Research ArticleJoint High-Resolution Range and DOA Estimation via MUSICMethod Based on Virtual Two-Dimensional Spatial Smoothing forOFDM Radar

Rui Zhang ,1 Ying-Hui Quan ,1 Sheng-Qi Zhu,1 Lei Yang ,2 Ya-chao Li ,1

and Meng-Dao Xing 1

1National Laboratory of Radar Signal Processing, Xidian University, Shaanxi 710071, China2Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to Ying-Hui Quan; quanyinghui@126.com

Received 29 June 2018; Accepted 24 September 2018; Published 21 November 2018

Academic Editor: Stefano Selleri

Copyright © 2018 Rui Zhang 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.

For the purpose of target parameter estimation of the orthogonal frequency-division multiplexing (OFDM) radar, a high-resolutionmethod of joint estimation on range and direction of arrival (DOA) based on OFDM array radar is proposed in this paper. It beginswith the design and analysis of an echo model of OFDM array radar. Since there is no coupling between range and angle parameterestimation for a narrow-band signal, a method which exploits the data of one snapshot to estimate the range and angle of the targetby means of multiple signal classification (MUSIC) based on virtual two-dimensional spatial smoothing in range and angledimensions is devised. The proposed method is capable of joint estimating the range and DOA of the target in a high resolutionunder a single snapshot circumstance. Simulation experiments demonstrate the validity of the proposal.

1. Introduction

Recently, with the development of hardware technique forcomplicated waveform generator, it is possible to configurethe radar waveform flexibly [1]. Multicarrier signal has beenwidely applied to radar [2, 3], because it can provide moreinformation about targets. Orthogonal frequency-divisionmultiplexing (OFDM) radar modulates traditional single-carrier transmitting signal to subcarrier signals one by one,and then, it transmits the superposed signal, which greatlyimproves the performance of radar system and attracts thou-sands of scholars’ wide attention.

OFDM radar has the following advantages [4]: flexiblewaveform design, using orthogonal subcarrier signal as sub-channel which improves the utilization ratio of the spectrum,enhancing the ability of resisting the pulse noise and channelfading, with large time-bandwidth product, and easy to beprocessed digitally. Reference [5] firstly considers the subcar-rier phase modulation into multicarrier structure of OFDMradar; the spectrum utilization ratio can be improved. In

[6], the concept of OFDM applied to the radar system is dis-cussed in detail. Meanwhile, the framework of the OFDMradar system is proposed and the simulation analysis of eachmodule is carried out. It has been proved that OFDM radarhas strong abilities of antinoise and antijamming in [6].The theory of Doppler signal processing for the OFDM radarsignal is proposed in [7]. However, Doppler processing onreceiver based on correlation function usually encountersthe troublesome Doppler ambiguity. Tigrek et al. [8] andGarmatyuk et al. [9] make a feasible study about the influenceof OFDM signal on radar imaging and establishment of thecommunication system network, which lays the foundationof the integration of radar and communication. UsingOFDM radar signal to eliminate Doppler ambiguity is pre-sented in [10], and the echoed signal model is establishedin vector expression. However, it does not take angle infor-mation of the target into consideration. Reference [11] uti-lizes the property that different carrier waveforms ofOFDM radar have different responses to the target signal toimprove the frequency diversity and solve the problem of

HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 6012426, 9 pageshttps://doi.org/10.1155/2018/6012426

multipath reflection. Reference [12] proposes an algorithm ofjoint estimation on direction of departure (DOD) and direc-tion of angle (DOA) based on multiple-input multiple-output (MIMO) radar. The complexity of the algorithm isdecreased by the dimension reduction, and it is suitable toestimate DOD and DOA under the condition of a singlesnapshot and multiple targets. A DOA estimation methodbased on subspace is put forward in [13], providing a funda-mental for time-varying DOA estimation. References [14, 15]transform the DOA model based on sparse representation ofcovariance matrix into sparse representation model of a sin-gle vector, and the effect of imprecise estimation on noisepower is mitigated by the dimension reduction. In [16], amethod which determines the range and angle of target ispresented, by sending double pulses based on traditional fre-quency diverse array radar.

