Tensor …downloads.hindawi.com/journals/mpe/2020/5720189.pdftarget through spectral peak search. At present, there are some solutions to this problem. In [16], a nonlinear FDA radarisproposed,butthismethodneedstochangethearray

Research ArticleTensor-Based Angle and Range Estimation Method in MonostaticFDA-MIMO Radar

Tengxian Xu1 Yongqin Yang 1 Mengxing Huang 1 Han Wang2 Di Wu1 and Qu Yi3

1State Key Laboratory of Marine Resource Utilization in South China SeaSchool of Information and Communication Engineering Haikou 570228 China2College of Physical Science and Technology Yichun University Yichun 336000 China3College of Physics and Electronic Engineering Hainan Normal University Haikou 571158 China

Correspondence should be addressed to Yongqin Yang yangyqhainanueducn and Mengxing Huang huangmx09163com

Received 3 July 2020 Accepted 15 July 2020 Published 12 August 2020

Guest Editor Wang Zheng

Copyright copy 2020 Tengxian Xu et al is 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

In the paper joint angle and range estimation issue for monostatic frequency diverse array multiple-input multiple-output (FDA-MIMO) is proposed and a tensor-based framework is addressed to solve it e proposed method exploits the multidimensionalstructure of matched filters in FDA-MIMO radar Firstly stack the received data to form a third-order tensor so that themultidimensional structure information of the received data can be acquired en the steering matrices contain the angle andrang information are estimated by using the parallel factor (PARAFAC) decomposition Finally the angle and range are achievedby utilizing the phase characteristic of the steering matrices Due to exploiting the multidimensional structure of the received datato further suppress the effect of noise the proposed method performs better in angle and range estimation than the existingalgorithms based on ESPRIT simulation results can prove the proposed methodrsquos effectiveness

1 Introduction

Multiple-input multiple-output (MIMO) radar was firstproposed in [1ndash3] which is a key research point in todayrsquosradar field In MIMO radar all the antennas are omnidi-rectional and the transmitted signals are orthogonal to eachother which can achieve the waveform diversity gain in thereceiver side Currently there are two main types of MIMOradars namely collocated MIMO radar [4ndash6] and statisticalMIMO radar [7 8] e statistical MIMO radar is composedwith separated transmit and receive antennas for obtainingboth the waveform and spatial diversity gain In contrast inorder to improve the estimation performance the collocatedMIMO radar closely places the transmitting and receivingarrays to form a virtual array that has large aperture In thepast few years parameter estimation including direction ofarrival (DOA) direction of departure (DOD) and Dopplerfrequency became a hot topic and investigated by a lot ofresearchers In the common subspace-based methods suchas multiple signal classification algorithm (MUSIC) and

estimation of signal parameters via rotational invariancetechniques (ESPRIT) MIMO radar has used them to achieveDOD and DOA estimation [9ndash14] which can obtain thedesired performance with reasonable SNR and snapshots Inaddition a joint angle and Doppler frequency estimationmethod is investigated which can be used to track thetargets However it is noted that these methods for MIMOradar with narrow-band signals cannot achieve the ranginformation which is very important for target localizationin practice

FDA radar can get the distance information of the target[15] ere is one difference between the FDA radar and thetraditional phased array radar and for the FDA radar thetransmitting frequency of each transmitting antenna has afrequency increment so that it can draw the relevant beampattern improve the degree of freedom in space and thenachieve the joint angle and distance estimation about targetin the FDA radar However the angle and range of the targetin the beam domain of the FDA radar are mixed so it cannotdirectly obtain the angle and distance information of the

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 5720189 8 pageshttpsdoiorg10115520205720189

target through spectral peak search At present there aresome solutions to this problem In [16] a nonlinear FDAradar is proposed but this method needs to change the arrayelement position in actual operation which is difficult toachieve In [17 18] a method based on subarrays to achieveangle and distance decoupling is proposed but this methodhas the problem of distance ambiguity In addition in [19]the target positioning method of the coprime array FDAradar is studied and in [20 21] the nested FDA parameterestimation method is studied

