SAR Radio Frequency Interference Suppression Based on Kurtosis-Guided Attention Network

Highlights

What are the main findings?

• A kurtosis-guided attention network (KANet) is proposed, which utilizes temporal kurtosis with an attention mechanism to guide the network to focus on interference-corrupted regions.
• Proposing a quantitative evaluation framework for phase fidelity to address phase variations in SAR interference suppression.

What are the implication of the main findings?

• The proposed method addresses the challenge of distinguishing complex RFI from high-energy ground backscatter based solely on signal energy, thereby improving the interference suppression performance.
• This work validates the method’s high phase fidelity during interference removal via quantitative evaluation, benefiting phase-sensitive SAR applications.

Abstract

Radio-frequency interference (RFI) severely degrades the imaging quality of synthetic aperture radar (SAR), especially when the interference energy is strongly coupled with ground backscatter in both the time and frequency domains. Existing algorithms typically rely on energy contrast or component decomposition in transform domains, which limits their ability to cleanly separate complex RFI from high-power echoes. Exploiting the fact that kurtosis is insensitive to ground clutter and background noise, this paper proposes an interference suppression network based on the temporal kurtosis guidance mechanism. Specifically, a statistical prior vector capturing the non-Gaussian characteristics of RFI is constructed using kurtosis in the time–frequency domain and is integrated into a multi-scale attention mechanism, allowing the network to more effectively concentrate on interfered regions. Meanwhile, a systematic framework is established for the quantitative assessment of phase fidelity in the reconstruction of complex-valued SAR echoes. On this basis, by exploiting the strong generalization capability and high processing efficiency of data-driven models, the proposed network achieves improved RFI separation and enhanced reconstruction accuracy of underlying scene features. Ablation experiments validated that the design of a kurtosis-guided module can reduce the mean square error (MSE) loss by 14.87% compared to the basic model. Furthermore, regarding the phase fidelity, the correlation coefficient between the suppressed signal and the original true signal reached 0.99. Finally, GF-3 satellite data are used to further demonstrate the effectiveness and practicality of the proposed method.

1. Introduction

Synthetic Aperture Radar (SAR) is an active microwave remote sensing system that achieves two-dimensional high-resolution imaging by transmitting wideband signals and utilizing the synthetic aperture principle to overcome physical aperture limitations. Its all-day, all-weather observation capabilities are significantly superior to those of traditional remote sensing techniques, such as optical and infrared sensors [1,2,3,4]. However, during the acquisition of target information, SAR systems are inevitably susceptible to interference from other in-band electromagnetic radiation sources, known as Radio Frequency Interference (RFI). RFI is typically much stronger than the target’s scattering signals and can mask target information, severely degrading imaging quality and the accuracy of information extraction [5]. Figure 1 illustrates an example of RFI-corrupted data from the GF-3 satellite. In this scenario, the RFI covers both land and ocean regions, where the energy of the stronger part of ground echoes is comparable to that of the interference, making it difficult to distinguish them based solely on energy threshold values. Consequently, developing efficient and robust RFI suppression methods that can eliminate interference while maximally preserving the original radar signal information has become an important scientific problem and engineering challenge in the field of SAR research.

Existing interference suppression methods can be broadly categorized into non-parametric, parametric, semi-parametric, and deep learning methods. Non-parametric methods operate by transforming the data to other domains where interference components can be more easily separated, such as notch filtering [6,7,8], adaptive filtering [9,10], and subspace projection filtering [11,12,13]. These methods are computationally simple and easy to implement, but their suppression performance heavily depends on the separability between interference and the radar signal. If their respective features overlap, the filtering process will also remove useful echoes, resulting in an irreversible loss of target’s information [14]. Parametric methods achieve interference suppression by modeling the echo signal [15,16,17]. For instance, to address the complex motion-induced non-stationarity, Zhang et al. [17] constructed an accurate echo model for FDA-MIMO SAR and utilized adaptive compensation to suppress mainlobe deceptive jamming. However, their performance bound is jointly determined by the complexity of the model and the precision of parameter estimation, and their suppression efficacy may degrade significantly in complex interference scenarios. In contrast to the previous two categories, semi-parametric methods leverage the specific attributes of different signal components, employing iterative optimization of a loss function to achieve matrix decomposition. This approach effectively separates interference from the mixed signal while preserving the radar signal. Existing semi-parametric models primarily include sparse low-rank models and weighted matrix decomposition [18,19,20,21,22]. Joy et al. [18] separated the RFI signal matrix from the SAR echo matrix by constructing low-rank models in the range and azimuth directions. Liu et al. [19], jointly using transform-domain characteristics, constructed a sparse model for wideband interference in the time-frequency domain and designed a joint estimation algorithm to simultaneously achieve RFI suppression and signal-of-interest recovery within an optimization framework. Huang et al. [20] employed a matrix decomposition algorithm that uses a weighted strategy to approximate the rank function, as well as an algorithm that uses the rank’s upper bound as a prior constraint, to suppress narrowband RFI in SAR systems. Although these semi-parametric methods can achieve good results, they require dataset-specific hyperparameter tuning and iterative computation. This results in high computational complexity and poor generalization, which is particularly pronounced in large-scale data processing.

With the rapid development of deep learning (DL) technology, researchers are increasingly applying DL methods to the SAR field, which include many neural network-based interference suppression techniques. Neural networks can automatically extract hierarchical features of targets from images and possess strong generalization capabilities: once trained, they can be directly transferred for application in similar scenes. Fan et al. [23] employed a deep convolutional neural network (CNN) for interference detection in the time-frequency domain, subsequently using deep residual networks to extract and reconstruct the time-frequency features of the target signal. Xu et al. [24] utilized a generative adversarial network (GAN) to map input echoes containing wideband interference to their corresponding counterparts without interference, thereby achieving interference suppression. Tao et al. [25] proposed a hybrid method fusing sparse low-rank models with stacked recurrent neural network (RNN) to achieve RFI extraction and suppression. Compared to traditional approaches, DL-based interference suppression methods yield better performance, but they also face several limitations. On one hand, existing suppression networks focus on the high energy or texture of the RFI, but their performance often degrades when ground targets and interference are highly coupled within the same signal pulse, in which case new criteria need to be introduced to distinguish between the two components. On the other hand, most existing evaluations of interference suppression efficacy are primarily concerned with amplitude errors in the resulting signal or image, while ignoring the critical information contained in the signal phase, and a quantitative framework for assessing phase fidelity is still lacking.

To address these issues, this paper proposes an interference suppression Network featuring an Attention mechanism guided by signal kurtosis characteristics (KANet). The core innovation of this method lies in its physics-informed attention design. For RFI suppression tasks, common attention mechanisms compute attention weights solely based on image domain features extracted from the input data, such as texture and energy distributions. However, when RFI and high energy ground echoes exhibit similar characteristics in the image domain, these mechanisms may fail to effectively distinguish between them. In contrast, the proposed kurtosis-guided attention incorporates statistical properties of the signal. Specifically, kurtosis is a fourth-order moment that captures the non-Gaussian nature of RFI. This statistical information at the signal-level is inherently inaccessible to conventional attention mechanisms, allowing the proposed approach to precisely locate RFI-corrupted regions even when RFI and ground echoes appear similar in image domain.

The main contributions of this paper can be concluded as follows:

• Firstly, considering the non-Gaussian nature of interference, the principle of kurtosis difference between SAR echoes and RFI signals in time-frequency domain is numerically derived, providing a theoretical basis for the proposed method.
• Secondly, the distribution of RFI in the echo pulse is mapped to a multi-scale kurtosis vector, guiding the network to increase attention to the interfered time-frequency units, achieving adaptive feature weighting.
• Thirdly, a fidelity evaluation framework based on the phase information of the reconstructed signal is established, evaluating the phase recovery results from three perspectives: phase preservation error, dispersion, and correlation coefficient.
• Finally, the proposed method is comprehensively validated experimentally based on semi-physical simulation data and real spaceborne SAR data.

Experimental results demonstrate that, compared with existing deep learning-based interference suppression algorithms, the proposed method effectively integrates traditional interference detection metrics with deep neural networks. By incorporating a kurtosis-guided attention mechanism, it can effectively separate long-duration interference from high-energy ground echoes. Furthermore, the method ensures both reconstruction accuracy and phase fidelity of the signal after interference suppression, which clearly highlights its superiority.

2. Theoretical Foundation

2.1. Formulation of SAR Echoes

Assuming the SAR system transmits a linear frequency modulation (LFM) signal, it can be expressed as Equation (1):

$$S_{tx}\left(t\right)=\mathrm{rect}(t/T_p)exp\left[j2\pi (f_{0}t+k_r{t}^2/2)\right]$$

(1)

In Equation (1), $f_0$ is the center frequency, $k_r$ is the chirp rate, $T_p$ is the pulse width, and t is the fast time. For a SAR system, the received echo signal consists of a superposition of the target signal, the interference signal, and the noise signal. The noise signal primarily stems from environment, whereas the interference signal originates from other in-band radiation sources. Typically, the energy of the interference signal is greater than that of the target signal and the noise signal. The interfered echo signal can be modeled as Equation (2):

$$S_{rx}(m,n)=S_t(m,n)+S_n(m,n)+S_i(m,n)$$

(2)

where m is the azimuth pulse index, n is the range time unit index, $S_t$ is the target signal, $S_n$ is the noise signal, and $S_i$ is the interference signal.

