Research ArticleDOA Estimation for Multiple Targets in MIMO Radar withNonorthogonal Signals

Zhenxin Cao1 Peng Chen 1 Zhimin Chen2 and Yi Jin3

1State Key Laboratory of Millimeter Waves Southeast University Nanjing 210096 China2School of Electronic and Information Engineering Shanghai Dianji University Shanghai 201306 China3Xirsquoan Branch of China Academy of Space Technology Xirsquoan 710100 China

Correspondence should be addressed to Peng Chen chenpengdspseueducn

Received 27 November 2017 Revised 26 June 2018 Accepted 8 July 2018 Published 15 July 2018

Academic Editor Raffaele Solimene

Copyright copy 2018 Zhenxin Cao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper addresses the direction of arrival (DOA) estimation problem in the colocated multiple-input multiple-output (MIMO)radar with nonorthogonal signals The maximum number of targets that can be estimated is theoretically derived as rank119877119904119873where 119873 denotes the number of receiving antennas and 119877119904 is the cross-correlation matrix of the transmitted signals Thereforewith the rank-deficient cross-correlationmatrix the maximum number that can be estimated is less than the radar with orthogonalsignals Then a multiple signal classification- (MUSIC-) based algorithm is given for the nonorthogonal signals Furthermore theDOA estimation performance is also theoretically analyzed by the Carmer-Rao lower bound Simulation results show that thenonorthogonality degrades the DOA estimation performance only in the scenario with the rank-deficient cross-correlationmatrix

1 Introduction

In the multiple-input multiple-output (MIMO) radar sys-tem [1ndash3] the waveform diversity can be used to improvethe target detection and estimation performance [4ndash8] Inthe existing works many methods have been proposed toestimate the target direction of arrival (DOA) For examplein the colocated MIMO radar system a reduced-dimensiontransformation is used to reduce the complexity in the DOAestimation based on the estimation of signal parametersvia rotational invariance technique (ESPRIT) [9] a com-putationally efficient DOA estimation algorithm is givenfor the monostatic MIMO radar based on the covariancematrix reconstruction in [10] a joint DOA and direction ofdeparture (DOD) estimation method based on ESPRIT isproposed in [11] Additionally in the coprime MIMO radarsystem a reduced-dimension multiple signal classification(MUSIC) algorithm is proposed [12] for bothDOA andDODestimation a combined unitary ESPRIT-based algorithm isgiven for the DOA estimation [13] In the bistatic MIMOradar system a joint DOD and DOA estimation method is

also proposed in the scenario with an unknown spatiallycorrelated noise [14]

In the existing papers about the MIMO radar systemsmost papers are about the orthogonal waveforms to esti-mate the DOA such as the waveforms adopted in [15ndash20]However it is difficult to generate the orthogonal signals inthe practical radar systems and the number of orthogonalwaveforms is much less than that of nonorthogonal wave-forms Therefore the traditional DOA estimation methodsand results with the orthogonal signalsmust also bemodifiedIn [21] the nonorthogonal waveforms are first proposedand the method with prewhitening processing is proposed in[22] However to the best of our knowledge the theoreticalanalysis of the nonorthogonal waveforms in theMIMO radarsystems has not yet been addressed in the papers

In this paper the DOA estimation problem for multipletargets is addressed and the system model of MIMO radarwith nonorthogonal signals is given Then the maximumnumber of targets that can be estimated is derived accord-ing to the cross-correlation matrix of transmitted signalsand a MUSIC algorithm for the nonorthogonal signals is

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6465856 7 pageshttpsdoiorg10115520186465856

2 Mathematical Problems in Engineering

also proposed Furthermore the Carmer-Rao lower bound(CRLB) of DOA estimation with the nonorthogonal signalsis also theoretically derived and shows that the estimationperformance with nonorthogonal signals is only degraded inthe scenario with rank-deficient cross-correlation matrix Tosummarize we make the following contributions

(i) The MIMO radar system model with nonorthogo-nal waveforms with considering the nonorthogonalwaveforms the system model of MIMO radar withDOA estimation is given

(ii) The theoretical maximum number of targets withthe subspace-based DOA estimation methods themaximum number of targets that can be estimated istheoretically derived

(iii) The MUSIC-based method and CRLB for DOAestimation to show the DOA estimation perfor-mance with nonorthogonal waveforms the MUSIC-based DOA estimation method is proposed and thecorresponding CRLB is theoretically derived

Notations 1119873 stands for a 119873 times 1 vector with all entriesbeing 1 119868119873 denotes an 119873 times 119873 identity matrix Esdot denotesthe expectation operation CN(119886119861) denotes the complexGaussian distributionwith themean being 119886 and the variancematrix being 119861 sdot 2 otimes ⊙ Trsdot vecsdot (sdot)119879 and (sdot)119867 denotethe ℓ2 norm the Kronecker product the Hadamard productthe trace of a matrix the vectorization of a matrix the matrixtranspose and the Hermitian transpose respectively

2 System Model

In this paper the colocated MIMO radar [8 23] is adoptedwhere the antenna numbers of transmitter and receiver are119872 and 119873 respectively We consider the DOA estimationproblem for 119870 far-field targets For the 119872 transmittingantennas 119904119898(119905) denotes the waveform in the 119898th antenna(119898 = 0 1 119872minus1) In the traditional MIMO radar systemthe waveforms 119904119898(119905) are orthogonal and int

119905119904119898(119905)1199041198981015840(119905)119889119905 =

0 (119898 = 1198981015840) However in this paper the nonorthogonalwaveforms are adopted so we have int

119905119904119898(119905)1199041198981015840(119905)119889119905 = 0 (119898 =

1198981015840) Additionally 119875 pulses are transmitted in the MIMOradar and the waveforms between pulses are the sameThenduring the 119901th pulse (119901 = 0 1 119875minus1) the signal in the 119899threceiving antenna (119899 = 0 1 119873 minus 1) can be expressed as

119903119899 (119901 119905) = 119872minus1sum119898=0

119870minus1sum119896=0

120572119896 (119901) 119904119898 (119905) 119890minus119895(2120587120582)(119898119889119879+119899119889119877) cos 120579119896+ 119908119899 (119901 119905)

(1)

where

120579 ≜ (1205790 120579119870minus1)119879 (2)

where 120579119896 denotes the DOA of the 119896th target (119896 = 0 1 119870minus1) and 119908119899(119901 119905) denotes the additive white Gaussian noise

(AWGN) with the variance being 1205902119899 120582 denotes the wave-length of the transmitted waveform 119889119879 and 119889119877 denote theantenna spacing in transmitter and receiver respectively120572119896(119901) denotes the fading coefficient of the 119896th targetrsquos radarcross section (RCS) during the 119901th pulse We assume that120572119896(119901) is a type of Swerling II RCS and follows the independentand identical distribution between pulses After the matchedfiltering processing for the 119898th signal during the 119901th pulsewe can obtain the sampled signal as

119903119899119901119898 ≜ int119905119903119899 (119901 119905) 119904119867119898 (119905) 119889119905 =

119872minus1sum1198981015840=0

119870minus1sum119896=0

120572119896 (119901)sdot 119890minus119895(2120587120582)(1198981015840119889119879+119899119889119877) cos 120579119896 int

1199051199041198981015840 (119905) 119904119867119898 (119905) 119889119905

+ int119905119908119899 (119901 119905) 119904119867119898 (119905) 119889119905

(3)

After the simplifications 119903119899119901119898 can be rewritten as

119903119899119901119898 = 119887119879119899 (120579) diag 120572 (119901)119860119879119877119904 [119898] + 119908119899119898 (119901) (4)

where 119877119904 denotes the cross-correlation matrix of the trans-mitted signals The 119898th row and 1198981015840th column entry of 119877119904 isdefined as

1198771199041198981198981015840 ≜ int119905119904119898 (119905) 1199041198671198981015840 (119905) 119889119905 (5)

and119877119904[119898] denotes the119898th column of119877119904 diag120572(119901) denotesa diagonal matrix with all the entries from the vector 120572(119901)and the steering vector in the transmitter is defined as

119886 (120579119896) ≜ [119890minus119895(2120587120582)0119889119879 cos 120579119896 119890minus119895(2120587120582)(119872minus1)119889119879 cos 120579119896]119879 (6)

The other symbols are defined as follows

119860 ≜ [119886 (1205790) 119886 (1205791) 119886 (120579119870minus1)] 119908119899119898 (119901) ≜ int

119905119908119899 (119901 119905) 119904119867119898 (119905) 119889119905

120572 (119901) ≜ [1205720 (119901) 120572119870minus1 (119901)]119879 119887119899 (120579) ≜ [119890minus119895(2120587120582)119899119889119877 cos 1205790 119890minus119895(2120587120582)119899119889119877 cos 120579119870minus1]119879

(7)

Since the nonorthogonal signals are transmitted in theMIMO radar the off-diagonal entries of 119877119904 are not all zeros

We can collect all the received signals in (4) into a matrixR(119901) as

119877 (119901) = 119861diag 120572 (119901)119860119879119877119904 +119882 (119901) (8)

where the 119899th row and 119898th column of 119877(119901) is 119903119899119901119898 and the119899th row and119898th column of119882(119901) is 119908119899119898(119901)119861 ≜ [1198870 (120579) 119887119873minus1 (120579)]119879 = [119887 (1205790) 119887 (120579119870minus1)] (9)

where the steering vector in the receiver is defined as

119887 (120579119896) ≜ [119890minus119895(2120587120582)0119889119877 cos 120579119896 119890minus119895(2120587120582)(119873minus1)119889119877 cos 120579119896]119879 (10)

In this paper we consider the DOA estimation withorthogonal waveforms and the system model is given in (8)where the unknownDOAs are collected bymatricesA andB

Mathematical Problems in Engineering 3

3 DOA Estimation

With the received signal119877(119901) theDOAcan be estimated andthe estimation performance with the nonorthogonal signalsis analyzed in this section The vectorization form of (8) canbe expressed as

119903 (119901) ≜ vec 119877 (119901) = 119879120572 (119901) + 119908 (119901) (11)

where

119879 ≜ (119877119879119904 119860 otimes 1119873) ⊙ (1119872 otimes 119861)= [119877119879119904 119886 (1205790) otimes 1119873 119877119879119904 119886 (120579119870minus1) otimes 1119873]

⊙ [1119872 otimes 119887 (1205790) 1119872 otimes 119887 (120579119870minus1)]= [1199050 119905119870minus1]

(12)

and otimes denotes the Kronecker product ⊙ denotes theHadamard product (entry-wise product)

119908 (119901) ≜ vec 119882 (119901) (13)

and

119905119896 ≜ 119877119879119904 119886 (120579119896) otimes 119887 (120579119896) (14)

denotes the 119896th column of 119879 Moreover the noise 119908(119901)follows the zero-mean Gaussian distribution with the covari-ance matrix being

E 119908 (119901)119908119867 (119901) = 1205902119899 (119877119879119904 otimes 119868119873) (15)

In the following proposition we will theoretically derivethe maximum number of targets that can be estimated by thenonorthogonal waveforms in the MIMO radar system

Proposition 1 In the colocated MIMO radar system with119872 transmitting antennas and 119873 receiving antennas themaximum number of targets in the DOA estimation usingthe subspace-based methods is rank119877119904119873 where 119877119904 denotesthe cross-correlation matrix of transmitted signals Sincerank119877119904 le 119872 the maximum number rank119877119904119873 le 119872119873

Proof (1) When 119877119904 is full rank rank119877119904 = 119872 we have

rank 1205902119899 (119877119879119904 otimes 119868119873) = 119872119873 (16)

Therefore the matrix (119877119879119904 otimes 119868119873) is invertible and the AWGNcan be obtained by the following prewhitening processing

119910 (119901) ≜ 119876119903 (119901) = 119904119879 + 119911 (119901) (17)

where

119876 ≜ (119877119879119904 otimes 119868119873)minus12 (18)

denotes the prewhitening matrix and

119904119879 ≜ 119876119879120572 (119901) (19)

119911(119901) denotes the AWGN with the variance matrix being1205902119899119868119872119873 Then the cross-correlation matrix of 119904119879 can beobtained as

119877119879 ≜ E 119904119879119904119867119879 (20)

and the dimension of signal subspace can be calculated as

119863119879 ≜ rank 119877119879le min 119872119873 rank 119877119879119904 119860 rank 119861 119870 (21)

With119870 le 119872119873 we have

119863119879 le min min 119872119870min 119873119870 119870 = 119870 le 119872119873 (22)

Therefore when the subspace-based methods such asMUSIC algorithm are adopted to estimate the DOA themaximum number of targets that can be estimated is 119870 le119872119873

