# Two-Dimensional Transceiver Beamforming for Mainlobe Jamming Suppression with FDA-MIMO Radar

Authors: Pengfei Wan; Guisheng Liao; Jingwei Xu; Guimei Zheng
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

With the rapid development of electronic warfare technology, the airborne electronic counter measures (ECM) system can generate mainlobe jamming using range gate pull-off (RGPO) strategy, which brings serious performance degradation of target tracking for the tracking and guidance radar. In this study, a two-dimensional transceiver beamforming approach is proposed to suppress the mainlobe jamming with frequency diverse array using multiple-input multiple-output (FDA-MIMO) radar. The mainlobe jamming signal differs from the real target echo in the joint transmit and receive domain due to the range dependence of FDA beampattern. The amplitude of RGPO signal is greater than the amplitude of real target echo. Thus, the transceiver beampattern can be designed to null out the jamming while maintaining the real target. The jamming suppression performance is studied in consideration of practical range constraint of RGPO. Simulation results are provided to verify the effectiveness of the proposed approach.


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## Body

## 1. Introduction

Tracking and guidance radar plays an important role in national defense applications [1–4]. It provides sufficient antijamming ability against the jammers with the low side lobe antenna technique and large time-bandwidth products. However, with the development of electronic interference technique in the advanced weapons, tracking and guidance radar encounters extremely hostile environment in the mainlobe [5–7]. For example, the electronic counter measures (ECM) system has been developed to generate strong jamming in the mainlobe [8–10], which becomes a great challenge for the traditional phased array radar systems.The mainlobe jamming is not easy to implement and also difficult to suppress. Especially, the multidimensional modulation deceptive jamming signal from the mainlobe seriously affects the performance of the radar system [11–14]. Deceptive jamming intercepts the radiation signal of radar by airborne electronic support measures (ESM) and modulates the range and speed in multiple dimensions, and then a deceptive jamming pattern similar to the real radar detection waveform is generated, that is, the false target is generated by the way of “intercept-modulation-forward,” which can make the radar system mistakenly regard the false target as the real target. Therefore, deceptive jamming has serious consequences such as increased false alarm, missing of real target, and extremely heavy computational burden [15–17].RGPO is an effective technique for deceptive jamming of radar range information. Because it has the advantages of low interference power and strong flexibility, it has become a hot research topic in recent years [18, 19]. Greco et al. [20] studied the working mechanism of RGPO and analyzed the influence of delay quantization based on digital radio frequency memory (DRFM) on jamming signals. Öztürk et al. [21] adopted the RGPO of bidirectional false target, which can effectively resist the pulse leading edge or trailing edge tracking technology adopted by radar. Xie et al. [22] proposed a range gate RGPO method based on bidirectional false targets, which is verified by evaluating radar measurements. In the study by Rui-xing and Jian-yun [23], a jamming power compensation technique was proposed to improve the success rate of range gate RGPO, and the output peak value of pulse compression was used to evaluate the RGPO effect. Xue and Yang [24] optimized the realization mode of range gate RGPO and put forward a method to improve the effect of range gate RGPO by frequency shift technology.In order to counter the RGPO, this study presents a method of countering the range gate RGPO based on frequency diverse array (FDA)-multiple-input multiple-output (MIMO) radar [25]. It is a new radar system that combines frequency diversity array and MIMO radar [26–29]. Because of the multiantenna transmission single frequency step system, it forms a three-dimensional range-angle-time-dependent pattern in the far field, and the research shows that the range-angle dependence of FDA’s transmission pattern is different from that of traditional radar [30]. However, in order to make full use of this characteristic, it is necessary to effectively separate the transmitter signal, and MIMO radar technology is an effective means to obtain the freedom of transmission [31]. Zhang and Xie [32] extracted the phase difference of adjacent array elements by analyzing the influence of each link in radar signal processing and realized the suppression of false targets. In [33], the antijamming ability was improved by joint optimization of transmission polarization and transmission frequency step interval. Reference [34] adopted a method based on eigenvector to improve jamming suppression ability.This study, according to the analysis of the principle of RGPO, takes advantage of the characteristic that the amplitude of RGPO signal is larger than the amplitude of the real target echo, a method is proposed for the FDA-MIMO radar to eliminate RGPO corresponding to the large eigenvalue, which improves the antijamming ability of the radar system and keeps the stable tracking.The structure of this article is as follows: Section2 introduces the signal model and the fundamentals of FDA-MIMO radar followed by the introduction of the algorithm to eliminate the jamming signal with range constraint in Section 3. Subsequently, simulation and analysis are given in Section 4. Finally, Section 5 draws a conclusion and summarizes this study.Notation:⊗ and ⊙ denote Kronecker product and Hadamard product, respectively. The letter j≜−1 represents the imaginary unit. The transpose and conjugate transpose of a matrix or vector are denoted by ·T and ·H. Boldfaced lowercase letters such as x represent a vector, boldfaced uppercase letters such as R denote a matrix, and italic letters such as a represent a scalar. For the vector x, we use xn to denote the nth element of vector x. For matrix R, we use Rm,n to denote the element of R in the mth row and the nth column. Finally, a,b indicates a closed interval in real number space.