2.2. Formulation of RFI Signals

The interference signals discussed in this paper refer to unintentional interference, as opposed to intentional jamming aimed at actively disrupting SAR systems. Based on the relative bandwidth of the interference signal with respect to the SAR signal, RFI can be classified as narrowband interference (NBI) and wideband interference (WBI). This section introduces three representative RFI types and provides their mathematical expressions. These formulations are consistent with the widely recognized interference models adopted in the literature [14,21,23], thereby ensuring the physical fidelity of our RFI simulation. These include single-tone interference as an example of NBI, and linear frequency modulation interference and sinusoidal modulation interference as examples of WBI.

The expression for narrowband interference is given by Equation (3):

$$I_{NB}(m,n)=\sum _{l=1}^L{A}_l\left(m\right)·exp(j2\pi f_{l}n+j{\varphi }_l)$$

(3)

where L is the total number of interference signals, $A_l\left(m\right)$ represents the amplitude of each single-tone interference, $f_l$ represents the frequency of each interference, and ${\varphi }_l$ is its initial phase.

The expression for linear frequency modulation wideband interference is given by Equation (4):

$$I_{WBCM}(m,n)=\sum _{l=1}^L{B}_l\left(m\right)·exp(j2\pi f_{l}n+j\pi {\gamma }_l{n}^2)$$

(4)

where L is the total number of interference signals, $B_l\left(m\right)$ represents the amplitude of each interference, $f_l$ represents the frequency of each interference, and ${\gamma }_l$ represents the chirp rate of the interference.

The expression for sinusoidal modulation wideband interference is given by Equation (5):

$$I_{WBSM}(m,n)=\sum _{l=1}^L{C}_l\left(m\right)·exp(j{\beta }_{l}sin(2\pi f_{l}n+{\varphi }_l))$$

(5)

where L is the total number of interference signals, $C_l\left(m\right)$ represents the amplitude of each interference, ${\beta }_l$ represents the modulation coefficient of each interference, $f_l$ represents the frequency of each interference, and ${\varphi }_l$ is its initial phase.

Figure 2 illustrates the frequency-domain and time-frequency-domain representations of the three aforementioned interference types. In each subfigure, the plot on the left shows the spectrum, and the plot on the right shows the time-frequency representation.

3. Kurtosis-Guided RFI Suppression Network

This chapter elaborates on the core principles and architectural design of the proposed kurtosis-guided attention-based interference suppression network (KANet). The main innovation of this method lies in the design of a kurtosis prior guidance module. Its effectiveness is based on the significant difference in the statistical distributions of SAR signals and RFI. This module aims to quantify the impulsive characteristics of the signal along the time dimension, thereby generating a statistical metric capable of accurately identifying RFI.

To clearly elucidate the method, the content of this chapter is organized as follows: First, we analyze the theoretical feasibility and practical effectiveness of using kurtosis as an indicator of RFI presence, providing the theoretical justification for the method. Building on this foundation, we then introduce the key technical details, including the overall network framework, the backbone architecture, the specific implementation of the kurtosis prior guidance module, and the data normalization strategy.

3.1. Analysis of the Kurtosis Prior

Specifically, the complex representation of the SAR signal and background noise exhibits quasi-Gaussian statistical characteristics. In contrast, most RFI exhibits non-Gaussian, impulsive, and transient characteristics, manifesting as short-duration, high-energy spikes. To effectively capture this disparity, this study employs instantaneous kurtosis as a key statistical measure. The kurtosis of the signal’s time-frequency spectrogram is calculated using a sliding time window to characterize the likelihood of RFI presence within each time unit. Kurtosis is a fourth-order moment that characterizes the peakedness and tailedness of a probability distribution. It is calculated as follows.

Given an input Short-Time Fourier Transform (STFT) time-frequency matrix $\mathbf{Y}$, as shown in Equation (6), with dimensions $F\times T$, where F is the number of frequency units and T is the number of time windows.

$$\mathbf{Y}=\left(\begin{array}{cccc}{y}_{1,1}& y_{1,2}& \dots & y_{1,T}\\ y_{2,1}& y_{2,2}& \dots & y_{2,T}\\ ⋮& ⋮& \ddots & ⋮\\ y_{F,1}& y_{F,2}& \dots & y_{F,T}\end{array}\right)$$

(6)

$${\mathbf{y}}_t={\left[\begin{array}{cccc}{y}_{1,t},& y_{2,t},& \dots & y_{F,t}\end{array}\right]}^T$$

(7)

Equation (7) represents the vector composed of signal amplitudes across all frequency units at time index t.

${\mu }_t$, the mean of the signal within the t-th time window is given by Equation (8):

$${\mu }_t={\displaystyle \frac{1}{F}}\sum _{f=1}^F{y}_{f,t}$$

(8)

The central second-order moment (variance), $m_{2,t}$, is given by Equation (9):

$$m_{2,t}={\displaystyle \frac{1}{F}}\sum _{f=1}^F{(y_{f,t}-{\mu }_t)}^2$$

(9)

The central fourth-order moment, $m_{4,t}$, is given by Equation (10):

$$m_{4,t}={\displaystyle \frac{1}{F}}\sum _{f=1}^F{(y_{f,t}-{\mu }_t)}^4$$

(10)

The kurtosis, $g_t$, within this time window is given by Equation (11):

$$g_t={\displaystyle \frac{m_{4,t}}{{\left(m_{2,t}\right)}^2}}-3$$

(11)

Kurtosis is defined as the standardized fourth-order moment minus 3. Since the standardized fourth-order moment of a standard Gaussian distribution is 3, this subtraction (of 3) serves to calibrate the kurtosis of the standard Gaussian distribution to zero.

To theoretically demonstrate why kurtosis can effectively identify RFI, we analyze two distinct cases: the interference-free SAR signal and the interference-corrupted signal. We then calculate the range of kurtosis values for each case.

Firstly, for an interference-free SAR signal, the echo $S_t\left(t\right)$ is traditionally modeled as the coherent superposition of numerous independent scatterers within a resolution cell. According to the Central Limit Theorem (CLT), this leads to a complex Gaussian representation of the echo and Rayleigh distribution for its amplitude. While the Rayleigh model provides an adequate description for homogeneous media, SAR echoes often exhibit pronounced spatial heterogeneity due to complex ground textures and land cover variability, resulting in heavy-tailed statistics that deviate substantially from the Rayleigh distribution. To accurately capture this behavior, we employ the K-distribution, which arises from a multiplicative combination of Rayleigh speckle and Gamma distributed texture [26,27,28]. By parameterizing the scene heterogeneity through its shape parameter, the K-distribution not only recovers the Rayleigh law as a limiting special case corresponding to homogeneous echoes, but also provides an excellent fit to the highly heterogeneous, heavy-tailed echoes observed over complex terrains.

After transforming the signal to the time-frequency domain via the STFT, we consider the signal’s amplitude distribution along the frequency axis within a fixed time window. The STFT is a linear transformation of the underlying random process, so within each time window the set of signal samples can be modeled as samples drawn from a K-distributed amplitude random process. Specifically, the statistical calculation is performed using the time-frequency amplitudes within a single time window. Consider the amplitude vector of the interference-free SAR signal across all F frequency units at the t-th time index. Let this vector be denoted as $\mathbf{x}={[x_1,x_2,\dots ,x_F]}^T$, which corresponds to the magnitude of ${\mathbf{y}}_t$ (defined in Equation (7)) in the absence of interference. The elements $x_i$ of this vector are modeled as independent realizations of a K-distributed random variable X. Its probability density function (PDF) under the K-distribution model is given by Equation (12):

$$f_X(x;\nu ,\alpha )={\displaystyle \frac{2}{\alpha \Gamma (\nu +1)}}{\left({\displaystyle \frac{x}{2\alpha }}\right)}^{\nu +1}{K}_{\nu }\left({\displaystyle \frac{x}{\alpha }}\right),x>0,\alpha >0,\nu >0,$$

(12)

where $\nu$ is the shape parameter controlling the degree of heterogeneity, $\alpha$ is the scale parameter, $\Gamma (·)$ is the Gamma function, and $K_{\nu }(·)$ denotes the modified Bessel function of the second kind of order $\nu$.

