An Active Deception Combined Jamming Identification Method Based on Waveform Modulation

Abstract

Jamming pattern identification is a crucial prerequisite for countering jamming. Combined jamming exhibits complex structures and diverse forms, making it difficult for traditional identification methods to extract suitable and stable features for effective discrimination. To address this challenge, this paper proposes a combined jamming identification method based on joint modulation of linear frequency modulation, phase coding and phase coding frequency modulation (LFM-PC-PCFM) waveforms. Building upon the time–frequency entropy features of combined interference, this method enhances the separability of jamming features in the radar-transmitted waveform dimension. The experiment employed the SVM classification algorithm based on particle swarm optimization for validation. Experiments demonstrate that the combined jamming recognition method under LFM-PC-PCFM waveform modulation achieves higher and more stable recognition accuracy than traditional LFM single-waveform modulation under jamming-to-noise ratios ranging from −10 dB to 30 dB.

1. Introduction

Nowadays, radar jamming patterns have become increasingly complex. With the rapid development of Digital Radio Frequency Memorization (DRFM), the performance of radar systems has been severely compromised [1,2,3]. Among these, the active deception signal is a type of jamming pattern generated by DRFM that can create false targets for radar receivers. These jamming mechanisms pose a significant threat to the target detection and tracking capabilities of radar systems [4,5]. With the continuous evolution of jamming techniques, combined jamming patterns composed of multiple interference styles have become the mainstream in jamming scenarios. Active deception jamming combinations are characterized by their complexity, flexible configuration, and difficulty in identification. For radar signal processing systems, mitigating flexible and complex combined jamming represents a major challenge. Jamming identification, as a prerequisite for modern radar systems to effectively mitigate jamming, correctly classifies jamming signals, tries to grasp the characteristics of signal sources, and enhances the effectiveness of mitigation techniques [6,7,8].

Jamming identification is a processing task performed in the front-end of radar receivers, and both domestic and international researchers have conducted extensive studies on this topic. To counter intermittent sampling forwarding jamming, the interfering party combines intermittent sampling repeated forwarding jamming with comb-shaped spectrum jamming to form composite jamming. This creates a multi-false target jamming effect in both the range dimensions of the target. It also proposes a non-uniform intermittent sampling jamming method [9,10]. In jamming recognition research, the mechanism of intermittent sampling relay jamming was analyzed, revealing the frequency-domain correlation dimension characteristics of intermittent sampling jamming under three relay modes (direct relay, repeated relay, cyclic relay) [11,12]. The fractional-order Fourier transform and short-time Fourier transform are commonly used methods for time–frequency analysis of jamming signals. By integrating these transform techniques, multi-domain fusion recognition patterns combining time domain, frequency domain, and time–frequency domain analysis have also become a common approach to enhance jamming detection accuracy [13,14,15,16]. To address speed, distance, and angle spoofing jamming, literature established mathematical models for spoofing jamming. By utilizing the dual-spectrum characteristics of the jamming signal in the frequency domain and a support vector machine classifier, these models enable the identification and classification of the three jamming patterns [17,18,19,20]. For speed, distance, and speed-distance towing-type jamming, this study proposes an jamming type perception method based on wavelet decomposition, time–frequency image distribution, and third-order Renyi entropy. Classification is performed using statistical decision trees, Bayesian classifiers, and other techniques [21,22,23]. Finally, for these two novel types of false target jamming including spectral diffusion jamming and slice combination jamming, identification can also be achieved through time–frequency image binarization features and Zernike moment features [24,25,26,27].

With the rapid advancement of artificial intelligence, various types of neural networks have also been employed in the identification of jamming signals. Convolutional neural networks and their variants, such as complex-valued convolutional neural networks (CV-CNNs), residual convolutional neural networks (Residual CNNs), and dual-cascade convolutional neural networks, achieve high-precision intelligent recognition of suppression-type jamming and deception-type jamming [28,29,30,31]. Some researchers have also proposed a JRNet jamming recognition network that identifies combined jamming formed by suppressed jamming, demonstrating stable performance under low jamming-to-noise ratio conditions [32]. Other studies have incorporated transfer learning and twin networks into convolutional neural networks to reduce training time and enhance recognition performance under small sample conditions [33]. However, the aforementioned jamming recognition primarily targets a single interference pattern, and these methods may not be applicable to complex active deception combined jamming patterns.

The identification of active combined jamming represents a challenging issue in practical jamming pattern recognition. Current research trends in combined jamming identification emphasize the extraction of effective jamming features and the improvement of classification algorithms to enhance recognition performance. Actual jamming patterns are characterized by high dynamics and incompleteness. Even after extensive in-database training, it remains challenging to rapidly extract effective features in out-of-database scenarios. Zero-Memory Incremental Learning Networks address this issue by training efficient feature extractors [34]. However, this process requires extensive training on in-house samples. The lightweight ML-SNet network reduces algorithmic complexity and incorporates an attention mechanism to enhance the network’s feature extraction capabilities [35]. In practical scenarios, small targets are easily masked by stronger jamming signals due to their weak signal scattering capabilities. For small-target scenarios, combining the small-sample jamming recognition network (JR-TFSAD) with temporal–frequency self-attention and global knowledge distillation enables jamming recognition using global features. This approach demonstrates superior efficiency and performance compared to conventional methods [36]. Research has optimized the lightweight YOLOv8 algorithm and applied it to open-set composite jamming recognition [37].

Feature extraction plays a crucial role in the identification of combined jamming. However, in complex, dynamic jamming environments, system performance tends to degrade as data complexity increases. Motivated by these challenges, this paper presents a method for extracting jamming features based on multi-waveform modulation. The proposed method operates by modulating distinct sub-pulses within a coherent processing interval with multiple waveforms. This design compels jammers to intercept the entire set of waveforms, thereby generating diverse, controllable jamming patterns. Unlike conventional methods that passively analyze combined jamming at the receiver, our approach actively induces it at the transmitter by emitting modulated waveforms. As jamming characteristics vary across different waveforms, a consistent set of features is extracted from each jamming modulated by different waveforms. Experiments demonstrate that the multi-waveform-controlled feature extraction method can effectively enhance the performance of combined jamming recognition.