In [18] FDA-MIMO radar with both frequency andwaveform diversity is proposed for the first time whichmakes use of the unique advantages of FDA radar andMIMO radar which not only has excellent target detectionperformance in target detection but also has a high freedomof the spatial degree it can realize the joint estimation of thetarget angle and range [17 22ndash24] At present there aremany kinds of FDA-MIMO radar target positioning algo-rithms such as MUSIC methods and ESPRITmethods butthe MUSIC method involves a spectral peak search andresults in a high algorithm complexity Although the ESPRITmethod can be applied to reduce the computational com-plexity and obtain the angle and range information of thetarget through rotation invariance this method is notsuitable for the phase ambiguity situation

e advantages of FDA are developed based on thecharacteristics of the multidimensional structure of thesignal e abovementioned traditional matrix processingmethod cannot make good use of the multidimensionalcharacteristics which limits their performance [25] To takeadvantage of the multidimensional structural characteristicsof the signal the PARAFAC decomposition is used for angleestimation in unknown target localization by modellingtensor signals [26ndash28] At present there are few studies ontensor-based FDA MIMO target estimation methods epaper proposes the algorithm based on parallel factorizationin monostatic FDA-MIMO radar is method first estab-lishes a third-order tensor signal model and uses parallelfactorization to obtain the transmission directionmatrix andthe reception direction matrix [29 30] en obtain theangle and distance information of the target by taking thephase of the directionmatrix and finally realize the angle anddistance estimation

e proposed method compares the achievement withtraditional estimation of signal parameters via rotationalinvariance techniques (ESPRIT) method [31] Unitary ES-PRIT method [32] and CramerndashRao Bound (CRB) epaper structure is divided into multiple parts Section 2introduces the tensor-based data model Section 3 deducesthe method of angle estimation and distance estimationeCRB of the target estimationmethod in FDA-MIMO radar isshown in Section 4 e Section 5 shows the simulationresults

Notation otimes ⊙ and deg represent the Kronecker productKhatrindashRao product and Hadamard product respectively(middot)T (middot)H (middot)minus 1 (middot)lowast and (middot)+ stand for transpose conju-gate-transpose inverse conjugate and pseudoinverse re-spectively Dn(A) represents a diagonal matrix whichcomposed by the nth row ofA middotF represents the Frobenius

norm IK indicates a K times K identity matrix angle (middot) andmin (middot) represent the phase angle of each element and theminimum element in the array Re(middot) denotes the real partoperator xmnl is denoted as a third-order tensor

2 Tensor-BasedDataModelofMonostaticFDA-MIMO Radar

e paper is based on a narrowband monostatic FDA-MIMO radar It is composed by M-element transmittingantennas and N-element receiving antennas Figure 1 showsthe structure of monostatic FDA-MIMO radar e trans-mitting array and receiving array are placed together so thedirection of arrival (DOA) is the same as the direction ofdeparture (DOD) In general the transmitting array andreceiving array are uniform linear arrays (ULA) and areseparated by half a wavelength In the paper there are K far-field targets by default which are independent of each otherBased on the first element of the transmitting antenna thefrequency of the mth element transmitting antenna is de-rived as

fm f0 +(m minus 1)Δf (1)

where the reference frequency is denoted by f0 and Δfdenotes frequency increase of the transmitting antennaMoreover Δf≪f0

After the matched filter the receiving data can berearranged as

x(t) at θ1 r1( 1113857otimes ar θ1( 1113857 at θ2 r2( 1113857otimes ar θ2( 1113857 at θk rk( 11138571113858

otimes ar θk( 11138571113859s(t) + n(t)

(2)

where n(t) denotes an MN times 1 noise vector that isthe additional white Gaussian noisevector at(θk rk) [1 e(j2π(dλ0)sin(θk)minus j2π(2Δfc)rk)

e(j2π(dλ0)(Mminus 1)sin(θk)minus j2π(2Δfc)(Mminus 1)rk)]T and ar(θk) [1

e(j2πf0dsin(θkc)) e(j2πf0(Nminus 1)dsin(θkc))]T stand for thetransmit steering vector and the receive steering vectorrespectively and X [x(1) x(2) x(L)] stands for thereceived data of L samples which can be expressed as

X

X1

X2

⋮

XN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

AT ⊙AR1113858 1113859BT

ARD1 AT( 1113857

ARD2 AT( 1113857

⋮

ARDN AT( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BT (3)

where B [b(1) b(2) b(L)]T isin CLtimesK the transmittingdirection matrix is defined as AT [at(θ1 r1)

at(θ2 r2) at(θk rk)] isin CMtimesK and the receiving direc-tion matrix is defined as AR [ar(θ1) ar(θ2)

ar(θk)] isin CNtimesKAT ⊙AR stands for KhatrindashRao product SoXn is expressed as

Xn ARDn AT( 1113857BT n 1 2 N (4)