The raw n-th-order moment of the K-distributed $\mathbf{x}$, $E\left[X^n\right]$, can be computed in closed form as in Equation (13):

$$E\left[X^n\right]={\displaystyle \frac{2^n{\alpha }^n\Gamma \left(0.5n+1\right)\Gamma \left(0.5n+\nu +1\right)}{\Gamma (\nu +1)}}.$$

(13)

The kurtosis of the K-distributed interference-free signal, $g_k$, is defined analogously to Equation (11), and can be written in terms of raw moments as in Equation (14):

$$g_k={\displaystyle \frac{E\left[X^4\right]-4E\left[X^3\right]E\left[X\right]+6E\left[X^2\right]E{\left[X\right]}^2-3E{\left[X\right]}^4}{{\left(E\left[X^2\right]-E{\left[X\right]}^2\right)}^2}}-3.$$

(14)

By substituting Equation (13) with $n=1, 2, 3, 4$ into Equation (14), we obtain an explicit expression of the kurtosis of signal X. After straightforward algebra, the scale parameter $\alpha$ cancels out and the resulting kurtosis depends only on $\nu$, as given in Equation (15):

$$\begin{array}{cc}\hfill g_k\left(\nu \right)& ={\displaystyle \frac{32(\nu +1)(\nu +2)\Gamma {(\nu +1)}^4-12\pi \Gamma {\left(\nu +\frac{3}{2}\right)}^2\Gamma {(\nu +1)}^2-3{\pi }^2\Gamma {\left(\nu +\frac{3}{2}\right)}^4}{{\left[4(\nu +1)\Gamma {(\nu +1)}^4-\pi \Gamma {\left(\nu +\frac{3}{2}\right)}^2\Gamma {(\nu +1)}^2\right]}^2}}-3.\hfill \end{array}$$

(15)

This result shows that the kurtosis $g_k$ of the interference-free SAR signal becomes an explicit function of the parameter $\nu$. For homogeneous or mildly heterogeneous scenes (large $\nu$), $g_k\left(\nu \right)$ approaches the Rayleigh limit and remains a small positive value; for more heterogeneous backgrounds, $g_k\left(\nu \right)$ increases accordingly but still remains within a relatively small range in practice. For example, using the closed-form expression in Equation (15), we obtain $g_k\left(0.1\right)\approx 3.64$ for a highly heterogeneous scene, whereas for $\nu =100$ the value drops to $g_k\left(100\right)\approx 0.25$. Therefore, for an interference-free SAR signal, the calculated kurtosis remains a small value close to zero.

Next, we consider the case where the SAR signal is superimposed with RFI, and calculate the kurtosis of the corrupted signal within a fixed time window. Assume the signal within this time window has F total samples along the frequency axis. The amplitude vector of the interference-free SAR signal, denoted as $\mathbf{x}={[x_1,x_2,\dots ,x_F]}^T$, consists of samples that follow the K-distribution described above. Concurrently, assume an RFI with an intensity far greater than that of the SAR signal corrupts M of the F frequency units. The signal amplitudes at these M interfered frequency samples are defined as a vector $\mathbf{A}={[A_1,A_2,\dots ,A_M]}^T$. For analytical simplicity, we assume the RFI is a single-tone interference affecting only one frequency unit (i.e., $M=1$, thus $\mathbf{A}$ simplifies to a scalar A). Therefore, the amplitude vector of the SAR signal superimposed with RFI can be expressed as $\mathbf{z}={[x_1,x_2,\dots ,x_{i-1},A,x_{i+1},\dots ,x_F]}^T$. We now calculate the kurtosis of the combined signal $\mathbf{z}$.

The raw n-th-order moment about the origin of the mixed signal $\mathbf{z}$, $E\left[Z^n\right]$, is given by Equation (16):

$$E\left[Z^n\right]={\displaystyle \frac{{\sum }_{i=1}^{F-1}{x_i}^n+A^n}{F}}={\displaystyle \frac{F-1}{F}}E\left[X^n\right]+{\displaystyle \frac{A^n}{F}},$$

(16)

where $E\left[X^n\right]$ is the raw n-th-order moment about the origin for the K-distribution, calculated as shown in Equation (13). According to the kurtosis calculation in Equation (11), we need to compute the central fourth-order moment and second-order moment of $\mathbf{z}$. The central fourth-order moment is given by Equation (17):

$$m_{4,z}={\displaystyle \frac{1}{F}}{\sum }_{i=1}^F{(z_i-{\mu }_z)}^4=E\left[Z^4\right]-4E\left[Z^3\right]E\left[Z\right]+6E\left[Z^2\right]E{\left[Z\right]}^2-3E{\left[Z\right]}^4$$

(17)

The $E\left[Z^n\right]$ (for $n=1, 2, 3, 4$) are calculated by substituting Equation (13) into Equation (16). These moments are then substituted into Equation (17) to yield a result expressed in terms of F, A, $\nu$ and $\alpha$, as given by Equation (18):

$$\begin{array}{c}{m}_{4,z}=\left({\displaystyle \frac{F-1}{F}}E\left[X^4\right]+{\displaystyle \frac{A^4}{F}}\right)-4\left({\displaystyle \frac{F-1}{F}}E\left[X^3\right]+{\displaystyle \frac{A^3}{F}}\right)\left({\displaystyle \frac{F-1}{F}}E\left[X\right]+{\displaystyle \frac{A}{F}}\right)\hfill \\ +6\left({\displaystyle \frac{F-1}{F}}E\left[X^2\right]+{\displaystyle \frac{A^2}{F}}\right){\left({\displaystyle \frac{F-1}{F}}E\left[X\right]+{\displaystyle \frac{A}{F}}\right)}^2-3{\left({\displaystyle \frac{F-1}{F}}E\left[X\right]+{\displaystyle \frac{A}{F}}\right)}^4\hfill \end{array}$$

(18)

where $E\left[X\right]$, $E\left[X^2\right]$, $E\left[X^3\right]$, and $E\left[X^4\right]$ follow from the K-distribution model in Equation (13).

The central second-order moment of Z is given by Equation (19):

$$m_{2,z}={\displaystyle \frac{1}{F}}{\sum }_{i=1}^F{(z_i-{\mu }_z)}^2=E\left[Z^2\right]-E{\left[Z\right]}^2$$

(19)

Similarly, by substituting $E\left[Z^2\right]$ and $E\left[Z\right]$ (obtained from Equation (16) with $E\left[X^n\right]$ given by Equation (13)), we derive a result expressed in terms of F, A, $\nu$ and $\alpha$, as given by Equation (20):

$$m_{2,z}=\left({\displaystyle \frac{F-1}{F}}E\left[X^2\right]+{\displaystyle \frac{A^2}{F}}\right)-{\left({\displaystyle \frac{F-1}{F}}E\left[X\right]+{\displaystyle \frac{A}{F}}\right)}^2$$

(20)

Finally, the kurtosis of signal $\mathbf{Z}$ is calculated as shown in Equation (21):

$$g_z(F,A,\nu ,\alpha )={\displaystyle \frac{\begin{array}{c}\left(\frac{F-1}{F}E\left[X^4\right]+\frac{A^4}{F}\right)-4\left(\frac{F-1}{F}E\left[X^3\right]+\frac{A^3}{F}\right)\left(\frac{F-1}{F}E\left[X\right]+\frac{A}{F}\right)\hfill \\ +6\left(\frac{F-1}{F}E\left[X^2\right]+\frac{A^2}{F}\right){\left(\frac{F-1}{F}E\left[X\right]+\frac{A}{F}\right)}^2-3{\left(\frac{F-1}{F}E\left[X\right]+\frac{A}{F}\right)}^4\hfill \end{array}}{{\left(\left(\frac{F-1}{F}E\left[X^2\right]+\frac{A^2}{F}\right)-{\left(\frac{F-1}{F}E\left[X\right]+\frac{A}{F}\right)}^2\right)}^2}}-3$$

(21)

where $E\left[X^n\right]$ are again given by the K-distribution moments in Equation (13).

To evaluate Equation (21), we substitute practical data to estimate the kurtosis of the interference-corrupted signal. We set $F=256$ and in this case, each time window of the time-frequency signal contains 256 sampling points along the frequency axis. Since $g_z$ depends on the background statistics through $(\nu ,\alpha )$ as well as on the interference amplitude A, we consider a representative parameter setting with $\nu =3$ and $\alpha =1$. Subsequently, let A be k times the mean amplitude of the SAR signal. Based on a comparative analysis of the amplitudes of interference-corrupted signals in real-world SAR data, we set the range of k to be between 3 and 10. Substituting the values of $F=256$, $\nu =3$, $\alpha =1$ and $A=kE\left[X\right]$ into Equation (21) demonstrates that the kurtosis of the interference-corrupted signal is significantly increased, reaching a magnitude that is typically more than 5 to 10 times higher than that of the interference-free signal (given in Equation (15)). For instance, with $k=10$, numerical evaluation yields $g_z\approx 55$, indicating more than an order-of-magnitude increase. Evidently, unlike the interference-free SAR signal, the kurtosis value increases rapidly when a time window is superimposed with interference. It is also worth noting that although the above derivation assumes a single-tone interference ($M=1$) for analytical simplicity, the conclusion generalizes to multi-tone RFI ($M>1$). Since RFI typically exhibits sparsity in the time-frequency domain ($M\ll F$), the signal retains distinct heavy-tailed characteristics even with increased M. Consequently, the kurtosis remains significantly higher than that of the interference-free background, ensuring the metric’s robustness against diverse RFI forms.

