Research ArticleA Novel Waveform Design Method for Shift-FrequencyJamming Confirmation

Chang Zhou ,1 Zhen-Bo Zhu,2 and Zi-Yue Tang2

1The Graduate Management Group, Air Force Early Warning Academy, Huangpu Avenue, Wuhan, China2The First Department, Air Force Early Warning Academy, Huangpu Avenue, Wuhan, China

Correspondence should be addressed to Chang Zhou; zc_radar@sohu.com

Received 31 January 2018; Accepted 19 April 2018; Published 2 July 2018

Academic Editor: Pierfrancesco Lombardo

Copyright © 2018 Chang Zhou et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Shift-frequency jamming is generally used to form range false targets for ground-based early-warning radar systems; the frequencyshift 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 novelsparse Doppler-sensitive waveform is designed based on the ambiguity function theory, where the basic idea is to design awaveform sensitive to a specific Doppler but insensitive to other Dopplers; therefore, this waveform can recognize the specificDoppler of the target unambiguously. To apply the designed waveform in practice, the detection and estimation processing flowis provided based on the waveform diversity technique and the family of the sparse Doppler-sensitive waveforms. Simulationexperiments are presented to validate the efficiency of the proposed method, and we conclude that the advantage of this methodis that it can be used to confirm the specific Doppler of the target unambiguously with few pulses even under the condition of alow PRF.

1. Introduction

Linear frequency-modulated (LFM) signals are widely usedin radar systems due to their large time-bandwidth productand large Doppler tolerance [1]. Taking advantage of therange-Doppler coupling characters of LFM signals, shift-frequency jamming [2, 3] has been extensively studied, as thisapproach can form false targets ahead of true targets byadjusting the parameters of the frequency shift even if theradar adopts the working mode of the frequency agility orpulse repetition frequency (PRF) agility. Since the frequencyshift modulation of the jammer is similar to the Dopplermodulation of the moving target, it is even more challengingto recognize shift-frequency jamming. However, the fre-quency shift amount of the jamming is much larger thanthe Doppler of the moving target in ground-based early-warning radar. For example, when the L-band radar operat-ing wavelength λ ≥ 15 cm, the speed of a conventional air-craft is less than Mach (Ma) 3; thus, the Doppler frequencyis less than 5.4 kHz, while the L-band signal bandwidth is

several MHz. To form a false target with at least one rangecell offset, the minimum frequency shift should be many tensof kHz, which can be considered much larger than the Dopp-ler of moving targets. Therefore, accurate estimation of thefrequency shift amount or Doppler frequency can enableshift-frequency interference recognition and suppression inground-based early-warning radar. However, for low- andmedium-PRF radars, it is challenging to obtain the actualDoppler because of the velocity ambiguity. To address thischallenge, the PRF agility technology is typically used forvelocity ambiguity resolution [1] in practice, while theexisting ambiguity resolution algorithm has the followingshortcomings [4, 5]: (1) the algorithm requires a relativelyhigh Doppler resolution in each PRF even in the presence ofnoise pollution and quantization errors, (2) the maximumunambiguous Doppler improvement is limited, and (3) thecomputation complexity is expensive, particularly in the caseof multiple moving targets. Therefore, this study attempts toachieve the frequency shift estimation from the perspective ofwaveform design.

HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 1569590, 13 pageshttps://doi.org/10.1155/2018/1569590

Many waveform design studies have focused on the lowautocorrelation sidelobe [6–10], which can improve thedetection performance for weak targets. The research hot-spots have developed from the design of binary-phasesequences to polyphase sequences and from fixed-lengthsequences to arbitrary-length sequences. In recent years,related algorithms can produce unimodular sequences ofthe lengthN = 106 or even longer, with favourable autocorre-lation properties [8]. In addition, several new applications,such as the sparse frequency waveform [11, 12], weightedsidelobe waveform [8], and orthogonal signal [6], have beendeveloped. The autocorrelation of the waveform is equivalentto the zero-Doppler cut of the ambiguity function [13];despite the fact that a satisfactory autocorrelation sidelobewaveform cannot guarantee the detection performance ofmoving targets, the design methods can be easily generalizedfor a proper ambiguity function cut design.

To detect a moving target, other possible Doppler cuts ofthe ambiguity function must be considered. In other words,we must consider the ambiguity function for a nonzeroDoppler. The studies of [14–16] focused on synthesizing anarbitrary desired ambiguity function. Although extensivework has been performed, there is no universal method tosolve this problem because it is challenging to determinewhether the desired ambiguity function can be synthesized;in addition, the process is time-consuming.

Therefore, to reduce complexity, some scholars haverelied on satisfying the partial constraints, such as ensuringa clear area near the origin [6] and minimizing the integralsidelobe level (ISL) in a certain area [17]. More specifically,the output response in certain range-Doppler areas, whichcover the interferences, should be as small as possible,whereas the output response in certain range-Doppler areaswith targets must ensure a level that is as high as possible[18]. In [19–21], the authors addressed the waveform designin the presence of coloured Gaussian disturbance noise andsolved the problem through semidefinite relaxation toachieve the optimal detection performance. For the unknownDoppler, an algorithm to guarantee that the minimumDoppler matches the output maximum has been proposed[22, 23], but the signal bandwidth and ISL were not con-sidered. The unimodular quadratic programme (UQP)and computational approaches to tackle the UQP weresummarized in [24]. In [25], the clutter model was estab-lished, and a slow-time ambiguity function design wasperformed, which is a more intuitive way to implementthe moving target detection (MTD) response. This prob-lem 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 thisproblem and obtain improved performance. However, theambiguity design method still addresses the velocity ambi-guity problem, and few references consider the bandwidthof the coding signal, which further limits the applicationin radar.