(2) When 119877119904 is not full rank119877119904 lt 119872 the prewhiteningmatrix in (17) is irreversible and cannot be adopted so wemust find other prewhitening matrices Since the covariancematrix of noise is a positive-semidefinite normal matrix wehave the following decomposition

119877119879119904 otimes 119868119873 = 119880119863119880119867 (23)

where119880 is a unitary matrix composed by the eigenvectors of119877119879119904 otimes 119868119873 and 119863 is a nonnegative real diagonal matrix withthe diagonal entries being the eigenvalues of 119877119879119904 otimes 119868119873 Bycollecting the nonzero entries of 119863 into a vector 119889 isin C119911times1

and the corresponding eigenvectors into a matrix119880119889 where

119911 ≜ rank 119877119879119904 otimes 119868119873 = 119873 rank 119877119904 (24)

we can obtain the following prewhitening matrix

1198761015840 ≜ [diagminus12 119889 0119911times(119872119873minus119911)] [1198801198891198800]119867 (25)

where 1198800 are the matrices of eigenvectors corresponding tothe zero eigenvalues Then we have the following prewhiten-ing processing

1199101015840 (119901) = 1198761015840119903 (119901) = 1198761015840119879120572 (119901) + 1199111015840 (119901) (26)

where the noise is

1199111015840 (119901) ≜ 1198761015840119908 (119901) (27)

and the covariance matrix of noise is

E 1199111015840 (119901) 1199111015840119867 (119901) = 1205902119899119868119911 (28)

Then the cross-correlation matrix of the echoed signals canbe obtained as

1198771015840119879 ≜ [1198761015840119879120572 (119901)] [1198761015840119879120572 (119901)]119867 (29)

and the corresponding rank is

1198631015840119879 ≜ rank 1198771015840119879 le min 119911 119870 (30)

4 Mathematical Problems in Engineering

With rank119877119904 lt 119872 the maximum number of targets thatcan be estimated is rank119877119904119873 so when the nonorthogonalsignals reduce the rank of the cross-correlation matrix 119877119904the maximum number of targets that can be estimated alsodecreases

With Proposition 1 we find that the maximum numberof targets that can be estimated is rank119877119904119873 so with 119870 ltrank119877119904119873 we propose a MUSIC-based method to estimatethe DOAs After the prewhitening processes the MUSICspatial spectrum can be obtained as

119904 (120579) = 1119888119867 (120579) 119878119873119878119867119873119888 (120579) (31)

where

119888 (120579) ≜ 1198761015840 [119877119879119904 119886 (120579) otimes 119887 (120579)] (32)

and 119878119873 is the matrix of noise subspace formulated by the(119872119873minus119870) eigenvectors corresponding to theminimumeigen-values of the received signals 1199101015840(119901) after the prewhiteningThe DOAs can be estimated as

= (1205790 1205791 120579119870minus1)119879 (33)

by selecting the peak positions of the MUSIC spatial spec-trum

4 Carmeacuter-Rao Lower Bound

A vector 119903 can be formulated by collecting the signals fromall receiving antennas and pulses

119903 = 119868119875 otimes 119879120572 + 119908 (34)

where

119903 ≜ [119903119879 (0) 119903119879 (1) 119903119879 (119875 minus 1)]119879 (35)

120572 ≜ [120572119879 (0) 120572119879 (1) 120572119879 (119875 minus 1)]119879 (36)

119908 ≜ [119908119879 (0) 119908119879 (1) 119908119879 (119875 minus 1)]119879 (37)

By assuming that the fading coefficients follow the zero-meanGaussian distribution

120572 sim CN (0119875119870 1205722119868119875119870) (38)

119903 also follows the zero-mean Gaussian distribution

119903 sim 119891 (119903 120579) = 1120587119872119873119875 det (119862119903) 119890minus119903119867119862minus1

119903119903 (39)

with the covariance matrix being

119862119903 ≜ 1205902119899119868119875 otimes (119877119879119904 otimes 119868119873) + 1205722119868119875 otimes 119879119879119867 (40)

With the receiving signal 119903 the DOA can be estimatedby the MUSIC algorithm proposed in this paper and theestimation performance is bounded by the CRLB

E 10038171003817100381710038171003817120579 minus 1003817100381710038171003817100381722 ge 119892 (120579) ≜ Tr 119865minus1 (41)

1 2 3 4 51

2

3

4

5

075

08

085

09

095

1

Figure 1 The illustration of 119877119904

where 119892(120579) denotes the CRLB of DOA estimation and 119865denotes the Fisher informationmatrix (FIM)The entry in the119894th row and 119895th column (119894 = 0 1 119870minus1 119895 = 0 1 119870minus1)of the FIM can be obtained as

119865119894119895 = minusE1205972 ln119891 (119903 120579)120597120579119894120597120579119895 (42)

and the detailed expressions of FIM are given in theAppendix Similarity with the orthogonal waveforms wehave 119877119904 = 119868119872 and the corresponding CRLB can be alsoobtained

5 Simulation Results

In this section the simulation results are given and theparameters are set as follows the numbers of transmittingantennas receiving antennas and pulses are119872 = 5119873 = 10and 119875 = 10 respectively the carrier frequency is 10GHz thewavelength is 120582 = 003 m the antenna spacing is 119889119879 = 119889119877 =1205822 In Figure 2 the MUSIC algorithm is used to estimatethe DOA of 3 targets where the DOAs are 01881 10461and 14071 respectivelyWith the signal-to-noise ratio (SNR)being 25 dB the MUSIC algorithm is used to estimate theDOA and 119877119904 is shown in Figure 1 After 105 Monte Carloexperiments the root-mean-square error (RMSE) of DOAestimation can be obtained as 00103 The rank of waveformsadopted in Figure 2 is119872 but the rank of waveforms adoptedin Figure 3 is 1 Therefore the estimation performance isdegraded in the scenario with rank-deficient waveforms

In the scenario with nonorthogonal signals Figure 4shows the DOA estimation performance using the MUSICmethod with different types of transmitted signals includingthe orthogonal signals the full-rank nonorthogonal signals(rank119877119904 = 5) and the rank-deficient nonorthogonal signals(rank119877119904 = 1) As shown in this figure with increasing theSNR of received signals the DOA estimation performance isimproved and approaches the CRLBs with both orthogonaland nonorthogonal signals However the nonorthogonalsignals achieveworse estimation performance comparedwith

Mathematical Problems in Engineering 5

0 02 04 06 08 1 12 14 15Angle (rad)

minus20

minus15

minus10

minus5

0

5

10

15

20Sp

atia

l spe

ctru

m (d

B)

MUSIC spatial spectrumTarget DOAs

Figure 2 The MUSIC algorithm to estimate DOA (full rank)

0 02 04 06 08 1 12 14 16Angle (rad)

minus50

0

50

100

150

200

Spat

ial s

pect

rum

(dB)

MUSIC spatial spectrumTarget DOAs

Figure 3 The MUSIC algorithm to estimate DOA (rank = 1)

the orthogonal signals especially in the scenario with rank-deficient waveforms Moreover we also show the DOA esti-mation performance using the state-of-art direction findingmethod In [24] by discretizing the detection area into gridsthe simultaneous orthogonal matching pursuit (SOMP) isproposed and the DOA estimation can be formulated asa sparse reconstruction problem Figure 5 shows the DOAestimation with different waveforms and the better estima-tion performance can be achieved by both the full-rank andthe orthogonal waveforms compared with the rank-deficientwaveform Therefore the nonorthogonality degrades theDOA estimation performance only in the scenario with rank-deficient waveforms

In Figure 6 we show the DOA estimation performancewith different numbers of targets and the MUSIC method

SNR (dB)0 5 10 15 20 25 30 35 40 45

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank=5)RMSE (Non-Orthogonal signals rank=5)CRLB (Non-Orthogonal signals rank=1)RMSE (Non-Orthogonal signals rank=1)

10minus4

10minus3

10minus2

10minus1

100

DO

A es

timat

ion

(RM

SE an

d CR

LB)

Figure 4 DOA estimation performance with different values ofSNR (MUSIC)

SNR (dB)0 5 10 15 20 25 30 35

DO

A es

timat

ion

(RM

SE an

d CR

LB)

006

008

01

012

014

016

018

02

RMSE (Orthogonal signals)RMSE (Non-Orthogonal signals rank=5)RMSE (Non-Orthogonal signals rank=1)

Figure 5 DOA estimation performance with different values ofSNR (SOMP)

is adopted where SNR is 30 dB As shown in this figurethe estimation performance is degraded by more targetsand full-rank cross-correlation matrix can achieve betterestimation performance than the rank-deficient one Theestimation performance can achieve the CRLB in the sce-nario with fewer targets Therefore in the DOA estimationscenario with multiple targets the nonorthogonality givesworse estimation performance especially with the rank-deficient nonorthogonal signals In the future work aboutwaveform design ofMIMO radar systems when the full-rankwaveforms are adopted the DOA estimation performance

6 Mathematical Problems in Engineering

Target Number1 2 3 4 5 6 7

DO

A es

timat

ion

(RM

SE an

d CR

LB)

0

005

01

015

02

025

03

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank = 5)RMSE (Non-Orthogonal signals rank = 5)CRLB (Non-Orthogonal signals rank = 1)RMSE (Non-Orthogonal signals rank = 1)

Figure 6 DOA estimation performance with different numbers oftargets

cannot be degraded greatly so we can choose the full-ranknonorthogonal waveforms with more freedom

6 Conclusion

In this paper the DOA estimation problem with nonorthog-onal signals has been addressed in the MIMO radar systemThe maximum number of targets which is limited by therank of the cross-correlation matrix has been analyzedand given Then the MUSIC algorithm for nonorthogonalsignals has also been given and the CRLB for nonorthogonalsignals has been derived Simulation results show that theestimation performance is mainly degraded by the rank-deficient cross-correlation matrix of the transmitted signalsFuture work will focus on the waveform design using the full-rank nonorthogonal waveforms in the MIMO radar systems

Appendix

The Detailed Expressions of FIM

The entry in the 119894th row and 119895th column of the FIM can beexpressed as

119865119894119895 = minusE1205972 ln119891 (119903 120579)120597120579119894120597120579119895 = Tr119862minus1119903 120597119862119903120597120579119895 119862

minus1119903

120597119862119903120597120579119894 + Tr119862minus1119903 1205972119862119903120597120579119894120597120579119895 (A1)

where 120597119862119903120597120579119894 can be simplified as

120597119862119903120597120579119894 = 1205722119868119875 otimes 120597119879119879119867120597120579119894

= 1205722119868119875 otimes (120597119879120597120579119894119879119867 + 119879120597119879119867120597120579119894 )

(A2)

where

120597119879120597120579119894 = [0119872119873times(119894minus1) 120597119905119894120597120579119894 0119872119873times(119870minus119894)] (A3)

120597119905119894120597120579119894 = 119877119879119904

120597119886 (120579119894)120597120579119894 otimes 119887 (120579119894) + 119877119879119904 119886 (120579119894) otimes 120597119887 (120579119894)120597120579119894 (A4)

Additionally we can obtain

120597119886 (120579119894)120597120579119894 = 1198952120587120582 119889119879 sin 120579119894 (120578119872 ⊙ 119886 (120579119894)) (A5)

And

120578119872 ≜ [0 1 119872 minus 1]119879 (A6)

Using the same method the derivative of 119887(120579119894) can be alsoobtained as

120597119887 (120579119894)120597120579119894 = 1198952120587120582 119889119877 sin 120579119894 (120578119873 ⊙ 119887 (120579119894)) (A7)

The second-order derivative of 119862119903 can be expressed as

1205972119862119903120597120579119894120597120579119895= 1205722119868119875 otimes [120597 ((120597119879120597120579119894)119879119867)120597120579119895 + 120597 (119879 (120597119879119867120597120579119894))120597120579119895 ]= 1205722119868119875

otimes [ 1205972119879120597120579119894120597120579119895119879119867 + 120597119879120597120579119894

120597119879119867120597120579119895 + 120597119879120597120579119895120597119879119867120597120579119894 + 119879 1205972119879119867120597120579119894120597120579119895]

=

0119872119873119875 119894 = 1198951205722119868119875 otimes (12059721199051198941205972120579119894 119905119867119894 + 2 120597119905119894120597120579119894

120597119905119867119894120597120579119894 + 11990511989412059721199051198671198941205972120579119894 ) 119894 = 119895

(A8)

where

12059721199051198941205972120579119894 =120597 [119877119879119904 (120597119886 (120579119894) 120597120579119894) otimes 119887 (120579119894)]120597120579119894+ 120597 [119877119879119904 119886 (120579119894) otimes (120597119887 (120579119894) 120597120579119894)]120597120579119894

= 119877119879119904 1205972119886 (120579119894)1205972120579119894 otimes 119887 (120579119894) + 2119877119879119904 120597119886 (120579119894)120597120579119894

otimes 120597119887 (120579119894)120597120579119894 + 119877119879119904 119886 (120579119894) otimes 1205972119887 (120579119894)1205972120579119894