To validate the practical effectiveness of the proposed theory, verification was performed using real-world GF-3 SAR data. Figure 3 presents the analysis across four distinct scenarios, displaying SAR images alongside time-frequency domain spectrograms and their corresponding time-sequenced kurtosis curves. The results demonstrate a precise temporal alignment between kurtosis peaks and RFI presence. In the interference-corrupted cases (Scenarios 1, 2, and 3), the kurtosis values increase significantly, exhibiting a strong response to RFI. In contrast, Scenario 4 confirms the metric’s robustness against complex ground echoes: in the absence of RFI, the kurtosis curve remains stable at a low level despite the presence of strong terrain backscatter. This verifies that the kurtosis metric effectively targets impulsive interference while remaining insensitive to typical Gaussian-like background echoes.

This experiment strongly demonstrates that the kurtosis calculation is insensitive to Gaussian-like signals, such as ground echoes, and responds strongly only to RFI. It can serve as a highly sensitive and robust RFI localization metric. In summary, a kurtosis value close to zero indicates the absence of RFI within that time window; conversely, a large positive kurtosis value clearly indicates RFI contamination. Therefore, this kurtosis value can be used to guide the attention mechanism, directing the network to focus on RFI-affected regions and thereby improving suppression efficacy.

3.2. Overall Framework

The overall workflow of the RFI suppression method in this paper is illustrated in Figure 4. First, the input signal is transformed into the time-frequency domain, and the amplitude information is divided into blocks along the time axis and sent into the network after passing through a min-max normalization strategy. Second, to enhance the network’s RFI suppression performance by leveraging the information of the SAR signal’s physical characteristics, we designed a kurtosis prior guidance module. This module quantifies the non-Gaussianity of the RFI by calculating the signal’s kurtosis in a time window-wise manner, and maps the kurtosis to an attention weight through a three-layer fully connected network. This weight undergoes adaptive pooling to match the feature map dimensions at different network levels, thereby performing feature weighting. Subsequently, guided by this module, the interference suppression network adaptively suppress the RFI. It then reconstructs the interference-free amplitude spectrum, which undergoes a de-normalization process. This spectrum is combined with the phase information to reconstruct the complex time-frequency domain signals. Finally, this representation is transformed back into the time domain via the inverse short-time Fourier transform (ISTFT), completing the RFI suppression process.

3.3. Backbone Network

This section details the backbone network of the proposed method. The core objective of the network is to effectively transform the input time-frequency matrix to achieve RFI suppression and reconstruct the SAR signals. To this end, the backbone must satisfy two key design requirements:

• It must possess strong multi-scale feature extraction capabilities to capture the complex patterns of diverse RFI types.
• It must be able to effectively separate RFI features from the features of the useful signal and accurately reconstruct the interference-free signal.

Figure 5 illustrates the overall architecture of the backbone network.

As shown in Figure 5, the proposed backbone networks consists of three parts: an encoder, a bottleneck module, and a decoder. In the encoder, the input time-frequency domain amplitude spectrum first passes through an initial convolutional layer for downsampling and shallow feature extraction. Subsequently, the feature maps pass sequentially through multiple encoding stages composed of residual units to progressively extract deeper, more abstract features. In these stages, the network employs strided convolution to reduce spatial resolution thereby increasing the receptive field, and introduces dilated convolution in the deeper stages to further expand the receptive field without additional spatial downsampling. The bottleneck module is responsible for channel compression and information integration of the deepest features from the encoder, refining semantic information in preparation for the decoding path. In the decoder, the network progressively restores spatial resolution through upsampling operations. After each upsampling step, features from the corresponding encoder level are fused via skip connections, effectively combining shallow spatial details with deep semantic information. Finally, the decoder restores the feature map to its original size and outputs the single-channel, reconstructed amplitude spectrum. The network employs the MSE function as its loss function. To clearly elucidate the specific operations of each module, the network structure is defined by the following equations.

The operation of the initial feature extraction layer is given by Equation (22):

$$E_1=Encode{r}_1\left(STFT\left(X\right)\right)$$

(22)

where $STFT\left(X\right)$ is the input signal in the time-frequency domain.

The calculation for the i-th encoding stage is given by Equation (23):

$$E_i=Encode{r}_i\left(E_{i-1}\right)$$

(23)

where $E_i$ is the output of i-th encoding stage, and $Encode{r}_i$ is i-th encoder.

The operation of the bottleneck module is given by Equation (24):

$$Z=Align\left(Squeeze\left(E_5\right)\right)$$

(24)

The $Squeeze$ convolutional layer performs feature compression, and the $Align$ convolutional layer performs channel alignment.

Equation (25) is the calculation of upsampling and skip connection:

$$F=Concat(Upsample\left(Z\right),E_3)$$

(25)

The decoder output is given by Equation (26):

$$Y=Sigmoid\left(Decoder\right(F\left)\right)$$

(26)

The decoder output is then passed through a Sigmoid activation function to yield the final output.

The schematic diagram of the residual unit in Figure 5 is shown in Figure 6.

3.4. Kurtosis-Guided Attention Module

The core of the proposed method is the time window-wise kurtosis calculation performed on the time-frequency domain signal. This operation transforms the 2D time-frequency matrix into a 1D kurtosis prior vector. Each element in this vector corresponds precisely to a time window, and its value directly reflects the likelihood of RFI presence within that window. This kurtosis prior vector, therefore, provides precise guidance for the attention mechanism, transforming the interference suppression task from an unconstrained search problem into a prior-guided optimization problem. This enhances the network’s suppression efficacy and accuracy.

To convert this guidance signal into attention weights that the network can utilize, we designed an attention weight generation module based on a Multi-Layer Perceptron. This module takes the aforementioned kurtosis prior vector as input and maps it to an attention weight vector. This mapping is achieved through a linear layer, a batch normalization layer, and a Sigmoid activation function. The specific architecture is illustrated in Figure 7.

Within this module, the Linear layer is responsible for feature transformation; the Batch Normalization (BN) layer is used to accelerate convergence and training; and the final Sigmoid activation function ensures the output attention weight vector has values in the [0, 1] range.

In our multi-scale attention implementation, the attention mechanism is flexibly embedded at multiple levels of the encoder. Given a feature map output by an intermediate layer, the dimension of the attention weight vector is first adjusted via adaptive average pooling to match the dimensions of the feature map. Subsequently, the adjusted weight vector ${\mathbf{k}}^{\prime }$ is broadcast across the channel and frequency dimensions and multiplied element-wise with the input feature map, thereby achieving adaptive feature enhancement:

$$F_{out}=F_{in}\odot Broadcast\left({\mathbf{k}}^{\prime }\right)$$

(27)

Through the operation in Equation (27), the network is guided to assign higher weights to feature regions exhibiting kurtosis anomalies, thereby achieving precise RFI suppression. A schematic diagram of this module is shown in Figure 8.

3.5. Data Normalization Strategy

The data normalization strategy employed in this paper is a paired Min-Max normalization, which is applied to the input interference-corrupted data and its corresponding ground truth data. For any given value x in the features, the normalization is calculated as shown in Equation (28):

$$x^{\prime }={\displaystyle \frac{x-min\left(X\right)}{max\left(X\right)-min\left(X\right)}}$$

(28)

where $min\left(X\right)$ is the minimum value of the data, $max\left(X\right)$ is the maximum value, and $x^{\prime }$ is the normalized value. Traditional independent normalization method calculates separate scaling factors for the interference-corrupted input data and the ground truth data. Since the presence of interference will change the $max\left(X\right)$ of the data, this approach causes the background signal (which is common to both) to be scaled inconsistently. This leads to the erroneous amplification of the background signal during network inference.

To circumvent this issue, we apply a joint processing approach to each sample pair, which consists of the interfered data and its corresponding ground truth. Specifically, we first identify the maximum and minimum values across both the input and target data within that pair. These shared extrema are then used to normalize both the interfered data and the ground truth data. This method ensures that the input and the target within each training pair share a uniform linear scaling baseline. This allows the network to focus on learning the task of interference suppression, rather than compensating for the scale discrepancies introduced by inconsistent normalization, thereby guaranteeing the accuracy of the final output.

4. Experiments and Analysis

4.1. Dataset and Experimental Setting

The training of neural networks relies on large-scale paired samples, i.e., input signals containing interference and their corresponding interference-free ground truths. However, in real world SAR observation, obtaining clean, interference-free echo data for interference-corrupted data is often infeasible. This constitutes the primary challenge in data acquisition.