In the present study, a sparse Doppler-sensitive wave-form is designed that can be used to confirm the specificDoppler of the target unambiguously. By designing a differ-ent waveform with a different specific Doppler to confirm

the different target Doppler, the large shift-frequency inter-ference can be identified and suppressed.

The remainder of this paper is organized as follows. Sec-tion 2 presents the problem statement. Section 3 examinesthe signal spectrum based on the optimal detection criterionfor a specific Doppler, and the waveform design algorithm isdiscussed. Subsequently, the method of Doppler confirma-tion processing is proposed based on the designed waveformin Section 4. The simulation experiments are presented inSection 5. Finally, Section 6 presents the conclusions.

2. Problem Statement

The ambiguity function is the most intuitive description ofthe signal output performance for different Dopplers. Tofacilitate the design of the coding signal, the discrete ambigu-ity function is expressed as [1]

χd n, ξk =〠m

x m x∗ m − n exp jξkm

=〠ω

X∗ ω X ω − ξk exp jωn ,1

where n denotes the discrete delay series, ξk is the discreteDoppler frequency, x m is the discrete coded signal, andX ω represents the discrete Fourier transform (DFT) ofx m .

With regard to the zero-Doppler cut χ τ, 0 of N arbi-trary complex coded signals, the maximum value is N andmaximum position is χ 0, 0 . For the other Doppler cuts,however, the shape is unknown. Assuming that the specificDoppler is ξK , to design a waveform that is sensitive to a spe-cific Doppler but insensitive to the others, the ambiguity fig-ure must satisfy three shape constraints, as shown in Figure 1.

(1) The specific Doppler cut (when ξk = ξK ) should existas a clear peak value χ q, ξK as the mainlobe, whereq ∈ − N − 1 , − N − 2 ,… , N − 1 represents theposition of the mainlobe (peak value) in this cut,and this cut can be set according to the actualrequirements.

(2) The sidelobe of the specific Doppler cut (when ξk =ξK and n ≠ q) should be as small as possible.

(3) Other Doppler cuts (when ξk ≠ ξK and ξk ≠ ξ0)should not have significant peak values.

3. Proposed Method of Waveform Design

Based on the shape constraints for the ambiguity function ofthe sparse Doppler-sensitive waveform described in Section2, the detailed waveform design method is discussed in thispart. In Section 3.1, the sidelobe characteristics of the specificDoppler cut are analysed, and the relationship between thesidelobe of the ambiguity figure cut and the signal spectrumis deduced. Section 3.2 further builds the overall spectrumrepresentation of the specific Doppler cut, and the signalspectrum is solved under the optimal detection criterion forthe specific Doppler cut. In the end, combining the deduced

2 International Journal of Antennas and Propagation

signal spectrum with the other Doppler cut constraints, theconstant modulus waveform is designed in Section 3.3.

3.1. Analysis of the Sidelobe Characteristics of the SpecificDoppler Cut (When ξk = ξK ). It is customary to discuss thesidelobe when describing the correlation properties of thewaveform. For a specific ξK , the discrete ambiguity functioncut χd n, ξK is equivalent to the cross-correlation [27] ofx n exp jξKn and x n , which is generally expressed asr n, ξK . To conform to the customary representation, wedefine the ISL on the specific Doppler ξK (where q representsthe position of the mainlobe in this cut) as

ISLξK = 〠N−1

n=− N−1n≠q

χ2d n, ξK = 〠

N−1

n=− N−1n≠q

r2 n, ξK 2

It can be verified that

ISLξK =12N 〠

2N

p=1〠N−1

n=− N−1r n, ξK exp −jωpn − L exp −jωpq

2

− r q, ξK − L 2,3

where L is a complex number. Detailed derivations are givenin Appendix A.

Since L could be an arbitrary complex number, to sim-plify the above formula, let L = r q, ξK ; substituting this terminto (3) leads to

ISLξK =12N 〠

2N

p=1〠N−1

n=− N−1r n, ξK exp −jωpn

− r q, ξK exp −jωpq

2 4

For example, if ξK = 0, q = 0, and L = r q, ξK =N , then

ISLξK =12N 〠

2N

p=1〠N−1

n=− N−1r n, 0 exp −jωpn −N

2

5

Equation (5) has been widely used to design the autocor-relation waveform [8] with low ISL. The derivation of thisarticle has relatively extensive adaptability.

According to the relationship between the power spec-trum and cross-correlation function, the following equationholds, and more derivations are obtained in Appendix B.

〠N−1

n=− N−1r n, ξK exp −jωpn

= 〠N

m=1x m exp jξKm exp −jωpm

⋅ 〠N

m=1x m exp −jωpm

∗

,

6

where ωp = 2π/2N ⋅ p, p = 1, 2,… , 2N , and ξK = 2π/N ⋅K , K = 1,… ,N .