In this paper, aiming at target parameter estimation inthe OFDM array system, the characteristic of received signalis analyzed and a high resolution method of joint estimationon the range and DOA of the target in OFDM array radar isproposed. Under the narrow-band signal condition, sincethere is no coupling between the range and angle of the targetin the echoed signal of the OFDM radar, the target informa-tion is included in the range and angle steering vector,respectively, which can be utilized to jointly estimate therange and angle of signal source. In single snapshot case, ajoint high resolution estimation method for the range andthe angle of signal source can be obtained by means of themultiple signal classification (MUSIC) algorithm and virtualtwo-dimensional spatial smoothing technique. In order toevaluate the performance of the proposed method in thispaper, the root-mean-square errors (RMSE) of the estimatesof the range and angle parameters are examined. Numericalsimulation experiments are designed to prove the effective-ness of the proposed method.

2. Echoed Signal Model of OFDM Radar

Taking linear array as an example, it is assumed that thenumber of array elements is M, the distance between twoelements is d, the propagation speed of electromagneticwave is c, and signal bandwidth is B. Considering phasedarray radar and MIMO (multiple-input and multiple-

output) radar system, there are P far-field narrow-band sig-nals in space. If the longest time which signal spends ontraversing array aperture is much shorter than reciprocalof signal bandwidth, M − 1 d /c≪ 1/B; the signal can becalled “narrow-band signal.”

Under narrow-band signal condition, the time delaybetween different array elements can be expressed as a for-mula about phase difference of signal. The response of differ-ent array elements to the same signal is to have differentphase shifts.

Consider a uniform linear array (ULA) including M ele-ments, where each of them transmits OFDM waveform.There are N subcarrier channels in the OFDM waveform,and the transmitting signal can be expressed as

p t = 〠N−1

n=0An exp j2π f c + nΔf t exp jφn , 1

where An is amplitude, φn is the initial phase of the nth sub-carrier, Δf denotes the frequency interval of two adjacentsubcarriers, and f c is the central frequency. Furthermore, inorder to ensure the orthogonality between subcarriers, Δfneeds to meet with T = 1/Δf , where T is the radar pulse dura-tion. Figure 1 shows the allocation of transmitting antennasand transmitting signals on each element. All the antennastransmit the same signal at the same time. And the signalof each subcarrier channel can be expressed as

pn t = An exp j2π f c + nΔf t exp jφn , n = 0⋯N − 12

Under narrow-band circumstance and for arbitrary tar-get signal in space, the downconverted echoes of the OFDMarray radar can be written as

s t = 〠N−1

n=0exp j2πnΔf t 1 − 2v

c−2Rc

− j2πf c2Rc

− j2πf ct2vc

exp jφn aT θ ,3

p(t)

p0(t)p1(t) pN−1(t) pN−1(t) pN−1(t)

p(t) p(t)

p1(t)p0(t) p1(t)p0(t)

Figure 1: Allocation of transmitting antennas.

2 International Journal of Antennas and Propagation

where R is the target distance, v is the target radial velocity, cis the speed of light, d is the element interval of ULA, θdenotes DOA of the single point target, and M stands forthe number of array elements. Note that t is sampled dis-cretely, i.e., t = nT/N = n/NΔf , where n denotes the samplingindex. And a θ is steering vector,

a θ = 1, exp j2πd sin θ

λ,⋯, exp j2π sin θ M − 1

λ

T

,

4

where λ is wave length.The signal model in the discretization form can be writ-

ten as

s = ψΓβAφaT θ , 5

where s denotes a N ×M data matrix of discreted echoedsignal, and

ψ = exp −j2πf c2Rc

, 6

Γ = diag 1, γ, γ2,… , γN−1 , 7

γ = exp −j2πf c2vc

1NΔf , 8

β =

1 1 1 ⋯ 11 β β2 βN−1

1 β2 β4 β2 N−1

⋯ ⋱ ⋮

1 βN−1 β2 N−1 ⋯ β N−1 2

, 9

β = exp j2π 1 − 2vc

1N

, 10

A = diag 1, α, α2,… , αN−1 , 11

α = exp −j2πΔf 2Rc

, 12

φT = exp jφ0 , exp jφ1 ,⋯, exp jφN−1 , 13

a θ = 1, exp j2πd sin θ

λ,⋯, exp j2π sin θ M − 1

λ

T

14

Assuming 2v/c≪ 1/N , and β is with the same dimen-sion as the inverse discrete Fourier transform (IDFT)matrix, where ⋅ T stands for the matrix transpose. Theuseful information about range and angle is contained ina θ , Γ, and A. In order to obtain information about rangeand angle, the signal phase φ needs to be compensated.Firstly, β should be compensated by Fourier transformmatrix F; then, next step is phase compensation [10].