To address this issue, this study employed the semi-physical simulation dataset SarTF, constructed and publicly released by Shen et al. [29], for model training and evaluation. This dataset generates paired samples by injecting simulated RFI into interference-free SAR signals. Its background signals are derived from two data types: simluated multi-point target echos and real X-band airborne SAR data. By controlling the intensity of the injected interference, the Signal-to-Interference Ratio (SIR) of all samples was set within the range of −10 dB to −5 dB. Finally, both the interference-corrupted signals and the original signals were transformed into the time-frequency domain to constitute the final paired training data. The SarTF dataset contains a total of 9747 sample pairs and is divided into a training set and a test set at an 80%/20% ratio. Figure 9 shows the time-frequency spectrograms of sample data from the training set, where (a) shows the interference-corrupted data, and (b) shows the corresponding interference-free data.

The raw data underwent a preprocessing pipeline to be converted into network inputs. Specifically, the complex echoes were transformed into the time-frequency domain using the STFT with a window length of 256 and an overlap of 255. Subsequently, a paired Min-Max normalization strategy was applied to map the amplitude values to $[0, 1]$. The time-frequency matrix was then segmented along the time axis into blocks of size $256\times 256$. For the network output, the matrices were denormalized using the corresponding input scaling parameters and reassembled to reconstruct the complete time-frequency matrix.

During the training of the network proposed in this paper, we established the hyperparameter setting to ensure optimal model performance. The batch size was set to 16, balancing training efficiency with good gradient estimation quality. For optimization, we selected the Adam optimizer for gradient backpropagation and parameter updates, as its adaptive learning rate mechanism effectively handles sparse gradients. A step decay strategy was adopted for the learning rate schedule: the initial learning rate was set to $2\times {10}^{-4}$ and was halved every 4 training epochs with a decay factor of 0.5. This progressive adjustment facilitates model fine-tuning in the later training stages. Specific to the architecture of the kurtosis prior-guided module, we set key parameters: the kurtosis-guided attention mechanism was applied to the Encoder 2, Encoder 3 and Encoder 4 stages, the attention visualization interval was set to 50 batches, and gradient clipping was enabled to prevent training instability. The experimental results show that after 15 training epochs, the proposed network achieved excellent interference suppression performance, thus validating the effectiveness of the designed hyperparameter configuration.

For the experimental setup, we designed multiple experiments to verify the method’s performance. First, we took interference-free airborne and spaceborne SAR data from several scenarios and corrupted them with complex interference, which was a composite of the three types mentioned in Section 2.2. The proposed method was then used to suppress the interference in this interference-corrupted data. Several representative algorithms from the SAR RFI suppression field were selected for comparison, including time-frequency domain notching [30], as well as other neural networks such as PISNet [29] and FDNet [31]. Second, we performed validation on real-world data, selecting an RFI-corrupted scene from the GF-3 satellite to verify the suppression performance in a real-world scenario. Subsequently, we conducted a phase analysis and phase fidelity assessment of the network’s output using simulated data. The detailed parameters of the data used in the experiments are listed in

Table 1. Detailed Sensor Parameters and Scene Descriptions of the Experimental Data.

Parameter	Airborne SAR	GF-3 Satellite
Band	X-band	C-band
Range Sampling Rate	548.571 MHz	66.667 MHz
PRF	533.330 Hz	1414.333 Hz
Bandwidth	480 MHz	60 MHz
Imaging Mode	Stripmap	Stripmap
Scene Types	Agriculture	Coast, Ocean, Mountain
Number of Scenes	2	3

.

4.2. Evaluation Metrics

For a comprehensive quantitative analysis of the experimental results, this section details the evaluation metrics employed in this work, including their physical significance and mathematical definitions. Based on their focus of evaluation, these metrics are categorized into two main classes: amplitude and image-domain metrics, and phase fidelity metrics.

4.2.1. Amplitude Reconstruction Metrics

This paper employs Peak Signal-to-Noise Ratio (PSNR), Structural Similarity (SSIM), Image Contrast (IC), and Image Entropy (IE) as the metrics to evaluate interference suppression performance. Suppose the size of the image to be evaluated is $N_a\times N_r$, where X denotes the interference-free image and $\widehat{X}$ denotes the reconstructed image output by the suppression algorithm.

PSNR is one of the most widely used metrics for assessing image reconstruction quality. It evaluates the degree of distortion by calculating the mean squared error (MSE) between the corresponding pixels of the ground truth image and the processed image. A higher PSNR value indicates that the processed image is closer to the ground truth image at the pixel level and suffers from less distortion. It is expressed in decibels (dB).

First, the MSE between the two images is calculated, as given in Equation (29):

$$\mathrm{MSE}={\displaystyle \frac{1}{N_a\times N_r}}\sum _{i=1}^{N_a}{\sum _{j=1}^{N_r}[X(i,j)-\widehat{X}(i,j)]}^2$$

(29)

The PSNR is then calculated from the MSE, as shown in Equation (30), where $MA{X}_I$ is the maximum possible value for the image pixels.

$$\mathrm{PSNR}=10·{log}_{10}({\displaystyle \frac{{\left(MA{X}_I\right)}^2}{\mathrm{MSE}}})$$

(30)

SSIM is a metric that evaluates the similarity of structural information between images, which aligns well with human visual perception. SSIM provides a comprehensive evaluation of the similarity between two images based on three components: luminance, contrast, and structure. Its value ranges from −1 to 1, where a value closer to 1 indicates a higher degree of structural and perceptual similarity between the two images. The calculation formula of SSIM is shown in Equation (31):

$$\mathrm{SSIM}={\left[l(m,n)\right]}^{\alpha }{\left[c(m,n)\right]}^{\beta }{\left[s(m,n)\right]}^{\gamma }$$

(31)

where $l(m,n)$ is the luminance comparison term at pixel $(m,n)$, $c(m,n)$ is the contrast comparison term, and $s(m,n)$ is the structure comparison term. The parameters $\alpha ,\beta ,$ and $\gamma$ are positive constants.

IC primarily measures the degree of difference between bright and dark regions in an image, as well as its dynamic range. It ranges from 0 to 1, where a value closer to 1 indicates stronger contrast between light and dark areas. In interference suppression scenarios, strong interference typically raises the noise floor in darker regions, thereby reducing the image contrast. An effective suppression algorithm can restore the contrast to a level approaching that of the ground truth image upon interference removal. In this paper, the IC is calculated as follows: a local window is defined around each pixel, and the maximum intensity value $I_{max}$ and minimum intensity value $I_{min}$ within that window are computed. These values are then used in Equation (32) to calculate the local contrast, $C_k$. k represents the window index, and K is the total number of windows. Finally, the local contrasts from all windows are averaged to yield the final contrast value.

$$C_k={\displaystyle \frac{I_{max}-I_{min}}{I_{max}+I_{min}}}$$

(32)

$$\mathrm{IC}={\displaystyle \frac{1}{K}}\sum _{k=1}^K{C}_k$$

(33)

IE is used to measure the uncertainty of the image information content or the pixel grayscale distribution. Because RFI injects pseudo-random information into the image, causing the pixel distribution to be chaotic and raising the entropy value, therefore the change in entropy can reflect the effect of interference suppression. Effective suppression should be able to remove the interference information, make the pixel distribution restore order, and thus lower the image entropy. The lower the entropy value, it indicates that the pixel distribution is more concentrated, and the uncertainty is smaller. Its calculation formula is as follows:

$$\mathrm{IE}=-\sum _{k=0}^{Q-1}p\left(k\right)·{log}_{2}p\left(k\right)$$

(34)

where Q is the total number of gray levels, and $p\left(k\right)$ is the probability of occurrence of gray level k in the image.

4.2.2. Phase Fidelity Metrics

To quantitatively evaluate the difference between the phase after interference suppression processing and the ground truth phase, this paper employs Mean Absolute Difference (MAD), Standard Deviation (SD), Root Mean Square Error (RMSE), and Pearson Correlation Coefficient (PCC) as phase fidelity metrics. Assume the ground truth phase is ${\varphi }_{true}$, the estimated phase is ${\varphi }_{est}$, and the total number of data points is $N_p$.

MAD measures the average magnitude of the error between the estimated phase (${\varphi }_{est}$) and the ground truth phase (${\varphi }_{true}$), indicating the degree of systematic offset. It reflects the expected value of the overall error. A smaller MAD value signifies that the phases of the two signals are closer on average. Its calculation is given by Equation (35):

$$\mathrm{MAD}={\displaystyle \frac{1}{N_p}}\sum _{i=1}^{N_p}\left|{\varphi }_{est}\left(i\right)-{\varphi }_{true}\left(i\right)\right|$$

(35)

SD measures the degree of dispersion within the phase error sequence, which reflects how closely the errors are distributed around their mean. A lower SD value indicates less random fluctuation in the error, signifying higher phase consistency. It is calculated as given in Equation (36):

$$\mathrm{SD}=\sqrt{{\displaystyle \frac{1}{N_p-1}}\sum _{i=1}^{N_p}{({\varphi }_{est}\left(i\right)-{\varphi }_{true}\left(i\right)-{\mu }_{err})}^2}$$

(36)

where ${\mu }_{err}$ is the mean of the phase error sequence.