## 2. Fundamentals of FDA-MIMO Radar

It is considered that the FDA-MMO radar system is an isometric linear array composed ofM transmitting antenna elements and N receiving antenna elements [35]. Under the condition of ignoring the antenna element pattern and array error, the transmitting and receiving antenna elements are omni-directional radiation, which are identical and uniform. The transmission signal form of the mth transmitting unit can be written as [36](1)smt=recttTpφmtexpj2πfmt,where t is the elapsed time of pulse propagation since the start of the pulse, rectt/Tp=1,0≤t≤TP0,else is the pulse modulation function, φmt is the baseband modulation signal corresponding to the mth transmitting unit [37], and fm is the transmitting frequency corresponding to the mth transmitting unit:(2)fm=f0+m−1Δf,m=1,2,...,M,where f0 is the frequency of the reference array element (the first array element) and Δf is the frequency offset between array elements.Assuming that there is a target at a certain positionR,θ in space, the echo from the mth transmitting antenna unit received by the nth receiving antenna unit can be written as(3)xs,m,nt−τm,n=βs0rectt−τm,nTpφmt−τm,nexpj2πf0t−τm,n,where βs0 represents the complex coefficient of the target echo including the full link of radar transmitting and receiving, τm,n=τ0−dm−1cosθ/c−dn−1cosθ/c represents the echo delay difference corresponding to the mth transmitting unit and the nth receiving unit, and d is the interelement spacing. Because the working frequency of each transmitting element of FDA-MIMO radar is different, when equation (3) expresses the approximate model under the assumption of far-field narrowband, the phase term introduced by frequency stepping cannot be ignored. When equation (1) is brought into equation (3), we can get(4)xs,m,nt−τ0≈βs0rectt−τ0Tpexpjϕmt−τ0expj2πΔfm−1t−τm,nexpj2πf0t−τm,n,where τ0=2R/c is the reference delay of the target echo. The target echo received by the nth receiving antenna unit can be approximately written as(5)xs,nt−τ0≈∑m=1Mβs0rectt−τ0Tpexpjϕmt−τ0expj2πΔfm−1t−τm,nexpj2πf0t−τm,n.The target echo is amplified and matched filtered, and the range unit where the target is located can express the signal as a concise form:(6)s=βsaR,θ⊗bθ,where βs represents the complex coefficient of the target echo after pulse compression; aR,θ and bθ are the transmit and receive steering vectors of the target, respectively, and ⊗ is the Krnoecker product:(7)aR,θ=arR⊙aθθ=1,exp−j4πΔfRc,...,exp−j4πΔfM−1RcT⊙1.expj2πdsinθλ,...,expj2πM−1dsinθλ=1,exp−j4πΔfRc+j2πdλcosθ,⋯,exp−j4πΔfRcM−1+j2πdλM−1cosθT,(8)bθ=1,expj2πdλcosθ,⋯,expj2πdλN−1cosθT,where ⊙ denotes the Hadamard product, arR and aθθ mean the launch range and launch angle steering vectors, respectively [38], and T is the transpose operator. As can be seen from equation (7), compared with the traditional radar, the range guidance vector of FDA-MIMO radar contains the range information R of the target signal, and its range guidance vector is correlated with angle and range in two dimensions. Because of the two-dimensional correlation between angle and range, FDA-MIMO radar has the ability to distinguish targets with different ranges in the transmitting space, that is, it can distinguish different targets on the close range gate with the same angle, which provides great practical value for radar to counter the jamming from mainlobe.Assuming that airborne ESM on space far-field target intercepts radar tracking signal and releases self-defense RGPO, the intercepted radar signal is stored and transmitted with the delay to form a false target jamming signal in the fast time dimension. The jamming signal is stronger than the target echo, and the signal form corresponding to themth transmitting unit and the nth receiving unit can be expressed as the signal form of the mth transmitting unit and the nth receiving unit:(9)xj,m,nt−τj≈βj,prectt−τjTpexpjϕmt−τjexpj2πΔfm−1t−τj,m,nexpj2πf0t−τj,m,n,where τj=Ri/c is the reference delay of the RGPO jamming signal generated by the jammer and τj,m,n=τj−dm−1cosθ/c−dn−1cosθ/c represents the echo delay difference between the mth transmitting unit and the nth receiving unit.As can be seen from equation (9), the pull-off jamming signal of the range map is completely consistent with the target echo. The time delay between the target echo and jamming is different. After the above analysis of RGPO, the jamming delay signal τj and the real target echo delay time τ0 are located in the same range gate. According to the traditional radar processing method, the radar range tracking center will be greatly affected. It is necessary to combine the radar prior knowledge and jamming characteristics to design antijamming. The specific analysis is introduced in the next section. For the convenience of description, the output signal form after matched filtering is given after considering the target signal, jamming, and noise comprehensively(10)xt=st+jt+nt=βstaR,θ⊗bθ+βjtaRj,θ⊗bθ+nt.Among them,βst=βsδt−τ0 is the time delay corresponding to the range gate where the target located is τ0 and βjt=βjδt−τj is the time delay of the jammer generating RGPO is τj.