Mathematical Problems in Engineering 7

1205972119886 (120579119894)1205972120579119894 = 1198952120587120582 119889119879 120597 sin 120579119894120578119872 ⊙ 119886 (120579119894)120597120579119894= 1198952120587120582 119889119879120578119872

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) 1205972119887 (120579119894)1205972120579119894 = 1198952120587120582 119889119877120578119873

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) (A9)

Conflicts of Interest

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

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (Grants nos 61471117 and61601281) the Open Program of the State Key Labora-tory of Millimeter Waves (Southeast University) (Grant noZ201804) and the Fundamental Research Funds for theCentral Universities (Southeast University)

References

[1] P Chen C Qi and L Wu ldquoAntenna placement optimisationfor compressed sensing-based distributed MIMO radarrdquo IETRadar Sonar amp Navigation vol 11 no 2 pp 285ndash293 2017

[2] J Li P Stoica L Xu and W Roberts ldquoOn parameter identifi-ability of MIMO radarrdquo IEEE Signal Processing Letters vol 14no 12 pp 968ndash971 2007

[3] I Bekkerman and J Tabrikian ldquoTarget detection and localiza-tion using MIMO radars and sonarsrdquo IEEE Transactions onSignal Processing vol 54 no 10 pp 3873ndash3883 2006

[4] J Li and P Stoica ldquoMIMO radar with colocated antennasrdquo IEEESignal Processing Magazine vol 24 no 5 pp 106ndash114 2007

[5] P Chen and L Wu ldquoSystem optimization for temporal corre-lated cognitive radar with EBPSK-based MCPC signalrdquo Math-ematical Problems in Engineering vol 2015 Article ID 302083pp 1ndash10 2015

[6] P Chen L Wu and C Qi ldquoWaveform optimization for targetscattering coefficients estimation under detection and peak-to-average power ratio constraints in cognitive radarrdquo CircuitsSystems and Signal Processing vol 35 no 1 pp 163ndash184 2016

[7] M Haghnegahdar S Imani S A Ghorashi and E MehrshahildquoSINR Enhancement in Colocated MIMO Radar Using Trans-mit Covariance Matrix Optimizationrdquo IEEE Signal ProcessingLetters vol 24 no 3 pp 339ndash343 2017

[8] P Chen L Zheng X Wang H Li and L Wu ldquoMoving targetdetection using colocated MIMO radar on multiple distributedmoving platformsrdquo IEEE Transactions on Signal Processing vol65 no 17 pp 4670ndash4683 2017

[9] X Zhang andDXu ldquoLow-complexity ESPRIT-basedDOAesti-mation for colocated MIMO radar using reduced-dimensiontransformationrdquo IEEE Electronics Letters vol 47 no 4 pp 283-284 2011

[10] Y Zhang G Zhang and X Wang ldquoComputationally efficientDOA estimation for monostatic MIMO radar based on covari-ancematrix reconstructionrdquo IEEEElectronics Letters vol 53 no2 pp 111ndash113 2017

[11] D Oh Y-C Li J Khodjaev J-W Chong and J-H Lee ldquoJointestimation of direction of departure and direction of arrival formultiple-input multiple-output radar based on improved jointESPRIT methodrdquo IET Radar Sonar amp Navigation vol 9 no 3pp 308ndash317 2015

[12] Y Jia X Zhong Y Guo and W Huo ldquoDOA and DODestimation based on bistatic MIMO radar with co-prime arrayrdquoin Proceedings of the 2017 IEEE Radar Conference RadarConf2017 pp 0394ndash0397 Seattle Wash USA May 2017

[13] J Li D Jiang and X Zhang ldquoDOA estimation based oncombined unitary ESPRIT for coprime MIMO radarrdquo IEEECommunications Letters vol 21 no 1 pp 96ndash99 2017

[14] H Jiang J-K Zhang and K M Wong ldquoJoint DOD and DOAestimation for bistatic MIMO radar in unknown correlatednoiserdquo IEEE Transactions on Vehicular Technology vol 64 no11 pp 5113ndash5125 2015

[15] A Khabbazibasmenj A Hassanien S A Vorobyov and M WMorency ldquoEfficient transmit beamspace design for search-freebased DOA estimation in MIMO radarrdquo IEEE Transactions onSignal Processing vol 62 no 6 pp 1490ndash1500 2014

[16] A Hassanien and S A Vorobyov ldquoTransmit energy focusingfor DOA estimation in MIMO radar with colocated antennasrdquoIEEE Transactions on Signal Processing vol 59 no 6 pp 2669ndash2682 2011

[17] P Chen P Zhan and L Wu ldquoClutter estimation based oncompressed sensing in bistatic MIMO radarrdquo IET Radar Sonaramp Navigation vol 11 no 2 pp 285ndash293 2017

[18] M L Bencheikh and Y Wang ldquoJoint DOD-DOA estimationusing combined ESPRIT-MUSIC approach in MIMO radarrdquoIEEE Electronics Letters vol 46 no 15 pp 1081ndash1083 2010

[19] A Hassanien S A Vorobyov and A KhabbazibasmenjldquoTransmit radiation pattern invariance in MIMO radar withapplication to DOA estimationrdquo IEEE Signal Processing Lettersvol 22 no 10 pp 1609ndash1613 2015

[20] X Zhang L Xu and D Xu ldquoDirection of departure (DOD)and direction of arrival (DOA) estimation inMIMO radar withreduced-dimension MUSICrdquo IEEE Communications Lettersvol 14 no 12 pp 1161ndash1163 2010

[21] M Cattenoz and S Marcos ldquoAdaptive processing for MIMOradar realistic non perfectly orthogonal waveformsrdquo in Proceed-ings of the 2014 IEEE Radar Conference RadarCon 2014 pp1323ndash1328 Cincinnati Ohio USA May 2014

[22] G Zheng ldquoDOA estimation inMIMO radar with non-perfectlyorthogonal waveformsrdquo IEEE Communications Letters vol 21no 2 pp 414ndash417 2017

[23] P Chen C Qi L Wu and X Wang ldquoEstimation of extendedtargets based on compressed sensing in cognitive radar systemrdquoIEEE Transactions on Vehicular Technology vol 66 no 2 pp941ndash951 2017

[24] J-F Determe J Louveaux L Jacques and F Horlin ldquoOn thenoise robustness of simultaneous orthogonalmatching pursuitrdquoIEEE Transactions on Signal Processing vol 65 no 4 pp 864ndash875 2017

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Mathematical Problems in Engineering 3

3 DOA Estimation

With the received signal119877(119901) theDOAcan be estimated andthe estimation performance with the nonorthogonal signalsis analyzed in this section The vectorization form of (8) canbe expressed as

119903 (119901) ≜ vec 119877 (119901) = 119879120572 (119901) + 119908 (119901) (11)

where

119879 ≜ (119877119879119904 119860 otimes 1119873) ⊙ (1119872 otimes 119861)= [119877119879119904 119886 (1205790) otimes 1119873 119877119879119904 119886 (120579119870minus1) otimes 1119873]

⊙ [1119872 otimes 119887 (1205790) 1119872 otimes 119887 (120579119870minus1)]= [1199050 119905119870minus1]

(12)

and otimes denotes the Kronecker product ⊙ denotes theHadamard product (entry-wise product)

119908 (119901) ≜ vec 119882 (119901) (13)

and

119905119896 ≜ 119877119879119904 119886 (120579119896) otimes 119887 (120579119896) (14)

denotes the 119896th column of 119879 Moreover the noise 119908(119901)follows the zero-mean Gaussian distribution with the covari-ance matrix being

E 119908 (119901)119908119867 (119901) = 1205902119899 (119877119879119904 otimes 119868119873) (15)

In the following proposition we will theoretically derivethe maximum number of targets that can be estimated by thenonorthogonal waveforms in the MIMO radar system

Proposition 1 In the colocated MIMO radar system with119872 transmitting antennas and 119873 receiving antennas themaximum number of targets in the DOA estimation usingthe subspace-based methods is rank119877119904119873 where 119877119904 denotesthe cross-correlation matrix of transmitted signals Sincerank119877119904 le 119872 the maximum number rank119877119904119873 le 119872119873

Proof (1) When 119877119904 is full rank rank119877119904 = 119872 we have

rank 1205902119899 (119877119879119904 otimes 119868119873) = 119872119873 (16)

Therefore the matrix (119877119879119904 otimes 119868119873) is invertible and the AWGNcan be obtained by the following prewhitening processing

119910 (119901) ≜ 119876119903 (119901) = 119904119879 + 119911 (119901) (17)

where

119876 ≜ (119877119879119904 otimes 119868119873)minus12 (18)

denotes the prewhitening matrix and

119904119879 ≜ 119876119879120572 (119901) (19)

119911(119901) denotes the AWGN with the variance matrix being1205902119899119868119872119873 Then the cross-correlation matrix of 119904119879 can beobtained as

119877119879 ≜ E 119904119879119904119867119879 (20)

and the dimension of signal subspace can be calculated as

119863119879 ≜ rank 119877119879le min 119872119873 rank 119877119879119904 119860 rank 119861 119870 (21)

With119870 le 119872119873 we have

119863119879 le min min 119872119870min 119873119870 119870 = 119870 le 119872119873 (22)

Therefore when the subspace-based methods such asMUSIC algorithm are adopted to estimate the DOA themaximum number of targets that can be estimated is 119870 le119872119873

(2) When 119877119904 is not full rank119877119904 lt 119872 the prewhiteningmatrix in (17) is irreversible and cannot be adopted so wemust find other prewhitening matrices Since the covariancematrix of noise is a positive-semidefinite normal matrix wehave the following decomposition

119877119879119904 otimes 119868119873 = 119880119863119880119867 (23)

where119880 is a unitary matrix composed by the eigenvectors of119877119879119904 otimes 119868119873 and 119863 is a nonnegative real diagonal matrix withthe diagonal entries being the eigenvalues of 119877119879119904 otimes 119868119873 Bycollecting the nonzero entries of 119863 into a vector 119889 isin C119911times1

and the corresponding eigenvectors into a matrix119880119889 where

119911 ≜ rank 119877119879119904 otimes 119868119873 = 119873 rank 119877119904 (24)

we can obtain the following prewhitening matrix

1198761015840 ≜ [diagminus12 119889 0119911times(119872119873minus119911)] [1198801198891198800]119867 (25)

where 1198800 are the matrices of eigenvectors corresponding tothe zero eigenvalues Then we have the following prewhiten-ing processing

1199101015840 (119901) = 1198761015840119903 (119901) = 1198761015840119879120572 (119901) + 1199111015840 (119901) (26)

where the noise is

1199111015840 (119901) ≜ 1198761015840119908 (119901) (27)

and the covariance matrix of noise is

E 1199111015840 (119901) 1199111015840119867 (119901) = 1205902119899119868119911 (28)

Then the cross-correlation matrix of the echoed signals canbe obtained as

1198771015840119879 ≜ [1198761015840119879120572 (119901)] [1198761015840119879120572 (119901)]119867 (29)

and the corresponding rank is

1198631015840119879 ≜ rank 1198771015840119879 le min 119911 119870 (30)

4 Mathematical Problems in Engineering

With rank119877119904 lt 119872 the maximum number of targets thatcan be estimated is rank119877119904119873 so when the nonorthogonalsignals reduce the rank of the cross-correlation matrix 119877119904the maximum number of targets that can be estimated alsodecreases

With Proposition 1 we find that the maximum numberof targets that can be estimated is rank119877119904119873 so with 119870 ltrank119877119904119873 we propose a MUSIC-based method to estimatethe DOAs After the prewhitening processes the MUSICspatial spectrum can be obtained as

119904 (120579) = 1119888119867 (120579) 119878119873119878119867119873119888 (120579) (31)

where

119888 (120579) ≜ 1198761015840 [119877119879119904 119886 (120579) otimes 119887 (120579)] (32)

and 119878119873 is the matrix of noise subspace formulated by the(119872119873minus119870) eigenvectors corresponding to theminimumeigen-values of the received signals 1199101015840(119901) after the prewhiteningThe DOAs can be estimated as

= (1205790 1205791 120579119870minus1)119879 (33)

by selecting the peak positions of the MUSIC spatial spec-trum

4 Carmeacuter-Rao Lower Bound

A vector 119903 can be formulated by collecting the signals fromall receiving antennas and pulses

119903 = 119868119875 otimes 119879120572 + 119908 (34)

where

119903 ≜ [119903119879 (0) 119903119879 (1) 119903119879 (119875 minus 1)]119879 (35)

120572 ≜ [120572119879 (0) 120572119879 (1) 120572119879 (119875 minus 1)]119879 (36)