2. Active Deception Jamming Model Under Different Waveforms

This section constructs a multi-pulse model featuring hybrid modulation of three radar transmission waveforms: linear frequency modulation (LFM), phase encoding (PC), and phase encoding frequency modulation (PCFM). It derives the principles underlying the design of these three radar waveforms and identifies the specific jamming patterns generated under each waveform control scheme. The time–frequency domain differences in jamming patterns across the three radar transmission waveforms are compared. This paper assumes a jammer model that can intercept any radar transmission signal but cannot discriminate between different waveform types (i.e., a “blind-forwarding” jammer).

2.1. Radar-Transmitted Waveform

This section describes the generation mechanism of hybrid-modulated pulse waveforms, where Equation (1) represents the linearly frequency-modulated waveform, Equation (3) represents the phase-coding waveform, and Equation (9) represents the phase-coding frequency-modulated waveform. Equation (10) is the signal model for the above waveform hybrid modulation.

In the time domain, the mathematical expression for a typical linear-frequency-modulated signal is as follows:

$$S_{LFM}\left(t\right)=u\left(t\right)e^{j2\pi f_{0}t}=rect\left({\displaystyle \frac{t}{T}}\right)e^{j2\pi (f_{0}t+{\displaystyle \frac{1}{2}}K{t}^2)}=s\left(t\right)e^{j2\pi f_{0}t}$$

(1)

$${\psi }_{LFM}\left(t\right)=2\pi \left(f_{0}t+{\displaystyle \frac{1}{2}}K{t}^2\right)-{\displaystyle \frac{T}{2}}\le t\le {\displaystyle \frac{T}{2}}$$

(2)

where $rect\left({\displaystyle \frac{t}{T}}\right)$ denotes a rectangular pulse signal with a width of T, which also represents the duration of the linear-frequency-modulated signal. $f_0$ indicates the center frequency of the signal. $K={\displaystyle \frac{B}{T}}$ indicates the frequency modulation coefficient. B is the signal bandwidth. $s\left(t\right)$ is the re-enveloping of the signal $S_{LFM}\left(t\right)$. And ${\psi }_{LFM}\left(t\right)$ indicates the instantaneous phase of the LFM signal.

Phase-coded signals employ binary encoding of signal phase, with commonly used code sequences including Barker codes, m-sequences, L-sequences, and others. Their signal model is represented as [38]:

$$S_{PC}\left(t\right)=a\left(t\right)e^{j{\psi }_{PC}\left(t\right)}{e}^{j2\pi f_{0}t}$$

(3)

The re-envelope of the signal is $a\left(t\right)e^{{j\psi }_{PC}\left(t\right)}$, where ${\psi }_{PC}\left(t\right)$ denotes the phase modulation function. For a binary-coded signal, its values are either 0 or $\pi$, represented by the binary sequence $\left\{C_k=+1,-1\right\}$. When the envelope of a binary-coded signal is a rectangular signal, its complex envelope $u\left(t\right)=a\left(t\right)e^{{j\psi }_{PC}\left(t\right)}$ can be written in the following form:

$$u\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{N}}}{\displaystyle \sum _{k=0}^{N-1}}{C}_{k}g\left(t-k{\tau }_0\right)\hfill & 0\le t\le N{\tau }_0\hfill \\ 0\hfill & \mathrm{others}\hfill \end{array}\right.$$

(4)

where N denotes the code length, ${\tau }_0$ denotes the pulse width of the subpulse, and $g\left(t\right)$ denotes the subpulse function.

Phase encoding frequency modulation signals represent a novel type of frequency modulation (FM)-based coded signal, achieved by encoding the modulation frequency of a frequency modulation signal. Let us assume that the phase form function of the waveform is $S_{PCFM}\left(t\right)=\left\{exp\left(j{\varphi }_1\left(t\right)\right)\right\}$. Then ${\varphi }_1\left(t\right)$ can be expressed as [39]:

$${\varphi }_1\left(t\right)={\int }_0^t\left[\sum _{n=1}^N{a}_n{g}_1\left(t^{\prime }-\left(n-1\right)T_{\mathrm{p}}\right)\right]d{t}^{\prime }+{\overline{\varphi }}_1$$

(5)

Among these, $a_n\left(n=1,2\dots ,N\right)$ denotes the first-order encoding of the waveform phase, $g_1\left(t\right)$ represents the pulse-shaping filter whose integral $\left[0,T_p\right]$ over time equals 1, and ${\overline{\varphi }}_1$ indicates the initial phase. Let the expression within the brackets in Equation (5) be represented by the following equation:

$${\xi }_1\left(t\right)=\sum _{n=1}^N{a}_n{g}_1\left(t^{\prime }-\left(n-1\right)T_{\mathrm{p}}\right)$$

(6)

where ${\xi }_1\left(t\right)$ represents the frequency value of the change, so the phase can be expressed as:

$${\varphi }_1\left(t\right)={\int }_0^t\left[{\xi }_1\left(t\right)\right]d{t}^{\prime }+{\overline{\varphi }}_1$$

(7)

The above represents the first-order form of the coded signal, while the phase-coded frequency-modulated waveform constitutes its second-order form. Let ${\xi }_2\left(t\right)={\displaystyle \sum _{n=1}^N}{b}_n{g}_2\left(t^{\prime }-\left(n-1\right)T_{\mathrm{p}}\right)$ denote the modulation frequency of the frequency-modulated signal. Then the phase expression for the PCFM waveform can be derived as:

$${\varphi }_2\left(t\right)={\int }_0^t{\int }_0^t{\xi }_2\left(t^{″}\right)d{t}^{″}d{t}^{\prime }+{\int }_0^t{\overline{\omega }}_{2}d{t}^{\prime }+{\overline{\varphi }}_2$$