## 3. Principle and Method of Anti-RGPO for FDA-MIMO Radar

In Section2, the real target echo and jamming signal in FDA-MIMO radar system are studied. The echo of the target releasing self-defense jamming is completely consistent with the jamming signal in angle dimension, but its range dimension is slightly deviated. Whether the range dimension deviation can be effectively used to suppress the jamming is the key to resist this kind of mainlobe jamming. Based on the analysis of jamming mechanism and mathematical model, combined with radar signal and information processing flow, this section expounds the application of FDA-MIMIO radar against RGPO in range dimension and gives the antijamming conditions and methods.


### 3.1. Mechanism of RGPO

RGPO is a kind of self-defense jamming. Usually, the airborne ESM system intercepts the radar radiation signal after finding that it is tracked and locked by the tracking and guidance radar and forwards the jamming signal with a certain delay through fast storage, so that the range tracking gate center of the enemy radar deviates from the real target and locks on the released false target, thus tracking the lost technical method. In the actual radar system, after tracking the target, the angle, range, and speed of the target can be predicted with high data rate, and the target position at the next moment can be interception in a certain range, which is called wave gate. If the jamming signal deviates greatly from the real target and exceeds the wave gate range, it may be eliminated as outliers or cannot form an effective jamming track in the data processing stage, and the jamming effect would not be achieved.RGPO can be divided into front-gate-pull-off jamming and back-gate-pull-off jamming according to different delay time functions [39]. For the jamming in front of the wavefront, the forwarding delay of the jamming signal gradually decreases for the radar tracking system, which results in an “illusion” that the false target is gradually approaching the radar system relative to the real target. For the back-gate-pull-off jamming, the delay of jamming signal forwarding increases gradually, resulting in a “phenomenon” that the range the false target is gradually apart from the real target in radar system. As for that front-gate-pull-off jamming, one or more pulse repetition stages need to be delayed. If the radar adopts frequency agility technology, the jamming effect would not be achieved. For back-gate-pull-off jamming, the time delay of the jamming signal needs to be within the same range gate as the target echo. If the radar adopts leading edge tracking technology, it can also effectively resist deceptive jamming. However, for the radar system, the time delay of the real echo received is inaccurate, and it is impossible to accurately determine the jamming style it is, so a new antijamming technology is needed to deal with the pull-off jamming with different ranges. This study focuses on countering the back-gate-pull-off jamming, and the proposed algorithm is also suitable for the front-gate-pull-off jamming.For RGPO, it is generally divided into three stages: interception stage, pull-off stage, and stop stage:Interception stage: after intercepting the tracking signal, the airborne ESM system stores and quickly forwards a jamming signal. Usually, the time delay of the jamming signal needs to basically coincide with the target echo in time dimension.As represents the amplitude of the target echo and Aj represents the amplitude of the jamming signal Aj/As≈1.3∼1.5Pull-off stage: in order to make the range gate center of radar deviate from the real echo of the target and avoid two echo peaks at the same time, ESM system needs to gradually increase the delay time of forwarding every time it intercepts a radar tracking signal, so that the range gate center gradually leaves the target position until the range gate deviates from the target echo by a predetermined range.Stopping stage: when the ESM system judges that the radar has deviated from the real target by enough range from the center of the wave gate, it stops radiating jamming signals, which leads to the radar losing the target or increasing the tracking error, so it is necessary to search and find the target again.If the above three steps are repeated, the radar can get rid of the tracking of the target or increase the tracking error of the radar.