119908 ≜ [119908119879 (0) 119908119879 (1) 119908119879 (119875 minus 1)]119879 (37)

By assuming that the fading coefficients follow the zero-meanGaussian distribution

120572 sim CN (0119875119870 1205722119868119875119870) (38)

119903 also follows the zero-mean Gaussian distribution

119903 sim 119891 (119903 120579) = 1120587119872119873119875 det (119862119903) 119890minus119903119867119862minus1

119903119903 (39)

with the covariance matrix being

119862119903 ≜ 1205902119899119868119875 otimes (119877119879119904 otimes 119868119873) + 1205722119868119875 otimes 119879119879119867 (40)

With the receiving signal 119903 the DOA can be estimatedby the MUSIC algorithm proposed in this paper and theestimation performance is bounded by the CRLB

E 10038171003817100381710038171003817120579 minus 1003817100381710038171003817100381722 ge 119892 (120579) ≜ Tr 119865minus1 (41)

1 2 3 4 51

2

3

4

5

075

08

085

09

095

1

Figure 1 The illustration of 119877119904

where 119892(120579) denotes the CRLB of DOA estimation and 119865denotes the Fisher informationmatrix (FIM)The entry in the119894th row and 119895th column (119894 = 0 1 119870minus1 119895 = 0 1 119870minus1)of the FIM can be obtained as

119865119894119895 = minusE1205972 ln119891 (119903 120579)120597120579119894120597120579119895 (42)

and the detailed expressions of FIM are given in theAppendix Similarity with the orthogonal waveforms wehave 119877119904 = 119868119872 and the corresponding CRLB can be alsoobtained

5 Simulation Results

In this section the simulation results are given and theparameters are set as follows the numbers of transmittingantennas receiving antennas and pulses are119872 = 5119873 = 10and 119875 = 10 respectively the carrier frequency is 10GHz thewavelength is 120582 = 003 m the antenna spacing is 119889119879 = 119889119877 =1205822 In Figure 2 the MUSIC algorithm is used to estimatethe DOA of 3 targets where the DOAs are 01881 10461and 14071 respectivelyWith the signal-to-noise ratio (SNR)being 25 dB the MUSIC algorithm is used to estimate theDOA and 119877119904 is shown in Figure 1 After 105 Monte Carloexperiments the root-mean-square error (RMSE) of DOAestimation can be obtained as 00103 The rank of waveformsadopted in Figure 2 is119872 but the rank of waveforms adoptedin Figure 3 is 1 Therefore the estimation performance isdegraded in the scenario with rank-deficient waveforms

In the scenario with nonorthogonal signals Figure 4shows the DOA estimation performance using the MUSICmethod with different types of transmitted signals includingthe orthogonal signals the full-rank nonorthogonal signals(rank119877119904 = 5) and the rank-deficient nonorthogonal signals(rank119877119904 = 1) As shown in this figure with increasing theSNR of received signals the DOA estimation performance isimproved and approaches the CRLBs with both orthogonaland nonorthogonal signals However the nonorthogonalsignals achieveworse estimation performance comparedwith

Mathematical Problems in Engineering 5

0 02 04 06 08 1 12 14 15Angle (rad)

minus20

minus15

minus10

minus5

0

5

10

15

20Sp

atia

l spe

ctru

m (d

B)

MUSIC spatial spectrumTarget DOAs

Figure 2 The MUSIC algorithm to estimate DOA (full rank)

0 02 04 06 08 1 12 14 16Angle (rad)

minus50

0

50

100

150

200

Spat

ial s

pect

rum

(dB)

MUSIC spatial spectrumTarget DOAs

Figure 3 The MUSIC algorithm to estimate DOA (rank = 1)

the orthogonal signals especially in the scenario with rank-deficient waveforms Moreover we also show the DOA esti-mation performance using the state-of-art direction findingmethod In [24] by discretizing the detection area into gridsthe simultaneous orthogonal matching pursuit (SOMP) isproposed and the DOA estimation can be formulated asa sparse reconstruction problem Figure 5 shows the DOAestimation with different waveforms and the better estima-tion performance can be achieved by both the full-rank andthe orthogonal waveforms compared with the rank-deficientwaveform Therefore the nonorthogonality degrades theDOA estimation performance only in the scenario with rank-deficient waveforms

In Figure 6 we show the DOA estimation performancewith different numbers of targets and the MUSIC method

SNR (dB)0 5 10 15 20 25 30 35 40 45

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank=5)RMSE (Non-Orthogonal signals rank=5)CRLB (Non-Orthogonal signals rank=1)RMSE (Non-Orthogonal signals rank=1)

10minus4

10minus3

10minus2

10minus1

100

DO

A es

timat

ion

(RM

SE an

d CR

LB)

Figure 4 DOA estimation performance with different values ofSNR (MUSIC)

SNR (dB)0 5 10 15 20 25 30 35

DO

A es

timat

ion

(RM

SE an

d CR

LB)

006

008

01

012

014

016

018

02

RMSE (Orthogonal signals)RMSE (Non-Orthogonal signals rank=5)RMSE (Non-Orthogonal signals rank=1)

Figure 5 DOA estimation performance with different values ofSNR (SOMP)

is adopted where SNR is 30 dB As shown in this figurethe estimation performance is degraded by more targetsand full-rank cross-correlation matrix can achieve betterestimation performance than the rank-deficient one Theestimation performance can achieve the CRLB in the sce-nario with fewer targets Therefore in the DOA estimationscenario with multiple targets the nonorthogonality givesworse estimation performance especially with the rank-deficient nonorthogonal signals In the future work aboutwaveform design ofMIMO radar systems when the full-rankwaveforms are adopted the DOA estimation performance

6 Mathematical Problems in Engineering

Target Number1 2 3 4 5 6 7

DO

A es

timat

ion

(RM

SE an

d CR

LB)

0

005

01

015

02

025

03

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank = 5)RMSE (Non-Orthogonal signals rank = 5)CRLB (Non-Orthogonal signals rank = 1)RMSE (Non-Orthogonal signals rank = 1)

Figure 6 DOA estimation performance with different numbers oftargets

cannot be degraded greatly so we can choose the full-ranknonorthogonal waveforms with more freedom

6 Conclusion

In this paper the DOA estimation problem with nonorthog-onal signals has been addressed in the MIMO radar systemThe maximum number of targets which is limited by therank of the cross-correlation matrix has been analyzedand given Then the MUSIC algorithm for nonorthogonalsignals has also been given and the CRLB for nonorthogonalsignals has been derived Simulation results show that theestimation performance is mainly degraded by the rank-deficient cross-correlation matrix of the transmitted signalsFuture work will focus on the waveform design using the full-rank nonorthogonal waveforms in the MIMO radar systems

Appendix

The Detailed Expressions of FIM

The entry in the 119894th row and 119895th column of the FIM can beexpressed as

119865119894119895 = minusE1205972 ln119891 (119903 120579)120597120579119894120597120579119895 = Tr119862minus1119903 120597119862119903120597120579119895 119862

minus1119903

120597119862119903120597120579119894 + Tr119862minus1119903 1205972119862119903120597120579119894120597120579119895 (A1)

where 120597119862119903120597120579119894 can be simplified as

120597119862119903120597120579119894 = 1205722119868119875 otimes 120597119879119879119867120597120579119894

= 1205722119868119875 otimes (120597119879120597120579119894119879119867 + 119879120597119879119867120597120579119894 )

(A2)

where

120597119879120597120579119894 = [0119872119873times(119894minus1) 120597119905119894120597120579119894 0119872119873times(119870minus119894)] (A3)

120597119905119894120597120579119894 = 119877119879119904

120597119886 (120579119894)120597120579119894 otimes 119887 (120579119894) + 119877119879119904 119886 (120579119894) otimes 120597119887 (120579119894)120597120579119894 (A4)

Additionally we can obtain

120597119886 (120579119894)120597120579119894 = 1198952120587120582 119889119879 sin 120579119894 (120578119872 ⊙ 119886 (120579119894)) (A5)

And

120578119872 ≜ [0 1 119872 minus 1]119879 (A6)

Using the same method the derivative of 119887(120579119894) can be alsoobtained as

120597119887 (120579119894)120597120579119894 = 1198952120587120582 119889119877 sin 120579119894 (120578119873 ⊙ 119887 (120579119894)) (A7)

The second-order derivative of 119862119903 can be expressed as

1205972119862119903120597120579119894120597120579119895= 1205722119868119875 otimes [120597 ((120597119879120597120579119894)119879119867)120597120579119895 + 120597 (119879 (120597119879119867120597120579119894))120597120579119895 ]= 1205722119868119875

otimes [ 1205972119879120597120579119894120597120579119895119879119867 + 120597119879120597120579119894

120597119879119867120597120579119895 + 120597119879120597120579119895120597119879119867120597120579119894 + 119879 1205972119879119867120597120579119894120597120579119895]

=

0119872119873119875 119894 = 1198951205722119868119875 otimes (12059721199051198941205972120579119894 119905119867119894 + 2 120597119905119894120597120579119894

120597119905119867119894120597120579119894 + 11990511989412059721199051198671198941205972120579119894 ) 119894 = 119895

(A8)

where

12059721199051198941205972120579119894 =120597 [119877119879119904 (120597119886 (120579119894) 120597120579119894) otimes 119887 (120579119894)]120597120579119894+ 120597 [119877119879119904 119886 (120579119894) otimes (120597119887 (120579119894) 120597120579119894)]120597120579119894

= 119877119879119904 1205972119886 (120579119894)1205972120579119894 otimes 119887 (120579119894) + 2119877119879119904 120597119886 (120579119894)120597120579119894

otimes 120597119887 (120579119894)120597120579119894 + 119877119879119904 119886 (120579119894) otimes 1205972119887 (120579119894)1205972120579119894

Mathematical Problems in Engineering 7

1205972119886 (120579119894)1205972120579119894 = 1198952120587120582 119889119879 120597 sin 120579119894120578119872 ⊙ 119886 (120579119894)120597120579119894= 1198952120587120582 119889119879120578119872

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) 1205972119887 (120579119894)1205972120579119894 = 1198952120587120582 119889119877120578119873

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) (A9)

Conflicts of Interest

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

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (Grants nos 61471117 and61601281) the Open Program of the State Key Labora-tory of Millimeter Waves (Southeast University) (Grant noZ201804) and the Fundamental Research Funds for theCentral Universities (Southeast University)

References

[1] P Chen C Qi and L Wu ldquoAntenna placement optimisationfor compressed sensing-based distributed MIMO radarrdquo IETRadar Sonar amp Navigation vol 11 no 2 pp 285ndash293 2017

[2] J Li P Stoica L Xu and W Roberts ldquoOn parameter identifi-ability of MIMO radarrdquo IEEE Signal Processing Letters vol 14no 12 pp 968ndash971 2007

[3] I Bekkerman and J Tabrikian ldquoTarget detection and localiza-tion using MIMO radars and sonarsrdquo IEEE Transactions onSignal Processing vol 54 no 10 pp 3873ndash3883 2006

[4] J Li and P Stoica ldquoMIMO radar with colocated antennasrdquo IEEESignal Processing Magazine vol 24 no 5 pp 106ndash114 2007

[5] P Chen and L Wu ldquoSystem optimization for temporal corre-lated cognitive radar with EBPSK-based MCPC signalrdquo Math-ematical Problems in Engineering vol 2015 Article ID 302083pp 1ndash10 2015

[6] P Chen L Wu and C Qi ldquoWaveform optimization for targetscattering coefficients estimation under detection and peak-to-average power ratio constraints in cognitive radarrdquo CircuitsSystems and Signal Processing vol 35 no 1 pp 163ndash184 2016

[7] M Haghnegahdar S Imani S A Ghorashi and E MehrshahildquoSINR Enhancement in Colocated MIMO Radar Using Trans-mit Covariance Matrix Optimizationrdquo IEEE Signal ProcessingLetters vol 24 no 3 pp 339ndash343 2017

[8] P Chen L Zheng X Wang H Li and L Wu ldquoMoving targetdetection using colocated MIMO radar on multiple distributedmoving platformsrdquo IEEE Transactions on Signal Processing vol65 no 17 pp 4670ndash4683 2017

[9] X Zhang andDXu ldquoLow-complexity ESPRIT-basedDOAesti-mation for colocated MIMO radar using reduced-dimensiontransformationrdquo IEEE Electronics Letters vol 47 no 4 pp 283-284 2011

[10] Y Zhang G Zhang and X Wang ldquoComputationally efficientDOA estimation for monostatic MIMO radar based on covari-ancematrix reconstructionrdquo IEEEElectronics Letters vol 53 no2 pp 111ndash113 2017

