# A Novel Waveform Design Method for Shift-Frequency Jamming Confirmation

Authors: Chang Zhou; Zhen-Bo Zhu; Zi-Yue Tang
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

Shift-frequency jamming is generally used to form range false targets for ground-based early-warning radar systems; the frequency shift value of such interference is larger than the Doppler shift value of the moving target, and the key element to suppress the shift-frequency jamming is the frequency shift value estimation. However, in the low- or medium-pulse repetition frequency (PRF) mode, it is challenging to estimate the accurate frequency shift due to the velocity ambiguity. To solve this problem, a novel sparse Doppler-sensitive waveform is designed based on the ambiguity function theory, where the basic idea is to design a waveform sensitive to a specific Doppler but insensitive to other Dopplers; therefore, this waveform can recognize the specific Doppler of the target unambiguously. To apply the designed waveform in practice, the detection and estimation processing flow is provided based on the waveform diversity technique and the family of the sparse Doppler-sensitive waveforms. Simulation experiments are presented to validate the efficiency of the proposed method, and we conclude that the advantage of this method is that it can be used to confirm the specific Doppler of the target unambiguously with few pulses even under the condition of a low PRF.


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## Body

## 1. Introduction

Linear frequency-modulated (LFM) signals are widely used in radar systems due to their large time-bandwidth product and large Doppler tolerance [1]. Taking advantage of the range-Doppler coupling characters of LFM signals, shift-frequency jamming [2, 3] has been extensively studied, as this approach can form false targets ahead of true targets by adjusting the parameters of the frequency shift even if the radar adopts the working mode of the frequency agility or pulse repetition frequency (PRF) agility. Since the frequency shift modulation of the jammer is similar to the Doppler modulation of the moving target, it is even more challenging to recognize shift-frequency jamming. However, the frequency shift amount of the jamming is much larger than the Doppler of the moving target in ground-based early-warning radar. For example, when the L-band radar operating wavelength λ≥15 cm, the speed of a conventional aircraft is less than Mach (Ma) 3; thus, the Doppler frequency is less than 5.4 kHz, while the L-band signal bandwidth is several MHz. To form a false target with at least one range cell offset, the minimum frequency shift should be many tens of kHz, which can be considered much larger than the Doppler of moving targets. Therefore, accurate estimation of the frequency shift amount or Doppler frequency can enable shift-frequency interference recognition and suppression in ground-based early-warning radar. However, for low- and medium-PRF radars, it is challenging to obtain the actual Doppler because of the velocity ambiguity. To address this challenge, the PRF agility technology is typically used for velocity ambiguity resolution [1] in practice, while the existing ambiguity resolution algorithm has the following shortcomings [4, 5]: (1) the algorithm requires a relatively high Doppler resolution in each PRF even in the presence of noise pollution and quantization errors, (2) the maximum unambiguous Doppler improvement is limited, and (3) the computation complexity is expensive, particularly in the case of multiple moving targets. Therefore, this study attempts to achieve the frequency shift estimation from the perspective of waveform design.Many waveform design studies have focused on the low autocorrelation sidelobe [6–10], which can improve the detection performance for weak targets. The research hotspots have developed from the design of binary-phase sequences to polyphase sequences and from fixed-length sequences to arbitrary-length sequences. In recent years, related algorithms can produce unimodular sequences of the length N=106 or even longer, with favourable autocorrelation properties [8]. In addition, several new applications, such as the sparse frequency waveform [11, 12], weighted sidelobe waveform [8], and orthogonal signal [6], have been developed. The autocorrelation of the waveform is equivalent to the zero-Doppler cut of the ambiguity function [13]; despite the fact that a satisfactory autocorrelation sidelobe waveform cannot guarantee the detection performance of moving targets, the design methods can be easily generalized for a proper ambiguity function cut design.To detect a moving target, other possible Doppler cuts of the ambiguity function must be considered. In other words, we must consider the ambiguity function for a nonzero Doppler. The studies of [14–16] focused on synthesizing an arbitrary desired ambiguity function. Although extensive work has been performed, there is no universal method to solve this problem because it is challenging to determine whether the desired ambiguity function can be synthesized; in addition, the process is time-consuming.Therefore, to reduce complexity, some scholars have relied on satisfying the partial constraints, such as ensuring a clear area near the origin [6] and minimizing the integral sidelobe level (ISL) in a certain area [17]. More specifically, the output response in certain range-Doppler areas, which cover the interferences, should be as small as possible, whereas the output response in certain range-Doppler areas with targets must ensure a level that is as high as possible [18]. In [19–21], the authors addressed the waveform design in the presence of coloured Gaussian disturbance noise and solved the problem through semidefinite relaxation to achieve the optimal detection performance. For the unknown Doppler, an algorithm to guarantee that the minimum Doppler matches the output maximum has been proposed [22, 23], but the signal bandwidth and ISL were not considered. The unimodular quadratic programme (UQP) and computational approaches to tackle the UQP were summarized in [24]. In [25], the clutter model was established, and a slow-time ambiguity function design was performed, which is a more intuitive way to implement the moving target detection (MTD) response. This problem has also been solved based on the maximum-block-improvement (MBI) method [26], and the majorization-minimization (MM) method was used in [18] to solve this problem and obtain improved performance. However, the ambiguity design method still addresses the velocity ambiguity problem, and few references consider the bandwidth of the coding signal, which further limits the application in radar.In the present study, a sparse Doppler-sensitive waveform is designed that can be used to confirm the specific Doppler of the target unambiguously. By designing a different waveform with a different specific Doppler to confirm the different target Doppler, the large shift-frequency interference can be identified and suppressed.The remainder of this paper is organized as follows. Section2 presents the problem statement. Section 3 examines the signal spectrum based on the optimal detection criterion for a specific Doppler, and the waveform design algorithm is discussed. Subsequently, the method of Doppler confirmation processing is proposed based on the designed waveform in Section 4. The simulation experiments are presented in Section 5. Finally, Section 6 presents the conclusions.