Under the interference of noise the signal model isupdated as

2 Mathematical Problems in Engineering

1113957Xn ARDn AT( 1113857BT+ Vn n 1 2 N (5) where Vn stands for the received noise of the nth slice By the

characteristics of the tensor model equation (4) is expanded by

xmnl 1113944K

k1AR(n k)AT(m k)B(l k) m 1 M n 1 N l 1 L (6)

where AT(m k) is defined as the (m k) element of thematrixAT andAR(n k) is defined as the (n k) element of thematrix AR Another term for the PARAFAC decompositionof a third-order tensor is trilinear decomposition Figure 2shows the specific decomposition process

Under the definition of PARAFAC decompositionequation (6) can be regarded as a trilinear model withsymmetrical properties So we can get Ym

BDm(AR)ATT m 1 2 M and Zl ATDl(B)AT

R

l 1 2 L By the symmetry of the trilinear decompo-sition the matrices Y and Z are constructed as

Y

Y1

Y2

⋮

YM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

AR ⊙B1113858 1113859ATT (7)

Z

Z1

Z2

⋮ZL

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ B⊙AT1113858 1113859AT

R (8)

We apply the tensor signal model and parallel factordecomposition to the monostatic FDA-MIMO radar andderive the PARAFAC-based angle and range estimationmethod In the following section we will derive the methodin more detail

3 The Proposed Method

31 Parallel FactorDecomposition In this section accordingto the trilinear alternating least square (TALS) method we

can estimate the transmitting direction matrix and the re-ceiving direction matrix e TALS method can be exploredfor data analysis in trilinear models

According to [33] the estimation of the directionmatrices can be obtained by the parallel factor decompo-sition e least squares (LS) update of equation (3) iswritten by

minATARB

1113957X minus AT ⊙AR1113858 1113859BT

F (9)

where the noise signal is denoted by 1113957X e LS update ofmatrix B is written by

1113954BT

1113954AT ⊙ 1113954AR1113960 1113961+ 1113957X (10)

where 1113954AT and 1113954AR stand for the estimates ofAT andAR whichhave previously obtained respectively e LS fitting ofequation (7) can be written as

minATARB

1113957Y minus AR ⊙B1113858 1113859ATT

F

(11)

where the noise signal is denoted by 1113957Y e LS update ofmatrix AT is written by

1113954AT

T 1113954AR ⊙ 1113954B1113960 1113961+ 1113957Y (12)

where 1113954AR and 1113954B on behalf of the estimates ofAR andBwhichhave previously obtained respectively Similarly the LSfitting of equation (8) can be written as

minATARB

1113957Z minus B⊙AT1113858 1113859ATR

F

(13)

where the noise signal is denoted by 1113957Z e LS update ofmatrix AR is

θ θ

d d

f0 f1

M transmitter antennas N receiver antennas

Targets

fM

Figure 1 Monostatic FDA-MIMO radar

Mathematical Problems in Engineering 3

1113954AT

R 1113954B⊙ 1113954AT1113960 1113961+ 1113957Z (14)

where 1113954B and 1113954AT on behalf of the estimates ofB andAT whichhave previously obtained respectively

According to equations (10) (12) and (14) B AT andAR are given by the LS e LS update does not stop untilconvergence and the constraint condition can be expressedas 1113957X minus [1113954AT ⊙ 1113954AR]1113954B

T2F le 10minus 10

For the received noise signals according to the trilineardecomposition we can get the estimated parameter ma-trices 1113954AT ATΛM1 + N1 1113954AR ARΛM2 + N2 and1113954B BΛM3 + N3 where Λ is a permutation matrix N1 N2and N3 stand for the matrices in correspondence to esti-mation errors and M1 M2 and M3 denote the diagonalscaling matrices and are subject to M1M2M3 IK e in-herent scale ambiguity and permutation of trilinear de-composition and normalization can be used to eliminatescale ambiguity

32 Angle Estimation From the trilinear decomposition inthe previous section we can get estimates of the directionmatrices 1113954AT and 1113954AR by the TALS method

It can be seen from the signal model that the steervector of 1113954AR is only related to the direction of arrival(DOA) and we can get the estimated value of the anglethrough 1113954AR

In the paper the receive array can be preset as half-wavelengths spaced uniform linear arrays (ULA) so thereceive steering vector of θk is

ar θk( 1113857 1 ejπ sin θk( ) e

jπ(Nminus 1)sin θk( )1113876 1113877T

(15)