RMSE reflects both the systematic offset and random fluctuation of the error by calculating its root mean square. Moreover, due to the squaring operation, RMSE is more sensitive to large error values. A smaller RMSE value indicates a better match between the estimated phase and the ground truth. It is calculated as given in Equation (37):

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N_p}}\sum _{i=1}^{N_p}{({\varphi }_{est}\left(i\right)-{\varphi }_{true}\left(i\right))}^2}$$

(37)

PCC does not directly measure the absolute magnitude of the error; instead, it evaluates the linear correlation between the two signals. It ranges from −1 to +1. A PCC value closer to +1 signifies a stronger positive correlation between the estimated phase and the ground truth phase, indicating that the signals’ overall structural features and patterns are more similar. Its calculation is given by Equation (38):

$$\mathrm{PCC}={\displaystyle \frac{{\sum }_{i=1}^{N_p}({\varphi }_{est}\left(i\right)-{\mu }_{est})({\varphi }_{true}\left(i\right)-{\mu }_{true})}{\sqrt{{\sum }_{i=1}^{N_p}{({\varphi }_{est}\left(i\right)-{\mu }_{est})}^2}\sqrt{{\sum }_{i=1}^{N_p}{({\varphi }_{true}\left(i\right)-{\mu }_{true})}^2}}}$$

(38)

where ${\mu }_{est}$ is the mean of the estimated phase, and ${\mu }_{true}$ is the mean of the ground truth phase.

4.3. Experimental Results

4.3.1. Comparative Results on Simulated Data

The simulation experiment uses three distinct scenes of interference-free SAR data as background, including airborne SAR data (Scene 1 and Scene 2) and one scene of spaceborne SAR data from GF-3 satellite (Scene 3). To simulate a complex interference environment, we introduced simulated composite RFI into these data using a dynamic generation strategy. This composite interference is segmented along the azimuth direction, comprising three sequential modes: (1) NBI only, (2) WBI only, (3) a hybrid of NBI and WBI. The specific interference parameters are detailed in

Table 2. Simulated Interference Parameters for Different Scenarios.

Interference Type	Parameter	Value (Scene 1)	Value (Scene 2)	Value (Scene 3)
NBI	Bandwidth	<10 MHz	<10 MHz	<10 MHz
	Frequency	4.5–7 GHz	6–8 GHz	3–6.5 GHz
	Count	3	2	2
LFM WBI	Bandwidth	100–300 MHz	130–200 MHz	80–270 MHz
	Frequency	3–10 GHz	6.2–8.3 GHz	4.5–9 GHz
	Chirp Rate	$0.26\times {10}^{14}\sim 1.1\times {10}^{14}$	$0.3\times {10}^{14}\sim 0.8\times {10}^{14}$	$0.38\times {10}^{14}\sim 0.6\times {10}^{14}$
	Count	3	2	2
Sinusoidal Modulation WBI	Bandwidth	25–100 MHz	50 MHz	30–60 MHz
	Frequency	0.2–1.1 MHz	0.77 MHz	0.4–0.8 MHz
	Modulation Coefficient	3–15	25	15–30
	Initial Phase	$0\sim \pi$	$\pi /4$	$\pi /4\sim 3\pi /4$
	Count	2	1	2

. It is crucial to note that the values listed in Table 2 represent the sampling ranges rather than fixed combinations. During the dataset generation, the specific parameters for each interference instance were randomly sampled from these continuous ranges. Furthermore, these diverse interference patterns are introduced at varying locations along the azimuth and range dimensions. The injected RFI signals are generated based on the standard mathematical models defined in Equations (3)–(5) in Section 2.2, ensuring that the simulated interference accurately reflects the characteristics of real-world RFI.

The interference-corrupted SAR data was processed using the proposed method and several comparison methods. SAR imaging was subsequently performed on each of the processed outputs. The resulting images are show in Figure 10. In Figure 10, each row represents a distinct experimental scenario. The correspondence between the columns (from left to right) and the respective methods is detailed in the figure caption. Scene 1 features an area traversed by a river; Scene 2 covers a mixed agricultural and built-up area; and Scene 3 depicts a land–sea interface. These regions span diverse and highly heterogeneous scene backscatter types, providing a representative set of challenging scenarios for assessing the robustness and generalization ability of the RFI suppression methods.

For a quantitative analysis of the experimental results,

Table 3. Metric Results for the Comparative Experiment (shown in Figure 10). The best results are highlighted in bold. For the IC metric, values in parentheses indicate the relative error rate with respect to the Ground Truth (lower is better).

Scene	Method	PSNR (dB) ↑	SSIM ↑	IC (Error %) ↓
Scene 1	Ground Truth	—	—	0.9063 (Ref)
	Interfered Image	16.34	0.4739	0.7846 (13.43%)
	Notch	22.74	0.7924	0.9034 (0.32%)
	PISNet	26.39	0.9006	0.9044 (0.21%)
	FDNet	27.14	0.9041	0.9048 (0.17%)
	Proposed Method	27.92	0.9205	0.9049 (0.15%)
Scene 2	Ground Truth	—	—	0.9068 (Ref)
	Interfered Image	16.11	0.7062	0.8597 (5.19%)
	Notch	17.44	0.4042	0.8967 (1.11%)
	PISNet	28.43	0.9377	0.9045 (0.25%)
	FDNet	28.54	0.9380	0.9049 (0.21%)
	Proposed Method	29.28	0.9471	0.9045 (0.25%)
Scene 3	Ground Truth	—	—	0.2925 (Ref)
	Interfered Image	13.27	0.2079	0.2092 (28.48%)
	Notch	15.20	0.3047	0.3206 (9.61%)
	PISNet	23.88	0.7598	0.2659 (9.09%)
	FDNet	24.32	0.7824	0.2701 (7.66%)
	Proposed Method	24.34	0.8249	0.2896 (0.99%)

Note: The symbols ↑ and ↓ indicate that higher values are better and lower values are better, respectively.

lists the detailed performance metrics (PSNR, SSIM, and IC, as described in Section 4.2), which were calculated using the interference-free ground truth SAR images as the reference.

In addition to the final SAR imaging results, the processing results in the time-frequency domain are also presented. Figure 11 shows the time-frequency domain processing results for a single pulse subjected to three different interference types.

A comprehensive analysis of the aforementioned results reveals that while the traditional time-frequency domain notching method can eliminate the main lobe energy of RFI, its approach of directly setting the amplitude to zero also removes the useful signal overlapping with the interference band. This causes significant spectral information loss, and this over-suppression phenomenon inevitably leads to a loss of detail and texture in the final SAR image. In contrast, the the deep learning-based methods demonstrate superior signal fidelity. DL-based methods are able to effectively suppress RFI’s intensity to a level approaching that of the background signal, thereby greatly reducing the impact of RFI on the imaging results.The proposed method, FDNet, and PISNet all effectively preserve most of the target echo information while suppressing interference. However, careful observation reveals that PISNet’s and FDNet’s results contain more residual interference, indicating that its suppression is incomplete. Furthermore, a slight attenuation of ground echoes is observed in the FDNet’s results, suggesting that it inadvertently suppresses portions of the original SAR signal, leading to signal energy loss.

In summary, the proposed method exhibits superior performance. As shown in Table 3, our method achieves the best metrics (PSNR and SSIM) across all three experimental scenarios. This indicates that the imaging results from the proposed method are the closest to the ground truth image at both the pixel and structural levels. Furthermore, regarding the IC metric, the analysis shows that the introduction of RFI causes a significant drop in image contrast compared to the ground truth, and suppression processing restores the contrast to some extent. The IC metric of the proposed method is the best among these methods, reaching a level closest to that of the ground truth image. These comparative results preliminarily verify the superior comprehensive suppression performance of our method.

4.3.2. Validation Results on Measured Data

To validate the performance of the proposed method in real-world scenarios, we applied it to an interference-corrupted SAR data acquired over Sanya by the GF-3 satellite. Figure 12 shows the suppression result of the proposed method for a representative pulse from the data:

In Figure 12a, the top plot shows the time-frequency domain spectrogram of the original data, while the bottom plot shows the spectrogram of the data after interference suppression. A comparison of the plots before and after processing reveals that the WBI in the −20 to −10 MHz band has been removed, while the surrounding SAR signal maintains good continuity. For a more precise evaluation, we compared the frequency spectrum of the signal before and after processing, as shown in Figure 12b. The results clearly show that the frequency band where the interference is concentrated has been effectively suppressed, while the spectral morphology of the remaining bands remains largely consistent with the original signal.