(8)

where ${\varphi }_2\left(t\right)$ denotes the phase of the PCFM waveform, and ${\overline{\omega }}_2$ and ${\overline{\varphi }}_2$ represent the second-order frequency and phase, respectively. Therefore, the PCFM waveform expression is:

$$S_{PCFM}\left(t\right)=\left\{exp\left[j\left({\int }_0^t{\int }_0^t{\xi }_2\left(t^{″}\right)d{t}^{″}d{t}^{\prime }+{\int }_0^t{\overline{\omega }}_{2}d{t}^{\prime }+{\overline{\varphi }}_2\right)\right]\right\}$$

(9)

Designing hybrid-modulated radar pulse signals combining linear-frequency-modulated waveform, phase-encoded waveform, and phase-encoded frequency-modulated waveform. The three waveforms serve as subpulse signals under multi-pulse conditions. Within a coherent processing interval, these waveforms are cyclically alternated as modulation signals for the subpulses. The signal model is as follows:

$$\begin{array}{cc}\hfill & S_{\mathrm{signal}}=\sum _{m=1}^{N}rect\left({\displaystyle \frac{t-m{T}_s}{T_p}}\right)e^{j2\pi (f_0\left(t-m{T}_s\right)+{\displaystyle \frac{1}{2}}K{\left(t-m{T}_s\right)}^2)}\hfill \\ \hfill & +\sum _{m=1}^{N}rect\left({\displaystyle \frac{t-m{T}_s-T_r}{T_p}}\right)a\left(t-m{T}_s-T_r\right)e^{j{\psi }_{PC}\left(t-m{T}_s-T_r\right)}{e}^{j2\pi f_0\left(t-m{T}_s-T_r\right)}\hfill \\ \hfill & +\sum _{m=1}^{N}rect\left({\displaystyle \frac{t-m{T}_s-2T_r}{T_p}}\right){\mathrm{e}}^{j{\varphi }_2\left(t-m{T}_s-2T_r\right)}\hfill \end{array}$$

(10)

where $T_s$ represents a coherent process interval, $T_r$ denotes the subpulse repetition period, and $T_p$ indicates the subpulse duration. Based on the signal model, the schematic diagrams of pulse signals under three modulation schemes are shown in Figure 1, representing linear frequency modulation (LFM), phase encoding modulation (PC), and phase encoding frequency modulation (PCFM), respectively.

2.2. Multi-Waveform Modulation Active Deception Jammer Combination

In this chapter, Equation (11) indicates the mathematical model of intermittent sampling and direct forwarding jamming, Equation (13) indicates the signal model of comb spectrum jamming, Equation (15) indicates the signal model of spectrum spread jamming, and Equation (16) indicates the signal model of noise convolution jamming. The aforementioned four types of jamming are highly effective at deceiving radar systems and are difficult to identify. Therefore, this paper investigates combinations formed by these four categories of jamming.

Taking linear frequency modulation waveforms as an example, we analyze the signal model of active spoofing jamming. Intermittent sampling direct-forward jamming is generated when the Digital Radio Frequency Memory (DRFM) of the jammer samples and forwards the radar signal. Upon receiving radar signals, it processes and forwards them after sampling only for a brief period, then continues sampling subsequent pulses for processing and forwarding. The sampling and forwarding processes alternate in a time division manner. The mathematical model for intermittent sampling forwarding jamming is as follows [40]:

$$J_{ISDJ}\left(t\right)=\sum _{n=0}^{N-1}rect\left({\displaystyle \frac{t-{\tau }_J-n{T}_J-T_I}{T_I}}\right)·S_{LFM}\left(t-{\tau }_J-T_I\right)$$

(11)

Among these, ${\tau }_J$ represents the forwarding delay, $T_I$ denotes the sampling pulse width, $T_J$ indicates the sampling period, and $S_{LFM}$ refers to the linear-frequency-modulated signal transmitted by the radar.

Comb spectrum jamming involves time-domain product modulation of intercepted radar signals with comb spectrum signals, essentially constituting a multi-component frequency-shift jamming. The comb spectrum signal is the weighted sum of $M+1$ single-frequency continuous waves, with the following signal form:

$$x_{\mathrm{COMB}}\left(t\right)=\sum _{m=0}^M{a}_{m}exp\left(j2\pi f_{m}t\right)$$

(12)

where $a_m$ denotes the amplitude of the $m+1$ signals, and $f_m$ denotes the frequency of the $m+1$ signals. Multiplying this comb spectrum signal in the time domain by the radar signal yields comb spectrum jamming. After undergoing LFM radar pulse compression processing, the comb spectrum jamming outputs multiple single-frequency oscillations whose envelope is the weighted sum of $M+1$ $sinc$ functions. when $f_m>0$, the pulse pressure peak of the mth component will appear before the pulse pressure peak of the true target echo, forming a pre-emptive jamming effect. Its expression is:

$$J_{\mathrm{COMB}}\left(t\right)=\sum _{m=0}^M{a}_{m}rect\left({\displaystyle \frac{t}{T}}\right)exp\left[j2\pi \left(f_0+f_m\right)t+j\pi \mu t^2\right]$$

(13)

In Equation (13), T denotes the pulse duration, $f_0$ is the signal carrier frequency. μ is the frequency modulation coefficient of linear frequency modulation signals.