[11] D Oh Y-C Li J Khodjaev J-W Chong and J-H Lee ldquoJointestimation of direction of departure and direction of arrival formultiple-input multiple-output radar based on improved jointESPRIT methodrdquo IET Radar Sonar amp Navigation vol 9 no 3pp 308ndash317 2015

[12] Y Jia X Zhong Y Guo and W Huo ldquoDOA and DODestimation based on bistatic MIMO radar with co-prime arrayrdquoin Proceedings of the 2017 IEEE Radar Conference RadarConf2017 pp 0394ndash0397 Seattle Wash USA May 2017

[13] J Li D Jiang and X Zhang ldquoDOA estimation based oncombined unitary ESPRIT for coprime MIMO radarrdquo IEEECommunications Letters vol 21 no 1 pp 96ndash99 2017

[14] H Jiang J-K Zhang and K M Wong ldquoJoint DOD and DOAestimation for bistatic MIMO radar in unknown correlatednoiserdquo IEEE Transactions on Vehicular Technology vol 64 no11 pp 5113ndash5125 2015

[15] A Khabbazibasmenj A Hassanien S A Vorobyov and M WMorency ldquoEfficient transmit beamspace design for search-freebased DOA estimation in MIMO radarrdquo IEEE Transactions onSignal Processing vol 62 no 6 pp 1490ndash1500 2014

[16] A Hassanien and S A Vorobyov ldquoTransmit energy focusingfor DOA estimation in MIMO radar with colocated antennasrdquoIEEE Transactions on Signal Processing vol 59 no 6 pp 2669ndash2682 2011

[17] P Chen P Zhan and L Wu ldquoClutter estimation based oncompressed sensing in bistatic MIMO radarrdquo IET Radar Sonaramp Navigation vol 11 no 2 pp 285ndash293 2017

[18] M L Bencheikh and Y Wang ldquoJoint DOD-DOA estimationusing combined ESPRIT-MUSIC approach in MIMO radarrdquoIEEE Electronics Letters vol 46 no 15 pp 1081ndash1083 2010

[19] A Hassanien S A Vorobyov and A KhabbazibasmenjldquoTransmit radiation pattern invariance in MIMO radar withapplication to DOA estimationrdquo IEEE Signal Processing Lettersvol 22 no 10 pp 1609ndash1613 2015

[20] X Zhang L Xu and D Xu ldquoDirection of departure (DOD)and direction of arrival (DOA) estimation inMIMO radar withreduced-dimension MUSICrdquo IEEE Communications Lettersvol 14 no 12 pp 1161ndash1163 2010

[21] M Cattenoz and S Marcos ldquoAdaptive processing for MIMOradar realistic non perfectly orthogonal waveformsrdquo in Proceed-ings of the 2014 IEEE Radar Conference RadarCon 2014 pp1323ndash1328 Cincinnati Ohio USA May 2014

[22] G Zheng ldquoDOA estimation inMIMO radar with non-perfectlyorthogonal waveformsrdquo IEEE Communications Letters vol 21no 2 pp 414ndash417 2017

[23] P Chen C Qi L Wu and X Wang ldquoEstimation of extendedtargets based on compressed sensing in cognitive radar systemrdquoIEEE Transactions on Vehicular Technology vol 66 no 2 pp941ndash951 2017

[24] J-F Determe J Louveaux L Jacques and F Horlin ldquoOn thenoise robustness of simultaneous orthogonalmatching pursuitrdquoIEEE Transactions on Signal Processing vol 65 no 4 pp 864ndash875 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

With rank119877119904 lt 119872 the maximum number of targets thatcan be estimated is rank119877119904119873 so when the nonorthogonalsignals reduce the rank of the cross-correlation matrix 119877119904the maximum number of targets that can be estimated alsodecreases

With Proposition 1 we find that the maximum numberof targets that can be estimated is rank119877119904119873 so with 119870 ltrank119877119904119873 we propose a MUSIC-based method to estimatethe DOAs After the prewhitening processes the MUSICspatial spectrum can be obtained as

119904 (120579) = 1119888119867 (120579) 119878119873119878119867119873119888 (120579) (31)

where

119888 (120579) ≜ 1198761015840 [119877119879119904 119886 (120579) otimes 119887 (120579)] (32)

and 119878119873 is the matrix of noise subspace formulated by the(119872119873minus119870) eigenvectors corresponding to theminimumeigen-values of the received signals 1199101015840(119901) after the prewhiteningThe DOAs can be estimated as

= (1205790 1205791 120579119870minus1)119879 (33)

by selecting the peak positions of the MUSIC spatial spec-trum

4 Carmeacuter-Rao Lower Bound

A vector 119903 can be formulated by collecting the signals fromall receiving antennas and pulses

119903 = 119868119875 otimes 119879120572 + 119908 (34)

where

119903 ≜ [119903119879 (0) 119903119879 (1) 119903119879 (119875 minus 1)]119879 (35)

120572 ≜ [120572119879 (0) 120572119879 (1) 120572119879 (119875 minus 1)]119879 (36)

119908 ≜ [119908119879 (0) 119908119879 (1) 119908119879 (119875 minus 1)]119879 (37)

By assuming that the fading coefficients follow the zero-meanGaussian distribution

120572 sim CN (0119875119870 1205722119868119875119870) (38)

119903 also follows the zero-mean Gaussian distribution

119903 sim 119891 (119903 120579) = 1120587119872119873119875 det (119862119903) 119890minus119903119867119862minus1

119903119903 (39)

with the covariance matrix being

119862119903 ≜ 1205902119899119868119875 otimes (119877119879119904 otimes 119868119873) + 1205722119868119875 otimes 119879119879119867 (40)

With the receiving signal 119903 the DOA can be estimatedby the MUSIC algorithm proposed in this paper and theestimation performance is bounded by the CRLB

E 10038171003817100381710038171003817120579 minus 1003817100381710038171003817100381722 ge 119892 (120579) ≜ Tr 119865minus1 (41)

1 2 3 4 51

2

3

4

5

075

08

085

09

095

1

Figure 1 The illustration of 119877119904

where 119892(120579) denotes the CRLB of DOA estimation and 119865denotes the Fisher informationmatrix (FIM)The entry in the119894th row and 119895th column (119894 = 0 1 119870minus1 119895 = 0 1 119870minus1)of the FIM can be obtained as

119865119894119895 = minusE1205972 ln119891 (119903 120579)120597120579119894120597120579119895 (42)

and the detailed expressions of FIM are given in theAppendix Similarity with the orthogonal waveforms wehave 119877119904 = 119868119872 and the corresponding CRLB can be alsoobtained

5 Simulation Results

In this section the simulation results are given and theparameters are set as follows the numbers of transmittingantennas receiving antennas and pulses are119872 = 5119873 = 10and 119875 = 10 respectively the carrier frequency is 10GHz thewavelength is 120582 = 003 m the antenna spacing is 119889119879 = 119889119877 =1205822 In Figure 2 the MUSIC algorithm is used to estimatethe DOA of 3 targets where the DOAs are 01881 10461and 14071 respectivelyWith the signal-to-noise ratio (SNR)being 25 dB the MUSIC algorithm is used to estimate theDOA and 119877119904 is shown in Figure 1 After 105 Monte Carloexperiments the root-mean-square error (RMSE) of DOAestimation can be obtained as 00103 The rank of waveformsadopted in Figure 2 is119872 but the rank of waveforms adoptedin Figure 3 is 1 Therefore the estimation performance isdegraded in the scenario with rank-deficient waveforms

In the scenario with nonorthogonal signals Figure 4shows the DOA estimation performance using the MUSICmethod with different types of transmitted signals includingthe orthogonal signals the full-rank nonorthogonal signals(rank119877119904 = 5) and the rank-deficient nonorthogonal signals(rank119877119904 = 1) As shown in this figure with increasing theSNR of received signals the DOA estimation performance isimproved and approaches the CRLBs with both orthogonaland nonorthogonal signals However the nonorthogonalsignals achieveworse estimation performance comparedwith

Mathematical Problems in Engineering 5

0 02 04 06 08 1 12 14 15Angle (rad)

minus20

minus15

minus10

minus5

0

5

10

15

20Sp

atia

l spe

ctru

m (d

B)

MUSIC spatial spectrumTarget DOAs

Figure 2 The MUSIC algorithm to estimate DOA (full rank)

0 02 04 06 08 1 12 14 16Angle (rad)

minus50

0

50

100

150

200

Spat

ial s

pect

rum

(dB)

MUSIC spatial spectrumTarget DOAs

Figure 3 The MUSIC algorithm to estimate DOA (rank = 1)

the orthogonal signals especially in the scenario with rank-deficient waveforms Moreover we also show the DOA esti-mation performance using the state-of-art direction findingmethod In [24] by discretizing the detection area into gridsthe simultaneous orthogonal matching pursuit (SOMP) isproposed and the DOA estimation can be formulated asa sparse reconstruction problem Figure 5 shows the DOAestimation with different waveforms and the better estima-tion performance can be achieved by both the full-rank andthe orthogonal waveforms compared with the rank-deficientwaveform Therefore the nonorthogonality degrades theDOA estimation performance only in the scenario with rank-deficient waveforms

In Figure 6 we show the DOA estimation performancewith different numbers of targets and the MUSIC method

SNR (dB)0 5 10 15 20 25 30 35 40 45

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank=5)RMSE (Non-Orthogonal signals rank=5)CRLB (Non-Orthogonal signals rank=1)RMSE (Non-Orthogonal signals rank=1)

10minus4

10minus3

10minus2

10minus1

100

DO

A es

timat

ion

(RM

SE an

d CR

LB)

Figure 4 DOA estimation performance with different values ofSNR (MUSIC)

SNR (dB)0 5 10 15 20 25 30 35

DO

A es

timat

ion

(RM

SE an

d CR

LB)

006

008

01

012

014

016

018

02

RMSE (Orthogonal signals)RMSE (Non-Orthogonal signals rank=5)RMSE (Non-Orthogonal signals rank=1)

Figure 5 DOA estimation performance with different values ofSNR (SOMP)

is adopted where SNR is 30 dB As shown in this figurethe estimation performance is degraded by more targetsand full-rank cross-correlation matrix can achieve betterestimation performance than the rank-deficient one Theestimation performance can achieve the CRLB in the sce-nario with fewer targets Therefore in the DOA estimationscenario with multiple targets the nonorthogonality givesworse estimation performance especially with the rank-deficient nonorthogonal signals In the future work aboutwaveform design ofMIMO radar systems when the full-rankwaveforms are adopted the DOA estimation performance

6 Mathematical Problems in Engineering

Target Number1 2 3 4 5 6 7

DO

A es

timat

ion

(RM

SE an

d CR

LB)

0

005

01

015

02

025

03

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank = 5)RMSE (Non-Orthogonal signals rank = 5)CRLB (Non-Orthogonal signals rank = 1)RMSE (Non-Orthogonal signals rank = 1)

Figure 6 DOA estimation performance with different numbers oftargets

cannot be degraded greatly so we can choose the full-ranknonorthogonal waveforms with more freedom

6 Conclusion

In this paper the DOA estimation problem with nonorthog-onal signals has been addressed in the MIMO radar systemThe maximum number of targets which is limited by therank of the cross-correlation matrix has been analyzedand given Then the MUSIC algorithm for nonorthogonalsignals has also been given and the CRLB for nonorthogonalsignals has been derived Simulation results show that theestimation performance is mainly degraded by the rank-deficient cross-correlation matrix of the transmitted signalsFuture work will focus on the waveform design using the full-rank nonorthogonal waveforms in the MIMO radar systems

Appendix

The Detailed Expressions of FIM

The entry in the 119894th row and 119895th column of the FIM can beexpressed as

119865119894119895 = minusE1205972 ln119891 (119903 120579)120597120579119894120597120579119895 = Tr119862minus1119903 120597119862119903120597120579119895 119862

minus1119903

120597119862119903120597120579119894 + Tr119862minus1119903 1205972119862119903120597120579119894120597120579119895 (A1)

where 120597119862119903120597120579119894 can be simplified as

120597119862119903120597120579119894 = 1205722119868119875 otimes 120597119879119879119867120597120579119894

= 1205722119868119875 otimes (120597119879120597120579119894119879119867 + 119879120597119879119867120597120579119894 )

(A2)

where

120597119879120597120579119894 = [0119872119873times(119894minus1) 120597119905119894120597120579119894 0119872119873times(119870minus119894)] (A3)

120597119905119894120597120579119894 = 119877119879119904

120597119886 (120579119894)120597120579119894 otimes 119887 (120579119894) + 119877119879119904 119886 (120579119894) otimes 120597119887 (120579119894)120597120579119894 (A4)

Additionally we can obtain

120597119886 (120579119894)120597120579119894 = 1198952120587120582 119889119879 sin 120579119894 (120578119872 ⊙ 119886 (120579119894)) (A5)