The signal phase φ can be compensated by P, where P = diagexp −jφ0 , exp −jφ1 ,⋯, exp −jφN−1 gives the diago-

nal matrix. After the compensation, the expression of theechoed signal can be written according to (3) as

PFs = ψPFΓβAφaT θ 15

When the product of matrix F and Γ meets (15), it willcause cyclic shift in each row of F. In order to compensate forthe cyclic shift, Doppler compensation should be completedbefore the phase compensation.

2vcf c = εΔf , 16

where ε is an integer. Each column of the echoed signal in(5) is sequentially processed with Doppler compensation,β compensation, and phase compensation. Then, each col-umn of the data that has been compensated is rearrangedas aMN ×N data matrix. Therefore, the space-domain steer-ing vector a θ, R of OFDM radar echo signal includes direc-tion steering vector a θ in (14) and distance steering vectora R , that is,

a θ, R = a θ ⊗ a R , 17

where ⊗ denotes Kronecker product and a R can beexpressed as

a R = 1, exp −j2π 2RcΔf ,⋯, exp −j2π 2R

cN − 1 Δf

T

18

From (17), it can be noted that there is no couplingbetween the distance and angle parameter estimation forOFDM array radar echo signal under the condition ofnarrow-band signal, and therefore, the joint estimation onthe range and DOA can be carried out.

3. A Method of Joint Estimation on Range andAngle in High Resolution by Virtual Two-Dimensional Spatial Smoothing

It is well known that the Fourier-based beamforming methodapplies the Fourier transform to the echoed signal to estimateDOA. However, because Fourier transform is limited byRayleigh limitation, the performance of the beamforming iswith high side lobes and limited resolution. To this end, thehigh-resolution method based on the second-order statisticalproperties can overcome this limitation. The MUSIC (multi-ple signal classification) can make an asymptotic and unbi-ased angle estimation of the target in theory. Under acertain SNR (signal-noise ratio) threshold, the performanceof MUSIC is closed to ML (maximum likelihood) estimation.It utilizes the orthogonality between signal subspace andnoise subspace to construct spatial spectral function andsearches array manifold vector which is orthogonal to noisesubspace for the DOA estimation.

3International Journal of Antennas and Propagation

Under the condition of narrow-band signal, the dis-tance and angle parameter estimation of target echoed sig-nal in OFDM array radar have no coupling relationship.Therefore, a high-resolution estimation of range and anglecan be realized by exploiting the data obtained from a sin-gle snapshot. Since the traditional MUSIC algorithm needsmany snapshots to estimate covariance matrix, a novelmethod that uses a virtual two-dimensional spatial smooth-ing method to sample a single snapshot data is proposed inthis paper. It is capable of estimating the covariance matrixby using sampled data which has been smoothed in two-dimensional space.

3.1. The Sampling Method of Virtual Two-DimensionalSpatial Smoothing. Consider the compensated data of onesingle snapshot. After match filtering and signal separationprocessing, the data is rearranged into a N ×M matrix,and then smooth the matrix spatially by column, andfinally divide it into N −Ns + 1 × M −Ms + 1 subar-rays, and the dimension of a subarry is Ns ×Ms, whereNs denotes the row number of subarrays and Ms is thecolumn number of subarrays. Then, rearrange all thesubarrays into a NsMs × N −Ns + 1 M −Ms + 1 matrix.Figure 2 shows the virtual two-dimensional spatialsmoothing method.

The first subarray is

X1 t = aNs×Msθ, R S t +N1 t , 19

and the steering vector is aNs×Msθ, R = aMs

θ ⊗ aNsR .

The second subarray is

X2 t = aNs×Msθ, R DS t +N2 t , 20

where D = exp j2πd sin θ/λ is the rotation factor.

The N −Ns + 1th subarray is

XN−Ns+1 t = aNs×Msθ, R DN−NsS t +NN−Ns+1 21

The N −Ns + 1 × 1 + 1th subarray

XN−Ns+2 t = aNs×Msθ, R LS t +NN−Ns+2 t , 22

where L = exp −j2π 2R/c Δf is the rotation factor.The N −Ns + 1 × m − 1 + nth subarray is

X N−Ns+1 × m−1 +n t = aNs×Msθ, R Dn−1Lm−1S t

+N N−Ns+1 × m−1 +n t ,23

where 1 ≤m ≤M −Ms + 1, 1 ≤ n ≤N −Ns + 1, and Dn−1Lm−1

is the combined rotational factor,

Dn−1Lm−1S t = S t exp j2π sin θ n − 1

λ − 2π 2R/c m − 1 Δf = S′ t

24

Therefore, it is equivalent to translate each subarray spa-tially through above steps. The rest rotation factors can becombined into signal envelope S t . The data of each subar-ray can be regarded as sampling data X t which has beensampled N −Ns + 1 × M −Ms + 1 times, that is,