The generation process of Spectrum Spread Jamming (SMSP) is as follows: After the jammer receiver receives the radar transmission signal, it undergoes mixing, low-pass filtering, and analog-to-digital conversion before storing the data in the digital RF memory. The jamming signal consists of N sub-pulses, forming a fixed-duration signal with a frequency modulation slope N times that of the radar transmission signal. Based on the principle of SMSP generation, the first sub-pulse signal is:

$$\begin{array}{c}{J}_1\left(t\right)=exp\left(j\pi K^{\prime }{t}^2\right)\hfill \\ K^{\prime }=NK,0\le t\le {\displaystyle \frac{T}{N}}\hfill \end{array}$$

(14)

where $K^{\prime }$ represents the frequency modulation slope of the subpulse, which is N times the original modulation slope K, and T denotes the duration of the radar transmission signal. Repeating this subpulse signal N times yields the time-domain expression for the jamming signal:

$$J_{SMSP}\left(t\right)=\sum _{i=1}^{N-1}{J}_1\left(t-i{\displaystyle \frac{T}{N}}\right)=J_1\left(t\right)\otimes \sum _{i=1}^{N-1}\delta \left(t-i{\displaystyle \frac{T}{N}}\right)$$

(15)

In Equation (15), ⊗ represents convolution operation, and spectrum spread jamming involves shifting multiple subpulses N times, with a shift period of ${\displaystyle \frac{T}{N}}$.

Noise Convolution Jamming (NC), also known as Convolutional Modulated Agile Noise, is a type of convolutional modulation jamming based on Digital Radio Frequency Memory (DRFM) technology. The radar signal transmitted is received by the jammer, amplified, and filtered. It is then downconverted to an intermediate frequency signal and stored in the DRFM. After undergoing specific processing, the signal is upconverted back to the radar’s operating frequency band. On another channel, the radar signal undergoes reception and processing before controlling the noise unit to generate noise data of appropriate length and type. The two output signals undergo convolution within the convolver, ultimately producing a sophisticated noise jamming waveform. Its time-domain expression is:

$$J_{NC}\left(t\right)=S_{LFM}\left(t\right)\otimes \left[H\left(F_{band}\left(t\right)\otimes n\left(t\right)\right)\right]$$

(16)

In Equation (16), $F_{band}\left(t\right)$ is the bandpass filter, $H\left(.\right)$ is the hilbert transform, and $n\left(t\right)$ represents the gaussian noise.

Based on the principle of generating jamming by capturing radar waveforms using DRFM, active spoofing jamming all modulate captured waveforms. Therefore, by altering the waveform, the jamming pattern can be changed. For ease of representation, the following abbreviations are used for intermittent sampling direct forwarding jamming, comb spectrum jamming, spectrum spreading jamming, and noise convolution jamming (ISDJ, COMB, SMSP, and NC). Mechanisms of ISDJ, COMB, SMSP and NC jamming generation are shown in the Figure 2.

Jamming mechanisms remain consistent across different radar waveforms, and the jamming expressions for linear-frequency-modulated waveforms (LFMs), phase-coding waveforms (PC), and phase-coding frequency-modulated waveforms (PCFMs) are expressed as Equations (17)–(20), where $J_{ISD{J}_{-}LFM}$ represents ISDJ jamming under LFM waveforms, $J_{ISD{J}_{-}PC}$ represents ISDJ jamming under PC waveforms, $J_{ISD{J}_{-}PCFM}$ represents ISDJ jamming under PCFM waveforms, and the remaining jamming expressions are similar.

$$\left\{\begin{array}{c}{J}_{ISD{J}_{-}LFM}\left(t\right)={\displaystyle \sum _{n=0}^{N-1}}rect\left({\displaystyle \frac{t-{\tau }_J-n{T}_J-T_I}{T_I}}\right)·S_{LFM}\left(t-{\tau }_J-T_I\right)\hfill \\ J_{ISD{J}_{-}PC}\left(t\right)={\displaystyle \sum _{n=0}^{N-1}}rect\left({\displaystyle \frac{t-{\tau }_J-n{T}_J-T_I}{T_I}}\right)·S_{PC}\left(t-{\tau }_J-T_I\right)\hfill \\ J_{ISD{J}_{-}PCFM}\left(t\right)={\displaystyle \sum _{n=0}^{N-1}}rect\left({\displaystyle \frac{t-{\tau }_J-n{T}_J-T_I}{T_I}}\right)·S_{PCFM}\left(t-{\tau }_J-T_I\right)\hfill \end{array}\right.$$

(17)

$$\left\{\begin{array}{c}{J}_{COM{B}_{-}LFM}\left(t\right)={\displaystyle \sum _{m=0}^M}{a}_{m}exp\left(j2\pi f_{m}t\right)·S_{LFM}\left(t\right)\hfill \\ J_{COM{B}_{-}PC}\left(t\right)={\displaystyle \sum _{m=0}^M}{a}_{m}exp\left(j2\pi f_{m}t\right)·S_{PC}\left(t\right)\hfill \\ J_{COM{B}_{-}PCFM}\left(t\right)={\displaystyle \sum _{m=0}^M}{a}_{m}exp\left(j2\pi f_{m}t\right)·S_{PCFM}\left(t\right)\hfill \end{array}\right.$$

(18)

$$\left\{\begin{array}{c}{J}_{SMS{P}_{-}LFM}={\displaystyle \sum _{m=0}^{N-1}}\left[rect\left({\displaystyle \frac{t-{\tau }_j-m{T}_{\alpha }}{T_{\alpha }}}\right)·S_{LFM}\left(t-m{T}_{\alpha }\right)\right]\hfill \\ J_{SMS{P}_{-}PC}={\displaystyle \sum _{m=0}^{N-1}}\left[rect\left({\displaystyle \frac{t-{\tau }_j-m{T}_{\alpha }}{T_{\alpha }}}\right)·S_{PC}\left(t-m{T}_{\alpha }\right)\right]\hfill \\ J_{SMS{P}_{-}PCFL}={\displaystyle \sum _{m=0}^{N-1}}\left[rect\left({\displaystyle \frac{t-{\tau }_j-m{T}_{\alpha }}{T_{\alpha }}}\right)·S_{PCFM}\left(t-m{T}_{\alpha }\right)\right]\hfill \end{array}\right.$$

