution kernel $k_2$	3
AB group	Number of ABs $u_2$	6
	Number of output channels for ${\mathrm{ECB}}_{\mathrm{qk}}$ $\theta$	128
	Number of output channels for ${\mathrm{ECB}}_{\mathrm{v}}$ E	512
ECB	Length of depthwise convolution kernel ${\mathit{k}}_4$	31
EAB	Length of depthwise convolution kernel $k_3$	17

. We used Adam as the optimizer, with an initial learning rate of 0.001, and we employed the ReduceLROnPlateau strategy to adjust the learning rate. The learning rate was halved automatically if there was no decrease in the loss function over three consecutive epochs. The batch size was set to 6.

Curriculum learning [46,47], a classic training strategy, involves training the network initially on simpler data and gradually progressing to more complex data. In the task of interference suppression, especially under low SNR conditions where noise can destroy or even obscure both target echoes and interference signals, this poses significant challenges in feature extraction. In Section 2, we analyzed three types of interference: SMSP, ISRJ, and INUSRJ. Due to their periodicity and randomness differences, the interference complexity increases in turn. Curriculum learning is employed in the network training process to enhance training efficiency and network generalization. Based on SNR levels and interference types, samples are progressively added to the training set and validation set.

4.4. Experiments

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SMSP: Figure 6 shows the range–Doppler (RD) [48] results before and after SMSP interference suppression under SNR = −60 dB and JSR = 45 dB. The addition of strong SMSP interference and noise raised the RD clutter floor, obscuring the target’s range information, though a row of peaks remains evident at $f_d$ = 266.85 Hz. Figure 6c shows the RD result after AGCT interference suppression. It can be seen that the AGCT effectively eliminated most of the interference and noise and obtained clean RD results similar to Figure 6a. From the perspective of the evaluation metrics, $\zeta$ and $\psi$ of the multi-pulse received signal in Figure 6b and the multi-pulse estimated target echo in Figure 6c were 0.5268, 0.0389, 0.2009, and 0.9569, respectively. These demonstrate that the AGCT can effectively extract the characteristics of the target echo from the interference and noise background and achieve a relatively ideal recovery effect. In order to analyze the interference suppression effect more intuitively, the cell-averaging constant false alarm rate (CA-CFAR) algorithm was applied to the RD results of Figure 6a–c, with the detection results displayed in Figure 6d–f. In Figure 6e, many false target peaks were generated near the actual target but remained below the CFAR threshold, indicating that interference destroyed the radar’s detection capability. Conversely, Figure 6d,f both present distinct main lobes at the 0 $\mathsf{\mu }$s, with peak sidelobe ratios (PSLR) of −13.4388 dB and −12.1681 dB, respectively, both above the CFAR threshold, effectively detecting the target.

To further validate the interference suppression performance of the AGCT, Figure 7 shows the evaluation metrics of the AGCT and the MRDENet in the SMSP interference test set. Both networks showed performance degradation under low SNR conditions, due to high noise levels that obscured the interference signal and target echo, complicating the feature extraction. However, the interference suppression performance would be improved under high JSR conditions. SMSP interference has extremely strong periodicity. An increase in interference power would strengthen the characteristics of SMSP interference, which benefits the network’s extraction of interference features, thereby achieving better results. Comparatively, the AGCT consistently outperforms the MRDENet across most conditions, especially at low SNR levels. Here, $\chi$ higher than 0.8 was considered as effective detection. The lower bound of the AGCT to achieve effective detection is JSR + SNR = −30 dB, while the lower bound of the MRDENet is −20 dB. This shows that the AGCT can achieve effective interference suppression under harsher conditions.

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ISRJ: Figure 8 demonstrates the RD results before and after ISRJ suppression when SNR was −25 dB and JSR was 45 dB. From Figure 8b,c, it can be seen that the AGCT significantly mitigated the adverse effects of irrelevant signals, restoring the Doppler and range information of the target; $\zeta$ and $\psi$ of the multi-pulse estimated target echo were 0.2331 and 0.9069, respectively, which were improvements of 0.2936 and 0.8959, respectively, compared with the multi-pulse received signal. In Figure 8e, the actual target was obscured by the retransmission interference, so the CFAR algorithm could not detect the target. The PSLR of Figure 8f was −11.2026 dB, which was an increase of 2.2362 dB compared to Figure 8d, and both successfully achieved target detection.