X t = X1 t ,X2 t ,XN−Ns+1 t ,… ,

X N−Ns+1 × m−1 +n t ,… ,

X N−Ns+1 × M−Ms+1 t

= aMsθ ⊗ aNs

R S′ t +N t

= aNs×Msθ, R S′ t +N t

25

1 2 3

2

3

1 .. .. .. .. .. ..

.. .. .. .. .. ..

.. .. .. .. .. ..

......

......

......

......

......

......

......

......

......

.. .. .. .. .. ..

.. .. ..

.. .. .... .. ..

.. .. ..

......

......

......

Array element direction

Subc

arrie

r dire

ctio

n

1st subarray

2nd subarray

N-Ns + 2th subarray

Ms Ms + 1 M

Ns

Ns + 1

N

Figure 2: Virtual two-dimensional spatial smoothing method.

4 International Journal of Antennas and Propagation

It is reasonable to estimate covariance matrix by X twhich has been two-dimensional spatial smoothed, and it issuitable to realize the joint estimation of range and angle inhigh resolution.

3.2. Procedures of Estimating Range and Angle by a MUSICAlgorithm. In this section, we can estimate the range andangle of the target by applying the MUSIC algorithm intoX t , which has been smoothed in two-dimensional spacein Section 3.1. The detailed processing procedures of the pro-posed method are given as follows:

(1) The covariance matrix R̂ can be estimated fromX t , and

R̂ = 1N −Ns + 1 × M −Ms + 1 X t X t H 26

(2) Perform eigenvalue decomposition to covariancematrix R̂

(3) Sort eigenvalues and determine the number of signalsources P, then extract eigenvectors corresponding toNs ×Ms − P small eigenvalues to form the noisesubspace NNs×Ms−P

Ns×Ms

(4) Project two-dimensional search vector aNs×Msθ, R

onto noise subspace NNs×Ms−PNs×Ms

, and calculate spectralpeak

S θ, R = 1∑Ns×Ms

i=P+1 aHNs×Msθ, R vi

2 27

(5) The position θ, R of the spectral peak is angle andrange of the target

3.3. Performance Analysis of Joint Estimation on Range andAngle in High Resolution. In order to test the performanceof joint estimation on range and angle in high resolution, thissection proposes Cramer-Rao bound (CRB) of parameterestimation on range and angle in OFDM radar.

Consider that estimating distance and angle parametersof a single point source in the condition of Gaussian whitenoise. Assume that the value of signal-to-noise ratio is SNR,and Fisher information matrix corresponding to angle θand distance R can be written as

J =J11 J12

J21 J22, 28

then

J11 = 2 Re∂aN s×Ms

θ, R∂θ

H ∂aNs×Msθ, R

∂θ= 2 Re aθHaθ ,

29

where aθ = ∂aMsθ /∂θ ⊗ aNs

R and aR = aMsθ ⊗ ∂aNs

R /∂R. Similarly, there are

J12 = 2 Re aθHaR ,

J21 = 2 Re aRHaθ ,

J22 = 2 Re aRHaR ,

30

then CRB = SNR × J −1.

4. Results and Discussion

The basic parameters of OFDM array radar and targetparameters information used in the numerical simulationare shown in Table 1.

4.1. Performance Comparison of Range and Angle Estimationwith Multiple Targets. Aiming to verify the effectiveness ofrange and angle estimation method in OFDM radar,Figure 3 shows the results of target parameter estimation,respectively, using two-dimensional (range-angle) commonbeamforming and two-dimensional MUSIC algorithm. Insimulation experiments, assume that there are three nonco-herent point targets, and the angle is, respectively, 14°, 41°,and −18°. The distance is, respectively, 30,550m, 30,200m,and 29,700m, and SNR of three point targets is 10 dB. Otherparameters are shown in Table 1. Note that the experiment iscarried out in the case of Gaussian white noise.

From Figure 3, since the common beamforming is lim-ited by the aperture of the physical antenna, the resultantbeam width is wide and the ability of target resolution islow. Because the target data is sampled for only once, thenoise has a great influence on side lobe and target parameterestimation is effected badly by SNR. On the contrary, theMUSIC algorithm utilizes virtual two-dimensional smooth-ing technique. Although it loses certain array aperture anddistance “aperture,” it has a good performance of noise

Table 1: Simulation parameters of joint estimation on range andangle in high resolution.