Let ψr denote the phase of ar(θk) which can beexpressed as

ψr angle ar θk( 1113857( 1113857 0 π sin θk π(N minus 1)sin θk1113858 1113859T

(16)

After the receiving steering vector is well calibrated it isrepresented as 1113954ar(θk) and utilizes normalization to dispelscale ambiguity So 1113954ψr can be obtained from equation (16)e estimate of sin θk can be calculated by LS principle Wecan construct the LS fitting as

Gq 1113954ψ (17)

where the q isin C2times1 and 1113954ψ stand for the estimated vector andthe estimated phase of steering vector G is defined as

G

1 0

1 π

⋮ ⋮

1 π(N minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

isin CNtimes2 (18)

1113954qr is the LS solution of qr by equation (17) which stands forthe estimated vector and can be expressed as

1113954qr GTG1113872 1113873minus 1GT 1113954ψr (19)

e receive angle 1113954θk is derived from1113954θk sinminus 1

1113954qr(2)( 1113857 (20)

where 1113954qr(2) represents the element in the second row of thevector1113954qr

33 Range Estimation For the tensor-based data model therange and transmit angle of the uncorrelated target are bothincluded in the transmit steering vector So the transmitsteering vector can be expressed as

at θk rk( 1113857 at θk( 1113857 ∘ at rk( 1113857 (21)

whereat(θk) [1 e(minus j2πd sin(θkλ0)) e(minus j2πd(Mminus 1)sin(θkλ0))]T andat(rk) [1 e(minus j2π2Δf(rkc)) e(minus j2π2Δf(Mminus 1)rkc)]T denotethe transmitting angle steering vector and range steeringvector respectively

e transmit array is preset as half-wavelengths spacedULA so the transmitting steer vector of (θk rk) can bereduced as

at θk rk( 1113857 1 ejπ sin θkminus 4Δf rkc( )( ) e

jπ(Mminus 1) sin θkminus 4Δf rkc( )( )1113876 1113877T

(22)

In the previous section the direction of arrival (DOA) 1113954θk

has been obtained Referring to the derivation process in theprevious section and letting ψt represent the phase ofat(rk θk) which can be expressed as

ψt angle at θk rk( 1113857( 1113857

0 π sin θk minus 4Δfrk

c1113874 1113875 π(M minus 1) sin θk minus 4Δf

rk

c1113874 11138751113876 1113877

T

(23)

In the previous section we have obtained the estimatedreceive vector 1113954qr Similarly we can get 1113954qt by equation (17)

a1(3)

a1(1)

= + +a1

(2)

a2(1)

a2(3)

a2(2)

a3(3)

a3(1)

a3(2)

X

Figure 2 Schematic diagram of trilinear decomposition


which stands for the estimated transmit receive vector andcan be expressed as

1113954qt GTG1113872 1113873minus 1GT 1113954ψt (24)

e range 1113954rk of target is derived from

1113954rk c sin 1113954θk minus 1113954qt(2)1113872 1113873

4Δf (25)

where 1113954qt(2) represents the element in the second row of thevector 1113954qt

So far we already get the angle and range of the un-correlated target in the monostatic FDA-MIMO radar Wecan recap the key processes of the proposed method asfollows

(i) Step 1 according to equation (6) the received datais transformed into a third-order tensor model

(ii) Step 2 apply the trilinear decomposition principlewe can obtain the transmitting direction matrix 1113954AT

and the receiving direction matrix 1113954AR(iii) Step 3 take the phase 1113954qr and 1113954qt of 1113954AR and 1113954AT by

equations (16) and (23) respectively(iv) Step 4 the phase 1113954ψr of the receive direction matrix

was plugged into equation (17) and obtain 1113954qr by LSfitting then the receive angle 1113954θr can be calculated byequation (20)

(v) Step 5 similarly we can obtain1113954qt then the range 1113954rk

of target can be calculated by substituting the angleinto equation (25)

4 CRB of FDA-MIMO Radar

e input signal spectrum is defined as

Sx ASfAH

+ σ2IMN (26)

where Sf is the unknown signal spectrum matrix CRBformula can be expressed as [17 32]

CRB σ2

2LRe SfA

HSxASf1113872 1113873otimes1 1

1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠ ∘ DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(27)

where D [((za(θ1 r1))zθ1) ((za(θ1 r1))zr1) ]and PperpA IMN minus A(AHA)minus 1AH

When (AHASf)σ2 is large enough Sf and AHA are notclose to singularity and the signals are unknown the aboveformula can be approximated as an approximate CRB