Define Xzerof ill ωp =∑Nm=1x m exp −jωpm ; then, (6)

can be expressed as

〠N−1

n=− N−1r n, ξK exp −jωpn = Xzerof ill ωp − ξK X∗

zerof ill ωp

7

Substituting (7) into (4), we obtain

ISLξK =12N 〠

2N

p=1Xzerof ill ωp − ξK X∗

zerof ill ωp

− r q, ξK exp −jωpq2

8

Equation (8) is applicable to any ambiguity figure cutχd n, ξK , which is also the basis of the following deduction.

3.2. Analysis of the Signal Spectrum under the OptimalDetection Criterion for the Specific Doppler Cut (Whenξk = ξK ). The purpose of the waveform design is to achieveeffective detection. To improve the detection performanceof the specific Doppler ξK , we need to ensure that (1)χd q, ξK is as large as possible and (2) ISLξk should be assmall as possible.

Notably, since Xzerof ill ωp =∑Nm=1x m exp −jωpm ,

Xzerof ill ωp is the DFT of x 1 , x 2 ,… , x N , 0,… , 0N

.

After adding zeros, the shape of the ambiguity figure remainsthe same, and according to Parseval’s theorem, we can obtain

Doppler

Range

Specific doppler cut

Other doppler cuts

Zero doppler cut

q

Mainlobe

Sidelobe

0

Figure 1: Shape constraints of various Doppler cuts in the ambiguity figure.

3International Journal of Antennas and Propagation

χd q, ξK = r q, ξK = 〠N

p=1X∗ ωp X ωp − ξK exp jωpq

= 12 ⋅ 〠

2N

p=1X∗zerof ill ωp Xzerof ill ωp − ξK exp jωpq

9From (8) and (9), we find that the key to design the

sparse Doppler-sensitive waveform for the specific DopplerξK is Xzerof ill ωp − ξK X∗

zerof ill ωp . For the convenience ofcalculation, the complex envelope of Xzerof ill ωp − ξKX∗zerof ill ωp is designated Aωp ,ξK , and the phase is θωp ,ξK ,

that is, Xzerof ill ωp − ξK X∗zerof ill ωp = Aωp ,ξK exp jθωp ,ξK .

Simultaneously, the complex envelope of Xzerof ill ωp is des-ignated Aωp

and the phase isθωp, that is, Xzerof ill ωp = Aωp

exp jθωp; clearly, Aωp ,ξK ≥ 0 and Aωp

≥ 0.To fully characterize the signal spectrum after the Dopp-

ler shift, the bandwidth of the designed signal B is chosen ashalf of the sampling frequency f s. Thus, the signal spectrumdoes not alias after the frequency shift. If the total point num-ber 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 Doppler ξK is

max  P1 Aωp ,ξK , θωp ,ξK = χd q, ξK 2

= 〠2N

p=1Aωp ,ξK exp jθωp ,ξK exp jωpq

2

,

min  P2 Aωp ,ξK , θωp ,ξK = ISLξK

= 12N 〠

2N

p=1Aωp ,ξK exp jθωp ,ξK

−12 〠

2N

p=1Aωp ,ξK exp jθωp ,ξK exp jωpq

exp −jωpq

2

,

s t  θωp ,ξK = θωp−K− θωp

, Aωp ,ξK

= Aωp⋅ Aωp−K

Aωp= 0, p =N + 1,… , 2N ,

〠2N

p=1Aωp

2=N2

10

Equation (10) is a nonlinear biobjective function forwhich 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

max  P Aωp ,ξK , θωp ,ξK = P1 Aωp ,ξK , θωp ,ξK

+ λP2 Aωp ,ξK , θωp ,ξK ,11

where λ is the weight coefficient and the constraintsremain the same. In (11), P1’s weight should be largebecause if no target of the specific Doppler can be detectedresults from the low detection probability, even the lowestsidelobe is useless.

Then, we design the θωp ,ξK and Aωpseparately using the

iterative algorithm for the optimal solution. When Aωpis

fixed, then Aωp ,ξK is fixed according to Aωp ,ξK = Aωp⋅ Aωp−K

,

and a genetic algorithm [28] is employed to quantify thephase and identify the optimal θωp ,ξK . While θωp ,ξK is fixed,

the function degenerates to a real optimization problem withnonlinear multiconstraints, and this problem can be solvedthrough sequential quadratic programming [29]. If MATLAB(a commercial mathematics software produced by AmericanMathWorks company) is available, then Aωp

can be solved

using the MATLAB function “fmincon” [30].The iterative algorithm is described as follows.

3.3. Waveform Design Method. After discussing the signalspectrum for the specific Doppler ξK in Section 3.2, twofurther steps are required to complete the design of the signal:(1) designing the constant modulus waveform approachingthe signal spectrum and (2) establishing the constraintsnecessary to avoid other Doppler cuts existing at significantpeak values.

From Section 3.2, for the given specific Doppler ξKand 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), let F be the unit DFT matrix as follows:

F = 12N

f 0 f 1 ⋯ f 2N − 1 , 12

where f denotes the vector value functions, given by

f p = exp j2π2N p ⋅ 0 ⋯ exp j

2π2N p ⋅ 2N − 1

T

,

13

where ⋅ T denotes the transpose for matrices/vectors. Wedefine the unimodular signal vector X and the discrete spec-trum vector D as follows:

Requirements: sequence length N , weight coefficient λ1. Initialize: θωp ,ξK = −ωpq.

2. Repeat.3. Substitute θωp ,ξK to the total objective function (12).

4. Compute Aωpthrough sequential quadratic programming.

5. Compute Aωp ,ξK according to Aωp ,ξK = Aωp⋅ Aωp−K

.

6. Compute θωp ,ξK using the genetic algorithm.

7. Continue until convergence.

Algorithm 1: Signal spectrum under the condition of optimaldetection criteria for a specific Doppler ξK.

4 International Journal of Antennas and Propagation

X =

x 1x 2⋮

x N

0⋮

0 2N

,

D =

X ω1

X ω2

⋮

⋮

X ω2N

=

Aω1exp jθω1

Aω2exp jθω2

⋮

⋮

Aω2Nexp jθω2N

14

The optimal spectral constraint problem is

min  P X,D = FHX −D 2,s t   x n = 1, n = 1, 2,… ,N ,exp   jθωp

exp −jθωp−K= exp jθωp ,ξK ,

 p = K + 1,… ,N ,

15

where ⋅ H and ⋅ denote the conjugate transpose andFrobenius norm for matrices/vectors, respectively.

For step (2), to realize the other Dopplers’ insensitivity,since the other Doppler cuts can be regarded as the sidelobeof the entire ambiguity figure, the multicyclic original(multi-CAO) algorithm [6], which is typically used to mini-mize the discrete ambiguity figure sidelobe of the partialinterested area, is introduced to design the signal.

Define vector X = X1 ⋯XP N+Q−1 ×Q K−1 , where Q isthe number of interested range cell number and K is thespecific Doppler:

Xm

xm 1 0⋮ ⋱

⋮ xm 1xm N ⋮

⋱ ⋮

0 xm N N+Q−1 ×Q

, m = 1, 2,… , K − 1,

16

where xm n = x n ej2π n m−1 /N , m = 1, 2,… , K − 1, andn = 1, 2,… ,N . The number of Doppler cells of interest isK − 1. The algorithm is described in detail elsewhere [6].The sidelobe constraint is

min  P X,U = X − NU2,

s t   x n = 1, n = 1, 2,… ,N ,UHU = I

17

Adding (15) and (17) with their corresponding weights,the total objective function is

min  P X,X,U,D = FHX −D 2 + η X − NU2,

s t   x n = 1, n = 1, 2,… ,N ,exp   jθωp

exp −jθωp−K

= exp j θωp ,ξK , p = K + 1,… ,N ,

UHU = I,

18

where η is the weight coefficient.The waveform design algorithm is described as follows.

4. Application Method of the SpecificDoppler Confirmation

4.1. Waveform Property and Performance Evaluation. Toenable quantitative analysis of the designed waveform prop-erty, we consider the main characteristics of the waveform.In Section 3.2, the objective function P1 is configured witha larger weight since a high peak value level for the specificDoppler cut should be guaranteed to ensure the matchedfilter output level; thus, the properties of the optimal signalsolution of P1 can represent the main characteristics of thewaveform. The optimal solution of P1 is θωp ,ξK = −ωpq, which

can be written as exp jθωpexp −jθωp−K

= exp jωpq ; let-

ting Aωp= 1, the properties of the waveform are discussed

as follows:

(1) If the maximum position is q for the specific DopplerξK , then χd q, ξK is the maximum value for thisDoppler cut. Under this condition, we can derivethat χd lq, ξlK is also the maximum value for theDoppler ξlK , where l denotes any positive integer.More details are given in Appendix C.

(2) The possible maximum in the ambiguity figuremay appear in n, k , which satisfies the formulaKn − kq = ±d ⋅ 2N , which can be easily proven usingf3 and f4 in Appendix D. Figure 2 displays the shapeof f1, f2, f3, and f4 in Appendix D, where N = 50,K = 5, q = 1, and the first K phases of the frequencyare random. The conclusion can also be observedfrom Figure 1.

(3) For all points n, k satisfying Kn − kq = ±d ⋅ 2N , theamplitude χd q, ξK is modulated by the first Kphases, and the different first K phases may lead todifferent ambiguity figures. Figure 3 shows the ambi-guity function (D.5) in Appendix D, where N = 50,K = 5, and q = 1. Figures 3(a) and 3(b) show the case

5International Journal of Antennas and Propagation

in which the first K phases of the frequency arerandom, whereas Figures 3(c) and 3(d) show the casein which the first K phases of the frequency are equal.Thus, the results vary significantly according to thefirst K phases of the frequency.

In addition, Figures 3(c) and 3(d) are similar to the LFMsignal, which can be explained by the fact that the LFM canbe considered a form of designed signal in this study, whereK = 1 and q = 2. In contrast, if K is sufficiently large, thenthe designed waveform becomes a random sequence phase

5

4040

−40

2020−200

0

A(n

, k)

RangeDoppler

(a)

40

2

0

40−40

2020−200

B(n

, k)

RangeDoppler

(b)

40

5

0

40−40

2020−200

C(n

, k)

Range Doppler

(c)

4040

−4020

20−200

2468

18

D(n

, k)

Range Doppler

(d)

Figure 2: f1, f2, f3, and f4 in Appendix D (N = 50, and the first K phases of the frequency are random).

Requirements: sequence length N , weight coefficient η1. Initialize: x n N

n=1 is a randomly generated unimodular sequence; configure X and X.2. Repeat.3. For a fixed X, compute the first K complex values of θωp

according to θωp ,ξK = θωp−K− θωp

and FHX −D.

4. For a fixed X, the minimizer U is given by U =U2UH1 , where matrices U1 and U2 are taken from the economic SVD of XH , which is

XH =U1∑UH2 .

5. For a fixed D, x n = exp j arg fT n D .6. For a fixed U, the criterion can be written as

X − NU 2 = 〠N

n=1〠

K−1 Q

l=1μnlx n − vnl

2 = const − 2〠N

n=1Re 〠

K−1 Q

l=1μ∗nlvnl x∗ n ,

where const is a constant that does not depend on x n ; μnl are given by the elements of X that contain x n

μn1 ⋯ μn, K−1 Q = 1⋯ 1K−1

ej2π n/N ⋯ ej2π n/N

K−1⋯ ej2π n Q−1 /N ⋯ ej2π n Q−1 /N

K−1 1× K−1 Q

vnl are the elements of NU, whose positions are identical to those of μnl in X. The minimizer x n is

x n = exp j arg 〠K−1 Q

l=1μ∗nlvnl , n = 1,… , 2N

7. Summarize the results in step 5 and step 6; the minimizer x n is given by

x n = exp j arg η fTD + 1 − η 〠K−1 Q

l=1μ∗nlvnl

8. Continue until convergence.

Algorithm 2: Sparse Doppler-sensitive waveform design algorithm with a constant modulus constraint.

6 International Journal of Antennas and Propagation

encoding signal with an “acicular” ambiguity figure. In sum-mary, the sparse Doppler-sensitive waveform is a waveformfamily that includes the LFM signal and random sequencephase encoding signal.

In this section, we also discuss the Doppler resolution Δfof the technique. Suppose that the signal bandwidth is B, thesampling frequency is f s, and the time duration is T in eachpulse repetition period. The number of sampling points isN , where N = T/f s; therefore, Δf = f s/N = f s/ T ⋅ f s = 1/T .The maximum unambiguous Doppler increases from thepulse repetition frequency f r to the sampling frequency f s;thus, this parameter can be used to recognize the largeshift-frequency interference.

4.2. Application Method of Waveform Diversity for AuxiliaryDoppler Confirmation. Due to its inherent cyclicality, whichis discussed in depth in Section 4.1, the designed signal can-not be directly used in signal detection on its own. Thus, thesignal here is combined with the LFM signal to measure thespecific Doppler target simultaneously, as shown in the flowchart of the proposed application technique in Figure 4. Inthis technique, two types of radar working modes are given:normal mode and measuring mode.

In normal mode, the LFM signal is used to detect thetarget individually in the same manner as other regular radarmeasurements. When the specific Doppler ξK requires con-firmation, the signal can be switched to measuring mode.In this mode, the LFM signal and signals I and II with param-eters N , B, K , q1 and N , B, K , q2 , respectively, are trans-mitted in order, where N and B are the sampling pointsand bandwidth of the transmitting pulse, respectively, whichcan be identical to the LFM to confuse the enemy jammer. K

depends on the specific Doppler ξK , and q1 and q2 denote themaximum position of Doppler ξK cut for signal I and signalII, respectively. Pf 1 denotes the constant false alarm rate(CFAR) of transmitting the LFM signal, and Pf 2 is the CFARof transmitting signal I and signal II.

For a specific Doppler ξK , q1 determines the relativeposition of the target of Doppler ξK between the LFM sig-nal and signal I. Similarly, q2 determines the relative posi-tion of the target of Doppler ξK between the LFM signaland signal II. If the received signals from the transmittingsignals (i.e., the LFM signal and signals I and II) satisfy therelative range offset for the specific Doppler ξK , then theDoppler of the target can be confirmed as ξK . In short,this technique confirms the Doppler by comparing the rangeoffsets of various waveforms.

Further, the reason to use two additional signals tocomplete the measurement is that the sidelobe of thedesigned waveform is larger than the LFM signal; thus,Pf 2 is made to be relatively large to ensure high detectionprobability, and the overall false alarm can be reducedthrough the joint detection.

5. Numerical Experiments

Example 1. There are three targets; the Doppler of the truetarget 1 is 20kHz, the Doppler of false target 2 and falsetarget 3 is 30kHz and 40kHz (generated by shift-frequencyjamming), respectively, the wavelength is λ = 0 1 m, andthe signal-to-noise ratio (SNR) of all the targets is −5dBbefore pulse compression. For the LFM signal and signals Iand II, the bandwidth is 1MHz, the time width is 100μs,

40

20

0

2040

X(n

, k)

−40−2002040

RangeDoppler

(a)

−40

−20

0

20

40

10 20 30 40 50

1

0.5

0

Rang

e

Doppler

(b)

X(n

, k)

40

20

0

2040

−40−2002040

RangeDoppler

(c)

−40

−20

0

20

40

10 20 30 40 50

1

0.5

0

Rang

eDoppler

(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 frequencyare equal.