(19)

$$\left\{\begin{array}{c}{J}_{N{C}_{-}LFM}\left(t\right)=S_{LFM}\left(t\right)\otimes \left[H\left(F_{\mathrm{band}}\left(t\right)\otimes n\left(t\right)\right)\right]\\ J_{N{C}_{-}PC}\left(t\right)=S_{PC}\left(t\right)\otimes \left[H\left(F_{\mathrm{band}}\left(t\right)\otimes n\left(t\right)\right)\right]\\ J_{N{C}_{-}PCFM}\left(t\right)=S_{PCFM}\left(t\right)\otimes \left[H\left(F_{\mathrm{band}}\left(t\right)\otimes n\left(t\right)\right)\right]\end{array}\right.$$

(20)

In Equations (17)–(20), $T_I$ represents the sampling pulse width, $T_J$ is the sampling period, and ${\tau }_J$ is the forwarding delay. $f_m$ is the frequency-shift component, $a_m$ is the amplitude of the comb-filtered signal, and $T_{\alpha }$ is the time duration of a sampling segment. $F_{\mathrm{band}}\left(t\right)$ is the bandpass filter, and $H(.)$ is the Hilbert transform.

3. Jamming Combination Model and Feature Analysis

This section analyzes the combined forms of the aforementioned four types of interference and presents signal models for these jamming combinations. It further examines the differences in time–frequency characteristics of combined jamming under various waveform modulation conditions, extracting and studying their time–frequency entropy features.

3.1. Jamming Combination Analysis

Based on the aforementioned waveforms, jamming models are constructed by pairing active deception interference. The signal model for the combined jamming is:

$$J_{\mathrm{total}}\left(t\right)=J_1\left(t-{\tau }_1\right)+J_2\left(t-{\tau }_2\right)+n\left(t\right)$$

(21)

In Equation (21), ${\tau }_1$ denotes the time delay of intermittent sampling forwarding jamming, ${\tau }_2$ represents the delay of all forwarding-type interference. Then there exists an approximation ${\tau }_2={\tau }_1+T_p$, where $T_p$ represents the time difference between the two types of interference generated by the jammer.

The specific combined jamming expression is as follows:

$$J_{ISDJ+COMB}\left(t\right)=J_{ISDJ}\left(t-{\tau }_1\right)+J_{COMB}\left(t-{\tau }_2\right)+n\left(t\right)$$

(22)

$$J_{ISDJ+SMSP}\left(t\right)=J_{ISDJ}\left(t-{\tau }_1\right)+J_{SMSP}\left(t-{\tau }_2\right)+n\left(t\right)$$

(23)

$$J_{ISDJ+NC}\left(t\right)=J_{ISDJ}\left(t-{\tau }_1\right)+J_{NC}\left(t-{\tau }_2\right)+n\left(t\right)$$

(24)

$$J_{SMSP+COMB}\left(t\right)=J_{SMSP}\left(t-{\tau }_2\right)+J_{COMB}\left(t-{\tau }_2\right)+n\left(t\right)$$

(25)

$$J_{NC+COMB}\left(t\right)=J_{NC}\left(t-{\tau }_2\right)+J_{COMB}\left(t-{\tau }_2\right)+n\left(t\right)$$

(26)

$$J_{SMSP+NC}\left(t\right)=J_{SMSP}\left(t-{\tau }_2\right)+J_{NC}\left(t-{\tau }_2\right)+n\left(t\right)$$

(27)

Equations (22)–(27) are the six distinct combinations of active spoofing jamming. Among them, $J_{ISDJ+COMB}$ represents the combination of ISDJ and COMB jamming. $J_{ISDJ+SMSP}$ represents the combination of ISDJ and SMSP jamming. $J_{ISDJ+NC}$ represents the combination of ISDJ and NC jamming. $J_{SMSP+COMB}$ represents the combination of SMSP and COMB jamming. $J_{NC+COMB}$ represents the combination of NC and COMB jamming. $J_{SMSP+NC}$ represents the combination of SMSP and NC jamming. Equations (22)–(24) are the combination of intermittent sampling forwarding jamming and all forwarding-type jamming. Equations (25)–(27) are the combination of all forwarding-type jamming. Intermittent sampling jamming can occur more rapidly than full-forwarding interference, resulting in a time difference.

Time–frequency analysis is performed on the combined jamming. The short-time Fourier transform is commonly used for time–frequency analysis of signals. By selecting an appropriate window function, a small segment of the signal is extracted and subjected to a fourier transform, thereby compensating for the drawback of the standard fourier transform, which completely loses temporal information. The mathematical expression for this process is defined as:

$$STFT\left(t,f\right)={\int }_{-\infty }^{\infty }{J}_{\mathrm{total}}\left(t\right)·w\left(\tau -t\right)·e^{-j2\pi f\tau }d\tau$$

(28)

Among these, $J_{\mathrm{total}}\left(t\right)$ is the combined jamming, $w\left(\tau -t\right)$ is the window function centered at time, and $e^{-j2\pi f\tau }$ is the kernel of the Fourier transform.

For the aforementioned active combined jamming, the time–frequency characteristics of the combined jamming were analyzed using short-time Fourier transforms under three waveform modulation. Specifically, the linear-frequency-modulated waveform had a duration of 10 μs, a bandwidth of 50 MHz, and a phase encoding of 69-bit two-phase coding, while the linear-frequency-modulated coding employed 128-bit multi-phase coding. Within a single coherent processing time, the time–frequency plots for the combined jamming under the three waveforms are as follows:

Figure 3 shows the time–frequency analysis of six combined jamming patterns under linear frequency modulation waveform. Figure 4 and Figure 5 show the time–frequency analysis of active combined jamming under phase coding and phase coding frequency–modulated waveform modulation, respectively. The short-time Fourier transform employs a Hamming window with a length of 48 samples and an overlap length of 40 samples. Combined jamming exhibits distinct characteristics in both the time and frequency domains, essentially representing the superposition of the individual jamming patterns in their respective time–frequency domains.