CRBACR σ2

2LRe Sf otimes

1 1

1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠deg DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(28)

5 Simulation Results

In this section the effectiveness and advantages of ourproposed method can be proved by some numerical sim-ulations e ESPRITmethod and Unitary-ESPRITmethodcan be utilized to contrast with our proposed method Wedefault the monostatic FDA-MIMO radar with M 8transmitting antennas and N 8 receiving antennas in thispaper

Unless otherwise specified it is supposed that there areK 3 uncorrelated targets which are objected to far-fielde three targets are placed at (θ1 r1) (minus 15deg 05 km)(θ2 r2) (10deg 6 km) and (θ3 r3) (35deg 80 km) e per-formance of our proposed method can be appraised by theroot mean square error (RMSE) which can be expressed as

RMSEθ

1K

1T

1113944

K

k11113944

T

t1

1113954θkt θk1113872 11138732

11139741113972

RMSEr

1K

1T

1113944

K

k11113944

T

t11113954rkt rk1113872 1113873

2

11139741113972

(29)

where 1113954θkt and 1113954rkt are the estimation of DOA θk and range rk

of the kth target for the tthMonte Carlo trials respectivelyTdenotes the total amount of Monte Carlo trials and T 500is preset in this simulation

In addition the probability of successful detection isanother metric used to appraise the achievement of ourmethod which is defined by

PSD V

Ttimes 100 (30)

where V represents the number of successful estimates andthe criterion for the success of the angle and range exper-iments is that the absolute value of all experimental nu-merical errors are less than the minimum 01deg and 01 kmrespectively

We preset SNR 20 dB in the first simulation Figure 3shows the estimation results of our proposed methodFigure 3 shows the estimation of the range and angle of theuncorrelated targets are correct and the landing points arehighly concentrated It directly proves the stability andaccuracy of the proposed method

And then L denotes the number of snapshots and it ispreset to L 50 in this simulation We first investigate theliaison between RMSE and SNR of range estimation andangle estimation in the second simulation We use twocomparison methods which are the ESPRIT method [31]and the Unitary ESPRIT method [32] respectively Afterintroducing CRB they are compared with the proposedmethod Figures 4 and 5 correspond to the simulation Fromthe two figures it can be demonstrated that the estimatedperformance of the proposed method is better than theESPRIT method and the Unitary ESPRIT method In ad-dition the RMSE of the proposed method is closer to CRB


From the following we explore the relationshipbetween RMSE and snapshots of range estimation andangle estimation in the third simulation and the resultsare shown in Figures 6 and 7 respectively e SNR

10 dB is preset in this simulation Similar to the firstsimulation after introducing CRB we use the contrastmethods to compare with the proposed algorithm As thenumber of snapshots increase we can know from thefigures that the RMSE of three methods and CRB alldecrease So we conclude that the accuracy of themethods is improving and the proposed method has thebest performance

In the fourth simulation the relationship betweenprobability and SNR of angle and range probability

successful detection is obtained e number of snapshots ispreset to L 50 Figures 8 and 9 correspond to the fourthsimulation From two figures we can know the success rateof three methods can reach 100 with the increase of SNRMoreover with the increase of SNR the probability ofsuccessful detection also increases It can reach a detectionsuccess rate of 100 when the SNR reaches a sufficientlyhigh level this SNR is generally called the SNR thresholdObviously the threshold of the proposed method is thelowest among the three methods which also illustrates thesuperiority of the proposed method

ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)

e proposed methodESPRIT

Unitary ESPRITCRB

Figure 4 RMSE of angle estimation versus SNR

ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(km

)


Unitary ESPRITCRB

Figure 5 RMSE of range estimation versus SNR

ndash20 ndash10 0 10 20 30 40DOA (deg)

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Rang

e (m

)

Figure 3 e range and angle estimation performance of theproposed method with SNR 20 dB

RMSE

(deg)


Unitary ESPRITCRB

50 100 150 200 250 300 350 400 450 500Number of snapshots

10ndash2

10ndash1

Figure 6 RMSE of angle estimation versus the total amount ofsnapshots


6 Conclusions

In the paper we proposed a tensor-based range and angleestimation method in monostatic FDA-MIMO radar eproposed method uses the trilinear model to obtain thedirection matrices through PARAFAC decomposition andextracts the phase from the direction matrix to estimate thedistance and angle is method uses the multidimensionalinformation of the received data Compared with the sub-space methods such as ESPRIT and Unitary-ESPRITmethods the proposed method has the best performancee superiority of the proposed method can be verified bysimulation