The performance of ISRJ suppression under various conditions is displayed in Figure 9. Contrary to SMSP interference, higher ISRJ power leads to poorer suppression outcomes. The interference mechanism of ISRJ is the retransmission of intercepted target echo segments. The interference does not contain all target echo information, which limits its utility in aiding the recovery of target echo. Additionally, the truncation of multiple segments creates boundary effects that hinder the network’s feature extraction. Increased interference power negatively affects target detection post-interference suppression. The AGCT effectively suppressed ISRJ under conditions with SNR above −30 dB and JSR not exceeding 45 dB. Especially under the higher SNR conditions on the right side of Figure 9b, $\zeta$ and $\psi$ of the multi-pulse estimated target echo could reach levels below 0.05 and above 0.99, respectively. Under the same conditions, the MRDENet’s interference suppression was less effective than the AGCT’s. Due to ISRJ’s higher complexity compared to SMSP, both networks demonstrated less effective suppression of ISRJ than SMSP.

(3)

INUSRJ: Figure 10 shows the RD results related to INUSRJ suppression when the SNR was −25 dB and the JSR was 45 dB. It can be clearly seen from Figure 10c that the AGCT achieved a terrific interference suppression effect; $\zeta$ and $\psi$ of the multi-pulse estimated target echo were 0.2922 and 0.8191, respectively. Compared with the multi-pulse received signal, the improvements were 0.2302 and 0.8039, respectively. Contrary to the detection failure result in Figure 10e, the RD value at the 0 $\mathsf{\mu }$s in Figure 10f was higher than the CFAR threshold and was successfully detected, with a PSLR of −11.0158 dB. Unlike the results in Figure 6f, where the pulse compression closely restored the original pulse structure after suppressing SMSP interference, the outcomes in Figure 8f and Figure 10f exhibit various degrees of deformation. This indicates that remnants of ISRJ and INUSRJ still affected the multi-pulse estimated target echoes after undergoing AGCT interference suppression.

Comparing Figure 9 and Figure 11, it is observable that both networks experienced some performance degradation under INUSRJ relative to ISRJ. The performance under both interference types correlated negatively with the interference power, consistent with the analysis in Section 2. The random retransmission characteristic of INUSRJ increases interference complexity, naturally raising the difficulty for networks to fit the interference. From Figure 11c, the AGCT effectively suppressed interference under conditions where SNR was above −30 dB and JSR was not higher than 45 dB, similar to Figure 9c. However, $\psi$ in Figure 9b surpassed those in Figure 11b under the same conditions. In Figure 11f, the MRDENet could effectively suppress interference only under high SNR and low JSR conditions, which further verified the noise robustness of the AGCT.

The number of network parameters and the inference time of the AGCT were 7.05 M and 175 ms, respectively, while the number of network parameters and the inference time of the MRDENet were 20.00 M and 51 ms, respectively. Although the AGCT is lighter than the MRDENet, its sophisticated attention mechanisms lead to longer inference times. However, considering the balance between performance and efficiency, these time costs are worthwhile.

5. Conclusions

This paper proposes the AGCT to suppress intra-pulse retransmission interference in electronic countermeasure. The AGCT adopts the structure of a complex-valued network to fit the model of complex-valued signals. Its main structures include an encoder, mask estimator, and decoder. In the mask estimator, the DCB group focuses on extracting the more delicate multi-scale details of the interference. In contrast, the AB group combines inter-pulse and intra-pulse attention, to obtain the large-scale information of the interference. An ECB integrated into DCBs and ABs can further improve the feature extraction capability for strong interference. The interference is suppressed through residual learning, and the multi-pulse estimated target echo is obtained. Extensive experiments under various SNR and JSR conditions validated the effectiveness of the AGCT in suppressing multiple types of intra-pulse retransmission interference. For SMSP with an SNR of −60 dB and a JSR of 45 dB, the AGCT achieved a low RMSE of 0.5268 and a high correlation coefficient of 0.9569. ISRJ, with an SNR of −25 dB and a JSR of 45 dB, achieved an RMSE of 0.2331 and a correlation coefficient of 0.9069. INUSRJ, with an SNR of −25 dB and a JSR of 45 dB, achieved an RMSE of 0.2922 and a correlation coefficient of 0.8191. Under the same SNR and JSR conditions mentioned above, the AGCT and the MRDENet were used to suppress the three types of interference, and the detection probability of the former was 0.021, 0.840, and 0.857 higher than the latter, respectively. The AGCT demonstrated superior performance compared to the MRDENet, making it a promising solution for complex electromagnetic environments. In the future, we aim to further enhance the AGCT’s ability to handle a wider range of interference types.