## 2. Problem Statement

The ambiguity function is the most intuitive description of the signal output performance for different Dopplers. To facilitate the design of the coding signal, the discrete ambiguity function is expressed as [1]
(1)χdn,ξk=∑mxmx∗m−nexpjξkm=∑ωX∗ωXω−ξkexpjωn,where n denotes the discrete delay series, ξk is the discrete Doppler frequency, xm is the discrete coded signal, and Xω represents the discrete Fourier transform (DFT) of xm.With regard to the zero-Doppler cutχτ,0 of N arbitrary complex coded signals, the maximum value is N and maximum position is χ0,0. For the other Doppler cuts, however, the shape is unknown. Assuming that the specific Doppler is ξK, to design a waveform that is sensitive to a specific Doppler but insensitive to the others, the ambiguity figure must satisfy three shape constraints, as shown in Figure 1.
(1)
The specific Doppler cut (whenξk=ξK) should exist as a clear peak value χq,ξK as the mainlobe, where q∈−N−1,−N−2,…,N−1 represents the position of the mainlobe (peak value) in this cut, and this cut can be set according to the actual requirements.
(2)
The sidelobe of the specific Doppler cut (whenξk=ξK and n≠q) should be as small as possible.
(3)
Other Doppler cuts (whenξk≠ξK and ξk≠ξ0) should not have significant peak values.Figure 1
Shape constraints of various Doppler cuts in the ambiguity figure.


## 3. Proposed Method of Waveform Design

Based on the shape constraints for the ambiguity function of the sparse Doppler-sensitive waveform described in Section2, the detailed waveform design method is discussed in this part. In Section 3.1, the sidelobe characteristics of the specific Doppler cut are analysed, and the relationship between the sidelobe of the ambiguity figure cut and the signal spectrum is deduced. Section 3.2 further builds the overall spectrum representation of the specific Doppler cut, and the signal spectrum is solved under the optimal detection criterion for the specific Doppler cut. In the end, combining the deduced signal spectrum with the other Doppler cut constraints, the constant modulus waveform is designed in Section 3.3.


### 3.1. Analysis of the Sidelobe Characteristics of the Specific Doppler Cut (Whenξk=ξK)

It is customary to discuss the sidelobe when describing the correlation properties of the waveform. For a specificξK, the discrete ambiguity function cut χdn,ξK is equivalent to the cross-correlation [27] of xnexpjξKn and xn, which is generally expressed as rn,ξK. To conform to the customary representation, we define the ISL on the specific Doppler ξK (where q represents the position of the mainlobe in this cut) as
(2)ISLξK=∑n=−N−1n≠qN−1χd2n,ξK=∑n=−N−1n≠qN−1r2n,ξK.It can be verified that(3)ISLξK=12N∑p=12N∑n=−N−1N−1rn,ξKexp−jωpn−Lexp−jωpq2−rq,ξK−L2,where L is a complex number. Detailed derivations are given in Appendix A.SinceL could be an arbitrary complex number, to simplify the above formula, let L=rq,ξK; substituting this term into (3) leads to
(4)ISLξK=12N∑p=12N∑n=−N−1N−1rn,ξKexp−jωpn−rq,ξKexp−jωpq2.For example, ifξK=0, q=0, and L=rq,ξK=N, then
(5)ISLξK=12N∑p=12N∑n=−N−1N−1rn,0exp−jωpn−N2.Equation (5) has been widely used to design the autocorrelation waveform [8] with low ISL. The derivation of this article has relatively extensive adaptability.According to the relationship between the power spectrum and cross-correlation function, the following equation holds, and more derivations are obtained in AppendixB.
(6)∑n=−N−1N−1rn,ξKexp−jωpn=∑m=1NxmexpjξKmexp−jωpm⋅∑m=1Nxmexp−jωpm∗,where ωp=2π/2N⋅p, p=1,2,…,2N, and ξK=2π/N⋅K, K=1,…,N.DefineXzerofillωp=∑m=1Nxmexp−jωpm; then, (6) can be expressed as
(7)∑n=−N−1N−1rn,ξKexp−jωpn=Xzerofillωp−ξKXzerofill∗ωp.Substituting (7) into (4), we obtain
(8)ISLξK=12N∑p=12NXzerofillωp−ξKXzerofill∗ωp−rq,ξKexp−jωpq2.Equation (8) is applicable to any ambiguity figure cut χdn,ξK, which is also the basis of the following deduction.