Parameter Value

Center frequency f c 1GHz

Channel number (M) 10

Sub-carrier number (N) 10

Array element distance (d) λ/2Frequency interval Δf 100 kHz

Monte Carlo times (K) 100

Pulse duration time Tg 1/Δf

5International Journal of Antennas and Propagation

statistics. According to the results of power spectrum estima-tion, the side lobe is very low and SNR effect on target param-eter estimation is deducted. Additionally, the MUSICalgorithm can break the physical limitation of antenna aper-ture, and its resolution is higher than the common beam-forming method. Therefore, MUSIC has a better ability ofresolution in the estimation of range and angle.

4.2. Comparison about Spectrum Width of a Single PointTarget with a Single Snapshot. For quantitative descriptionof target parameter estimation accuracy, Figure 4 shows thecomparing results of using the two-dimensional commonbeamforming method and virtual two-dimensional smooth-ing spectrum estimation in a high-resolution method, respec-tively, to estimate the range and angle in OFDM radar.Simulation parameters are shown in Table 1.

According to simulation experiments, whether in angledomain or in range domain, the spectrum peaks of theMUSIC algorithm are sharper than that of the commonbeamforming method. The spectrum peaks of the MUSICalgorithm almost have no side lobes. However, the side lobesare high by using the common beamforming method. Theresults not only demonstrate the effectiveness of analysisabout echoed signal properties of OFDM radar but also showthat in a single snapshot case, the signal space that is esti-mated by the sampled data, which is obtained from a virtualtwo-dimensional smoothing method, is accurate. It is suc-cessful to break the limitation that the performance of theMUSIC algorithm depends on a large number of snapshots.Although a virtual two-dimensional smoothing method losesa part of apertures, the resolution ability of target parameterscan still reach 5 times more than that of the two-dimensionalcommon beamforming method, as shown in Figure 4.

−500Distance (m)Angle (°)

−500

50

0.2

0.4

Nor

mal

ized

pow

er sp

ectr

um

0.6

0.8

1

0500

0500

A0

50

(a) Frequency spectrum of normalized common beamforming

−500Distance (m)Angle (°)

−500

50

0.2

0.4

Nor

mal

ized

pow

er sp

ectr

um

0.6

0.8

1

0500

500 e (m)Angle (°−50

050

0500

(b) Frequency spectrum of normalized MUSIC

Figure 3: Spectrum comparison of two methods after normalized.

Common beamforming methodMUSIC method

−80 −60 −40 −20 0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Angle (°)

Nor

mal

ized

pow

er sp

ectr

um (d

B)

(a) Variation of spectral width with angle

0 500 1000 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Distance (m)

Nor

mal

ized

pow

er sp

ectr

um (d

B)

Common beamforming methodMUSIC method

(b) Variation of spectral width with distance

Figure 4: Spectrum comparison of two methods after normalized.

6 International Journal of Antennas and Propagation

4.3. The Performance Analysis of Joint Estimation on Rangeand Angle in High Resolution of a Single Point Target with aSingle Snapshot. Monte Carlo experiments are carried outto estimate angle θ and distance R. It is obvious that RMSE

of angle θ is RMSE θ = 1/K∑Km=1 θ∧m − θ 2, and RMSE

of distance R is RMSE R = 1/K∑Km=1 R∧

m − R 2, where

K is the number of Monte Carlo experiments, θ∧m and R∧m

are, respectively, the estimated angle and range value of mthMonte Carlo experiment, θ and R are, respectively, the trueangle and range of the target, and RMSE functions of θ and Rvary with SNR.

Exploiting the simulation parameters in Table 1 to simu-late how the RMSEs, which are estimated by using a high-resolution method of range and angle, varies with SNR.Then, compare the above simulation performance with theresults when using the ML method to estimate RMSE andthe outcomes that how Cramer-Rao bound varies withSNR, respectively. Figure 5 shows the comparison results.