And

120578119872 ≜ [0 1 119872 minus 1]119879 (A6)

Using the same method the derivative of 119887(120579119894) can be alsoobtained as

120597119887 (120579119894)120597120579119894 = 1198952120587120582 119889119877 sin 120579119894 (120578119873 ⊙ 119887 (120579119894)) (A7)

The second-order derivative of 119862119903 can be expressed as

1205972119862119903120597120579119894120597120579119895= 1205722119868119875 otimes [120597 ((120597119879120597120579119894)119879119867)120597120579119895 + 120597 (119879 (120597119879119867120597120579119894))120597120579119895 ]= 1205722119868119875

otimes [ 1205972119879120597120579119894120597120579119895119879119867 + 120597119879120597120579119894

120597119879119867120597120579119895 + 120597119879120597120579119895120597119879119867120597120579119894 + 119879 1205972119879119867120597120579119894120597120579119895]

=

0119872119873119875 119894 = 1198951205722119868119875 otimes (12059721199051198941205972120579119894 119905119867119894 + 2 120597119905119894120597120579119894

120597119905119867119894120597120579119894 + 11990511989412059721199051198671198941205972120579119894 ) 119894 = 119895

(A8)

where

12059721199051198941205972120579119894 =120597 [119877119879119904 (120597119886 (120579119894) 120597120579119894) otimes 119887 (120579119894)]120597120579119894+ 120597 [119877119879119904 119886 (120579119894) otimes (120597119887 (120579119894) 120597120579119894)]120597120579119894

= 119877119879119904 1205972119886 (120579119894)1205972120579119894 otimes 119887 (120579119894) + 2119877119879119904 120597119886 (120579119894)120597120579119894

otimes 120597119887 (120579119894)120597120579119894 + 119877119879119904 119886 (120579119894) otimes 1205972119887 (120579119894)1205972120579119894

Mathematical Problems in Engineering 7

1205972119886 (120579119894)1205972120579119894 = 1198952120587120582 119889119879 120597 sin 120579119894120578119872 ⊙ 119886 (120579119894)120597120579119894= 1198952120587120582 119889119879120578119872

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) 1205972119887 (120579119894)1205972120579119894 = 1198952120587120582 119889119877120578119873

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) (A9)

Conflicts of Interest

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

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (Grants nos 61471117 and61601281) the Open Program of the State Key Labora-tory of Millimeter Waves (Southeast University) (Grant noZ201804) and the Fundamental Research Funds for theCentral Universities (Southeast University)

References

[1] P Chen C Qi and L Wu ldquoAntenna placement optimisationfor compressed sensing-based distributed MIMO radarrdquo IETRadar Sonar amp Navigation vol 11 no 2 pp 285ndash293 2017

[2] J Li P Stoica L Xu and W Roberts ldquoOn parameter identifi-ability of MIMO radarrdquo IEEE Signal Processing Letters vol 14no 12 pp 968ndash971 2007

[3] I Bekkerman and J Tabrikian ldquoTarget detection and localiza-tion using MIMO radars and sonarsrdquo IEEE Transactions onSignal Processing vol 54 no 10 pp 3873ndash3883 2006

[4] J Li and P Stoica ldquoMIMO radar with colocated antennasrdquo IEEESignal Processing Magazine vol 24 no 5 pp 106ndash114 2007

[5] P Chen and L Wu ldquoSystem optimization for temporal corre-lated cognitive radar with EBPSK-based MCPC signalrdquo Math-ematical Problems in Engineering vol 2015 Article ID 302083pp 1ndash10 2015

[6] P Chen L Wu and C Qi ldquoWaveform optimization for targetscattering coefficients estimation under detection and peak-to-average power ratio constraints in cognitive radarrdquo CircuitsSystems and Signal Processing vol 35 no 1 pp 163ndash184 2016

[7] M Haghnegahdar S Imani S A Ghorashi and E MehrshahildquoSINR Enhancement in Colocated MIMO Radar Using Trans-mit Covariance Matrix Optimizationrdquo IEEE Signal ProcessingLetters vol 24 no 3 pp 339ndash343 2017

[8] P Chen L Zheng X Wang H Li and L Wu ldquoMoving targetdetection using colocated MIMO radar on multiple distributedmoving platformsrdquo IEEE Transactions on Signal Processing vol65 no 17 pp 4670ndash4683 2017

[9] X Zhang andDXu ldquoLow-complexity ESPRIT-basedDOAesti-mation for colocated MIMO radar using reduced-dimensiontransformationrdquo IEEE Electronics Letters vol 47 no 4 pp 283-284 2011

[10] Y Zhang G Zhang and X Wang ldquoComputationally efficientDOA estimation for monostatic MIMO radar based on covari-ancematrix reconstructionrdquo IEEEElectronics Letters vol 53 no2 pp 111ndash113 2017

[11] D Oh Y-C Li J Khodjaev J-W Chong and J-H Lee ldquoJointestimation of direction of departure and direction of arrival formultiple-input multiple-output radar based on improved jointESPRIT methodrdquo IET Radar Sonar amp Navigation vol 9 no 3pp 308ndash317 2015

[12] Y Jia X Zhong Y Guo and W Huo ldquoDOA and DODestimation based on bistatic MIMO radar with co-prime arrayrdquoin Proceedings of the 2017 IEEE Radar Conference RadarConf2017 pp 0394ndash0397 Seattle Wash USA May 2017

[13] J Li D Jiang and X Zhang ldquoDOA estimation based oncombined unitary ESPRIT for coprime MIMO radarrdquo IEEECommunications Letters vol 21 no 1 pp 96ndash99 2017

[14] H Jiang J-K Zhang and K M Wong ldquoJoint DOD and DOAestimation for bistatic MIMO radar in unknown correlatednoiserdquo IEEE Transactions on Vehicular Technology vol 64 no11 pp 5113ndash5125 2015

[15] A Khabbazibasmenj A Hassanien S A Vorobyov and M WMorency ldquoEfficient transmit beamspace design for search-freebased DOA estimation in MIMO radarrdquo IEEE Transactions onSignal Processing vol 62 no 6 pp 1490ndash1500 2014

[16] A Hassanien and S A Vorobyov ldquoTransmit energy focusingfor DOA estimation in MIMO radar with colocated antennasrdquoIEEE Transactions on Signal Processing vol 59 no 6 pp 2669ndash2682 2011

[17] P Chen P Zhan and L Wu ldquoClutter estimation based oncompressed sensing in bistatic MIMO radarrdquo IET Radar Sonaramp Navigation vol 11 no 2 pp 285ndash293 2017

[18] M L Bencheikh and Y Wang ldquoJoint DOD-DOA estimationusing combined ESPRIT-MUSIC approach in MIMO radarrdquoIEEE Electronics Letters vol 46 no 15 pp 1081ndash1083 2010

[19] A Hassanien S A Vorobyov and A KhabbazibasmenjldquoTransmit radiation pattern invariance in MIMO radar withapplication to DOA estimationrdquo IEEE Signal Processing Lettersvol 22 no 10 pp 1609ndash1613 2015

[20] X Zhang L Xu and D Xu ldquoDirection of departure (DOD)and direction of arrival (DOA) estimation inMIMO radar withreduced-dimension MUSICrdquo IEEE Communications Lettersvol 14 no 12 pp 1161ndash1163 2010

[21] M Cattenoz and S Marcos ldquoAdaptive processing for MIMOradar realistic non perfectly orthogonal waveformsrdquo in Proceed-ings of the 2014 IEEE Radar Conference RadarCon 2014 pp1323ndash1328 Cincinnati Ohio USA May 2014

[22] G Zheng ldquoDOA estimation inMIMO radar with non-perfectlyorthogonal waveformsrdquo IEEE Communications Letters vol 21no 2 pp 414ndash417 2017

[23] P Chen C Qi L Wu and X Wang ldquoEstimation of extendedtargets based on compressed sensing in cognitive radar systemrdquoIEEE Transactions on Vehicular Technology vol 66 no 2 pp941ndash951 2017

[24] J-F Determe J Louveaux L Jacques and F Horlin ldquoOn thenoise robustness of simultaneous orthogonalmatching pursuitrdquoIEEE Transactions on Signal Processing vol 65 no 4 pp 864ndash875 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

0 02 04 06 08 1 12 14 15Angle (rad)

minus20

minus15

minus10

minus5

0

5

10

15

20Sp

atia

l spe

ctru

m (d

B)

MUSIC spatial spectrumTarget DOAs

Figure 2 The MUSIC algorithm to estimate DOA (full rank)

0 02 04 06 08 1 12 14 16Angle (rad)

minus50

0

50

100

150

200

Spat

ial s

pect

rum

(dB)

MUSIC spatial spectrumTarget DOAs

Figure 3 The MUSIC algorithm to estimate DOA (rank = 1)

the orthogonal signals especially in the scenario with rank-deficient waveforms Moreover we also show the DOA esti-mation performance using the state-of-art direction findingmethod In [24] by discretizing the detection area into gridsthe simultaneous orthogonal matching pursuit (SOMP) isproposed and the DOA estimation can be formulated asa sparse reconstruction problem Figure 5 shows the DOAestimation with different waveforms and the better estima-tion performance can be achieved by both the full-rank andthe orthogonal waveforms compared with the rank-deficientwaveform Therefore the nonorthogonality degrades theDOA estimation performance only in the scenario with rank-deficient waveforms

In Figure 6 we show the DOA estimation performancewith different numbers of targets and the MUSIC method

SNR (dB)0 5 10 15 20 25 30 35 40 45

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank=5)RMSE (Non-Orthogonal signals rank=5)CRLB (Non-Orthogonal signals rank=1)RMSE (Non-Orthogonal signals rank=1)

10minus4

10minus3

10minus2

10minus1

100

DO

A es

timat

ion

(RM

SE an

d CR

LB)

Figure 4 DOA estimation performance with different values ofSNR (MUSIC)

SNR (dB)0 5 10 15 20 25 30 35

DO

A es

timat

ion

(RM

SE an

d CR

LB)

006

008

01

012

014

016

018

02

RMSE (Orthogonal signals)RMSE (Non-Orthogonal signals rank=5)RMSE (Non-Orthogonal signals rank=1)

Figure 5 DOA estimation performance with different values ofSNR (SOMP)

is adopted where SNR is 30 dB As shown in this figurethe estimation performance is degraded by more targetsand full-rank cross-correlation matrix can achieve betterestimation performance than the rank-deficient one Theestimation performance can achieve the CRLB in the sce-nario with fewer targets Therefore in the DOA estimationscenario with multiple targets the nonorthogonality givesworse estimation performance especially with the rank-deficient nonorthogonal signals In the future work aboutwaveform design ofMIMO radar systems when the full-rankwaveforms are adopted the DOA estimation performance

6 Mathematical Problems in Engineering

Target Number1 2 3 4 5 6 7

DO

A es

timat

ion

(RM

SE an

d CR

LB)

0

005

01

015

02

025

03

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank = 5)RMSE (Non-Orthogonal signals rank = 5)CRLB (Non-Orthogonal signals rank = 1)RMSE (Non-Orthogonal signals rank = 1)

Figure 6 DOA estimation performance with different numbers oftargets

cannot be degraded greatly so we can choose the full-ranknonorthogonal waveforms with more freedom

6 Conclusion

In this paper the DOA estimation problem with nonorthog-onal signals has been addressed in the MIMO radar systemThe maximum number of targets which is limited by therank of the cross-correlation matrix has been analyzedand given Then the MUSIC algorithm for nonorthogonalsignals has also been given and the CRLB for nonorthogonalsignals has been derived Simulation results show that theestimation performance is mainly degraded by the rank-deficient cross-correlation matrix of the transmitted signalsFuture work will focus on the waveform design using the full-rank nonorthogonal waveforms in the MIMO radar systems

Appendix

The Detailed Expressions of FIM

The entry in the 119894th row and 119895th column of the FIM can beexpressed as

119865119894119895 = minusE1205972 ln119891 (119903 120579)120597120579119894120597120579119895 = Tr119862minus1119903 120597119862119903120597120579119895 119862

minus1119903

120597119862119903120597120579119894 + Tr119862minus1119903 1205972119862119903120597120579119894120597120579119895 (A1)

where 120597119862119903120597120579119894 can be simplified as

120597119862119903120597120579119894 = 1205722119868119875 otimes 120597119879119879119867120597120579119894

= 1205722119868119875 otimes (120597119879120597120579119894119879119867 + 119879120597119879119867120597120579119894 )

(A2)

where

120597119879120597120579119894 = [0119872119873times(119894minus1) 120597119905119894120597120579119894 0119872119873times(119870minus119894)] (A3)