### 3.2. Analysis of the Signal Spectrum under the Optimal Detection Criterion for the Specific Doppler Cut (Whenξk=ξK)

The purpose of the waveform design is to achieve effective detection. To improve the detection performance of the specific DopplerξK, we need to ensure that (1) χdq,ξK is as large as possible and (2) ISLξk should be as small as possible.Notably, sinceXzerofillωp=∑m=1Nxmexp−jωpm, Xzerofillωp is the DFT of x1,x2,…,xN,0,…,0⏟N. After adding zeros, the shape of the ambiguity figure remains the same, and according to Parseval’s theorem, we can obtain
(9)χdq,ξK=rq,ξK=∑p=1NX∗ωpXωp−ξKexpjωpq=12⋅∑p=12NXzerofill∗ωpXzerofillωp−ξKexpjωpq.From (8) and (9), we find that the key to design the sparse Doppler-sensitive waveform for the specific Doppler ξK is Xzerofillωp−ξKXzerofill∗ωp. For the convenience of calculation, the complex envelope of Xzerofillωp−ξKXzerofill∗ωp is designated Aωp,ξK, and the phase is θωp,ξK, that is, Xzerofillωp−ξKXzerofill∗ωp=Aωp,ξKexpjθωp,ξK. Simultaneously, the complex envelope of Xzerofillωp is designated Aωp and the phase isθωp, that is, Xzerofillωp=Aωpexpjθωp; clearly, Aωp,ξK≥0 and Aωp≥0.To fully characterize the signal spectrum after the Doppler shift, the bandwidth of the designed signalB is chosen as half of the sampling frequency fs. Thus, the signal spectrum does not alias after the frequency shift. If the total point number is 2N, we suppose that Aωp=0 for p=N+1,…,2N in this study.Therefore, the objective function to obtain the optimal detecting performance for the DopplerξK is
(10)maxP1Aωp,ξK,θωp,ξK=χdq,ξK2=∑p=12NAωp,ξKexpjθωp,ξKexpjωpq2,minP2Aωp,ξK,θωp,ξK=ISLξK=12N∑p=12NAωp,ξKexpjθωp,ξK−12∑p~=12NAωp~,ξKexpjθωp~,ξKexpjωp~qexp−jωpq2,s.t.θωp,ξK=θωp−K−θωp,Aωp,ξK=Aωp⋅Aωp−KAωp=0,p=N+1,…,2N,∑p=12NAωp2=N2.Equation (10) is a nonlinear biobjective function for which it is challenging to obtain an analytic solution. First, we adopt the method of weighted sums to solve the problem, and the total objective function is
(11)maxPAωp,ξK,θωp,ξK=P1Aωp,ξK,θωp,ξK+λP2Aωp,ξK,θωp,ξK,where λ is the weight coefficient and the constraints remain the same. In (11), P1’s weight should be large because if no target of the specific Doppler can be detected results from the low detection probability, even the lowest sidelobe is useless.Then, we design theθωp,ξK and Aωp separately using the iterative algorithm for the optimal solution. When Aωp is fixed, then Aωp,ξK is fixed according to Aωp,ξK=Aωp⋅Aωp−K, and a genetic algorithm [28] is employed to quantify the phase and identify the optimal θωp,ξK. While θωp,ξK is fixed, the function degenerates to a real optimization problem with nonlinear multiconstraints, and this problem can be solved through sequential quadratic programming [29]. If MATLAB (a commercial mathematics software produced by American MathWorks company) is available, then Aωp can be solved using the MATLAB function “fmincon” [30].The iterative algorithm is described as follows.Algorithm 1: Signal spectrum under the condition of optimal detection criteria for a specific Doppler ξK.
Requirements: sequence lengthN, weight coefficient λ
1. Initialize:θωp,ξK=−ωpq.
2. Repeat.
3. Substituteθωp,ξK to the total objective function (12).
4. ComputeAωp through sequential quadratic programming.
5. ComputeAωp,ξK according to Aωp,ξK=Aωp⋅Aωp−K.
6. Computeθωp,ξK using the genetic algorithm.
7. Continue until convergence.


### 3.3. Waveform Design Method

After discussing the signal spectrum for the specific DopplerξK in Section 3.2, two further steps are required to complete the design of the signal: (1) designing the constant modulus waveform approaching the signal spectrum and (2) establishing the constraints necessary to avoid other Doppler cuts existing at significant peak values.From Section3.2, for the given specific Doppler ξK and mainlobe position q, we obtain θωp,ξK and Aωp. While θωp,ξK=θωp−K−θωp according to (11), knowledge of the first K phases is required to obtain the complete θωp.For step (1), letF be the unit DFT matrix as follows:
(12)F=12Nf0f1⋯f2N−1,where f denotes the vector value functions, given by
(13)fp=expj2π2Np⋅0⋯expj2π2Np⋅2N−1T,where ⋅T denotes the transpose for matrices/vectors. We define the unimodular signal vector X~ and the discrete spectrum vector D as follows:
(14)X~=x1x2⋮xN0⋮02N,D=Xω1Xω2⋮⋮Xω2N=Aω1expjθω1Aω2expjθω2⋮⋮Aω2Nexpjθω2N.The optimal spectral constraint problem is(15)minPX~,D=FHX~−D2,s.t.xn=1,n=1,2,…,N,expjθωpexp−jθωp−K=expjθωp,ξK,p=K+1,…,N,where ⋅H and ⋅ denote the conjugate transpose and Frobenius norm for matrices/vectors, respectively.For step (2), to realize the other Dopplers’ insensitivity, since the other Doppler cuts can be regarded as the sidelobe of the entire ambiguity figure, the multicyclic original (multi-CAO) algorithm [6], which is typically used to minimize the discrete ambiguity figure sidelobe of the partial interested area, is introduced to design the signal.Define vectorX=X1⋯XPN+Q−1×QK−1, where Q is the number of interested range cell number and K is the specific Doppler:
(16)Xmxm10⋮⋱⋮xm1xmN⋮⋱⋮0xmNN+Q−1×Q,m=1,2,…,K−1,where xmn=xnej2πnm−1/N, m=1,2,…,K−1, and n=1,2,…,N. The number of Doppler cells of interest is K−1. The algorithm is described in detail elsewhere [6]. The sidelobe constraint is
(17)minPX,U=X−NU2,s.t.xn=1,n=1,2,…,N,UHU=I.Adding (15) and (17) with their corresponding weights, the total objective function is
(18)minPX~,X,U,D=FHX~−D2+ηX−NU2,s.t.xn=1,n=1,2,…,N,expjθωpexp−jθωp−K=expjθωp,ξK,p=K+1,…,N,UHU=I,where η is the weight coefficient.The waveform design algorithm is described as follows.Algorithm 2: Sparse Doppler-sensitive waveform design algorithm with a constant modulus constraint.
Requirements: sequence lengthN, weight coefficient η
1. Initialize:xnn=1N is a randomly generated unimodular sequence; configure X~ and X.
2. Repeat.
3. For a fixedX~, compute the first K complex values of θωp according to θωp,ξK=θωp−K−θωp and FHX~−D.
4. For a fixedX, the minimizer U is given by U=U2U1H, where matrices U1 and U2 are taken from the economic SVD of XH, which is XH=U1∑U2H.
5. For a fixedD, xn=expjargfTnD.
6. For a fixedU, the criterion can be written as
X−NU2=∑n=1N∑l=1K−1Qμnlxn−vnl2=const−2∑n=1NRe∑l=1K−1Qμnl∗vnlx∗n,
where const is a constant that does not depend onxn; μnl are given by the elements of X that contain xn
μn1⋯μn,K−1Q=1⋯1⏟K−1ej2πn/N⋯ej2πn/N⏟K−1⋯ej2πnQ−1/N⋯ej2πnQ−1/N⏟K−11×K−1Q.
vnl are the elements of NU, whose positions are identical to those of μnl in X. The minimizer xn is
xn=expjarg∑l=1K−1Qμnl∗vnl,n=1,…,2N.
7. Summarize the results in step 5 and step 6; the minimizerxn is given by
xn=expjargηfTD+1−η∑l=1K−1Qμnl∗vnl.
8. Continue until convergence.