7International Journal of Antennas and Propagation

and the sampling frequency is 2MHz. To measure differentDopplers, we set different K and q values to match differentDopplers based on Section 5. The square law detection is usedwith Pf 1 = 10−6 and Pf 2 = 10−3. Signal parameters and the cor-responding target positions in theory are shown in Table 1.Figures 5 and 6 show the ambiguity figure of the sparseDoppler-sensitive waveform with the parameters set to K =3 and q1 = 2. Figure 5 shows that when the Doppler numberis 1 or 2, the ambiguity cut has no significant peak output,whereas when the Doppler number is 3, corresponding to awaveform design parameter of K = 3, the ambiguity cut hasa prominent mainlobe and relatively low sidelobe. In addi-tion, the multiple Doppler number of 3 also has a peak out-put, which is displayed in Figure 6; the results prove theconclusion derived in Section 4.1, and the graph can fullyrepresent the sparsity of the waveform.

Clearly, the speed of a normal aircraft is less than Ma 3,and the Doppler should be less than 20.4 kHz; thus, targets2 and 3 must be false targets. While MTD processing isused, the Doppler of signals 2 and 3 ranges from zero toseveral hundred Hz due to the velocity ambiguity (the

PRF of early-warning radar is typically 200–500Hz), andthe false targets cannot be effectively identified. Therefore,a correct measurement is the key to recognize large shift-frequency interference.

Figure 7 shows the received results of transmitting theLFM signal and signals I and II after pulse compression forthree targets. The relative range offsets between LFM, signalI, and signal II for all targets are consistent with Table 1.When transmitting signal I and signal II with K = 3, onlytarget 2 displays a peak output response since the specificDoppler is 30 kHz in this case. In addition, the designedsignal (signals I and II) has a higher sidelobe than theLFM signal. Consequently, the detection performance isworse than that of the LFM signal.

TransmittingLFM signal

Transmittingsignal I

Transmittingsignal II

Pulse compressionCFAR (Pf1)

Pulse compressionCFAR (Pf2)

Pulse compressionCFAR (Pf2)

Output thetarget range

Calculate the range offset of the same targetbetween different waveforms

Range offset matching process

Confirm whether the target is thespecific doppler

Normal mode Measuring mode

Figure 4: Flow chart of the proposed application technique.

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 signal1 02 × 10−4 s

(204)1 105 × 10−4 s

(221)1 215 × 10−4 s

(243)

Signal I(K = 3, q1 = 2) None

1 085 × 10−4 s(217)

None

Signal II(K = 3, q2 = 3) None

1 09 × 10−4 s(218)

None

0 50 100 150 200 250 300 350 4000

0.51

1

0 50 100 150 200 250 300 350 4000

0.51

2

0 50 100 150 200 250 300 350 4000

0.51

Dop

pler

num

er3

0 50 100 150 200 250 300 350 4000

0.51

4

0 50 100 150 200 250 300 350 4000

0.51

5

0 50 100 150 200 250 300 350 4000

0.51

Coding number

6

Figure 5: Ambiguity figure cut of Doppler numbers 1 to 6.

8 International Journal of Antennas and Propagation

Figure 8 represents the received results of transmittingthe LFM and signals I and II after CFAR detection. InFigure 8, the range offset between the different waveformsfor target 2 corresponds to the theoretical offset of the spe-cific Doppler at 30 kHz, and the Doppler of target 2 can bedetermined to be 30 kHz. Although a high Pf2

may causefalse alarm points when transmitting signal I and signalII individually, the entire CFAR can remain at a relativelylow level since the correct confirming results require jointoffset detection matching, which means that the targetmust be detected in the corresponding positions of threefigures simultaneously, which is challenging for any falsealarm points.

In addition, the simulation results also show one advan-tage of the method relative to PRF agility: if the SNR is suffi-cient to allow detection, then the confirmation can beperformed with notably few pulses.

Example 2. In this simulation, we analyse the robustness ofthe method. The parameter setting is identical to that of

Example 1, and the results are obtained using 1 × 106 MonteCarlo runs.

Figure 9 shows the probability of correct confirmationversus the SNR of the targets before pulse compression;the results correspond to target 2. The probability of correctconfirmation indicates that the likelihood of target 2 beingaccurately confirmed is 30 kHz. The normal mode is set asa reference group that represents the probability of target 2being detected by the LFM signal.

The curves in Figure 9 highlight that the probabilityincreases when the SNR increases, while the probabilitydecreases when Pf2

decreases. When Pf2is relatively large,

such as 10−3, the correct confirmation probability is nearlyequal to the detection probability of the normal mode, whichmeans that the target can be correctly confirmed as long asthe target can be detected by the LFM signal. When the

Coding number

Dop

pler

num

ber

195 200 205 210 215 220 225 230 235

5

10

15

20

25

20

40

60

80

100

120

140

160

180

Figure 6: Contour map of the ambiguity figure.

0.6 0.8 1 1.2 1.4 1.6 1.8 ×10−40

100200

Time (s)

Am

plitu

de (V

)A

mpl

itude

(V)

Am

plitu

de (V

)

0.6 0.8 1 1.2 1.4 1.6 1.8 ×10−40

100200

Time (s)

0.6 0.8 1 1.2 1.4 1.6 1.8 ×10−40

100200

Time (s)

Target 2

Target 2

Target 2

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.