### 3.2. Mathematical Model of RGPO

RGPO is mainly aimed at the radar working in tracking mode. Because the tracking filter has started to work, the radar has a priori information about the position and speed of the target, and through this priori information, the three-dimensional information of the target in the next working cycle can be predicted. In order to get rid of radar tracking, the target releases RGPO. Assuming that the pull-off range of RGPO to the center of echo isΔR at the i pulse repetition interval (PRI) of radar, the time delay of radar receiving jamming signal is(11)τj=τs,i+Δτj,i.In the equation,τj is the delay time of the jamming signal [40], τs,i is the delay time of the real target echo, Δτj,i=ΔR/c is the time corresponding to the pull-off jamming range of the i frame, and τj,i>τj,i−1. The pull-off range gradually increases, as shown in Figure 1.Figure 1
Schematic diagram of RGPO.The time delay informationΔτj,i of false target caused by range gate RGPO jamming can be expressed as(12)Δτj,i=02vi−mcorai−m2c,Pull−off stop,0≤i<m Interception stage,m≤i<n Pull−off stage,n≤i<TStopping stage.Among them,a represents the pull-off acceleration in the process of uniform acceleration pull-off and v is the pull-off speed in the process of uniform acceleration pull-off. Figure 2 is a schematic diagram of time domain waveform of RGPO signal.Figure 2
Schematic diagram of time domain waveform of RGPO signal.


### 3.3. Jamming Suppression Method

It can be concluded that the RGPO has the following characteristics:Feature 1: in order to deviate the center of radar gate, the amplitude of RGPO signal must be greater than the amplitude of real target echoFeature 2: because the jamming signal and the target echo are basically consistent in time dimension during the interception stage, the jamming suppression processing cannot be carried out in time domain during the interception stageFeature 3: during the pull-off stage, because the jamming signal gradually deviates from the target echo, the delay time of the jamming signal is different from that of the target real echo in time dimensionAssuming that the jamming parameter isRj,θj and the transmitting space frequency and receiving space corresponding to the jamming signal are, respectively,(13)fTj=−2ΔfRjc+dλcosθj,(14)fRj=dλcosθj.Assume that the parameter of the target isRs,θs, where θs=θj, Rs=Rj−ΔR, and ΔR are the pull-off range.Then, the transmitting spatial frequency and receiving spatial frequency corresponding to the target scattering signal are(15)fTs=−2ΔfRsc+dλcosθs,(16)fRs=dλcosθs.It can be seen from the above equation that for RGPO, the receiving spatial frequency is completely consistent with the backscattered signal of the target, but the transmitting spatial frequency is different. In order to ensure the difference of transmitting spatial frequencies, the jamming suppression algorithm proposed in this study is mainly completed during the pull-off stage.Because the release time of RGPO is that the radar has been working in the target tracking state, the relevant prior information of the target has been obtained, including the distance and angle of the target. It is assumed that the predicted position of the radar for the target isRy,θy, and the target and jamming position are extended and constrained, so that the jamming and target signals fall into the constrained range. The specific method is to set the range tracking accuracy of the radar to be σR. Centered on the target position predicted by the radar, and search inside Rl=±3σR, namely:(17)Rs∈Ry−Rl,Ry+Rl=RhRj∈Ry−Rl,Ry+Rl=Rh,where Rl is the range constraint value and Rh is the range in the constructed target steering vector.According to the working principle of RGPO, assuming that the jammer releases a pull-off signal that is greater than the target echo amplitude, the covariance matrixRX of the constrained azimuth echo can be expressed by eigenvector:(18)RX=∑k=12λkukukH+∑k=3N2λkukukH.The first term of equality coordinatesuk is the eigenvector corresponding to the signal subspace and λk is the eigenvalue corresponding to the signal subspace. Under the ideal condition of not considering false alarm, because the echo and jamming signal are independent, two large eigenvalues can be obtained after the eigendecomposition of the received covariance matrix. Because the signal strength of pull-off jamming is stronger than that of target echo. The eigenvalues are arranged in sequence from large to small, the jamming signal corresponding to the largest eigenvalue is eliminated, and the rest is the range corresponding to the target signal, and then the transmission angular frequency compensation is carried out according to the range of the target signal. The transmission spatial frequencies of the compensated target signal and the pull-off jamming signal are as follows:(19)f⌢Ts=dλcosθs,(20)f˜Tj,p=−2ΔfΔRc+dλcosθj.After compensation, the pull-off jamming signal can be clearly distinguished from the target signal, as shown in Figure3.Figure 3
Distribution of real target and false target in transmitting-receiving two-dimensional spatial frequency domain after range pull-off.As shown in the figure, the real target is distributed diagonally in the transmit-receive spatial frequency, and the pull-off jamming needs to deviate from the center of the gate, which is obviously different from the real target in the transmit spatial frequency.Based on the analysis of real target and RGPO signal characteristics, combined with the characteristics of FDA-MIMO radar, two-dimensional beamforming technology is used to suppress pull-off jamming signal. Its beamformer weights can be expressed as(21)wMF=af^T⊗bf^R.After the target range compensation, the weights of the two-dimensional filter are independent of the range parameters, but only related to the angle parameters of radar detection. However, the weights of two-dimensional filters are affected by the following two factors:(1)
The accuracy of radar prediction of target range, which would cause the loss of matching output(2)
The accuracy of radar angle estimation, which directly affects the accuracy of weightsThe adaptive beamformer based on the minimum lossless response criterion can overcome the above influence factors, which can be expressed as(22)minwAMFEwAMFHx˜2s.t.wAMFHaf^T⊗bf^R=1.where x˜ is the compensated data. After adaptive beamforming, the influence of range gate pull-off jamming can be effectively suppressed.In this study, the flow of the specific method proposed is shown in Figure4:Step 1: use the radar’s predicted value of the target range dimension to constrain the target steering vector according to equation (16), and set the constraint range to within ±3σR the radar tracking accuracy, so that the real target and the jamming signal are both within the constrained steering vector range;Step 2: perform eigendecomposition on the covariance matrixRX, including the target echo and the jamming signal, and use the characteristic that the jamming signal is stronger than the target echo to eliminate the jamming signal range corresponding to the large eigenvalue to obtain the true range of the target;Step 3: after obtaining the true range of the real target, compensate the launch angle frequency, distinguish the jamming signal and the target echo in the launch-receive spatial frequency, use the MVDR criterion to form an adaptive filter, complete the range compensation in the launch dimension, and complete the target track.Figure 4
Flowchart of FDA-MIMO radar anti-RGPO.