Next, we demonstrate the performance of the proposed method on this real-world data through its imaging results. Figure 13 illustrates the performance of methods when processing a full real-world scene. As shown in Figure 13a, this scene is corrupted by large-area RFI, which manifest as bright stripes running through the image and severely affects its interpretability.

For a detailed evaluation, we selected a local region rich in land cover features and significantly affected by interference for a magnified analysis (Figure 13b). In this region, the strong RFI stripes mask most of the land cover textures. Subsequently, the original echo data from this region was processed for interference suppression and re-imaged using the proposed method and the comparison methods. Figure 13c–f respectively show the imaging results after processing by time-frequency domain notching, PISNet, FDNet, and the proposed method.

A visual comparison of the imaging results in Figure 13 reveals that, in the real-world data scenario, the time-frequency domain notching performs poorly, whose result losses lots of land cover information, rendering the imaging result uninterpretable. In contrast, the DL-based methods effectively suppress the interference stripes: the uniformity of the background region is restored, and the overall visual quality is significantly improved, thereby revealing the land cover structures that were originally masked by interference. A further comparison of the three DL-based methods shows that the proposed method provides the best preservation of original details while effectively filtering the interference. Its resulting land cover features are the closest to the ground truth, demonstrating superior fidelity.

Subsequently, two No-Reference metrics (as mentioned in Section 4.2), Image Contrast (IC) and Image Entropy (IE) were used to quantitatively analyze the imaging results. The results are presented in

Table 4. Metric Results for the Real-world Data Experiment (shown in Figure 13). The best valid results are highlighted in bold.

Method	IC ↑	IE ↓
Interfered Image	0.2543	7.6627
Time-Frequency Notching	0.2319	6.2006
PISNet	0.2656	7.2297
FDNet	0.2670	7.3432
Proposed Method	0.2732	7.1838

Note: The symbols ↑ and ↓ indicate that higher values are better and lower values are better, respectively.

:

Table 4 provides the quantitative evaluation results based on real-world data. Regarding the IC, the original interference-corrupted image (Figure 13b) exhibits low IC because the interference raises the noise floor in darker regions. After suppression, the image contrast is improved, among which the proposed method achieves the highest IC value, indicating that its restored image has the clearest luminance definition. In terms of the IE, RFI injects pseudo-random information into the image, disrupting the pixel distribution and raising the entropy value, and the image entropy is reduced after suppression. The drastic entropy reduction from the time-frequency domain notching method is attributed to the excessive removal of land cover information and is therefore not indicative of good performance. Among the three DL-based methods, the image processed by the proposed method achieves the best entropy reduction rate, indicating that it removed the most redundant interference information and that the restored image structure is the most well-structured.

The results from the real-world data experiment further verify the superior comprehensive performance of the proposed method in suppressing interference in real-world, complex scenarios.

4.3.3. Ablation Experiment

To verify the effectiveness of the kurtosis-guided attention mechanism, we conducted an ablation experiment on the kurtosis-guided attention module. This experiment compares the baseline network (without the module) with the proposed method (which includes the module). Both networks were trained on the same training set. The results are shown in Figure 14, and the specific values are presented in

Table 5. Ablation Study Results for the Kurtosis-Guided Attention Module. The proposed method (with the module) is compared against the baseline. The best performance metrics are highlighted in bold.

Method	MSE Loss ↓	PSNR (dB) ↑	SSIM ↑
Baseline Network	0.000854	31.2568	0.9640
Proposed Method	0.000727	32.5152	0.9736
Improvement Rate	14.87%	4.03%	1.00%

Note: The symbols ↑ and ↓ indicate that higher values are better and lower values are better, respectively. Values in italics represent the calculated percentage improvement of the proposed method compared to the baseline.

.

The quantitative results of the ablation study (Table 5) clearly confirm the effectiveness of the kurtosis-guided attention module. The network with this module outperformed the baseline network on three key metrics: Mean Squared Error (MSE), PSNR, and SSIM, achieving a PSNR improvement of over 1 dB and a 14.87% reduction in MSE. This result demonstrates that leveraging the kurtosis prior to guide the network’s attention is an effective design. It significantly enhances the model’s ability to distinguish between interference and the useful signal, thereby improving the final suppression performance and reconstruction quality.

4.3.4. Evaluation of Phase Fidelity

Phase information fidelity is critical for SAR applications such as interferometric measurements. This section investigates the proposed method’s ability to preserve the original signal phase while concurrently suppressing RFI. Furthermore, an evaluation framework is introduced for the quantitative analysis of phase differences before and after RFI suppression. This study utilizes semi-physical simulation data based on real airborne SAR echoes, into which simulated RFI is injected following the methodology described in Section 4. Specifically, we compare the phases of three distinct signals: the interference-free ground truth, the interference-corrupted signal, and the signal recovered by the proposed suppression method. Figure 15 illustrates the time-frequency spectrograms and the unwrapped phase of these three signals. Figure 15a presents the time-frequency spectrograms for the three signals, which are, from top to bottom, the ground truth signal, the interfered signal, and the interference-suppressed signal. Figure 15b compares the corresponding unwrapped phases, where the true phase is marked by the blue line, the phase of the interfered signal by the red line, and the phase of the suppressed signal by the green line.

The comparison result presented in Figure 15 clearly reveals that the presence of RFI induces a substantial deviation between the phase of the interfered signal and the ground truth phase. However, after processing by the proposed method, the signal phase is significantly restored, with both its overall structure and detailed features demonstrating a high degree of consistency with the ground truth phase.

The quantitative evaluation results are presented in

Table 6. Phase Fidelity Evaluation Results for Different Interference Scenarios. The units for MAD, SD, and RMSE are radians. The best results are highlighted in bold.

Scenario	Signal State	MAD ↓	SD ↓	RMSE ↓	PCC ↑
NBI	Interfered	2659.571	1502.496	3054.628	0.905
	Processed	70.080	36.664	76.393	0.980
WBI	Interfered	689.955	702.168	860.066	−0.684
	Processed	28.933	16.542	33.309	0.994
NBI+WBI	Interfered	821.557	408.719	917.609	0.429
	Processed	17.243	20.290	20.408	0.990

Note: The symbols ↑ and ↓ indicate that higher values are better and lower values are better, respectively.

, where the units of MAD, SD, and RMSE are radians. The data indicate that the phase error metrics of the processed signal were significantly reduced compared with those of the interfered signal. For instance, under the WBI+NBI scenario, the MAD decreased from 821.557 to 17.243, corresponding to a 97.9% reduction. Based on both qualitative visual comparisons and quantitative analyses, the experiment demonstrates that the proposed method can effectively suppress RFI while preserving the original phase characteristics of the SAR signal effectively, which is crucial for subsequent data processing and practical applications.

5. Discussion

The experimental results presented in Section 4 demonstrate that the proposed method achieves superior interference suppression performance compared to competing methods. This improvement is largely attributed to the introduction of the kurtosis-guided attention mechanism, which effectively distinguishes RFI from SAR echoes based on their statistical discrepancies. In this section, we provide a comprehensive analysis of the proposed method, covering statistical modeling of SAR echoes, sensitivity analysis of STFT parameters for kurtosis calculation, computational complexity, and performance under low SIR conditions.

5.1. Statistical Modeling and Applicability Analysis of K-Distribution

An important theoretical premise of this work is the modeling of SAR echoes using the K-distribution. While the Rayleigh distribution is classically used to model the amplitude of SAR echoes in homogeneous regions based on the Central Limit Theorem, it is theoretically insufficient for capturing the heavy-tailed characteristics observed in heterogeneous scenes such as urban areas or land-sea interfaces [27].

To verify the applicability of the K-distribution to real SAR data, we selected a representative heterogeneous region from the experimental dataset and performed a fitting analysis on the amplitude distribution of the SAR echoes, employing both the K-distribution and the Rayleigh distribution for a comparative evaluation. The parameters for the K-distribution (shape parameter $\nu$ and scale parameter $\alpha$) were estimated using a two-step approach: initial values were calculated via the Method of Moments (MoM) [32] and subsequently refined via nonlinear least-squares fitting. The Rayleigh distribution parameter ($\sigma$) was estimated using the Maximum Likelihood Estimator (MLE).

As shown in Figure 16a, the selected region covers a complex transition area containing both land and water features. The corresponding amplitude distribution fitting results are presented in Figure 16b. The empirical histogram exhibits a distinct heavy tail, a characteristic manifestation of heterogeneous scattering. The standard Rayleigh fit fails to capture this behavior, significantly underestimating the probability of high-amplitude echoes. In contrast, the K-distribution fit aligns closely with the empirical data. The estimated shape parameter $\nu =0.10$ is close to zero, quantitatively corroborating the strong heterogeneity of the scene. This experiment strongly supports the physical validity of using the K-distribution to characterize the SAR echoes in complex scenarios.