## 3.1. Analysis of the Sidelobe Characteristics of the Specific Doppler Cut (Whenξk=ξK)

It is customary to discuss the sidelobe when describing the correlation properties of the waveform. For a specificξK, the discrete ambiguity function cut χdn,ξK is equivalent to the cross-correlation [27] of xnexpjξKn and xn, which is generally expressed as rn,ξK. To conform to the customary representation, we define the ISL on the specific Doppler ξK (where q represents the position of the mainlobe in this cut) as
(2)ISLξK=∑n=−N−1n≠qN−1χd2n,ξK=∑n=−N−1n≠qN−1r2n,ξK.It can be verified that(3)ISLξK=12N∑p=12N∑n=−N−1N−1rn,ξKexp−jωpn−Lexp−jωpq2−rq,ξK−L2,where L is a complex number. Detailed derivations are given in Appendix A.SinceL could be an arbitrary complex number, to simplify the above formula, let L=rq,ξK; substituting this term into (3) leads to
(4)ISLξK=12N∑p=12N∑n=−N−1N−1rn,ξKexp−jωpn−rq,ξKexp−jωpq2.For example, ifξK=0, q=0, and L=rq,ξK=N, then
(5)ISLξK=12N∑p=12N∑n=−N−1N−1rn,0exp−jωpn−N2.Equation (5) has been widely used to design the autocorrelation waveform [8] with low ISL. The derivation of this article has relatively extensive adaptability.According to the relationship between the power spectrum and cross-correlation function, the following equation holds, and more derivations are obtained in AppendixB.
(6)∑n=−N−1N−1rn,ξKexp−jωpn=∑m=1NxmexpjξKmexp−jωpm⋅∑m=1Nxmexp−jωpm∗,where ωp=2π/2N⋅p, p=1,2,…,2N, and ξK=2π/N⋅K, K=1,…,N.DefineXzerofillωp=∑m=1Nxmexp−jωpm; then, (6) can be expressed as
(7)∑n=−N−1N−1rn,ξKexp−jωpn=Xzerofillωp−ξKXzerofill∗ωp.Substituting (7) into (4), we obtain
(8)ISLξK=12N∑p=12NXzerofillωp−ξKXzerofill∗ωp−rq,ξKexp−jωpq2.Equation (8) is applicable to any ambiguity figure cut χdn,ξK, which is also the basis of the following deduction.