## 3.1. Mechanism of RGPO

RGPO is a kind of self-defense jamming. Usually, the airborne ESM system intercepts the radar radiation signal after finding that it is tracked and locked by the tracking and guidance radar and forwards the jamming signal with a certain delay through fast storage, so that the range tracking gate center of the enemy radar deviates from the real target and locks on the released false target, thus tracking the lost technical method. In the actual radar system, after tracking the target, the angle, range, and speed of the target can be predicted with high data rate, and the target position at the next moment can be interception in a certain range, which is called wave gate. If the jamming signal deviates greatly from the real target and exceeds the wave gate range, it may be eliminated as outliers or cannot form an effective jamming track in the data processing stage, and the jamming effect would not be achieved.RGPO can be divided into front-gate-pull-off jamming and back-gate-pull-off jamming according to different delay time functions [39]. For the jamming in front of the wavefront, the forwarding delay of the jamming signal gradually decreases for the radar tracking system, which results in an “illusion” that the false target is gradually approaching the radar system relative to the real target. For the back-gate-pull-off jamming, the delay of jamming signal forwarding increases gradually, resulting in a “phenomenon” that the range the false target is gradually apart from the real target in radar system. As for that front-gate-pull-off jamming, one or more pulse repetition stages need to be delayed. If the radar adopts frequency agility technology, the jamming effect would not be achieved. For back-gate-pull-off jamming, the time delay of the jamming signal needs to be within the same range gate as the target echo. If the radar adopts leading edge tracking technology, it can also effectively resist deceptive jamming. However, for the radar system, the time delay of the real echo received is inaccurate, and it is impossible to accurately determine the jamming style it is, so a new antijamming technology is needed to deal with the pull-off jamming with different ranges. This study focuses on countering the back-gate-pull-off jamming, and the proposed algorithm is also suitable for the front-gate-pull-off jamming.For RGPO, it is generally divided into three stages: interception stage, pull-off stage, and stop stage:Interception stage: after intercepting the tracking signal, the airborne ESM system stores and quickly forwards a jamming signal. Usually, the time delay of the jamming signal needs to basically coincide with the target echo in time dimension.As represents the amplitude of the target echo and Aj represents the amplitude of the jamming signal Aj/As≈1.3∼1.5Pull-off stage: in order to make the range gate center of radar deviate from the real echo of the target and avoid two echo peaks at the same time, ESM system needs to gradually increase the delay time of forwarding every time it intercepts a radar tracking signal, so that the range gate center gradually leaves the target position until the range gate deviates from the target echo by a predetermined range.Stopping stage: when the ESM system judges that the radar has deviated from the real target by enough range from the center of the wave gate, it stops radiating jamming signals, which leads to the radar losing the target or increasing the tracking error, so it is necessary to search and find the target again.If the above three steps are repeated, the radar can get rid of the tracking of the target or increase the tracking error of the radar.