In a Gaussian white noise condition, the ML method isthe most optimal estimation algorithm. It is found fromFigure 5 that the performance of MUSIC is close to the MLmethod. That is to say, although the MUSIC algorithm issuboptimal in a Gaussian white noise case, its estimationaccuracy loss can be negligible. Also, the MUSIC algorithmhas a high resolution for the range and angle estimates. It isseen that both the performances of two algorithms are awfulwhen SNR is below −5 dB. The accuracy of two algorithms isclose to Cramer-Rao bound when SNR is from −5 dB to15 dB. Because with a low SNR, the estimation performanceof two methods is influenced greatly by SNR, that is to say,the application of two methods needs a certain SNR thresh-old; the two algorithms are not close to Cramer-Rao boundwhen SNR is above 15 dB for the reason that there is inherent

error caused by angle and distance increment. Therefore,under the condition of a certain SNR value, as long as the stepsize in search is small enough, the MUSIC algorithm canmake an accurate estimation of distance and angle.

4.4. Analysis of Speed and Computational Complexity forMUSIC. The first step of MUSIC is to compute covariancematrix R̂. According to Section 3.1, the dimension of X tis Ns × 1 so that the computational complexity of this stepisO N2

s . The second step is to perform eigenvalue decompo-sition. The computational complexity of eigenvalue decom-position is O N3

s [17]. The third step is to sort theeigenvalues which have been obtained from the second step,and the average computational complexity is O NsMs whenutilizing bubble sort [18]. The fourth step is to calculate spec-tral peaks and search. According to formula (27) above, thecomputational complexity of every spectral peak searchingis O NsM

2s . Aiming to two-dimensional searching of θ, R

and to meet the demand of real-time processing when it isput into practice, the proper computing choices includeFPGA (field programmable gate array) and GPU (graphicsprocessing units) for their parallel computing ability. Thereal-time processing speed depends on the processor typeand the range of spectral peak searching [19, 20].

5. Conclusion

In this paper, aiming at the problem of radar target parame-ter estimation in the OFDM array system, the property ofreceived signal has been analyzed and a method abouthigh-resolution joint estimation on range and angle inOFDM radar is proposed. Under narrow-band signal condi-tion, there is no coupling between the range and angle of atarget in the echoed signal. The information about the range

RMSE

(°)

10−1

100

101

−1 −5 0 5 10 15 20SNR (dB)

Cramer-Rao bound

ML methodMUSIC method

(a) Performance comparison of angle estimation

−1 −5 0 5 10 15 20

100

101

102

RMSE

(m)

SNR (dB)

Cramer-Rao bound

ML methodMUSIC method

(b) Performance comparison of range estimation

Figure 5: Performance comparison of range and angle joint estimation.

7International Journal of Antennas and Propagation

and angle is included, respectively, in its steering vector,which can be used to jointly estimate range and angle of thetarget. Under the single snapshot condition, combining theMUSIC algorithm with a virtual two-dimensional spatialsmoothing method can jointly estimate range and angle inhigh resolution. In the background of the Gaussian whitenoise, the accuracy of the proposed method is close to thatof the common beamforming method. Although the high-resolution method which utilizes a virtual two-dimensionalsmoothing method loses a part of aperture under the condi-tion of a single snapshot, it still has high resolution in dis-tance and angle. In the future, further investigation on thejoint high-resolution estimation of OFDM radar based onsparse recovery will be performed.

Abbreviations

OFDM: Orthogonal frequency-division multiplexingDOA: Direction of arrivalMUSIC: Multiple signal classificationDOD: Direction of departureRMSE: Root-mean-square errorsULA: Uniform linear arrayIDFT: Inverse discrete Fourier transformSNR: Signal-noise ratioML: Maximum likelihoodCRB: Cramer-Rao boundFPGA: Field programmable gate ArrayGPU: Graphics processing units.

Data Availability

We have presented some persuasive simulation results as fig-ures based on MATLAB. Requests for access to these detaileddata which used to support the findings of this study shouldbe made to Ying-Hui Quan with email addresses yhquan@-mail.xidian.edu.cn and quanyinghui@126.com.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

Rui Zhang, Ying-Hui Quan, and Sheng-Qi Zhu designed thealgorithm scheme. Rui Zhang and Ying-Hui Quan per-formed the experiments and analyzed the experiment results.Lei Yang, Ya-chao Li, and Meng-Dao Xing contributed to themanuscript drafting and critical revision. All authors readand approved the final manuscript.

Acknowledgments

This work is partially supported by the Fund for ForeignScholars in University Research and Teaching Programs(the 111 Project) (no. B18039). This work was supported bythe National Natural Science Foundation of China underGrant nos. 61772397, 61303035, and 61471283 and NaturalScience Foundation of Tianjin, China under Grant nos.16JCYBJC41200 and 20162898.

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