Furthermore, we discuss the applicability of the model across varying scenarios. We acknowledge that in real-world SAR data, the statistical characteristics of SAR signals may vary with factors such as frequency bands, resolutions, and scene types. Consequently, the empirical distribution parameters will fluctuate, and in some complex scenarios, the distribution may exhibit deviations from the ideal K-distribution model. However, the robustness of our method does not rely on a mathematically perfect distributional fit. The proposed method leverages the relative disparity in statistical properties (specifically kurtosis) between RFI and background echoes. As evidenced by the calculation results in Section 3.1, interference-free SAR echoes consistently exhibit low kurtosis values, regardless of whether the scene is highly heterogeneous (well-modeled by the K-distribution with a small shape parameter $\nu$) or homogeneous (approaching a Rayleigh distribution with a larger $\nu$). Conversely, the high-power and non-stationary nature of RFI signals causes a sharp rise in kurtosis, leading to values that are significantly higher than those of the background echoes. This pronounced statistical divergence ensures that the kurtosis-guided attention mechanism serves as a robust and broadly applicable indicator for RFI localization, maintaining its effectiveness even as SAR echo model varies under different operating conditions.

5.2. Sensitivity Analysis of STFT Parameters for Kurtosis Calculation

Section 3.1 detailed the temporal kurtosis calculation employed in our proposed method. Specifically, the raw data is transformed into the time-frequency domain via STFT, and the kurtosis is computed column-wise (i.e., for each time instance) on the resulting amplitude matrix. Since the STFT parameters determine the time-frequency resolution and the number of samples for statistical calculation, they inevitably influence the kurtosis calculation. For the STFT configuration in this work, we set the window length to 256 and the overlap ratio to 255/256. This subsection experimentally investigates the impact of these STFT settings on the kurtosis calculation, providing an engineering justification for our selection.

To evaluate the impact of STFT parameters, we selected a representative RFI-corrupted SAR signal and computed its kurtosis using various STFT configurations. Figure 17a illustrates the kurtosis curves for different window lengths with fixed overlap. Changing the window length from 256 to 128 is observed to result in a global decrease in the kurtosis values. This reduction reduces the statistical contrast between RFI and background echoes, which is detrimental to interference detection. Increasing the window length to 512 significantly enhances the kurtosis in strong interference regions (red dashed box), but suppresses the values in regions with weaker interference (blue dashed box), potentially compromising the detection of subtle RFI signals. Consequently, a window length of 256 was selected as the optimal setting to balance sensitivity and localization precision. Figure 17b illustrates the kurtosis curves for varying overlap settings under a constant window length. The results indicate that the kurtosis is relatively insensitive to the overlap parameter. Therefore, we selected a high overlap ratio primarily to ensure the temporal continuity of the generated STFT spectrograms.

5.3. Computational Complexity Analysis

In this section, we conducted an analysis of computational complexity. We compared the proposed KANet with comparison methods in terms of model size (Parameters), computational cost (FLOPs), processing latency, and training efficiency. The evaluation metrics of processing latency are defined as follows: “Inference Time” denotes the network forward-pass latency for a single $256\times 256$ time-frequency patch, while “Full Pipeline” measures the total processing duration for a single SAR signal pulse, encompassing the STFT, patch-wise inference, and ISTFT reconstruction. For training efficiency, we recorded the total training time required to complete 15 epochs and the peak GPU memory consumption during training with a batch size of 16. All experiments were performed on a workstation equipped with an NVIDIA GeForce RTX 4070 GPU (NVIDIA, Santa Clara, CA, USA). The quantitative comparison results are summarized in

Table 7. Computational Complexity Comparison of Different Methods.

Method	Params	FLOPs	Inference (ms)	Full Pipeline (ms)	Training Time (s)	GPU Mem (MB)
KANet (Proposed)	14.50 M	45.57 G	7.93	1886.00	2418.10	2116
PISNet	15.17 M	44.32 G	7.34	1816.64	2031.81	2008
FDNet	13.17 M	32.69 G	8.63	1847.83	2551.72	2821
Notch Filter	—	—	—	497.05	—	—

.

As indicated in Table 7, the traditional Notch Filter achieves the fastest processing speed due to its non-parametric nature. However, as demonstrated in Section 4, this efficiency comes at the cost of severe signal loss and image degradation, making it unsuitable for high-precision tasks. Among the DL-based methods, the proposed KANet demonstrates a competitive balance between performance and efficiency.

For processing latency, although the introduction of the kurtosis-guided attention module slightly increases the FLOPs, KANet achieves a single-patch inference time of 7.93 ms. Notably, despite FDNet having lower FLOPs, it exhibits higher latency than KANet, which may be attributed to its dual-branch architecture. Furthermore, the Full Pipeline times across all DL-based methods are comparable, suggesting that total runtime is dominated by the fixed overhead of pre-processing and post-processing stages.

Regarding training efficiency, the proposed KANet requires 2418 s for complete training, with a peak GPU consumption of 2116 MB. Compared to PISNet, the additional overhead introduced by the kurtosis-guided attention module remains modest, while FDNet demands more resources due to its design. These results confirm that the proposed KANet achieves superior suppression performance without imposing significant computational burden.

5.4. Performance Under Low SIR Conditions

To evaluate the limiting suppression capability of the proposed method, we conducted additional experiments under varying SIR conditions ranging from 0 dB to −20 dB. The experimental setup follows the description in Section 4 with composite RFI injected into the airborne SAR data. To ensure consistency, the interference spatial distribution and spectral characteristics were kept constant across all SIR levels, with only the interference amplitude scaled to achieve the target SIR values.

Figure 18 presents the imaging results for each SIR level in this experiment. The leftmost panel shows the ground truth SAR image without interference, serving as a reference for comparison. The remaining columns correspond to decreasing SIR values of 0 dB, −5 dB, −10 dB, −15 dB, and −20 dB, respectively. The top row depicts the SAR images contaminated by simulated composite RFI, where the interference manifests as distinct bright rectangular regions that become increasingly dominant as SIR decreases. The bottom row presents the images recovered by the proposed suppression method.

Table 8. Performance of the Proposed Method Under Different SIR Conditions.

SIR (dB)	Interfered Image		Suppressed Image
	PSNR (dB)	SSIM	PSNR (dB)	SSIM
0	15.87	0.6762	28.28	0.9426
−5	13.06	0.4781	25.80	0.8523
−10	11.17	0.2729	22.17	0.7753
−15	10.02	0.1338	20.13	0.6957
−20	9.38	0.0658	18.61	0.6002

presents the PSNR and SSIM metrics comparing the interference-corrupted images and the suppressed images against the ground truth at different SIR levels.

As shown in Table 8, the proposed method demonstrates consistent suppression performance across all tested SIR conditions. At 0 dB SIR, the method achieves a PSNR improvement of 12.41 dB and elevates the SSIM from 0.6762 to 0.9426, indicating excellent structural preservation. As the SIR decreases to −10 dB, the method still maintains substantial improvement with PSNR increasing by 11.00 dB and SSIM improving from 0.2729 to 0.7753. Under extreme conditions at −15 dB and −20 dB SIR, the suppression performance gradually degrades but remains effective. As visible in Figure 18, faint residual interference appear in the suppressed images at these extremely low SIR levels. At −20 dB SIR, the proposed method still achieves a PSNR gain of 9.23 dB, successfully recovering the main structural features of the SAR scene.

These results validate that the proposed method enables effective RFI suppression across a broad dynamic range of interference intensities. The method maintains excellent visual quality down to approximately −10 dB SIR, beyond which residual interference become visible but the overall scene structure remains well preserved. This experiment demonstrates the robustness and practical applicability of the proposed method operating in environments of severe RFI.

6. Conclusions

Addressing the challenge of RFI suppression in SAR systems, this paper proposes a deep learning network featuring an attention mechanism guided by signal statistics. The method calculates the kurtosis of time-frequency signals in a window-wise manner, which quantifies the non-Gaussianity and impulsive nature of RFI, then incorporated the kurtosis into a multi-scale attention to guide the network in suppressing interference. Furthermore, unlike most existing methods that predominantly focus on amplitude information, this study systematically analyzes the phase fidelity of the proposed method. We established a quantitative evaluation framework and verified, via simulation, that the method effectively filters interference without significantly compromising phase information.

However, as with other DL-based RFI suppression methods, the proposed method has certain limitations. Under extremely low SIR conditions, residual interference may remain in the suppressed results, as demonstrated in Section 5.4. Additionally, the generalization capability depends on the diversity of training data, and performance may be affected when the interference characteristics in practical applications differ significantly from those included in the training dataset.

Future research will focus on the following aspects: First, we aim to further refine the network architecture to enhance its accuracy and reliability under challenging conditions, such as low-SIR scenarios. Second, we plan to expand the training dataset to include more diverse interference, improving the model’s generalization capability. Third, we plan to conduct extended experiments on phase fidelity. Building on the current real-world data analysis, we will investigate the method’s impact on phase-sensitive SAR applications. Since these applications are inherently sensitive to phase errors, their performance can serve as an indirect yet practical metric for validating phase fidelity on measured SAR data.