## 3.2. Mathematical Model of RGPO

RGPO is mainly aimed at the radar working in tracking mode. Because the tracking filter has started to work, the radar has a priori information about the position and speed of the target, and through this priori information, the three-dimensional information of the target in the next working cycle can be predicted. In order to get rid of radar tracking, the target releases RGPO. Assuming that the pull-off range of RGPO to the center of echo isΔR at the i pulse repetition interval (PRI) of radar, the time delay of radar receiving jamming signal is(11)τj=τs,i+Δτj,i.In the equation,τj is the delay time of the jamming signal [40], τs,i is the delay time of the real target echo, Δτj,i=ΔR/c is the time corresponding to the pull-off jamming range of the i frame, and τj,i>τj,i−1. The pull-off range gradually increases, as shown in Figure 1.Figure 1
Schematic diagram of RGPO.The time delay informationΔτj,i of false target caused by range gate RGPO jamming can be expressed as(12)Δτj,i=02vi−mcorai−m2c,Pull−off stop,0≤i<m Interception stage,m≤i<n Pull−off stage,n≤i<TStopping stage.Among them,a represents the pull-off acceleration in the process of uniform acceleration pull-off and v is the pull-off speed in the process of uniform acceleration pull-off. Figure 2 is a schematic diagram of time domain waveform of RGPO signal.Figure 2
Schematic diagram of time domain waveform of RGPO signal.