3.2. Time–Frequency Entropy Feature Analysis

Entropy is a parameter in thermodynamics that measures the uniformity of energy distribution within a substance. Radar jamming signals, a combination of both jamming signals and noise signals, contain both deterministic and random components. Therefore, entropy theory provides a sound explanation for the complexity of radar jamming waveforms. When applied to signal processing, information entropy serves as a characteristic parameter for assessing the uncertainty in signal state distribution and the complexity of waveforms.

Assume the received signal sequence is $\left\{X_n\right\}=\left\{x_1,x_2,\dots ,x_n\right\}$, where $p_i$ represents the energy proportion of each point, denoted by $\left\{p_1, p_2,\dots ,p_n\right\}$, $0<p_i<1,$ $\left(i=1,2,\dots ,n\right)$, and ${\displaystyle \sum _{i=1}^n}{p}_i=1$. The information entropy is defined as:

$$ShEn=-\sum _{i=1}^n{p}_i{log}_a{p}_i$$

(29)

Due to a drawback in information entropy, while $p_i\to 0$, the increment of information $\mathsf{\Delta }I\left(p_i\right)\to \infty$, the definition of exponential entropy was introduced:

$$ExEn=\sum _{i=1}^n{p}_i·e^{(1-p_i)}$$

(30)

Norm entropy is introduced in the R-norm metric space. Let the signal sequence be $\left\{X_n\right\}=\left\{x_1,x_2,\dots ,x_n\right\}$, for discrete probability distributions $p=\left(p_1,p_2,\dots ,p_n\right)$, $0<p_i<1,$ $\left(i=1,2,\dots ,n\right)$, and ${\displaystyle \sum _{i=1}^n}{p}_i=1$. Then norm entropy is defined as [41]:

$$NoEn={\displaystyle \frac{R}{R-1}}\left(1-\sum _{i=1}^n{p}_i^R\right), 1<R<2$$

(31)

Arrange the time–frequency sequences of combined jamming into one-dimensional arrays and extract their time–frequency entropy features. For each jamming-to-noise ratio (JNR) ranging from −10 to 30 dB, perform 100 Monte Carlo simulations on the combined jamming and noise data. The mean value from these 100 simulations is selected as the combined jamming feature for that specific JNR.

The specific workflow of the LFM-PC-PCFM waveform joint control method is illustrated in Figure 6. The jammer modulates three distinct jamming patterns corresponding to three waveforms. The radar receives these patterns and performs feature extraction, separately deriving information entropy, exponential entropy, and norm entropy as joint entropy features. These features are then input into an SVM classifier optimized by a particle swarm optimization algorithm. And it uses classification accuracy as the performance metric for jamming identification.

Entropy features under a single waveform exhibit suboptimal separability. However, when three waveforms act simultaneously, their entropy features can compensate for the separability limitations inherent in single-waveform features. Furthermore, adopting a waveform domain perspective allows for more proactive enhancement of feature separability. As shown in Figure 7a,d,g, the information entropy features under the single LFM waveform exhibit low discriminative power when the jamming-to-noise ratio is low. However, with the incorporation of PC and PCFM waveforms, features that were indistinguishable within the LFM waveform domain can be effectively differentiated in the PC and PCFM waveforms. Therefore, under conditions of a low jamming-to-noise ratio, waveform modulation can enhance the distinctiveness of combined jamming characteristics, thereby improving the accuracy of combined jamming identification.

4. Simulations and Results

This section introduces a combined jamming identification method under an LFM-PC-PCFM waveform joint modulation and identification process, employing an enhanced SVM as the classifier for combined jamming. It primarily compares the accuracy of jamming identification under LFM-PC-PCFM joint modulation with that of traditional single LFM combined jamming identification. Simultaneously, the performance of combined jamming identification under LFM-PC-PCFM waveform modulation is validated.

4.1. Simulation Experiment

Numerous factors influence experimental outcomes, including waveform parameters and classification algorithm initialization results. Therefore, detailed numerical results are provided to demonstrate the method’s effectiveness. All simulations were conducted in MATLAB R2024a on a computer equipped with a 2.2 GHz i9-14900HX CPU and 32 GB RAM.

This paper assumes that each radar pulse waveform has a width of 10 μs and a bandwidth of 50 MHz. The sequence length N of the linear frequency modulation (LFM) waveform is $N=1000$. The phase coding sequence consists of a 69-bit binary code with a peak side lobe ratio of 4 dB. The phase coding frequency modulation sequence utilizes a 128-bit polynomial code [42]. Set parameters to simulate four types of single active disturbances: ISDJ, COMB, SMSP, and NC. Combine them through time-domain superposition to form composite forms. ISDJ jamming is sampled and forwarded immediately, arriving at the radar receiver earlier than the other three types of full-forwarding jamming. Due to the extremely short sampling and forwarding time of the jammer, it arrives at the receiver approximately one jamming duration ahead of schedule. Construct a combined jamming model. Perform a short-time Fourier transform on it to extract time–frequency domain entropy features from different waveforms as classification feature inputs. Considering factors such as parameter uncertainty in non-ideal signals and timing jitter, introduce perturbations to the following parameters. For Doppler mismatch, introduce a random Doppler offset $f_d$ for the combined interference, with a range of $[-B/10,B/10]$. The specific simulation parameters for transmission waveforms are shown in

Table 1. Radar-transmitted waveform parameters.

Transmitted Waveform	Parameters	Value Range
LFM	Pulse width (μs)	10
	Bandwidth (MHz)	50
PC	Code length	69
	Oversampling rate	3
PCFM	Code length	128
	Oversampling rate	3

. And the specific simulation parameters for the combined jamming are shown in

Table 2. Simulation parameters for radar jamming signals and their combinations.