Am

plitu

de (V

)A

mpl

itude

(V)

Am

plitu

de (V

)

0.6 0.8 1 1.2 1.4 1.6 1.8 ×10−40

0.51

1.5

Time (s)

0.6 0.8 1 1.2 1.4 1.6 1.8 ×10−40

0.51

1.5

Time (s)

0.6 0.8 1 1.2 1.4 1.6 1.8 ×10−40

0.51

1.5

Time (s)

False alarm point

False alarm point

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 lawdetection).

−20 −15 −10 −50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Normal mode

Cor

rect

estim

atio

n pr

obab

ilty

Pf2 = 10−3

Pf2 = 10−4

Pf2 = 10−5

Pf2 = 10−6

Figure 9: Probability of correct confirmation for target 2.

9International Journal of Antennas and Propagation

SNR before pulse compression is larger than −10dB, the cor-rect estimation probability approaches 1, demonstrating thatthe performance of the algorithm is robust even in the case oflow Pf2

. However, the performance varies substantially if theSNR is less than −10 dB; for example, when the SNR is equalto −15 dB, the probability difference between Pf2

= 10−3 andPf2

= 10−6 is predicted to be 0.5. Thus, we conclude that Pf2should be larger when the SNR is relatively low.

The results of Figures 10 and 11 also show the falsealarm probability of signal I and signal II, respectively, ver-sus the SNR of the targets before pulse compression. SincePf 2 represents only the false alarm probability in the caseof noise, the sidelobe of the designed waveform also affectsthe false alarm probability.

The false alarm probability is relatively high for a sin-gle waveform, and the curves highlight that a larger SNRleads to a higher false alarm probability. Although thewaveform diversity technology is exploited to reduce theoverall false alarm probability through joint detection, itis preferred to adopt smaller Pf 2 when the SNR is large (suchas SNR>−10dB) since this condition may cause false judge-ment of other target results from the false alarm points.

6. Conclusion

In this study, we investigated the problem of shift-frequencyinterference with radar, where the Doppler shift value esti-mation is the key for antijamming techniques, and a methodfor specific Doppler confirmation processing is providedfrom the perspective of the waveform. The main contribu-tions of this study are summarized as follows:

(i) The relationship between the ISL of an arbitraryDoppler ambiguity figure cut and the signal spec-trum is derived.

(ii) The optimal signal spectrum based on the optimaldetection criterion for a specific Doppler is obtained.

(iii) A sparse Doppler-sensitive waveform sensitive to aspecific Doppler but insensitive to other Dopplersis designed.

(iv) The application method of waveform diversityfor the specific Doppler confirmation in the fast-time domain is proposed based on the designedwaveform.

In addition, the Doppler number of the true targets isrelatively small in ground-based early-warning radar, and weneed not address the excessive spending of radar detectionresources to confirm much Doppler information using thismethod. Furthermore, the ratio of jamming to noise (JNR)is always sufficient to detect, and the confirmation can beperformed with notably few pulses relative to the PRF agility.

Appendix

A.

Proof. Equation (3) holds due to

〠2N

p=1〠N−1

n=− N−1r n, ξK exp −jωpn − L exp −jωpq

2

= 〠2N

p=1〠N−1

n=− N−1r n, ξK exp −jωpn

− 〠N−1

n=− N−1Lδn−q exp −jωp n − q exp −jωpn

2

= 〠2N

p=1〠N−1

n=− N−1r n, ξK − Lδn−q exp −jωpn

2

−20 −15 −10 −50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

SNR (dB)

False

alar

m p

roba

bilit

y

Pf2 = 10−3

Pf2 = 10−4Pf2 = 10−5

Pf2 = 10−6

Figure 10: Probability of false alarm detection (signal I).

False

alar

m p

roba

bilit

y

−20 −15 −10 −50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

SNR (dB)

Pf2 = 10−3

Pf2 = 10−4Pf2 = 10−5

Pf2 = 10−6

Figure 11: Probability of false alarm detection (signal II).

10 International Journal of Antennas and Propagation

= 2N ⋅ 〠N−1

n=− N−1r n, ξK − Lδn−q

2

= 2N ⋅ 〠N−1

n=− N−1n≠q

r2 n, ξK + r q, ξK − L 2

= 2N ⋅ ISLξK + r q, ξK − L 2 ,

A 1

where δK is the Kronecker delta:

δK =1, forK = 0,0, forK ≠ 0

A 2

Extracting the ISLξK to the left of the equation, we findthat (3) is valid.

B.

Proof. Equation (6) holds due to

〠N

m=1x m exp jξKm exp −jωpm ⋅ 〠

N

m=1x m exp −jωpm

∗

= 〠N

m=1x m exp jξKm exp −jωpm 〠

N

m=1x∗ m exp jωpm

= 〠N

m=1〠N

m=1x m exp jξKm exp −jωpm x∗ m exp jωpm

= 〠N

m=1〠N

m=1x m x∗ m exp jξKm ⋅ exp −jωp m −m

B 1

Let m −m = n, we can obtain

〠N

m=1〠N

m=1x m x∗ m exp jξKm ⋅ exp −jωp m −m

= 〠N−1

n=− N−1r n, ξK exp −jωpn

B 2

C.

Proof. This part discusses the periodicity of the signal. Thephase of the signal spectral X ωp is defined as ∠X ωp ,where p = 1, 2,… , 2N . If the given condition is X ωp − ξKX∗ ωp = Aωp ,ξK exp −jωpq + jφ , then

∠X ωp = ∠X ωp−K exp jωpq − jφ

= ∠X ωp−2K exp jωp−Kq − jφ exp jωpq − jφ

= ∠X ωp−lK exp jωp− K−1 q − jφ ⋯ exp jωpq − jφ

= ∠X ωp−lK exp jωplq exp −j2π2N

l − 1 Klq2 − jKφ ,

C 1

where l and K are given; exp −j 2π/2N l − 1 Klq/2 −jKφ , is a fixed value, and we can obtain

X ωp − ξlK X∗ ωp = Aωp ,ξlK exp −jωplq exp

−j2π2N

l − 1 Klq2 − jKφ

C 2

Although Aωp ,ξlK is not known, for χd n, ξlK =∑2Np=1

X∗ ωp X ωp − ξlK exp jωpn , the maximum value mustbe χd lq, ξlK .

D.

Proof. In this part, we derive the ambiguity function sectionof another Doppler k when Doppler K is configured. Sincep = 1, 2,… , 2N , we can write p = i + lK , where i = 1, 2,… ,K and l = 0, 1,… , 2N/K − 1; thus,

∠X ωp = ∠X ωi+lK = ∠X ωi+ l−1 K exp jωi+lKq − jφ

= ∠X ωi+ l−2 K exp jωi+ l−1 Kq − jφ exp jωi+lKq − jφ

= ∠X ωi exp jωi+Kq − jφ ⋯ exp jωi+lKq − jφ

= ∠X ωi exp j2π2N ⋅

2i + K + lK2 ⋅ lq exp −jlφ

D 1

Then, we must obtain the value of ∠X ωp − ξk . Let k =j +mK , where j = 1, 2,… , K and m = 0, 1,… , 2N/K − 1.Since the bandwidth of the signal is half of the samplingfrequency,

〠2N

p=1X∗ ωp X ωp − ξk exp jωpn

= 〠N

p=1X∗ ωp X ωp − ξk exp jωpn

D 2

According to the symmetrical characteristic, consideringonly the case of i + lK > j +mK , we discuss the cases of i > j,l ≥m and i ≤ j, l >m separately.

(1) When i > j, l ≥m

11International Journal of Antennas and Propagation

∠X ωp − ξk = ∠X ω i−j + l−m K

= ∠X ωi−j exp

j2π2N

2 i − j + l −m + 1 K2 l −m q

⋅ exp −j l −m φ

D 3

(2) When i ≤ j, l >m

∠X ωp − ξk

= ∠X ω i−j+K + l−m−1 K

= ∠X ωi−j+K exp

⋅ j2π2N

2 i − j + K + l −m K2 l −m − 1 q

⋅ exp −j l −m − 1 φ

D 4

The ambiguity function is as follows:

〠N

p=k+1X∗ ωp X ωp − ξk exp jωpn

= 〠j

i=1〠

2N/K −1

l=m+1∠X∗ ωi exp −j

2π2N ⋅

2i + K + lK2 ⋅ lq

exp jlφ ∠X ωi−j+K ⋅ exp

j2π2N

2 i − j + K + l −m K2 l −m − 1 q

exp −j l −m − 1 φ exp jωpn + 〠K

i=j+1〠

2N/K −1

l=m∠X∗ ωi

exp −j2π2N ⋅

2i + K + lK2 ⋅ lq exp jlφ ∠X ωi−j

⋅ exp j2π2N

2 i − j + l −m + 1 K2 l −m q

exp −j l −m φ exp jωpn = 〠j

i=1∠X∗ ωi ∠X ωi−j+K

exp j2π2N

2i − 2j + 2K −mK −m − 12 q

⋅ exp j2π2N in exp j m + 1 φ 〠

2N/K −1

l=m+1exp

j2π2N Kn − j +mK q l + 〠

K

i=j+1∠X∗ ωi ∠X ωi−j

exp j2π2N

−2mi + 2mj + Km2 − Km2 q

⋅ exp j2π2N in exp jmφ 〠

2N/K −1

l=mexp

⋅ j2π2N Kn − j +mK q l = f1 〠

2N/K −1

l=m+1exp

⋅ j2π2N Kn − kq l + f2 〠

2N/K −1

l=mexp j

2π2N Kn − kq l

= f1 ⋅ f3 + f2 ⋅ f4D 5

where

f1 = 〠j

i=1∠X∗ ωi ∠X ωi−j+K

⋅ exp j2π2N

2i − 2j + 2K −mK −m − 12 q

⋅ exp j2π2N in exp j m + 1 φ ,

f2 = 〠K

i=j+1∠X∗ ωi ∠X ωi−j

exp j2π2N

−2mi + 2mj + Km2 − Km2 q

⋅ exp j2π2N in exp jmφ ,

f3 = 〠2N/K −1

l=m+1exp j

2π2N Kn − j +mK q l ,

f4 = 〠2N/K −1

l=mexp j

2π2N Kn − j +mK q l

D 6

Data Availability

All data are provided in full in this paper.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

This work was partially supported by a grant from theNational Natural Science Foundation of China (61671469).

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