## 3.3. Jamming Suppression Method

It can be concluded that the RGPO has the following characteristics:Feature 1: in order to deviate the center of radar gate, the amplitude of RGPO signal must be greater than the amplitude of real target echoFeature 2: because the jamming signal and the target echo are basically consistent in time dimension during the interception stage, the jamming suppression processing cannot be carried out in time domain during the interception stageFeature 3: during the pull-off stage, because the jamming signal gradually deviates from the target echo, the delay time of the jamming signal is different from that of the target real echo in time dimensionAssuming that the jamming parameter isRj,θj and the transmitting space frequency and receiving space corresponding to the jamming signal are, respectively,(13)fTj=−2ΔfRjc+dλcosθj,(14)fRj=dλcosθj.Assume that the parameter of the target isRs,θs, where θs=θj, Rs=Rj−ΔR, and ΔR are the pull-off range.Then, the transmitting spatial frequency and receiving spatial frequency corresponding to the target scattering signal are(15)fTs=−2ΔfRsc+dλcosθs,(16)fRs=dλcosθs.It can be seen from the above equation that for RGPO, the receiving spatial frequency is completely consistent with the backscattered signal of the target, but the transmitting spatial frequency is different. In order to ensure the difference of transmitting spatial frequencies, the jamming suppression algorithm proposed in this study is mainly completed during the pull-off stage.Because the release time of RGPO is that the radar has been working in the target tracking state, the relevant prior information of the target has been obtained, including the distance and angle of the target. It is assumed that the predicted position of the radar for the target isRy,θy, and the target and jamming position are extended and constrained, so that the jamming and target signals fall into the constrained range. The specific method is to set the range tracking accuracy of the radar to be σR. Centered on the target position predicted by the radar, and search inside Rl=±3σR, namely:(17)Rs∈Ry−Rl,Ry+Rl=RhRj∈Ry−Rl,Ry+Rl=Rh,where Rl is the range constraint value and Rh is the range in the constructed target steering vector.According to the working principle of RGPO, assuming that the jammer releases a pull-off signal that is greater than the target echo amplitude, the covariance matrixRX of the constrained azimuth echo can be expressed by eigenvector:(18)RX=∑k=12λkukukH+∑k=3N2λkukukH.The first term of equality coordinatesuk is the eigenvector corresponding to the signal subspace and λk is the eigenvalue corresponding to the signal subspace. Under the ideal condition of not considering false alarm, because the echo and jamming signal are independent, two large eigenvalues can be obtained after the eigendecomposition of the received covariance matrix. Because the signal strength of pull-off jamming is stronger than that of target echo. The eigenvalues are arranged in sequence from large to small, the jamming signal corresponding to the largest eigenvalue is eliminated, and the rest is the range corresponding to the target signal, and then the transmission angular frequency compensation is carried out according to the range of the target signal. The transmission spatial frequencies of the compensated target signal and the pull-off jamming signal are as follows:(19)f⌢Ts=dλcosθs,(20)f˜Tj,p=−2ΔfΔRc+dλcosθj.After compensation, the pull-off jamming signal can be clearly distinguished from the target signal, as shown in Figure3.Figure 3
Distribution of real target and false target in transmitting-receiving two-dimensional spatial frequency domain after range pull-off.As shown in the figure, the real target is distributed diagonally in the transmit-receive spatial frequency, and the pull-off jamming needs to deviate from the center of the gate, which is obviously different from the real target in the transmit spatial frequency.Based on the analysis of real target and RGPO signal characteristics, combined with the characteristics of FDA-MIMO radar, two-dimensional beamforming technology is used to suppress pull-off jamming signal. Its beamformer weights can be expressed as(21)wMF=af^T⊗bf^R.After the target range compensation, the weights of the two-dimensional filter are independent of the range parameters, but only related to the angle parameters of radar detection. However, the weights of two-dimensional filters are affected by the following two factors:(1)
The accuracy of radar prediction of target range, which would cause the loss of matching output(2)
The accuracy of radar angle estimation, which directly affects the accuracy of weightsThe adaptive beamformer based on the minimum lossless response criterion can overcome the above influence factors, which can be expressed as(22)minwAMFEwAMFHx˜2s.t.wAMFHaf^T⊗bf^R=1.where x˜ is the compensated data. After adaptive beamforming, the influence of range gate pull-off jamming can be effectively suppressed.In this study, the flow of the specific method proposed is shown in Figure4:Step 1: use the radar’s predicted value of the target range dimension to constrain the target steering vector according to equation (16), and set the constraint range to within ±3σR the radar tracking accuracy, so that the real target and the jamming signal are both within the constrained steering vector range;Step 2: perform eigendecomposition on the covariance matrixRX, including the target echo and the jamming signal, and use the characteristic that the jamming signal is stronger than the target echo to eliminate the jamming signal range corresponding to the large eigenvalue to obtain the true range of the target;Step 3: after obtaining the true range of the real target, compensate the launch angle frequency, distinguish the jamming signal and the target echo in the launch-receive spatial frequency, use the MVDR criterion to form an adaptive filter, complete the range compensation in the launch dimension, and complete the target track.Figure 4
Flowchart of FDA-MIMO radar anti-RGPO.


## 4. Simulation Verification

In this section, the effectiveness of the proposed approach is verified by simulation examples. Without losing generality, assume that the radar receives the jamming signal radiated by the enemy when it has stably tracked the target, and the jammer intercepts the radar tracking signal and tows it to release the RGPO. The relevant parameters of simulation are given in Table1.Table 1
Radar simulation parameters.
ParameterParameter valueParameterParameter valueOperating frequency10 GHzPulse repetition frequency10 kHzSampling frequency5 MHzNumber of receiving array elements10Number of transmitting array elements10Receiving array element spacing0.015 mLaunching element spacing0.015 mRange tracking error40 mTarget state noise20 mTarget motion modelConstant velocity modelSNR10 dBTracking filterStandard KalmanTarget range10 kmTarget angle0°RGPO JNR15 dBTarget speed100 m/sPull-off time30 sRange pull-off speed10 m/s


### 4.1. Power Spectrum Analysis before and after Antijamming

Figure5 shows the simulation comparison results of echo before and after processing the antijamming approach proposed in the transmission-reception frequency domain. It can be seen from the simulation results that because the echo in FDA-MIMO system contains the range dimension information of the echo in the transmitting frequency domain, the radar cannot accurately distinguish the real target echo because the amplitude of the jamming signal is greater than that of the echo after receiving the RGPO signal, and the range between the jamming signal and the target echo is close during the pull-off stage. After constrained and large signal is proposed, the transmission frequency domain can be compensated, and the true echo of the target can be accurately detected. It should be noted that due to the target motion and radar detection error, the position of the compensated target may change slightly. Figure 6 shows the signal strength distribution after pulse compression in range dimension before and after antijamming. In this simulation, with receiving the jamming signal, the maximum amplitude of the output is about 28 dB and the range cell is 350 where the jamming is, after processing antijamming, the maximum amplitude of the output is about 25 dB and the range cell is 333 where the real target is.Figure 5
Power spectrum characteristics of real target and pull-off jamming. (a) Power spectrum pull-off by range. (b) Power spectrum after antijamming.
(a)(b)Figure 6
Output results of signal processing before and after antijamming.


### 4.2. Tracking Performance Analysis

The tracking performance before and after antijamming is analyzed. In order to accurately show the experimental results, it is assumed that the flying height of the target remains unchanged in the northeast coordinate system, and the tracking performance is mainly analyzed onX axis and Y axis. Figure 7 shows the change of the whole range segment of the range-dimensional tracking gate center before and after the radar is jammed by range pull-off. It can be seen from the example that after receiving the RGPO, the range tracking gate center of the radar would gradually deviate from the real target position, resulting in increased tracking error or even out of tolerance, and the target would be lost after the jamming signal disappears. Figure 8 shows the average tracking error of X axis and Y axis in the radar tracking process. It can be seen that the tracking error increases dramatically after receiving the RGPO. After filtering, the two-dimensional range error is about ±300 m, which is far greater than the radar tracking accuracy. After antijamming, the two-dimensional range error changes ±40 m, and the radar can keep stable tracking of the target.Figure 7
Change of wave gate center before and after radar jamming.Figure 8
The average change of tracking error before and after radar antijamming.


## 4.1. Power Spectrum Analysis before and after Antijamming

Figure5 shows the simulation comparison results of echo before and after processing the antijamming approach proposed in the transmission-reception frequency domain. It can be seen from the simulation results that because the echo in FDA-MIMO system contains the range dimension information of the echo in the transmitting frequency domain, the radar cannot accurately distinguish the real target echo because the amplitude of the jamming signal is greater than that of the echo after receiving the RGPO signal, and the range between the jamming signal and the target echo is close during the pull-off stage. After constrained and large signal is proposed, the transmission frequency domain can be compensated, and the true echo of the target can be accurately detected. It should be noted that due to the target motion and radar detection error, the position of the compensated target may change slightly. Figure 6 shows the signal strength distribution after pulse compression in range dimension before and after antijamming. In this simulation, with receiving the jamming signal, the maximum amplitude of the output is about 28 dB and the range cell is 350 where the jamming is, after processing antijamming, the maximum amplitude of the output is about 25 dB and the range cell is 333 where the real target is.Figure 5
Power spectrum characteristics of real target and pull-off jamming. (a) Power spectrum pull-off by range. (b) Power spectrum after antijamming.
(a)(b)Figure 6
Output results of signal processing before and after antijamming.


## 4.2. Tracking Performance Analysis

The tracking performance before and after antijamming is analyzed. In order to accurately show the experimental results, it is assumed that the flying height of the target remains unchanged in the northeast coordinate system, and the tracking performance is mainly analyzed onX axis and Y axis. Figure 7 shows the change of the whole range segment of the range-dimensional tracking gate center before and after the radar is jammed by range pull-off. It can be seen from the example that after receiving the RGPO, the range tracking gate center of the radar would gradually deviate from the real target position, resulting in increased tracking error or even out of tolerance, and the target would be lost after the jamming signal disappears. Figure 8 shows the average tracking error of X axis and Y axis in the radar tracking process. It can be seen that the tracking error increases dramatically after receiving the RGPO. After filtering, the two-dimensional range error is about ±300 m, which is far greater than the radar tracking accuracy. After antijamming, the two-dimensional range error changes ±40 m, and the radar can keep stable tracking of the target.Figure 7
Change of wave gate center before and after radar jamming.Figure 8
The average change of tracking error before and after radar antijamming.


## 5. Conclusion

Tracking and guidance radar is usually affected by the mainlobe jamming. In this study, focused on the principle and application of FDA-MIMO radar against range gate pull-off jamming, an effective way to solve the problem that the center of range gate deviates from the real target greatly due to pull-off jamming is provided on the basis of analyzing the principle and working process of RGPO. Based on two-dimensional transceiver beamforming and range eliminating, the jamming signal can be suppressed. The experimental results verify the effectiveness and feasibility of the proposed method from two important links of detection and tracking radar.



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*Source: 1265658-2022-07-08.xml*