120597119905119894120597120579119894 = 119877119879119904

120597119886 (120579119894)120597120579119894 otimes 119887 (120579119894) + 119877119879119904 119886 (120579119894) otimes 120597119887 (120579119894)120597120579119894 (A4)

Additionally we can obtain

120597119886 (120579119894)120597120579119894 = 1198952120587120582 119889119879 sin 120579119894 (120578119872 ⊙ 119886 (120579119894)) (A5)

And

120578119872 ≜ [0 1 119872 minus 1]119879 (A6)

Using the same method the derivative of 119887(120579119894) can be alsoobtained as

120597119887 (120579119894)120597120579119894 = 1198952120587120582 119889119877 sin 120579119894 (120578119873 ⊙ 119887 (120579119894)) (A7)

The second-order derivative of 119862119903 can be expressed as

1205972119862119903120597120579119894120597120579119895= 1205722119868119875 otimes [120597 ((120597119879120597120579119894)119879119867)120597120579119895 + 120597 (119879 (120597119879119867120597120579119894))120597120579119895 ]= 1205722119868119875

otimes [ 1205972119879120597120579119894120597120579119895119879119867 + 120597119879120597120579119894

120597119879119867120597120579119895 + 120597119879120597120579119895120597119879119867120597120579119894 + 119879 1205972119879119867120597120579119894120597120579119895]

=

0119872119873119875 119894 = 1198951205722119868119875 otimes (12059721199051198941205972120579119894 119905119867119894 + 2 120597119905119894120597120579119894

120597119905119867119894120597120579119894 + 11990511989412059721199051198671198941205972120579119894 ) 119894 = 119895

(A8)

where

12059721199051198941205972120579119894 =120597 [119877119879119904 (120597119886 (120579119894) 120597120579119894) otimes 119887 (120579119894)]120597120579119894+ 120597 [119877119879119904 119886 (120579119894) otimes (120597119887 (120579119894) 120597120579119894)]120597120579119894

= 119877119879119904 1205972119886 (120579119894)1205972120579119894 otimes 119887 (120579119894) + 2119877119879119904 120597119886 (120579119894)120597120579119894

otimes 120597119887 (120579119894)120597120579119894 + 119877119879119904 119886 (120579119894) otimes 1205972119887 (120579119894)1205972120579119894

Mathematical Problems in Engineering 7

1205972119886 (120579119894)1205972120579119894 = 1198952120587120582 119889119879 120597 sin 120579119894120578119872 ⊙ 119886 (120579119894)120597120579119894= 1198952120587120582 119889119879120578119872

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) 1205972119887 (120579119894)1205972120579119894 = 1198952120587120582 119889119877120578119873

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) (A9)

Conflicts of Interest

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

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (Grants nos 61471117 and61601281) the Open Program of the State Key Labora-tory of Millimeter Waves (Southeast University) (Grant noZ201804) and the Fundamental Research Funds for theCentral Universities (Southeast University)

References

[1] P Chen C Qi and L Wu ldquoAntenna placement optimisationfor compressed sensing-based distributed MIMO radarrdquo IETRadar Sonar amp Navigation vol 11 no 2 pp 285ndash293 2017

[2] J Li P Stoica L Xu and W Roberts ldquoOn parameter identifi-ability of MIMO radarrdquo IEEE Signal Processing Letters vol 14no 12 pp 968ndash971 2007

[3] I Bekkerman and J Tabrikian ldquoTarget detection and localiza-tion using MIMO radars and sonarsrdquo IEEE Transactions onSignal Processing vol 54 no 10 pp 3873ndash3883 2006

[4] J Li and P Stoica ldquoMIMO radar with colocated antennasrdquo IEEESignal Processing Magazine vol 24 no 5 pp 106ndash114 2007

[5] P Chen and L Wu ldquoSystem optimization for temporal corre-lated cognitive radar with EBPSK-based MCPC signalrdquo Math-ematical Problems in Engineering vol 2015 Article ID 302083pp 1ndash10 2015

[6] P Chen L Wu and C Qi ldquoWaveform optimization for targetscattering coefficients estimation under detection and peak-to-average power ratio constraints in cognitive radarrdquo CircuitsSystems and Signal Processing vol 35 no 1 pp 163ndash184 2016

[7] M Haghnegahdar S Imani S A Ghorashi and E MehrshahildquoSINR Enhancement in Colocated MIMO Radar Using Trans-mit Covariance Matrix Optimizationrdquo IEEE Signal ProcessingLetters vol 24 no 3 pp 339ndash343 2017

[8] P Chen L Zheng X Wang H Li and L Wu ldquoMoving targetdetection using colocated MIMO radar on multiple distributedmoving platformsrdquo IEEE Transactions on Signal Processing vol65 no 17 pp 4670ndash4683 2017

[9] X Zhang andDXu ldquoLow-complexity ESPRIT-basedDOAesti-mation for colocated MIMO radar using reduced-dimensiontransformationrdquo IEEE Electronics Letters vol 47 no 4 pp 283-284 2011

[10] Y Zhang G Zhang and X Wang ldquoComputationally efficientDOA estimation for monostatic MIMO radar based on covari-ancematrix reconstructionrdquo IEEEElectronics Letters vol 53 no2 pp 111ndash113 2017

[11] D Oh Y-C Li J Khodjaev J-W Chong and J-H Lee ldquoJointestimation of direction of departure and direction of arrival formultiple-input multiple-output radar based on improved jointESPRIT methodrdquo IET Radar Sonar amp Navigation vol 9 no 3pp 308ndash317 2015

[12] Y Jia X Zhong Y Guo and W Huo ldquoDOA and DODestimation based on bistatic MIMO radar with co-prime arrayrdquoin Proceedings of the 2017 IEEE Radar Conference RadarConf2017 pp 0394ndash0397 Seattle Wash USA May 2017

[13] J Li D Jiang and X Zhang ldquoDOA estimation based oncombined unitary ESPRIT for coprime MIMO radarrdquo IEEECommunications Letters vol 21 no 1 pp 96ndash99 2017

[14] H Jiang J-K Zhang and K M Wong ldquoJoint DOD and DOAestimation for bistatic MIMO radar in unknown correlatednoiserdquo IEEE Transactions on Vehicular Technology vol 64 no11 pp 5113ndash5125 2015

[15] A Khabbazibasmenj A Hassanien S A Vorobyov and M WMorency ldquoEfficient transmit beamspace design for search-freebased DOA estimation in MIMO radarrdquo IEEE Transactions onSignal Processing vol 62 no 6 pp 1490ndash1500 2014

[16] A Hassanien and S A Vorobyov ldquoTransmit energy focusingfor DOA estimation in MIMO radar with colocated antennasrdquoIEEE Transactions on Signal Processing vol 59 no 6 pp 2669ndash2682 2011

[17] P Chen P Zhan and L Wu ldquoClutter estimation based oncompressed sensing in bistatic MIMO radarrdquo IET Radar Sonaramp Navigation vol 11 no 2 pp 285ndash293 2017

[18] M L Bencheikh and Y Wang ldquoJoint DOD-DOA estimationusing combined ESPRIT-MUSIC approach in MIMO radarrdquoIEEE Electronics Letters vol 46 no 15 pp 1081ndash1083 2010

[19] A Hassanien S A Vorobyov and A KhabbazibasmenjldquoTransmit radiation pattern invariance in MIMO radar withapplication to DOA estimationrdquo IEEE Signal Processing Lettersvol 22 no 10 pp 1609ndash1613 2015

[20] X Zhang L Xu and D Xu ldquoDirection of departure (DOD)and direction of arrival (DOA) estimation inMIMO radar withreduced-dimension MUSICrdquo IEEE Communications Lettersvol 14 no 12 pp 1161ndash1163 2010

[21] M Cattenoz and S Marcos ldquoAdaptive processing for MIMOradar realistic non perfectly orthogonal waveformsrdquo in Proceed-ings of the 2014 IEEE Radar Conference RadarCon 2014 pp1323ndash1328 Cincinnati Ohio USA May 2014

[22] G Zheng ldquoDOA estimation inMIMO radar with non-perfectlyorthogonal waveformsrdquo IEEE Communications Letters vol 21no 2 pp 414ndash417 2017

[23] P Chen C Qi L Wu and X Wang ldquoEstimation of extendedtargets based on compressed sensing in cognitive radar systemrdquoIEEE Transactions on Vehicular Technology vol 66 no 2 pp941ndash951 2017

[24] J-F Determe J Louveaux L Jacques and F Horlin ldquoOn thenoise robustness of simultaneous orthogonalmatching pursuitrdquoIEEE Transactions on Signal Processing vol 65 no 4 pp 864ndash875 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

Target Number1 2 3 4 5 6 7

DO

A es

timat

ion

(RM

SE an

d CR

LB)

0

005

01

015

02

025

03

CRLB (Orthogonal signals)RMSE (Orthogonal signals)CRLB (Non-Orthogonal signals rank = 5)RMSE (Non-Orthogonal signals rank = 5)CRLB (Non-Orthogonal signals rank = 1)RMSE (Non-Orthogonal signals rank = 1)

Figure 6 DOA estimation performance with different numbers oftargets

cannot be degraded greatly so we can choose the full-ranknonorthogonal waveforms with more freedom

6 Conclusion

In this paper the DOA estimation problem with nonorthog-onal signals has been addressed in the MIMO radar systemThe maximum number of targets which is limited by therank of the cross-correlation matrix has been analyzedand given Then the MUSIC algorithm for nonorthogonalsignals has also been given and the CRLB for nonorthogonalsignals has been derived Simulation results show that theestimation performance is mainly degraded by the rank-deficient cross-correlation matrix of the transmitted signalsFuture work will focus on the waveform design using the full-rank nonorthogonal waveforms in the MIMO radar systems

Appendix

The Detailed Expressions of FIM

The entry in the 119894th row and 119895th column of the FIM can beexpressed as

119865119894119895 = minusE1205972 ln119891 (119903 120579)120597120579119894120597120579119895 = Tr119862minus1119903 120597119862119903120597120579119895 119862

minus1119903

120597119862119903120597120579119894 + Tr119862minus1119903 1205972119862119903120597120579119894120597120579119895 (A1)

where 120597119862119903120597120579119894 can be simplified as

120597119862119903120597120579119894 = 1205722119868119875 otimes 120597119879119879119867120597120579119894

= 1205722119868119875 otimes (120597119879120597120579119894119879119867 + 119879120597119879119867120597120579119894 )

(A2)

where

120597119879120597120579119894 = [0119872119873times(119894minus1) 120597119905119894120597120579119894 0119872119873times(119870minus119894)] (A3)

120597119905119894120597120579119894 = 119877119879119904

120597119886 (120579119894)120597120579119894 otimes 119887 (120579119894) + 119877119879119904 119886 (120579119894) otimes 120597119887 (120579119894)120597120579119894 (A4)

Additionally we can obtain

120597119886 (120579119894)120597120579119894 = 1198952120587120582 119889119879 sin 120579119894 (120578119872 ⊙ 119886 (120579119894)) (A5)

And

120578119872 ≜ [0 1 119872 minus 1]119879 (A6)

Using the same method the derivative of 119887(120579119894) can be alsoobtained as

120597119887 (120579119894)120597120579119894 = 1198952120587120582 119889119877 sin 120579119894 (120578119873 ⊙ 119887 (120579119894)) (A7)

The second-order derivative of 119862119903 can be expressed as

1205972119862119903120597120579119894120597120579119895= 1205722119868119875 otimes [120597 ((120597119879120597120579119894)119879119867)120597120579119895 + 120597 (119879 (120597119879119867120597120579119894))120597120579119895 ]= 1205722119868119875

otimes [ 1205972119879120597120579119894120597120579119895119879119867 + 120597119879120597120579119894

120597119879119867120597120579119895 + 120597119879120597120579119895120597119879119867120597120579119894 + 119879 1205972119879119867120597120579119894120597120579119895]

=

0119872119873119875 119894 = 1198951205722119868119875 otimes (12059721199051198941205972120579119894 119905119867119894 + 2 120597119905119894120597120579119894

120597119905119867119894120597120579119894 + 11990511989412059721199051198671198941205972120579119894 ) 119894 = 119895

(A8)

where

12059721199051198941205972120579119894 =120597 [119877119879119904 (120597119886 (120579119894) 120597120579119894) otimes 119887 (120579119894)]120597120579119894+ 120597 [119877119879119904 119886 (120579119894) otimes (120597119887 (120579119894) 120597120579119894)]120597120579119894

= 119877119879119904 1205972119886 (120579119894)1205972120579119894 otimes 119887 (120579119894) + 2119877119879119904 120597119886 (120579119894)120597120579119894

otimes 120597119887 (120579119894)120597120579119894 + 119877119879119904 119886 (120579119894) otimes 1205972119887 (120579119894)1205972120579119894