## 3.2. Analysis of the Signal Spectrum under the Optimal Detection Criterion for the Specific Doppler Cut (Whenξk=ξK)

The purpose of the waveform design is to achieve effective detection. To improve the detection performance of the specific DopplerξK, we need to ensure that (1) χdq,ξK is as large as possible and (2) ISLξk should be as small as possible.Notably, sinceXzerofillωp=∑m=1Nxmexp−jωpm, Xzerofillωp is the DFT of x1,x2,…,xN,0,…,0⏟N. After adding zeros, the shape of the ambiguity figure remains the same, and according to Parseval’s theorem, we can obtain
(9)χdq,ξK=rq,ξK=∑p=1NX∗ωpXωp−ξKexpjωpq=12⋅∑p=12NXzerofill∗ωpXzerofillωp−ξKexpjωpq.From (8) and (9), we find that the key to design the sparse Doppler-sensitive waveform for the specific Doppler ξK is Xzerofillωp−ξKXzerofill∗ωp. For the convenience of calculation, the complex envelope of Xzerofillωp−ξKXzerofill∗ωp is designated Aωp,ξK, and the phase is θωp,ξK, that is, Xzerofillωp−ξKXzerofill∗ωp=Aωp,ξKexpjθωp,ξK. Simultaneously, the complex envelope of Xzerofillωp is designated Aωp and the phase isθωp, that is, Xzerofillωp=Aωpexpjθωp; clearly, Aωp,ξK≥0 and Aωp≥0.To fully characterize the signal spectrum after the Doppler shift, the bandwidth of the designed signalB is chosen as half of the sampling frequency fs. Thus, the signal spectrum does not alias after the frequency shift. If the total point number is 2N, we suppose that Aωp=0 for p=N+1,…,2N in this study.Therefore, the objective function to obtain the optimal detecting performance for the DopplerξK is
(10)maxP1Aωp,ξK,θωp,ξK=χdq,ξK2=∑p=12NAωp,ξKexpjθωp,ξKexpjωpq2,minP2Aωp,ξK,θωp,ξK=ISLξK=12N∑p=12NAωp,ξKexpjθωp,ξK−12∑p~=12NAωp~,ξKexpjθωp~,ξKexpjωp~qexp−jωpq2,s.t.θωp,ξK=θωp−K−θωp,Aωp,ξK=Aωp⋅Aωp−KAωp=0,p=N+1,…,2N,∑p=12NAωp2=N2.Equation (10) is a nonlinear biobjective function for which it is challenging to obtain an analytic solution. First, we adopt the method of weighted sums to solve the problem, and the total objective function is
(11)maxPAωp,ξK,θωp,ξK=P1Aωp,ξK,θωp,ξK+λP2Aωp,ξK,θωp,ξK,where λ is the weight coefficient and the constraints remain the same. In (11), P1’s weight should be large because if no target of the specific Doppler can be detected results from the low detection probability, even the lowest sidelobe is useless.Then, we design theθωp,ξK and Aωp separately using the iterative algorithm for the optimal solution. When Aωp is fixed, then Aωp,ξK is fixed according to Aωp,ξK=Aωp⋅Aωp−K, and a genetic algorithm [28] is employed to quantify the phase and identify the optimal θωp,ξK. While θωp,ξK is fixed, the function degenerates to a real optimization problem with nonlinear multiconstraints, and this problem can be solved through sequential quadratic programming [29]. If MATLAB (a commercial mathematics software produced by American MathWorks company) is available, then Aωp can be solved using the MATLAB function “fmincon” [30].The iterative algorithm is described as follows.Algorithm 1: Signal spectrum under the condition of optimal detection criteria for a specific Doppler ξK.
Requirements: sequence lengthN, weight coefficient λ
1. Initialize:θωp,ξK=−ωpq.
2. Repeat.
3. Substituteθωp,ξK to the total objective function (12).
4. ComputeAωp through sequential quadratic programming.
5. ComputeAωp,ξK according to Aωp,ξK=Aωp⋅Aωp−K.
6. Computeθωp,ξK using the genetic algorithm.
7. Continue until convergence.


## 3.3. Waveform Design Method

After discussing the signal spectrum for the specific DopplerξK in Section 3.2, two further steps are required to complete the design of the signal: (1) designing the constant modulus waveform approaching the signal spectrum and (2) establishing the constraints necessary to avoid other Doppler cuts existing at significant peak values.From Section3.2, for the given specific Doppler ξK and mainlobe position q, we obtain θωp,ξK and Aωp. While θωp,ξK=θωp−K−θωp according to (11), knowledge of the first K phases is required to obtain the complete θωp.For step (1), letF be the unit DFT matrix as follows:
(12)F=12Nf0f1⋯f2N−1,where f denotes the vector value functions, given by
(13)fp=expj2π2Np⋅0⋯expj2π2Np⋅2N−1T,where ⋅T denotes the transpose for matrices/vectors. We define the unimodular signal vector X~ and the discrete spectrum vector D as follows:
(14)X~=x1x2⋮xN0⋮02N,D=Xω1Xω2⋮⋮Xω2N=Aω1expjθω1Aω2expjθω2⋮⋮Aω2Nexpjθω2N.The optimal spectral constraint problem is(15)minPX~,D=FHX~−D2,s.t.xn=1,n=1,2,…,N,expjθωpexp−jθωp−K=expjθωp,ξK,p=K+1,…,N,where ⋅H and ⋅ denote the conjugate transpose and Frobenius norm for matrices/vectors, respectively.For step (2), to realize the other Dopplers’ insensitivity, since the other Doppler cuts can be regarded as the sidelobe of the entire ambiguity figure, the multicyclic original (multi-CAO) algorithm [6], which is typically used to minimize the discrete ambiguity figure sidelobe of the partial interested area, is introduced to design the signal.Define vectorX=X1⋯XPN+Q−1×QK−1, where Q is the number of interested range cell number and K is the specific Doppler:
(16)Xmxm10⋮⋱⋮xm1xmN⋮⋱⋮0xmNN+Q−1×Q,m=1,2,…,K−1,where xmn=xnej2πnm−1/N, m=1,2,…,K−1, and n=1,2,…,N. The number of Doppler cells of interest is K−1. The algorithm is described in detail elsewhere [6]. The sidelobe constraint is
(17)minPX,U=X−NU2,s.t.xn=1,n=1,2,…,N,UHU=I.Adding (15) and (17) with their corresponding weights, the total objective function is
(18)minPX~,X,U,D=FHX~−D2+ηX−NU2,s.t.xn=1,n=1,2,…,N,expjθωpexp−jθωp−K=expjθωp,ξK,p=K+1,…,N,UHU=I,where η is the weight coefficient.The waveform design algorithm is described as follows.Algorithm 2: Sparse Doppler-sensitive waveform design algorithm with a constant modulus constraint.
Requirements: sequence lengthN, weight coefficient η
1. Initialize:xnn=1N is a randomly generated unimodular sequence; configure X~ and X.
2. Repeat.
3. For a fixedX~, compute the first K complex values of θωp according to θωp,ξK=θωp−K−θωp and FHX~−D.
4. For a fixedX, the minimizer U is given by U=U2U1H, where matrices U1 and U2 are taken from the economic SVD of XH, which is XH=U1∑U2H.
5. For a fixedD, xn=expjargfTnD.
6. For a fixedU, the criterion can be written as
X−NU2=∑n=1N∑l=1K−1Qμnlxn−vnl2=const−2∑n=1NRe∑l=1K−1Qμnl∗vnlx∗n,
where const is a constant that does not depend onxn; μnl are given by the elements of X that contain xn
μn1⋯μn,K−1Q=1⋯1⏟K−1ej2πn/N⋯ej2πn/N⏟K−1⋯ej2πnQ−1/N⋯ej2πnQ−1/N⏟K−11×K−1Q.
vnl are the elements of NU, whose positions are identical to those of μnl in X. The minimizer xn is
xn=expjarg∑l=1K−1Qμnl∗vnl,n=1,…,2N.
7. Summarize the results in step 5 and step 6; the minimizerxn is given by
xn=expjargηfTD+1−η∑l=1K−1Qμnl∗vnl.
8. Continue until convergence.


## 4. Application Method of the Specific Doppler Confirmation

### 4.1. Waveform Property and Performance Evaluation

To enable quantitative analysis of the designed waveform property, we consider the main characteristics of the waveform. In Section3.2, the objective function P1 is configured with a larger weight since a high peak value level for the specific Doppler cut should be guaranteed to ensure the matched filter output level; thus, the properties of the optimal signal solution of P1 can represent the main characteristics of the waveform. The optimal solution of P1 is θωp,ξK=−ωpq, which can be written as expjθωpexp−jθωp−K=expjωpq; letting Aωp=1, the properties of the waveform are discussed as follows:
(1)
If the maximum position isq for the specific Doppler ξK, then χdq,ξK is the maximum value for this Doppler cut. Under this condition, we can derive that χdlq,ξlK is also the maximum value for the Doppler ξlK, where l denotes any positive integer. More details are given in Appendix C.
(2)
The possible maximum in the ambiguity figure may appear inn,k, which satisfies the formula Kn−kq=±d⋅2N, which can be easily proven using f3 and f4 in Appendix D. Figure 2 displays the shape of f1, f2, f3, and f4 in Appendix D, where N=50, K=5, q=1, and the first K phases of the frequency are random. The conclusion can also be observed from Figure 1.
(3)
For all pointsn,k satisfying Kn−kq=±d⋅2N, the amplitude χdq,ξK is modulated by the first K phases, and the different first K phases may lead to different ambiguity figures. Figure 3 shows the ambiguity function (D.5) in Appendix D, where N=50, K=5, and q=1. Figures 3(a) and 3(b) show the case in which the first K phases of the frequency are random, whereas Figures 3(c) and 3(d) show the case in which the first K phases of the frequency are equal. Thus, the results vary significantly according to the first K phases of the frequency.Figure 2
f1, f2, f3, and f4 in Appendix D (N=50, and the first K phases of the frequency are random).
(a)
(b)
(c)
(d)Figure 3
Ambiguity function (N=50). (a, b) The first K phases of the frequency are random; (c, d) the first K phases of the frequency are equal.
(a)
(b)
(c)
(d)In addition, Figures3(c) and 3(d) are similar to the LFM signal, which can be explained by the fact that the LFM can be considered a form of designed signal in this study, where K=1 and q=2. In contrast, if K is sufficiently large, then the designed waveform becomes a random sequence phase encoding signal with an “acicular” ambiguity figure. In summary, the sparse Doppler-sensitive waveform is a waveform family that includes the LFM signal and random sequence phase encoding signal.In this section, we also discuss the Doppler resolutionΔf of the technique. Suppose that the signal bandwidth is B, the sampling frequency is fs, and the time duration is T in each pulse repetition period. The number of sampling points is N, where N=T/fs; therefore, Δf=fs/N=fs/T⋅fs=1/T. The maximum unambiguous Doppler increases from the pulse repetition frequency fr to the sampling frequency fs; thus, this parameter can be used to recognize the large shift-frequency interference.


### 4.2. Application Method of Waveform Diversity for Auxiliary Doppler Confirmation

Due to its inherent cyclicality, which is discussed in depth in Section4.1, the designed signal cannot be directly used in signal detection on its own. Thus, the signal here is combined with the LFM signal to measure the specific Doppler target simultaneously, as shown in the flow chart of the proposed application technique in Figure 4. In this technique, two types of radar working modes are given: normal mode and measuring mode.Figure 4
Flow chart of the proposed application technique.In normal mode, the LFM signal is used to detect the target individually in the same manner as other regular radar measurements. When the specific DopplerξK requires confirmation, the signal can be switched to measuring mode. In this mode, the LFM signal and signals I and II with parameters N,B,K,q1 and N,B,K,q2, respectively, are transmitted in order, where N and B are the sampling points and bandwidth of the transmitting pulse, respectively, which can be identical to the LFM to confuse the enemy jammer. K depends on the specific Doppler ξK, and q1 and q2 denote the maximum position of Doppler ξK cut for signal I and signal II, respectively. Pf1 denotes the constant false alarm rate (CFAR) of transmitting the LFM signal, and Pf2 is the CFAR of transmitting signal I and signal II.For a specific DopplerξK, q1 determines the relative position of the target of Doppler ξK between the LFM signal and signal I. Similarly, q2 determines the relative position of the target of Doppler ξK between the LFM signal and signal II. If the received signals from the transmitting signals (i.e., the LFM signal and signals I and II) satisfy the relative range offset for the specific Doppler ξK, then the Doppler of the target can be confirmed as ξK. In short, this technique confirms the Doppler by comparing the range offsets of various waveforms.Further, the reason to use two additional signals to complete the measurement is that the sidelobe of the designed waveform is larger than the LFM signal; thus,Pf2 is made to be relatively large to ensure high detection probability, and the overall false alarm can be reduced through the joint detection.


## 4.1. Waveform Property and Performance Evaluation

To enable quantitative analysis of the designed waveform property, we consider the main characteristics of the waveform. In Section3.2, the objective function P1 is configured with a larger weight since a high peak value level for the specific Doppler cut should be guaranteed to ensure the matched filter output level; thus, the properties of the optimal signal solution of P1 can represent the main characteristics of the waveform. The optimal solution of P1 is θωp,ξK=−ωpq, which can be written as expjθωpexp−jθωp−K=expjωpq; letting Aωp=1, the properties of the waveform are discussed as follows:
(1)
If the maximum position isq for the specific Doppler ξK, then χdq,ξK is the maximum value for this Doppler cut. Under this condition, we can derive that χdlq,ξlK is also the maximum value for the Doppler ξlK, where l denotes any positive integer. More details are given in Appendix C.
(2)
The possible maximum in the ambiguity figure may appear inn,k, which satisfies the formula Kn−kq=±d⋅2N, which can be easily proven using f3 and f4 in Appendix D. Figure 2 displays the shape of f1, f2, f3, and f4 in Appendix D, where N=50, K=5, q=1, and the first K phases of the frequency are random. The conclusion can also be observed from Figure 1.
(3)
For all pointsn,k satisfying Kn−kq=±d⋅2N, the amplitude χdq,ξK is modulated by the first K phases, and the different first K phases may lead to different ambiguity figures. Figure 3 shows the ambiguity function (D.5) in Appendix D, where N=50, K=5, and q=1. Figures 3(a) and 3(b) show the case in which the first K phases of the frequency are random, whereas Figures 3(c) and 3(d) show the case in which the first K phases of the frequency are equal. Thus, the results vary significantly according to the first K phases of the frequency.Figure 2
f1, f2, f3, and f4 in Appendix D (N=50, and the first K phases of the frequency are random).
(a)
(b)
(c)
(d)Figure 3
Ambiguity function (N=50). (a, b) The first K phases of the frequency are random; (c, d) the first K phases of the frequency are equal.
(a)
(b)
(c)
(d)In addition, Figures3(c) and 3(d) are similar to the LFM signal, which can be explained by the fact that the LFM can be considered a form of designed signal in this study, where K=1 and q=2. In contrast, if K is sufficiently large, then the designed waveform becomes a random sequence phase encoding signal with an “acicular” ambiguity figure. In summary, the sparse Doppler-sensitive waveform is a waveform family that includes the LFM signal and random sequence phase encoding signal.In this section, we also discuss the Doppler resolutionΔf of the technique. Suppose that the signal bandwidth is B, the sampling frequency is fs, and the time duration is T in each pulse repetition period. The number of sampling points is N, where N=T/fs; therefore, Δf=fs/N=fs/T⋅fs=1/T. The maximum unambiguous Doppler increases from the pulse repetition frequency fr to the sampling frequency fs; thus, this parameter can be used to recognize the large shift-frequency interference.


## 4.2. Application Method of Waveform Diversity for Auxiliary Doppler Confirmation

Due to its inherent cyclicality, which is discussed in depth in Section4.1, the designed signal cannot be directly used in signal detection on its own. Thus, the signal here is combined with the LFM signal to measure the specific Doppler target simultaneously, as shown in the flow chart of the proposed application technique in Figure 4. In this technique, two types of radar working modes are given: normal mode and measuring mode.Figure 4
Flow chart of the proposed application technique.In normal mode, the LFM signal is used to detect the target individually in the same manner as other regular radar measurements. When the specific DopplerξK requires confirmation, the signal can be switched to measuring mode. In this mode, the LFM signal and signals I and II with parameters N,B,K,q1 and N,B,K,q2, respectively, are transmitted in order, where N and B are the sampling points and bandwidth of the transmitting pulse, respectively, which can be identical to the LFM to confuse the enemy jammer. K depends on the specific Doppler ξK, and q1 and q2 denote the maximum position of Doppler ξK cut for signal I and signal II, respectively. Pf1 denotes the constant false alarm rate (CFAR) of transmitting the LFM signal, and Pf2 is the CFAR of transmitting signal I and signal II.For a specific DopplerξK, q1 determines the relative position of the target of Doppler ξK between the LFM signal and signal I. Similarly, q2 determines the relative position of the target of Doppler ξK between the LFM signal and signal II. If the received signals from the transmitting signals (i.e., the LFM signal and signals I and II) satisfy the relative range offset for the specific Doppler ξK, then the Doppler of the target can be confirmed as ξK. In short, this technique confirms the Doppler by comparing the range offsets of various waveforms.Further, the reason to use two additional signals to complete the measurement is that the sidelobe of the designed waveform is larger than the LFM signal; thus,Pf2 is made to be relatively large to ensure high detection probability, and the overall false alarm can be reduced through the joint detection.


## 5. Numerical Experiments

Example 1.
There are three targets; the Doppler of the true target 1 is 20 kHz, the Doppler of false target 2 and false target 3 is 30 kHz and 40 kHz (generated by shift-frequency jamming), respectively, the wavelength isλ=0.1 m, and the signal-to-noise ratio (SNR) of all the targets is −5 dB before pulse compression. For the LFM signal and signals I and II, the bandwidth is 1 MHz, the time width is 100 μs, and the sampling frequency is 2 MHz. To measure different Dopplers, we set different K and q values to match different Dopplers based on Section 5. The square law detection is used with Pf1=10−6 and Pf2=10−3. Signal parameters and the corresponding target positions in theory are shown in Table 1.Table 1
Signal parameters and target positions (in theory).
Signal (parameter)
Target 1 range (range cell)
Target 2 range (range cell)
Target 3 range (range cell)
LFM signal
1.02×10−4s (204)
1.105×10−4s (221)
1.215×10−4s (243)
Signal I(K=3, q1=2)
None
1.085×10−4s (217)
None
Signal II(K=3, q2=3)
None
1.09×10−4s (218)
NoneFigures5 and 6 show the ambiguity figure of the sparse Doppler-sensitive waveform with the parameters set to K=3 and q1=2. Figure 5 shows that when the Doppler number is 1 or 2, the ambiguity cut has no significant peak output, whereas when the Doppler number is 3, corresponding to a waveform design parameter of K=3, the ambiguity cut has a prominent mainlobe and relatively low sidelobe. In addition, the multiple Doppler number of 3 also has a peak output, which is displayed in Figure 6; the results prove the conclusion derived in Section 4.1, and the graph can fully represent the sparsity of the waveform.Figure 5
Ambiguity figure cut of Doppler numbers 1 to 6.Figure 6
Contour map of the ambiguity figure.Clearly, the speed of a normal aircraft is less than Ma 3, and the Doppler should be less than 20.4 kHz; thus, targets 2 and 3 must be false targets. While MTD processing is used, the Doppler of signals 2 and 3 ranges from zero to several hundred Hz due to the velocity ambiguity (the PRF of early-warning radar is typically 200–500 Hz), and the false targets cannot be effectively identified. Therefore, a correct measurement is the key to recognize large shift-frequency interference.Figure7 shows the received results of transmitting the LFM signal and signals I and II after pulse compression for three targets. The relative range offsets between LFM, signal I, and signal II for all targets are consistent with Table 1. When transmitting signal I and signal II with K=3, only target 2 displays a peak output response since the specific Doppler is 30 kHz in this case. In addition, the designed signal (signals I and II) has a higher sidelobe than the LFM signal. Consequently, the detection performance is worse than that of the LFM signal.Figure 7
Results of transmitting the LFM signal, signal I (K=3, q1=2) and signal II (K=3, q2=3) after the pulse compression.Figure8 represents the received results of transmitting the LFM and signals I and II after CFAR detection. In Figure 8, the range offset between the different waveforms for target 2 corresponds to the theoretical offset of the specific Doppler at 30 kHz, and the Doppler of target 2 can be determined to be 30 kHz. Although a high Pf2 may cause false alarm points when transmitting signal I and signal II individually, the entire CFAR can remain at a relatively low level since the correct confirming results require joint offset detection matching, which means that the target must be detected in the corresponding positions of three figures simultaneously, which is challenging for any false alarm points.Figure 8
Results of transmitting the LFM signal and signal I (K=3, q1=2) and signal II (K=3, q2=3) after CFAR (square law detection).In addition, the simulation results also show one advantage of the method relative to PRF agility: if the SNR is sufficient to allow detection, then the confirmation can be performed with notably few pulses.Example 2.
In this simulation, we analyse the robustness of the method. The parameter setting is identical to that of Example1, and the results are obtained using 1×106 Monte Carlo runs.Figure9 shows the probability of correct confirmation versus the SNR of the targets before pulse compression; the results correspond to target 2. The probability of correct confirmation indicates that the likelihood of target 2 being accurately confirmed is 30 kHz. The normal mode is set as a reference group that represents the probability of target 2 being detected by the LFM signal.Figure 9
Probability of correct confirmation for target 2.The curves in Figure9 highlight that the probability increases when the SNR increases, while the probability decreases when Pf2 decreases. When Pf2 is relatively large, such as 10−3, the correct confirmation probability is nearly equal to the detection probability of the normal mode, which means that the target can be correctly confirmed as long as the target can be detected by the LFM signal. When the SNR before pulse compression is larger than −10 dB, the correct estimation probability approaches 1, demonstrating that the performance of the algorithm is robust even in the case of low Pf2. However, the performance varies substantially if the SNR is less than −10 dB; for example, when the SNR is equal to −15 dB, the probability difference between Pf2=10−3 and Pf2=10−6 is predicted to be 0.5. Thus, we conclude that Pf2 should be larger when the SNR is relatively low.The results of Figures10 and 11 also show the false alarm probability of signal I and signal II, respectively, versus the SNR of the targets before pulse compression. Since Pf2 represents only the false alarm probability in the case of noise, the sidelobe of the designed waveform also affects the false alarm probability.Figure 10
Probability of false alarm detection (signal I).Figure 11
Probability of false alarm detection (signal II).The false alarm probability is relatively high for a single waveform, and the curves highlight that a larger SNR leads to a higher false alarm probability. Although the waveform diversity technology is exploited to reduce the overall false alarm probability through joint detection, it is preferred to adopt smallerPf2 when the SNR is large (such as SNR > −10 dB) since this condition may cause false judgement of other target results from the false alarm points.


## 6. Conclusion

In this study, we investigated the problem of shift-frequency interference with radar, where the Doppler shift value estimation is the key for antijamming techniques, and a method for specific Doppler confirmation processing is provided from the perspective of the waveform. The main contributions of this study are summarized as follows:(i)
The relationship between the ISL of an arbitrary Doppler ambiguity figure cut and the signal spectrum is derived.
(ii)
The optimal signal spectrum based on the optimal detection criterion for a specific Doppler is obtained.
(iii)
A sparse Doppler-sensitive waveform sensitive to a specific Doppler but insensitive to other Dopplers is designed.
(iv)
The application method of waveform diversity for the specific Doppler confirmation in the fast-time domain is proposed based on the designed waveform.In addition, the Doppler number of the true targets is relatively small in ground-based early-warning radar, and we need not address the excessive spending of radar detection resources to confirm much Doppler information using this method. Furthermore, the ratio of jamming to noise (JNR) is always sufficient to detect, and the confirmation can be performed with notably few pulses relative to the PRF agility.



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*Source: 1569590-2018-07-02.xml*