Jamming	Parameters	Value Range
ISDJ	JNR (dB)	−10 to 30
	Sampling width (μs)	0.5 to 1
	Sampling cycle (μs)	2
COMB	JNR (dB)	−10 to 30
	Frequency interval (MHz)	3
	Frequency-shifted (MHz)	5 to 20
SMSP	JNR (dB)	−10 to 30
	Sampling width (μs)	2
	Pulse repetition times	5
NC	JNR (dB)	−10 to 30
	Bandwidth (MHz)	5 to 45
	Sampling duration (μs)	10
ISDJ + COMB	JNR (dB)	−10 to 30
	Overlap time (μs)	0
	Sampling duration (μs)	5 to 25
ISDJ + SMSP	JNR (dB)	−10 to 30
	Overlap time (μs)	0
	Sampling duration (μs)	5 to 25
ISDJ + NC	JNR (dB)	−10 to 30
	Overlap time (μs)	0
	Sampling duration (μs)	5 to 25
COMB + SMSP	JNR (dB)	−10 to 30
	Overlap time (μs)	10
	Sampling duration (μs)	10 to 20
COMB + NC	JNR (dB)	−10 to 30
	Overlap time (μs)	10
	Sampling duration (μs)	10 to 20
SMSP + NC	JNR (dB)	−10 to 30
	Overlap time (μs)	10
	Sampling duration (μs)	10 to 20

.

Although the combination of lightweight neural networks with FPGA hardware acceleration improves recognition accuracy, network-based methods ultimately rely on data-driven training and are far more dependent on data than traditional SVM algorithms. Traditional algorithms focus on feature extraction; even with small sample sizes, it can still deliver good performance. The classifier in this paper employs a support vector machine optimized using particle swarm optimization (PSO-SVM). The particle swarm optimization algorithm is a type of computer-based intelligent optimization algorithm and belongs to the category of evolutionary algorithms. It also starts from random solutions and iteratively searches for the optimal solution. In the particle swarm optimization algorithm, the initial population size is set to 5. The SVM model’s penalty parameter c defines the tolerance for misclassification, with a range of $[0.1,100]$. The SVM kernel function parameter g defines the influence of individual samples on the model, also within the range $[0.1,100]$. The cross-validation fold count is set to 3. During the optimization iteration, each pair of particle parameter sets is applied to the SVM model. The resulting average accuracy serves as the particle’s fitness. The particle with the highest fitness is ultimately returned, representing the optimal combination of SVM model parameters.

4.2. Results Analysis

To validate that entropy features under multi-waveform modulation can enhance the classification performance of original features, experiments employed Monte Carlo simulations. Within a jamming-to-noise ratio (JNR) range of −10 to 30 dB, 100 samples were generated for each of six active composite jammers. This yielded a total of 600 samples across six jammer categories for each JNR level. For each sample, three types of entropy features were extracted under LFM waveform, phase-coding waveform, and phase-coding frequency-modulated waveform conditions. Classification was performed using the PSO-SVM algorithm, with a training-to-test ratio of 2:1. The performance of the entropy feature recognition method under LFM modulation was compared with that of the joint waveform modulation feature extraction method combining LFM-PC-PCFM.

This experiment employs two methods: traditional LFM waveform modulation, PC waveform modulation, PCFM waveform modulation and the joint LFM-PC-PCFM waveform modulation. Among these, LFM waveform modulation is the current mainstream approach for jamming identification research. Under this method, jamming features with good separability must be extracted for analysis, and the extracted features lack universality. The proposed LFM-PC-PCFM joint waveform modulation method enhances feature differentiation based on existing characteristics, enabling adaptation to complex combined jamming environments. Figure 8 shows a comparison of the accuracy of various combined jamming under different waveform modulation methods at 0 dB. It can be seen that at a low jamming-to-noise ratio of 0 dB, the LFM-PC-PCFM joint modulation method demonstrates superior average recognition performance against combined jamming.

Figure 9 shows the recognition accuracy variation with jamming-to-noise ratio of six types of combined jamming under LFM-PC-PCFM modulation and LFM modulation. Due to the inherent complexity of combined jamming, recognition accuracy is low and unstable under traditional LFM waveform modulation alone. Even under conditions with relatively high JNR, recognition performance remains unstable and even shows a declining trend. The identification of five types of combined jamming, including ISDJ + SMSP, ISDJ + NC, SMSP + COMB, COMB + NC and SMSP + NC, is particularly challenging under the modulation of the single LFM waveform. In contrast, the LFM-PC-PCFM joint modulation identification method demonstrates superior performance in identifying combined jamming and offers enhanced stability. When the JNR exceeds 5 dB, the recognition accuracy for the aforementioned six types of combined jamming consistently achieves over 98%.

The analysis focuses on the average classification accuracy of combined jamming under three jamming-to-noise ratio conditions: −10 dB, 0 dB and 10 dB. The average accuracy is calculated as the mean of the classification accuracy results from five independent replicate experiments. Under conditions of a low jamming-to-noise ratio of −10 dB, Figure 10a shows the confusion matrix for combined jamming recognition result at −10 dB under LFM waveform modulation. Figure 10b is the jamming recognition result at −10 dB under LFM-PC-PCFM waveform modulation. At a jamming-to-noise ratio of −10 dB, the LFM-PC-PCFM modulation scheme demonstrates a noticeable improvement over LFM modulation. At a jamming-to-noise ratio of 0 dB, as shown in Figure 10d, the interference classification accuracy for all six combined jamming patterns exceeds 90% under the LFM-PC-PCFM modulation interference identification method. Among these, the accuracy for the four combinations, including ISDJ + SMSP, COMB + SMSP, COMB + NC and SMSP + NC, reaches 100%. However, under a jamming-to-noise ratio of 0 dB, the best-performing jamming under the traditional LFM single-modulation recognition method was COMB + NC jamming, achieving an average accuracy of 100%. The worst-performing was COMB + SMSP jamming recognition, with an average accuracy of 40.74%. The traditional LFM modulation method exhibit limited combinatorial jamming characteristics due to its monotonous waveform, resulting in unstable classification performance. At a jamming-to-noise ratio of 10 dB, the LFM-PC-PCFM modulation method can completely distinguish between the combined jamming. However, LFM modulation method still exhibits classification errors in combined interference scenarios under a jamming-to-noise ratio of 10 dB.

The following is an analysis of the performance of the LFM-PC-PCFM joint modulation recognition method under partial waveform interception and non-Gaussian noise conditions. Non-Gaussian noise employs two types: Laplace noise and Rayleigh noise. Partial interception of waveform sets by jammers refers to the interception of one or two waveforms within the set. When the jammer intercepts only two of the waveforms, it corresponds to three partial interception possibilities: LFM-PC, LFM-PCFM, and PC-PCFM. Except for the intercepted waveform and the type of noise, the rest of the experimental environment and parameters are consistent with those described above.

Figure 11a and

Table 3. Accuracy rates of different JNRs under different interception types.

Type of Interception	0 dB	5 dB	15 dB	20 dB
LFM-PC-PCFM	100%	100%	100%	100%
LFM-PC	97%	98.5%	100%	100%
LFM-PCFM	90.5%	99.5%	99.5%	99%
PC-PCFM	94%	93%	99%	99.5%
LFM	61%	75.5%	78.5%	80.5%

illustrate the impact of partial interception types on the recognition performance of combined interference. When the JNR exceeds 5 dB, various partial interceptions have little effect on the accuracy of combined interference recognition. However, when the JNR falls below 0 dB, partial interception weakens the performance of the LFM-PC-PCFM joint modulation recognition method, though it still significantly outperforms the combined interference recognition performance of LFM modulation. Figure 11b and

Table 4. Accuracy under of different JNRs under different noise conditions.

Type of Modulation	Type of Noise	0 dB	5 dB	15 dB	20 dB
LFM-PC-PCFM	Gaussian	100%	99.5%	100%	100%
	Laplace	99.5%	99%	100%	100%
	Rayleigh	100%	100%	100%	100%
LFM	Gaussian	61%	81%	59.5%	78.5%
	Laplace	69.5%	87.5%	78.5%	85%
	Rayleigh	76.5%	85.5%	80%	81%

present the recognition performance of the combined interference recognition method using LFM-PC-PCFM joint modulation under different JNRs for Gaussian white noise, Laplace noise, and Rayleigh noise. Compared to the LFM modulation recognition method, this approach demonstrates superior and stable performance under Gaussian white noise and certain non-Gaussian noises.

The LFM-PC-PCFM combined modulation jamming identification method relies on specific radar system architectures for its engineering implementation. This method requires the radar transmitter to rapidly and accurately switch between three distinct subpulse modulation schemes within the coherent processing interval. This typically necessitates support from a software-defined radar platform. Its core requirements include:

• Hardware platform constraints: Direct Digital Frequency Synthesizers (DDFSs) or Arbitrary Waveform Generators (AWGs) must be capable of generating LFM, PC, and PCFM waveforms. Waveform switching time should be significantly shorter than the channel coherence time to avoid performance degradation caused by switching. The bandwidth and sampling rate of the system must simultaneously satisfy the Nyquist sampling theorem for the widest bandwidth among the three waveforms, with ample margin.
• Synchronization Requirements: The timing error between subpulses must be controlled within the nanosecond range to ensure the jammer captures the complete waveform set. This can be achieved through a highly stable system clock and precision digital delay circuits.
• Waveform Compatibility: The spectral characteristics of the three waveforms must align with the radar’s operational frequency band; Waveform parameters are designed using constrained optimization algorithms to enhance features while keeping metrics such as peak-to-average power ratio and modulation complexity within hardware-feasible limits; For modern radars already equipped with multi-waveform generation capabilities, this approach primarily optimizes waveform scheduling and signal processing algorithms.

5. Conclusions

In this paper, a combined jamming recognition method based on joint modulation of LFM-PC-PCFM waveforms is proposed. By designing waveform signal models for LFM-PC-PCFM and employing them as radar-transmitted waveforms, the dimensionality of combined jamming features is expanded, enhancing feature differentiation and improving recognition performance. Recognition capability is evaluated based on accuracy. Compared to traditional multi-domain feature extraction and recognition methods, this method not only reduces the complexity of feature extraction at the receiving end but also employs waveforms as the driving dimension for jamming identification, enabling radar to recognize jamming more proactively. To a certain extent, it mitigates the passivity of traditional jamming identification approaches. This paper employs the SVM optimized using the particle swarm optimization algorithm as a classifier for combined jamming. It demonstrates that under traditional classification algorithms, the LFM-PC-PCFM waveform modulation combined jamming identification method can effectively enhance the recognition performance of combined jamming. We extract the message entropy, the exponential entropy, and the norm entropy features as validation targets for the LFM-PC-PCFM waveform modulation method. Experiments comparatively analyzed the entropy feature differences between LFM-PC-PCFM and LFM waveform modulation, demonstrating that the LFM-PC-PCFM modulation approach effectively enhances feature separability. This method actively reduces the complexity of combined jamming feature analysis and extraction. This method is applicable to software-defined radar systems. By designing waveform generation algorithms and transmission scheduling parameters, it enables periodic transmission of waveforms in different formats.

This paper compares the LFM-PC-PCFM combined waveform modulation method with the traditional LFM single-waveform modulation method, investigating the recognition accuracy of combined jamming under jamming-to-noise ratios ranging from −10 dB to 30 dB. The combined jamming recognition approach using LFM-PC-PCFM joint waveform modulation marks the beginning of radar active recognition of combined jamming. By leveraging waveform dimensionality to actively expose jamming, it also provides a theoretical foundation for subsequent radar-jamming countermeasure games. The core premise of the proposed method is the assumption that jammers operate in a “blind forwarding” mode. When confronted with advanced jammers capable of intelligently identifying and adapting to radar waveforms, this approach may encounter challenges. Additionally, the recognition performance of this method may decline when processing feature sets with excessively high dimensions.