Mathematical Problems in Engineering 7

1205972119886 (120579119894)1205972120579119894 = 1198952120587120582 119889119879 120597 sin 120579119894120578119872 ⊙ 119886 (120579119894)120597120579119894= 1198952120587120582 119889119879120578119872

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) 1205972119887 (120579119894)1205972120579119894 = 1198952120587120582 119889119877120578119873

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) (A9)

Conflicts of Interest

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

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (Grants nos 61471117 and61601281) the Open Program of the State Key Labora-tory of Millimeter Waves (Southeast University) (Grant noZ201804) and the Fundamental Research Funds for theCentral Universities (Southeast University)

References

[1] P Chen C Qi and L Wu ldquoAntenna placement optimisationfor compressed sensing-based distributed MIMO radarrdquo IETRadar Sonar amp Navigation vol 11 no 2 pp 285ndash293 2017

[2] J Li P Stoica L Xu and W Roberts ldquoOn parameter identifi-ability of MIMO radarrdquo IEEE Signal Processing Letters vol 14no 12 pp 968ndash971 2007

[3] I Bekkerman and J Tabrikian ldquoTarget detection and localiza-tion using MIMO radars and sonarsrdquo IEEE Transactions onSignal Processing vol 54 no 10 pp 3873ndash3883 2006

[4] J Li and P Stoica ldquoMIMO radar with colocated antennasrdquo IEEESignal Processing Magazine vol 24 no 5 pp 106ndash114 2007

[5] P Chen and L Wu ldquoSystem optimization for temporal corre-lated cognitive radar with EBPSK-based MCPC signalrdquo Math-ematical Problems in Engineering vol 2015 Article ID 302083pp 1ndash10 2015

[6] P Chen L Wu and C Qi ldquoWaveform optimization for targetscattering coefficients estimation under detection and peak-to-average power ratio constraints in cognitive radarrdquo CircuitsSystems and Signal Processing vol 35 no 1 pp 163ndash184 2016

[7] M Haghnegahdar S Imani S A Ghorashi and E MehrshahildquoSINR Enhancement in Colocated MIMO Radar Using Trans-mit Covariance Matrix Optimizationrdquo IEEE Signal ProcessingLetters vol 24 no 3 pp 339ndash343 2017

[8] P Chen L Zheng X Wang H Li and L Wu ldquoMoving targetdetection using colocated MIMO radar on multiple distributedmoving platformsrdquo IEEE Transactions on Signal Processing vol65 no 17 pp 4670ndash4683 2017

[9] X Zhang andDXu ldquoLow-complexity ESPRIT-basedDOAesti-mation for colocated MIMO radar using reduced-dimensiontransformationrdquo IEEE Electronics Letters vol 47 no 4 pp 283-284 2011

[10] Y Zhang G Zhang and X Wang ldquoComputationally efficientDOA estimation for monostatic MIMO radar based on covari-ancematrix reconstructionrdquo IEEEElectronics Letters vol 53 no2 pp 111ndash113 2017

[11] D Oh Y-C Li J Khodjaev J-W Chong and J-H Lee ldquoJointestimation of direction of departure and direction of arrival formultiple-input multiple-output radar based on improved jointESPRIT methodrdquo IET Radar Sonar amp Navigation vol 9 no 3pp 308ndash317 2015

[12] Y Jia X Zhong Y Guo and W Huo ldquoDOA and DODestimation based on bistatic MIMO radar with co-prime arrayrdquoin Proceedings of the 2017 IEEE Radar Conference RadarConf2017 pp 0394ndash0397 Seattle Wash USA May 2017

[13] J Li D Jiang and X Zhang ldquoDOA estimation based oncombined unitary ESPRIT for coprime MIMO radarrdquo IEEECommunications Letters vol 21 no 1 pp 96ndash99 2017

[14] H Jiang J-K Zhang and K M Wong ldquoJoint DOD and DOAestimation for bistatic MIMO radar in unknown correlatednoiserdquo IEEE Transactions on Vehicular Technology vol 64 no11 pp 5113ndash5125 2015

[15] A Khabbazibasmenj A Hassanien S A Vorobyov and M WMorency ldquoEfficient transmit beamspace design for search-freebased DOA estimation in MIMO radarrdquo IEEE Transactions onSignal Processing vol 62 no 6 pp 1490ndash1500 2014

[16] A Hassanien and S A Vorobyov ldquoTransmit energy focusingfor DOA estimation in MIMO radar with colocated antennasrdquoIEEE Transactions on Signal Processing vol 59 no 6 pp 2669ndash2682 2011

[17] P Chen P Zhan and L Wu ldquoClutter estimation based oncompressed sensing in bistatic MIMO radarrdquo IET Radar Sonaramp Navigation vol 11 no 2 pp 285ndash293 2017

[18] M L Bencheikh and Y Wang ldquoJoint DOD-DOA estimationusing combined ESPRIT-MUSIC approach in MIMO radarrdquoIEEE Electronics Letters vol 46 no 15 pp 1081ndash1083 2010

[19] A Hassanien S A Vorobyov and A KhabbazibasmenjldquoTransmit radiation pattern invariance in MIMO radar withapplication to DOA estimationrdquo IEEE Signal Processing Lettersvol 22 no 10 pp 1609ndash1613 2015

[20] X Zhang L Xu and D Xu ldquoDirection of departure (DOD)and direction of arrival (DOA) estimation inMIMO radar withreduced-dimension MUSICrdquo IEEE Communications Lettersvol 14 no 12 pp 1161ndash1163 2010

[21] M Cattenoz and S Marcos ldquoAdaptive processing for MIMOradar realistic non perfectly orthogonal waveformsrdquo in Proceed-ings of the 2014 IEEE Radar Conference RadarCon 2014 pp1323ndash1328 Cincinnati Ohio USA May 2014

[22] G Zheng ldquoDOA estimation inMIMO radar with non-perfectlyorthogonal waveformsrdquo IEEE Communications Letters vol 21no 2 pp 414ndash417 2017

[23] P Chen C Qi L Wu and X Wang ldquoEstimation of extendedtargets based on compressed sensing in cognitive radar systemrdquoIEEE Transactions on Vehicular Technology vol 66 no 2 pp941ndash951 2017

[24] J-F Determe J Louveaux L Jacques and F Horlin ldquoOn thenoise robustness of simultaneous orthogonalmatching pursuitrdquoIEEE Transactions on Signal Processing vol 65 no 4 pp 864ndash875 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

1205972119886 (120579119894)1205972120579119894 = 1198952120587120582 119889119879 120597 sin 120579119894120578119872 ⊙ 119886 (120579119894)120597120579119894= 1198952120587120582 119889119879120578119872

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) 1205972119887 (120579119894)1205972120579119894 = 1198952120587120582 119889119877120578119873

⊙ (119886 (120579119894) cos 120579119894 + 120597119886 (120579119894)120597120579119894 sin 120579119894) (A9)

Conflicts of Interest

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

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (Grants nos 61471117 and61601281) the Open Program of the State Key Labora-tory of Millimeter Waves (Southeast University) (Grant noZ201804) and the Fundamental Research Funds for theCentral Universities (Southeast University)

References

[1] P Chen C Qi and L Wu ldquoAntenna placement optimisationfor compressed sensing-based distributed MIMO radarrdquo IETRadar Sonar amp Navigation vol 11 no 2 pp 285ndash293 2017

[2] J Li P Stoica L Xu and W Roberts ldquoOn parameter identifi-ability of MIMO radarrdquo IEEE Signal Processing Letters vol 14no 12 pp 968ndash971 2007

[3] I Bekkerman and J Tabrikian ldquoTarget detection and localiza-tion using MIMO radars and sonarsrdquo IEEE Transactions onSignal Processing vol 54 no 10 pp 3873ndash3883 2006

[4] J Li and P Stoica ldquoMIMO radar with colocated antennasrdquo IEEESignal Processing Magazine vol 24 no 5 pp 106ndash114 2007

[5] P Chen and L Wu ldquoSystem optimization for temporal corre-lated cognitive radar with EBPSK-based MCPC signalrdquo Math-ematical Problems in Engineering vol 2015 Article ID 302083pp 1ndash10 2015

[6] P Chen L Wu and C Qi ldquoWaveform optimization for targetscattering coefficients estimation under detection and peak-to-average power ratio constraints in cognitive radarrdquo CircuitsSystems and Signal Processing vol 35 no 1 pp 163ndash184 2016

[7] M Haghnegahdar S Imani S A Ghorashi and E MehrshahildquoSINR Enhancement in Colocated MIMO Radar Using Trans-mit Covariance Matrix Optimizationrdquo IEEE Signal ProcessingLetters vol 24 no 3 pp 339ndash343 2017

[8] P Chen L Zheng X Wang H Li and L Wu ldquoMoving targetdetection using colocated MIMO radar on multiple distributedmoving platformsrdquo IEEE Transactions on Signal Processing vol65 no 17 pp 4670ndash4683 2017

[9] X Zhang andDXu ldquoLow-complexity ESPRIT-basedDOAesti-mation for colocated MIMO radar using reduced-dimensiontransformationrdquo IEEE Electronics Letters vol 47 no 4 pp 283-284 2011

[10] Y Zhang G Zhang and X Wang ldquoComputationally efficientDOA estimation for monostatic MIMO radar based on covari-ancematrix reconstructionrdquo IEEEElectronics Letters vol 53 no2 pp 111ndash113 2017

[11] D Oh Y-C Li J Khodjaev J-W Chong and J-H Lee ldquoJointestimation of direction of departure and direction of arrival formultiple-input multiple-output radar based on improved jointESPRIT methodrdquo IET Radar Sonar amp Navigation vol 9 no 3pp 308ndash317 2015

[12] Y Jia X Zhong Y Guo and W Huo ldquoDOA and DODestimation based on bistatic MIMO radar with co-prime arrayrdquoin Proceedings of the 2017 IEEE Radar Conference RadarConf2017 pp 0394ndash0397 Seattle Wash USA May 2017

[13] J Li D Jiang and X Zhang ldquoDOA estimation based oncombined unitary ESPRIT for coprime MIMO radarrdquo IEEECommunications Letters vol 21 no 1 pp 96ndash99 2017

[14] H Jiang J-K Zhang and K M Wong ldquoJoint DOD and DOAestimation for bistatic MIMO radar in unknown correlatednoiserdquo IEEE Transactions on Vehicular Technology vol 64 no11 pp 5113ndash5125 2015

[15] A Khabbazibasmenj A Hassanien S A Vorobyov and M WMorency ldquoEfficient transmit beamspace design for search-freebased DOA estimation in MIMO radarrdquo IEEE Transactions onSignal Processing vol 62 no 6 pp 1490ndash1500 2014

[16] A Hassanien and S A Vorobyov ldquoTransmit energy focusingfor DOA estimation in MIMO radar with colocated antennasrdquoIEEE Transactions on Signal Processing vol 59 no 6 pp 2669ndash2682 2011

[17] P Chen P Zhan and L Wu ldquoClutter estimation based oncompressed sensing in bistatic MIMO radarrdquo IET Radar Sonaramp Navigation vol 11 no 2 pp 285ndash293 2017

[18] M L Bencheikh and Y Wang ldquoJoint DOD-DOA estimationusing combined ESPRIT-MUSIC approach in MIMO radarrdquoIEEE Electronics Letters vol 46 no 15 pp 1081ndash1083 2010

[19] A Hassanien S A Vorobyov and A KhabbazibasmenjldquoTransmit radiation pattern invariance in MIMO radar withapplication to DOA estimationrdquo IEEE Signal Processing Lettersvol 22 no 10 pp 1609ndash1613 2015

[20] X Zhang L Xu and D Xu ldquoDirection of departure (DOD)and direction of arrival (DOA) estimation inMIMO radar withreduced-dimension MUSICrdquo IEEE Communications Lettersvol 14 no 12 pp 1161ndash1163 2010

[21] M Cattenoz and S Marcos ldquoAdaptive processing for MIMOradar realistic non perfectly orthogonal waveformsrdquo in Proceed-ings of the 2014 IEEE Radar Conference RadarCon 2014 pp1323ndash1328 Cincinnati Ohio USA May 2014

[22] G Zheng ldquoDOA estimation inMIMO radar with non-perfectlyorthogonal waveformsrdquo IEEE Communications Letters vol 21no 2 pp 414ndash417 2017

[23] P Chen C Qi L Wu and X Wang ldquoEstimation of extendedtargets based on compressed sensing in cognitive radar systemrdquoIEEE Transactions on Vehicular Technology vol 66 no 2 pp941ndash951 2017

[24] J-F Determe J Louveaux L Jacques and F Horlin ldquoOn thenoise robustness of simultaneous orthogonalmatching pursuitrdquoIEEE Transactions on Signal Processing vol 65 no 4 pp 864ndash875 2017

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom