# A Method against Interrupted-Sampling Repeater Jamming Based on Energy Function Detection and Band-Pass Filtering

Authors: Hui Yuan; Chun-yang Wang; Xin Li; Lei An
Journal: International Journal of Antennas and Propagation
 (2017)

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## Abstract

Interrupted-sampling repeater jamming (ISRJ) is a new kind of coherent jamming to the large time-bandwidth linear frequency modulation (LFM) signal. Many jamming modes, such as lifelike multiple false targets and dense false targets, can be made through setting up different parameters. According to the “storage-repeater-storage-repeater” characteristics of the ISRJ and the differences in the time-frequency-energy domain between the ISRJ signal and the target echo signal, one new method based on the energy function detection and band-pass filtering is proposed to suppress the ISRJ. The methods mainly consist of two parts: extracting the signal segments without ISRJ and constructing band-pass filtering function with low sidelobe. The simulation results show that the method is effective in the ISRJ with different parameters.


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## Body

## 1. Introduction

Interrupted-sampling repeater jamming (ISRJ) [1] based on the digital radio frequency memory (DRFM) technology is a new kind of smart jamming, which aims especially at the linear frequency modulation (LFM) signals with large time-bandwidth product. The prominent characteristics of ISRJ are embodied in its antenna system and its manner of sampling and storing signals. ISRJ can be used in receive-transmit time-sharing antenna system as well as antenna system with two receive-transmit antennas which work asynchronously. Besides, sampling and storing the whole radar signal without distortion are not necessary. Therefore, the problems of high-speed sampling and receive-transmit antenna isolation are settled satisfactorily, which means great value of practical application on the occasions of small jammer-platform like missile-borne jammer.Compared with ISRJ, the Chopping and Interleaving (C&I) proposed in [2] by Sparrow and Cakilo can only be accomplished in the next or after several radar pulse repetition periods and the signal is continuous, while the ISRJ signal can be generated in the current radar pulse repetition period and the signal is intermittent, which leads the methods such as phase perturbation [3], adjusting the chirp rate [4], and pulse diversity [5–8] to failure. The fundamental ISRJ, whose repeat sampling interval is constant and the duty ratio is 50%, has many deficiencies. For example, the amplitudes of the subfalse targets decay rapidly and the main false target is always behind the true target on the self-defense occasion. For solving these problems, some further studies have been presented like interrupted-sampling and periodic repeater jamming [9], irregularly interrupted-sampling and repeater jamming [10], interrupted-sampling and nonuniform periodic repeater jamming [11], and interrupted-sampling and successive repeater jamming [12]. For sharing the common operating mode “storage-repeater-storage-repeater,” they are collectively called ISRJ in this paper. Compared with the ISRJ with fixed repeat sampling interval, interrupted-sampling and periodic repeater jamming and interrupted-sampling and successive repeater jamming can form a mass of lifelike false targets and have obvious advantages in fidelity and amount of false targets, flexibility of system, computation burden, and requirement of hardware environment [13]. The ISRJ with irregular sampling periods and interrupted-sampling and nonuniform periodic repeater jamming can achieve the desired jamming effect through setting up different parameters like sampling pulse width, delay time of repeater, pulse width of repeater, and repeat sampling interval. Moreover, the false targets with different amplitudes can be generated without changing the instantaneous transmit power which reduce the complexity of hardware system and the energy loss caused by the power switching frequently. Besides the studies on jamming methods, Feng made some researches on the performance of ISRJ on monopulse radar [14], dechirping radar [15], and the impact of ISJR on radar constant false alarm rate (CFAR) detection [16]. There are also many researches on 2-dimension (2D) ISRJ methods and their application on synthetic aperture radar (SAR) or inverse synthetic aperture radar (ISAR) [17–19].Due to the sensitivity of the military, there is only limited public research literature on electronic counter-countermeasure (ECCM) schemes with respect to the extensive researches on ISRJ. As a coherent repeater jamming method, ISRJ has both deception and blanket jamming effects under certain parameters and cannot be suppressed by traditional radar antijamming methods such as matched filter and coherent integration. Furthermore, ISRJ is mostly used as self-defense jamming, so the traditional sidelobe cancellation and sidelobe blanking are also useless. Aiming at the recognition of ISRJ, Jiang et al. [20] have put forward some recognition methods by using the differences on frequency spectrum of the jamming signal and the target signal, which is really important for further research on ECCM methods against ISRJ. According to the discontinuity in time domain of the ISRJ signal, Gong et al. [21] have proposed an effective ECCM scheme for eliminating the ISRJ-based false targets from the pulse compression result based on the time-frequency analysis method. The segments not disturbed by ISRJ are detected by finding the modulus minimum of the short-time Fourier transform (STFT) result, and the band-pass filter is constructed to filter out the ISRJ signal. However, Gong does not give a method to determine the length of the time unit in which only the target echo signal exists.In this paper, a new method against the ISRJ is proposed based on the “storage-repeater-storage-repeater” characteristics of the ISRJ and the differences in the time-frequency-energy domain between the ISRJ signal and the target signal. The new method does not need to make the time-frequency analysis such as short-time Fourier transform and only needs to compute the simple energy function of the radar signal. Then the signal segments not disturbed by the ISRJ are obtained by comparing the energy function with a threshold value. The rest of the paper is organized as follows. Section2 first reviews the signal models of the target echo signal and the ISRJ signal before and after the dechirping processing, respectively, and then analyzes their time-frequency-energy domain characteristics. Section 3 puts forward the antijamming method based on the energy function detection and band-pass filtering. The methods of extracting the signal segments without ISRJ, determining the threshold value, and building the band-pass filtering function with low sidelobes are presented. Section 4 carries out some simulations to demonstrate that the proposed ECCM method is effective in ISRJ with different parameters resulting in different jamming results including the multiple lifelike false targets and the intensively distributed false targets.


## 2. Signal Model and Properties on Time-Frequency-Energy Domain

### 2.1. Signal Model

Dechirping processing is one of the common methods of wideband LFM signal, which can convert the time delay of the target echo signal into frequency; that is, the target echo signal after dechirping is a single-frequency signal whose frequency is proportional to the target distance. Then, the pulse compression result can be obtained by doing the Fourier transform to the radar echo signal after dechirping processing. This method can not only keep the high range-resolution but also reduce the radar’s intermediate frequency (IF) bandwidth. ISRJ can also form effective multiple false targets to the dechirping wideband LFM pulse compression radar [16]. Furthermore, the prime component of the jamming signal’s pulse compression output can be ahead of the target by controlling the time delay of repeater, which means that the spatial distributions of the false targets are more flexible.Assume that the normalized LFM signal emitted by the radar is(1)st=rectt-T/2Texp⁡j2πfct+12kt2,where (2)recttT=1,-T2≤t≤T2,0,else represents a rectangular pulse with pulse width T and center location 0, fc is the radar carrier frequency, k=B/T represents the chirp rate, and B is the bandwidth of the signal. For the convenience of researching, assume that the true target used in this paper just has one scatter point and the distance between them is Rt. Then, the target echo signal can be denoted as(3)stt=Arrectt-T/2-τtT×exp⁡j2πfct-τt+12kt-τt2,where τt=2Rt/c is the time delay of the target echo signal, c is the velocity of light, and Ar=2PtG2λ2σ/4π3Rt4 is the amplitude of the target echo signal. Pt,G,λ,σ are the peak power of the radar signal, antenna gain, wavelength, and the target radar cross section (RCS), respectively.The reference signal of dechirping radar can be expressed by(4)sreft=rectt-Tref/2-τrefTref×exp⁡j2πfct-τref+12kt-τref2,where τref=2Rref/c,Rref is the reference range, Tref is the receiving window, and generally Tref>T.The dechirping processing consists of mixing radar signals and reference signal and low-pass filtering. Using the analytic signal, the target echo signal after dechirping can be denoted as(5)st_de=stt·sref∗t=Arrectt-T/2-τtT×exp⁡j2π-kτt-τreft-fcτt-τref+12kτt2-τref2.The basic idea of ISRJ is “storage-repeater-storage-repeater,” which means that the jammer firstly samples and stores parts of the radar signal and then transmits it once or more than once, uniformly or nonuniformly to the radar. Repeat like this until the end of the radar signal. Without loss of generality, we analyze the interrupted-sampling and periodic repeater jamming. Figure1 shows the jamming principle. Ts is the repeat sampling interval. τ is the sampling pulse width. Ts is M+1 times of τ. Then the interrupted-sampling and periodic repeater jamming can be described by(6)sjt=Aj∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ×exp⁡j2πfct-mτ-τt-τd+12kt-mτ-τt-τd2,where τd denotes the delay of the jammer and Aj=2PjGλ2/4πRt2 denotes the amplitude of the jamming signal. Pj is the transmit power of the jammer. Then the jamming signal after dechirping can be expressed by(7)sj_de=sjt·sref∗t=Aj∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτexp⁡j2π-kmτ+τt+τd-τreft-fcmτ+τt+τd-τref+12kmτ+τt+τd2-12kτref2.Figure 1
The principle of the interrupted-sampling and periodic repeater jamming.


### 2.2. Properties on Time-Frequency-Energy Domain

The instantaneous frequency of a signal is defined by [22] as the derivative of phase with respect to time.(8)f=12πdφtdt.Then according to (3), (5), (6), and (7), the instantaneous frequency of the target echo signal and the jamming signal before and after dechirping processing can be obtained as follows:(9)ft=12πrectt-T/2-τtT·ddtfct-τt+12kt-τt2=rectt-T/2-τtTfc+kt-τt,ft_de=12πrectt-T/2-τtT·ddt2π-kτt-τreft-fcτt-τref+12kτt2-τref2=rectt-T/2-τtT-kΔτt,Δτt=τt-τref,fj=12π∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ·ddt2πfct-mτ-τt-τd+12kt-mτ-τt-τd2=∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ·fc+kt-mτ-τt-τd,fj_de=12π∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ·ddt2π-kmτ+τt+τd-τreft-fcmτ+τt+τd-τref+12kmτ+τt+τd2-12kτref2=∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ-kΔτj,Δτj=mτ+τt+τd-τref.According to (9), we can realize the TF distribution of the target signal and the jamming signal before and after dechirping by Figure 2. It can be seen from Figure 2 that the TF distribution of the jamming signal is discontinuous no matter before or after dechirping because the ISRJ jammer does not transmit the jamming signal when it is receiving the radar signal, which is also the main advantage of ISRJ. However, in pulse duration, the target echo signal before and after dechirping is continuously changing with time. Especially after dechirping, the frequency of the target echo signal is invariable. Based on the great difference between the TF characteristics of the ISRJ signal and the target echo signal, we can build a band-pass filter using the target echo signal segments which are not disturbed by the ISRJ signal to eliminate most of the false targets.Figure 2
TF distributions of the target echo signal and the ISRJ signal.
(a)
Before dechirping processing
(b)
After dechirping processingUsing the energy function defined as the square of the signal’s modulus, we analyze the distribution characteristics of the radar signal in the time-energy domain under the jamming condition. The energy functions of the target echo signalst_de(t), the ISRJ signal sj_de(t), and the radar received signals x(t)=st_de(t)+sj_de(t) under the jamming condition after dechirping are(10)Ett=st_det2=Ar2rectt-T/2-τtT,(11)Ejt=sj_strecht2=Aj2∑n=0N-1rectt-τt-τd-1+M/2τ-nTsMτ,(12)Ext=sr_strecht+sj_strecht2=Ar2rectt-T/2-τtT+Aj2∑n=0N-1rectt-τt-τd-1+M/2τ-nTsMτ+2ArAj∑m=1M∑n=0N-1rectt-τt-τd-τ/2-mτ-nTsτ×cos⁡2πkmτ+τdt+φ,respectively, where φ=2πfc(mτ+τd)+1/2kτt2-1/2k(mτ+τt+τd)2.Forcos2πk(mτ+τd)t+φ≤1, Ext satisfies Aj-Ar2≤Ex(t)≤Aj+Ar2 during transmitting the jamming signals. Figure 3 shows the curves of the energy functions of the target echo signal, the ISRJ signal, and the radar received signals under the jamming condition after dechirping.Figure 3
Energy functions.It can be seen from Figure3 that, during the sampling duration, the energy is stable and small because the jammer does not transmit the jamming signal and only the target echo signal is received, while the energy is fluctuant quickly and strong during the time of transmitting the jamming signal.


## 2.1. Signal Model

Dechirping processing is one of the common methods of wideband LFM signal, which can convert the time delay of the target echo signal into frequency; that is, the target echo signal after dechirping is a single-frequency signal whose frequency is proportional to the target distance. Then, the pulse compression result can be obtained by doing the Fourier transform to the radar echo signal after dechirping processing. This method can not only keep the high range-resolution but also reduce the radar’s intermediate frequency (IF) bandwidth. ISRJ can also form effective multiple false targets to the dechirping wideband LFM pulse compression radar [16]. Furthermore, the prime component of the jamming signal’s pulse compression output can be ahead of the target by controlling the time delay of repeater, which means that the spatial distributions of the false targets are more flexible.Assume that the normalized LFM signal emitted by the radar is(1)st=rectt-T/2Texp⁡j2πfct+12kt2,where (2)recttT=1,-T2≤t≤T2,0,else represents a rectangular pulse with pulse width T and center location 0, fc is the radar carrier frequency, k=B/T represents the chirp rate, and B is the bandwidth of the signal. For the convenience of researching, assume that the true target used in this paper just has one scatter point and the distance between them is Rt. Then, the target echo signal can be denoted as(3)stt=Arrectt-T/2-τtT×exp⁡j2πfct-τt+12kt-τt2,where τt=2Rt/c is the time delay of the target echo signal, c is the velocity of light, and Ar=2PtG2λ2σ/4π3Rt4 is the amplitude of the target echo signal. Pt,G,λ,σ are the peak power of the radar signal, antenna gain, wavelength, and the target radar cross section (RCS), respectively.The reference signal of dechirping radar can be expressed by(4)sreft=rectt-Tref/2-τrefTref×exp⁡j2πfct-τref+12kt-τref2,where τref=2Rref/c,Rref is the reference range, Tref is the receiving window, and generally Tref>T.The dechirping processing consists of mixing radar signals and reference signal and low-pass filtering. Using the analytic signal, the target echo signal after dechirping can be denoted as(5)st_de=stt·sref∗t=Arrectt-T/2-τtT×exp⁡j2π-kτt-τreft-fcτt-τref+12kτt2-τref2.The basic idea of ISRJ is “storage-repeater-storage-repeater,” which means that the jammer firstly samples and stores parts of the radar signal and then transmits it once or more than once, uniformly or nonuniformly to the radar. Repeat like this until the end of the radar signal. Without loss of generality, we analyze the interrupted-sampling and periodic repeater jamming. Figure1 shows the jamming principle. Ts is the repeat sampling interval. τ is the sampling pulse width. Ts is M+1 times of τ. Then the interrupted-sampling and periodic repeater jamming can be described by(6)sjt=Aj∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ×exp⁡j2πfct-mτ-τt-τd+12kt-mτ-τt-τd2,where τd denotes the delay of the jammer and Aj=2PjGλ2/4πRt2 denotes the amplitude of the jamming signal. Pj is the transmit power of the jammer. Then the jamming signal after dechirping can be expressed by(7)sj_de=sjt·sref∗t=Aj∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτexp⁡j2π-kmτ+τt+τd-τreft-fcmτ+τt+τd-τref+12kmτ+τt+τd2-12kτref2.Figure 1
The principle of the interrupted-sampling and periodic repeater jamming.


## 2.2. Properties on Time-Frequency-Energy Domain

The instantaneous frequency of a signal is defined by [22] as the derivative of phase with respect to time.(8)f=12πdφtdt.Then according to (3), (5), (6), and (7), the instantaneous frequency of the target echo signal and the jamming signal before and after dechirping processing can be obtained as follows:(9)ft=12πrectt-T/2-τtT·ddtfct-τt+12kt-τt2=rectt-T/2-τtTfc+kt-τt,ft_de=12πrectt-T/2-τtT·ddt2π-kτt-τreft-fcτt-τref+12kτt2-τref2=rectt-T/2-τtT-kΔτt,Δτt=τt-τref,fj=12π∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ·ddt2πfct-mτ-τt-τd+12kt-mτ-τt-τd2=∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ·fc+kt-mτ-τt-τd,fj_de=12π∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ·ddt2π-kmτ+τt+τd-τreft-fcmτ+τt+τd-τref+12kmτ+τt+τd2-12kτref2=∑m=1M∑n=0N-1rectt-τ/2-τt-τd-mτ-nTsτ-kΔτj,Δτj=mτ+τt+τd-τref.According to (9), we can realize the TF distribution of the target signal and the jamming signal before and after dechirping by Figure 2. It can be seen from Figure 2 that the TF distribution of the jamming signal is discontinuous no matter before or after dechirping because the ISRJ jammer does not transmit the jamming signal when it is receiving the radar signal, which is also the main advantage of ISRJ. However, in pulse duration, the target echo signal before and after dechirping is continuously changing with time. Especially after dechirping, the frequency of the target echo signal is invariable. Based on the great difference between the TF characteristics of the ISRJ signal and the target echo signal, we can build a band-pass filter using the target echo signal segments which are not disturbed by the ISRJ signal to eliminate most of the false targets.Figure 2
TF distributions of the target echo signal and the ISRJ signal.
(a)
Before dechirping processing
(b)
After dechirping processingUsing the energy function defined as the square of the signal’s modulus, we analyze the distribution characteristics of the radar signal in the time-energy domain under the jamming condition. The energy functions of the target echo signalst_de(t), the ISRJ signal sj_de(t), and the radar received signals x(t)=st_de(t)+sj_de(t) under the jamming condition after dechirping are(10)Ett=st_det2=Ar2rectt-T/2-τtT,(11)Ejt=sj_strecht2=Aj2∑n=0N-1rectt-τt-τd-1+M/2τ-nTsMτ,(12)Ext=sr_strecht+sj_strecht2=Ar2rectt-T/2-τtT+Aj2∑n=0N-1rectt-τt-τd-1+M/2τ-nTsMτ+2ArAj∑m=1M∑n=0N-1rectt-τt-τd-τ/2-mτ-nTsτ×cos⁡2πkmτ+τdt+φ,respectively, where φ=2πfc(mτ+τd)+1/2kτt2-1/2k(mτ+τt+τd)2.Forcos2πk(mτ+τd)t+φ≤1, Ext satisfies Aj-Ar2≤Ex(t)≤Aj+Ar2 during transmitting the jamming signals. Figure 3 shows the curves of the energy functions of the target echo signal, the ISRJ signal, and the radar received signals under the jamming condition after dechirping.Figure 3
Energy functions.It can be seen from Figure3 that, during the sampling duration, the energy is stable and small because the jammer does not transmit the jamming signal and only the target echo signal is received, while the energy is fluctuant quickly and strong during the time of transmitting the jamming signal.


## 3. ECCM Scheme against ISRJ

### 3.1. Executing Steps of ECCM Scheme

According to the above analysis, an ECCM scheme against the ISRJ can be summarized by the following steps.Step 1.
Calculate energy function ofx(t) (see (12)).Step 2.
Determine a threshold valueγ.Step 3.
Extract the signal segments not disturbed by the ISRJ signal.
For any timet0 during the radar received signals, when Ex(t0)<γ, p(t0)=1; otherwise, p(t0)=0. Then the signal segments not disturbed by ISRJ signal can be expressed as(13)ht=xt·pt.Step 4.
Get the band-pass filtering functionH(f) by doing the Fourier transform on h(t) and then carry out the normalization process.(14)Hf=∫tht·e-j2πftdtmax⁡∫tht·e-j2πftdt.
Doing the Fourier transform onx(t), we can get the pulse compression result with ISRJ signal X(f).(15)Xf=∫txt·e-j2πftdt.Step 5 (filtering).
MultiplyingX(f) with the band-pass filtering function H(f), the ISRJ can be suppressed and then the final pulse compression result can be obtained as(16)Yf=Hf·Xf.


### 3.2. How to Determine the Threshold Value

Not considering the noise or when the noise is small enough and can be negligible, we can get thatAj-Ar2≤Ex(t)≤Aj+Ar2 according to (12). Then, when Aj-Ar2>Ar2, that is, Aj>2Ar (this is a necessary condition to form an effective jamming), we always can find a threshold value γ, satisfying Ar2<γ<Aj-Ar2. Then the target echo signal segments not disturbed by the ISRJ signal can be found by comparing Ex(t) with γ which can be set as(17)γ=meanenvminpExt,where envminp(Ex(t)) represents the minimum envelope of the energy function and mean(·) denotes the expectation operation.For(18)envminpExt=∑n=0N-1Ar2rectt-τ/2-nTsτ+Aj-Ar2rectt-1+M/2τ-nTsMτwe have(19)γ=meanenvminpExt=Ar2·τ+Aj-Ar2·Ts-τTs=Aj-Ar2-τTsAjAj-2Ar=Ar2+1-τTsAj2-2AjArand generally 0<τ/Ts≤1/2<1, so γ=mean(envminp(px(t))) satisfies Ar2<γ<Aj-Ar2.When the noise is not small enough and cannot be negligible, the envelope of the energy function is not constant any more. There will be large error using the above methods of determining the threshold value and extracting the signal segments not disturbed by the ISRJ signal. Then an improved method is proposed as follows:(1)
Calculating the maximum and the minimum envelopes of the energy function,envmaxp(Ex(t)) and envminp(Ex(t)), respectively(2)
Calculating the mean envelope of the maximum and the minimum envelopes:(20)envmoypExt=envmaxpExt+envminpExt2(3)
Comparingenvmoyp(Ex(t)) with γ: when envmoyp(Ex(t0))<γ,p(t0)=1; otherwise, p(t0)=0. According to (13), we can get the signal segments not disturbed by the ISRJ signalThis method can maximally avoid the influence of noise on extracting the signal segments not disturbed by the ISRJ signal.


### 3.3. The Improved Method of Designing the Band-Pass Filtering FunctionH(f)

Theoretically,p(t) should be a rectangular envelope pulse train with pulse duration τ and pulse repeat sampling interval Ts, which can be defined by(21)ht=xt·pt=st_det·pt,where p(t)=rectt-τ/2/τ∗∑n=0N-1δ(t-nTs),∗ denotes convolution operator.The frequency spectrums ofp(t) and st_de(t), respectively, are(22)Pf=∑n=0N-1τfssin⁡cnfsτe-jπnfsτδf-nfs=∑n=0N-1anδf-nfs,St_def=ArTsin⁡cf-ft_deTejφ0-πf-ft_deT,where φ0=-2πfc(τt-τref)+πk(τt2-τref2). Then the frequency spectrum of h(t) is(23)Hf=∑n=0N-1anSt_def-nfs,where fs=1/Ts.FromH(f), we can clearly see that the frequency fmax corresponding to the maximum value of H(f) is just ft_de. But there exist high sidelobes beside the mainlobe which may cause the result that there are still some strong false targets after filtering and the radar system still cannot detect the true target correctly.In this paper, the method of windowing in the frequency domain is designed as follows to suppress the sidelobes ofH(f).Step 1.
Choose a window functionw(t), such as hamming window. By doing Fourier transform on w(t), we get its frequency spectrum W(f) whose center frequency is located in zero.Step 2.
Constructing a one-dimensional optimization function,(24)gf0=∑fWf-f0·Hf.And find f0max that makes g(f0) get the maximum value.(25)f0max=maxf0gf0.Then, the window function in frequency domain can be obtained as(26)Waf=Wf-f0max.Step 3.
MultiplyingWa(f) and H(f) to get the filtering function with low sidelobes Ha(f),(27)Haf=Waf·Hf.


## 3.1. Executing Steps of ECCM Scheme

According to the above analysis, an ECCM scheme against the ISRJ can be summarized by the following steps.Step 1.
Calculate energy function ofx(t) (see (12)).Step 2.
Determine a threshold valueγ.Step 3.
Extract the signal segments not disturbed by the ISRJ signal.
For any timet0 during the radar received signals, when Ex(t0)<γ, p(t0)=1; otherwise, p(t0)=0. Then the signal segments not disturbed by ISRJ signal can be expressed as(13)ht=xt·pt.Step 4.
Get the band-pass filtering functionH(f) by doing the Fourier transform on h(t) and then carry out the normalization process.(14)Hf=∫tht·e-j2πftdtmax⁡∫tht·e-j2πftdt.
Doing the Fourier transform onx(t), we can get the pulse compression result with ISRJ signal X(f).(15)Xf=∫txt·e-j2πftdt.Step 5 (filtering).
MultiplyingX(f) with the band-pass filtering function H(f), the ISRJ can be suppressed and then the final pulse compression result can be obtained as(16)Yf=Hf·Xf.


## 3.2. How to Determine the Threshold Value

Not considering the noise or when the noise is small enough and can be negligible, we can get thatAj-Ar2≤Ex(t)≤Aj+Ar2 according to (12). Then, when Aj-Ar2>Ar2, that is, Aj>2Ar (this is a necessary condition to form an effective jamming), we always can find a threshold value γ, satisfying Ar2<γ<Aj-Ar2. Then the target echo signal segments not disturbed by the ISRJ signal can be found by comparing Ex(t) with γ which can be set as(17)γ=meanenvminpExt,where envminp(Ex(t)) represents the minimum envelope of the energy function and mean(·) denotes the expectation operation.For(18)envminpExt=∑n=0N-1Ar2rectt-τ/2-nTsτ+Aj-Ar2rectt-1+M/2τ-nTsMτwe have(19)γ=meanenvminpExt=Ar2·τ+Aj-Ar2·Ts-τTs=Aj-Ar2-τTsAjAj-2Ar=Ar2+1-τTsAj2-2AjArand generally 0<τ/Ts≤1/2<1, so γ=mean(envminp(px(t))) satisfies Ar2<γ<Aj-Ar2.When the noise is not small enough and cannot be negligible, the envelope of the energy function is not constant any more. There will be large error using the above methods of determining the threshold value and extracting the signal segments not disturbed by the ISRJ signal. Then an improved method is proposed as follows:(1)
Calculating the maximum and the minimum envelopes of the energy function,envmaxp(Ex(t)) and envminp(Ex(t)), respectively(2)
Calculating the mean envelope of the maximum and the minimum envelopes:(20)envmoypExt=envmaxpExt+envminpExt2(3)
Comparingenvmoyp(Ex(t)) with γ: when envmoyp(Ex(t0))<γ,p(t0)=1; otherwise, p(t0)=0. According to (13), we can get the signal segments not disturbed by the ISRJ signalThis method can maximally avoid the influence of noise on extracting the signal segments not disturbed by the ISRJ signal.


## 3.3. The Improved Method of Designing the Band-Pass Filtering FunctionH(f)

Theoretically,p(t) should be a rectangular envelope pulse train with pulse duration τ and pulse repeat sampling interval Ts, which can be defined by(21)ht=xt·pt=st_det·pt,where p(t)=rectt-τ/2/τ∗∑n=0N-1δ(t-nTs),∗ denotes convolution operator.The frequency spectrums ofp(t) and st_de(t), respectively, are(22)Pf=∑n=0N-1τfssin⁡cnfsτe-jπnfsτδf-nfs=∑n=0N-1anδf-nfs,St_def=ArTsin⁡cf-ft_deTejφ0-πf-ft_deT,where φ0=-2πfc(τt-τref)+πk(τt2-τref2). Then the frequency spectrum of h(t) is(23)Hf=∑n=0N-1anSt_def-nfs,where fs=1/Ts.FromH(f), we can clearly see that the frequency fmax corresponding to the maximum value of H(f) is just ft_de. But there exist high sidelobes beside the mainlobe which may cause the result that there are still some strong false targets after filtering and the radar system still cannot detect the true target correctly.In this paper, the method of windowing in the frequency domain is designed as follows to suppress the sidelobes ofH(f).Step 1.
Choose a window functionw(t), such as hamming window. By doing Fourier transform on w(t), we get its frequency spectrum W(f) whose center frequency is located in zero.Step 2.
Constructing a one-dimensional optimization function,(24)gf0=∑fWf-f0·Hf.And find f0max that makes g(f0) get the maximum value.(25)f0max=maxf0gf0.Then, the window function in frequency domain can be obtained as(26)Waf=Wf-f0max.Step 3.
MultiplyingWa(f) and H(f) to get the filtering function with low sidelobes Ha(f),(27)Haf=Waf·Hf.


## 4. Simulations

In this section, we analyze the performance of the proposed ECCM scheme to the ISRJ through some simulations. The simulation parameters employed are illustrated in Table1. Here, the system delay of the jammer is too small to consider.Table 1
Simulation parameters.
Parameter
Value
Peak power of radarPt/MW
1.5
Radar antenna gainG/dB
45
Wavelengthλ/cm
5
Noise temperatureTe/K
290
Noise figureF/dB
3
Radar lossL/dB
6
Bandwidth of radar LFM signalB/MHz
10
Radar pulse widthT/μs
128
Radar IF sampling frequencyfs/MHz
5
Target RCSσ/m2
1
Distance between target (jammer) and radarRt/km(self-defense jamming)
100
Transmit power of jammerPj/W
70
Jammer sampling pulse widthτ/μs
1
Jammer repeat sampling intervalTs/μs
2 and 10The influences of the jamming repeat sampling intervalTs and the duty ratio τ/Ts on the jamming performance of ISRJ have been studied in [22]. The distance between the false targets and the amplitudes of the false targets are determined by the jamming repeat sampling interval Ts and the duty ratio τ/Ts, respectively. Higher Ts produces false targets intensively distributed, and smaller Ts generates false targets sparsely distributed. There are only three to five strong false targets while the amplitudes of the high-order false targets attenuate rapidly under the high duty ratio conditions. A large number of false targets with slowly attenuated amplitudes and more serious power loss are generated under the small duty ratio conditions. Assume that the jamming sampling duration τ=1μs is invariable in the following simulations.WhenTs=2μs, that is, the duty ratio is 50%, the theoretical analysis and the simulation result shown in Figure 4 reveal that there exist three strong lifelike false targets resulting in a confusion that the radar system cannot judge which one is the true target. The x-axis in the simulation figures is represented by the relative distance to reference point as a result of the proportional relationship with the single-frequency target echo signal after dechirping.Figure 4
The compression result before ECCM (Ts=2μs).Figure5 shows the magnitude-frequency responses of the filters before and after windowing in frequency domain. It can be seen from Figure 5 that the sidelobes of the filtering function are suppressed by a wide margin through windowing in frequency domain.Figure 5
The band-pass filtering function (Ts=2μs).The suppression result with the filter not windowed is shown in Figure6. It can be seen that there still exist one strong false target for high sidelobes resulting in a confusion to the radar system. Figure 7 shows the suppression result with the windowed filter. From Figure 7, we can see that all false targets are removed for the extremely low sidelobes of the windowed filter.Figure 6
The jamming suppression result usingH(f) (Ts=2μs).Figure 7
The jamming suppression result usingHa(f) (Ts=2μs).WhenTs=10μs, that is, the duty ratio is 10%, the theoretical analysis and the simulation result shown in Figure 8 illustrate that the intensively distributed false targets are generated resulting in a confusion to the radar system.Figure 8
The compression result before ECCM (Ts=10μs).Similarly, Figure9 shows the magnitude-frequency response of the filters before and after windowing in frequency domain and has similar interpretation to Figure 5.Figure 9
The band-pass filtering function (Ts=10μs).The suppression result with the filtering function not windowed is shown in Figure10. It can be seen that there are still many false targets left surrounding the true target for high sidelobes. Figure 11 shows the suppression result with the windowed filtering function. From Figure 11, we can see that the suppression efficiency is obviously good for the extremely low sidelobes of the windowed filtering function.Figure 10
The jamming suppression result usingH(f) (Ts=10μs).Figure 11
The jamming suppression result usingHa(f) (Ts=10μs).Signal-to-noise ratio (SNR) and jamming-to-signal ratio (JSR) will affect the validity of the proposed ECCM method. SNR and JSR are defined as(28)SNR=10log10Ar2σ2,JSR=10log10Aj2Ar2,where σ2 represents the power of noise. Then under different JSR, assume that the peak value of the compression result after executing the ECCM scheme proposed above is the target and make the Monte-Carlo simulation 1000 times in every SNR. When the noise power is increasing gradually, Figure 12 shows the detection probability of the true target.Figure 12
Detection probability of the true target with different SNR under different JSR.From Figure12, it can be seen that the larger JSR, the larger detection probability of the true target. Under the condition of JSR=10dB,JSR=15dB and JSR=20dB, the true target can be detected with almost 100% probability when SNR>0dB,SNR>-5dB and SNR>-10dB, respectively. The larger JSR is beneficial to extract the signal segments not disturbed by the ISRJ signal and construct the band-pass filter exactly.


## 5. Conclusions

Based on the discontinuity characteristics of the ISRJ signal in TF domain and the great differences in time-energy domain, a simple and effective ECCM scheme against the ISRJ is proposed in this paper. Firstly, a simple energy function is defined and used to extract the target echo signal segments without jamming. It is not necessary to perform the complicated time-frequency decomposition. Then, the band-pass filtering function with low sidelobes is obtained by doing the Fourier transformation directly on the extracted signal and windowing in frequency domain. The simulation results show that the proposed method is effective not only in multiple lifelike false targets, but also in intensively distributed false targets generated through setting different jamming repeat sampling intervals. Moreover, this method is effective even with low SNR under certain JSR, and we can acquire the detection probability of almost 100% to the true target.



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*Source: 6759169-2017-03-30.xml*