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Hainan Provincial NaturalScience Foundation of China (no 2018CXTD336) andNational Natural Science Foundation of China (no61864002)

References

[1] E Fishler A Haimovich R Blum et al ldquoMIMO radar an ideawhose time has comerdquo in Proceedings of the IEEE Radarconference vol 7 pp 71ndash78 Philadelphia PA USA April2004

[2] H Krim and M Viberg ldquoTwo decades of array signal pro-cessing research the parametric approachrdquo IEEE SignalProcessing Magazine vol 13 no 4 pp 67ndash94 1996

[3] E Fishler A Haimovich R Blum et al ldquoPerformance ofMIMO radar systems advantages of angular diversityrdquo inProceeding of the 7irty-Eighth Asilomar Conference on Sig-nals Systems and Computer vol 1 pp 305ndash309 PacificGrove CA USA November 2004

[4] J Li and P Stoica ldquoMIMO Radar with colocated antennasrdquoIEEE Signal Processing Magazine vol 24 no 5 pp 106ndash1142007

[5] A A Gorji R armarasa T Kirubarajan et al ldquoOptimalantenna allocation in MIMO radars with collocated anten-nasrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 50 no 1 pp 542ndash558 2014

[6] A Hassanien and S A Vorobyov ldquoPhased-MIMO radar atradeoff between phased-array and MIMO radarsrdquo IEEE


Unitary ESPRIT

ndash15 ndash10 ndash5 0 5 10 15 20 25 30SNR (dB)

0

01

02

03

04

05

06

07

08

09

1

Ang

le p

roba

bilit

y of

succ

essfu

l det

ectio

n

Figure 8 Probability of angle successful detection versus SNR


Unitary ESPRIT


01

02

03

04

05

06

07

08

09

1

Rang

e pro

babi

lity

of su

cces

sful d

etec

tion

0

Figure 9 Probability of range successful detection versus SNR

RMSE

(km

)


Unitary ESPRITCRB


10ndash2

10ndash1

Figure 7 RMSE of range estimation versus the total amount ofsnapshots


Transactions Signal Processing vol 58 no 6 pp 3137ndash31512009

[7] E Fishler A Haimovich R S Blum L J Cimini D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-models anddetection performancerdquo IEEE Transactions on Signal Pro-cessing vol 54 no 3 pp 823ndash838 2006

[8] A Haimovich R Blum L Cimini et al ldquoMIMO radar withwidely separated antennasrdquo IEEE Signal Processing Magazinevol 25 no 1 pp 116ndash129 2008

[9] A J Fenn D H Temme W P Delaney et al ldquoe devel-opment of phased-array radar technologyrdquo Lincoln Labora-tory Journal vol 12 no 2 pp 321ndash340 2000

[10] Y Hua T K Sarkar D D Weiner et al ldquoAn L-shaped arrayfor estimating 2-D directions of wave arrivalrdquo IEEE Trans-actions on Antennas and Propagation vol 39 no 2pp 143ndash146 1991

[11] X Wang L Wan M Huang C Shen and K Zhang ldquoPo-larization channel estimation for circular and non-circularsignals in massive MIMO systemsrdquo IEEE Journal of SelectedTopics in Signal Processing vol 13 no 5 pp 1001ndash1016 2019

[12] H Wang L Wan M Dong K Ota and X Wang ldquoAssistantvehicle localization based on three collaborative base stationsvia SBL-based robust DOA estimationrdquo IEEE Internet of7ings Journal vol 6 no 3 pp 5766ndash5777 2019

[13] X Wang D Meng M Huang and L Wan ldquoReweightedregularized sparse recovery for DOA estimation with un-known mutual couplingrdquo IEEE Communications Lettersvol 23 no 2 pp 290ndash293 2019

[14] D Meng XWang M Huang LWan and B Zhang ldquoRobustweighted subspace fitting for DOA estimation via block sparserecoveryrdquo IEEE Communications Letters vol 24 no 3pp 563ndash567 2020

[15] P Antonik M C Wicks H D Griffiths et al ldquoFrequencydiverse array radarsrdquo in Proceedings of the 2006 IEEE RadarConference pp 215ndash217 Verona NY USA April 2006

[16] P F Sammartino C J Baker H D Griffiths et al ldquoFrequencydiverse MIMO techniques for radarrdquo IEEE Transactions onAerospace and Electronic Systems vol 49 no 1 pp 201ndash2222013

[17] W-Q Wang and H C So ldquoTransmit subaperturing for rangeand angle estimation in frequency diverse array radarrdquo IEEETransactions on Signal Processing vol 62 no 8 pp 2000ndash2011 2014

[18] W-Q Wang ldquoSubarray-based frequency diverse array radarfor target range-angle estimationrdquo IEEE Transactions onAerospace and Electronic Systems vol 50 no 4 pp 3057ndash3067 2014

[19] S Qin Y D Zhang M G Amin and F Gini ldquoFrequencydiverse coprime arrays with coprime frequency offsets formultitarget localizationrdquo IEEE Journal of Selected Topics inSignal Processing vol 11 no 2 pp 321ndash335 2017

[20] W-Q Wang and C Zhu ldquoNested array receiver with time-delayers for joint target range and angle estimationrdquo IETRadar Sonar amp Navigation vol 10 no 8 pp 1384ndash13932016

[21] S Qin and Y M D Zhang ldquoFrequency diverse array radar fortarget range-angle estimationrdquo International Journal ofComputations and Mathematics in Electrical vol 35 no 3pp 1257ndash1270 2016

[22] P F Sammartino C J Baker H D Griffiths et al ldquoRange-angle dependent waveformrdquo in Proceedindgs of the IEEERadar Conference pp 511ndash515 Washington DC USA May2010

[23] Y-Q Yang H Wang H-Q Wang S-Q Gu D-L Xu andS-L Quan ldquoOptimization of sparse frequency diverse arraywith time-invariant spatial-focusing beampatternrdquo IEEEAntennas and Wireless Propagation Letters vol 17 no 2pp 351ndash354 2018

[24] A M Yao W Wu D G Fang et al ldquoFrequency diverse arrayradar with time-dependent frequency offsetrdquo IEEE Antennasand Wireless Propagation Letters vol 13 pp 758ndash761 2014

[25] D Nion and N D Sidiropoulos ldquoAdaptive algorithms totrack the PARAFAC decomposition of a third-order tensorrdquoIEEE Transactions on Signal Processing vol 57 no 6pp 2299ndash2310 2009

[26] D Nion and N D Sidiropoulos ldquoTensor algebra and mul-tidimensional harmonic retrieval in signal processing forMIMO radarrdquo IEEE Transactions on Signal Processing vol 58no 11 pp 5693ndash5705 2010

[27] B Xu Y Zhao Z Cheng H Li et al ldquoA novel unitaryPARAFAC method for DOD and DOA estimation in bistaticMIMO radarrdquo Signal Processing vol 138 no 11 pp 273ndash2792017

[28] D Nion and N D Sidiropoulos ldquoA PARAFAC-basedtechnique for detection and localization of multiple targets ina MIMO radar systemrdquo in Proceedings of the IEEE interna-tional conference on Speech and Signal amp Processingpp 2077ndash2080 Taipei Taiwan April 2009

[29] F Wen X Xiong and Z Zhang ldquoAngle and mutual couplingestimation in bistatic MIMO radar based on PARAFACdecompositionrdquo Digital Signal Processing vol 65 pp 1ndash102017

[30] J F Li and X F Zhang ldquoA method for joint angle and arraygain-phase error estimation in bistatic multiple-input mul-tiple-output non-linear arraysrdquo IET Signal Processing vol 8no 2 pp 131ndash137 2014

[31] B Li W Bai G Zheng et al ldquoSuccessive ESPRIT algorithmfor joint DOA-range-polarization estimation with polariza-tion sensitive FDA-MIMO radarrdquo IEEE Access vol 6pp 36376ndash36382 2018

[32] F Liu X Wang M Huang et al ldquoA novel unitary ESPRITalgorithm formonostatic FDA-MIMO radarrdquo Sensors vol 20no 3 827 pages 2020

[33] X Zhang Z Xu and D Xu ldquoTrilinear decomposition-basedtransmit angle and receive angle estimation for multiple-inputmultiple-output radarrdquo IET Radar Sonar amp Navigationvol 5 no 6 pp 626ndash631 2011


target through spectral peak search At present there aresome solutions to this problem In [16] a nonlinear FDAradar is proposed but this method needs to change the arrayelement position in actual operation which is difficult toachieve In [17 18] a method based on subarrays to achieveangle and distance decoupling is proposed but this methodhas the problem of distance ambiguity In addition in [19]the target positioning method of the coprime array FDAradar is studied and in [20 21] the nested FDA parameterestimation method is studied

In [18] FDA-MIMO radar with both frequency andwaveform diversity is proposed for the first time whichmakes use of the unique advantages of FDA radar andMIMO radar which not only has excellent target detectionperformance in target detection but also has a high freedomof the spatial degree it can realize the joint estimation of thetarget angle and range [17 22ndash24] At present there aremany kinds of FDA-MIMO radar target positioning algo-rithms such as MUSIC methods and ESPRITmethods butthe MUSIC method involves a spectral peak search andresults in a high algorithm complexity Although the ESPRITmethod can be applied to reduce the computational com-plexity and obtain the angle and range information of thetarget through rotation invariance this method is notsuitable for the phase ambiguity situation

e advantages of FDA are developed based on thecharacteristics of the multidimensional structure of thesignal e abovementioned traditional matrix processingmethod cannot make good use of the multidimensionalcharacteristics which limits their performance [25] To takeadvantage of the multidimensional structural characteristicsof the signal the PARAFAC decomposition is used for angleestimation in unknown target localization by modellingtensor signals [26ndash28] At present there are few studies ontensor-based FDA MIMO target estimation methods epaper proposes the algorithm based on parallel factorizationin monostatic FDA-MIMO radar is method first estab-lishes a third-order tensor signal model and uses parallelfactorization to obtain the transmission directionmatrix andthe reception direction matrix [29 30] en obtain theangle and distance information of the target by taking thephase of the directionmatrix and finally realize the angle anddistance estimation

e proposed method compares the achievement withtraditional estimation of signal parameters via rotationalinvariance techniques (ESPRIT) method [31] Unitary ES-PRIT method [32] and CramerndashRao Bound (CRB) epaper structure is divided into multiple parts Section 2introduces the tensor-based data model Section 3 deducesthe method of angle estimation and distance estimationeCRB of the target estimationmethod in FDA-MIMO radar isshown in Section 4 e Section 5 shows the simulationresults

Notation otimes ⊙ and deg represent the Kronecker productKhatrindashRao product and Hadamard product respectively(middot)T (middot)H (middot)minus 1 (middot)lowast and (middot)+ stand for transpose conju-gate-transpose inverse conjugate and pseudoinverse re-spectively Dn(A) represents a diagonal matrix whichcomposed by the nth row ofA middotF represents the Frobenius

norm IK indicates a K times K identity matrix angle (middot) andmin (middot) represent the phase angle of each element and theminimum element in the array Re(middot) denotes the real partoperator xmnl is denoted as a third-order tensor

2 Tensor-BasedDataModelofMonostaticFDA-MIMO Radar

e paper is based on a narrowband monostatic FDA-MIMO radar It is composed by M-element transmittingantennas and N-element receiving antennas Figure 1 showsthe structure of monostatic FDA-MIMO radar e trans-mitting array and receiving array are placed together so thedirection of arrival (DOA) is the same as the direction ofdeparture (DOD) In general the transmitting array andreceiving array are uniform linear arrays (ULA) and areseparated by half a wavelength In the paper there are K far-field targets by default which are independent of each otherBased on the first element of the transmitting antenna thefrequency of the mth element transmitting antenna is de-rived as

fm f0 +(m minus 1)Δf (1)

where the reference frequency is denoted by f0 and Δfdenotes frequency increase of the transmitting antennaMoreover Δf≪f0

After the matched filter the receiving data can berearranged as

x(t) at θ1 r1( 1113857otimes ar θ1( 1113857 at θ2 r2( 1113857otimes ar θ2( 1113857 at θk rk( 11138571113858

otimes ar θk( 11138571113859s(t) + n(t)

(2)

where n(t) denotes an MN times 1 noise vector that isthe additional white Gaussian noisevector at(θk rk) [1 e(j2π(dλ0)sin(θk)minus j2π(2Δfc)rk)

e(j2π(dλ0)(Mminus 1)sin(θk)minus j2π(2Δfc)(Mminus 1)rk)]T and ar(θk) [1

e(j2πf0dsin(θkc)) e(j2πf0(Nminus 1)dsin(θkc))]T stand for thetransmit steering vector and the receive steering vectorrespectively and X [x(1) x(2) x(L)] stands for thereceived data of L samples which can be expressed as

X

X1

X2

⋮

XN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

AT ⊙AR1113858 1113859BT

ARD1 AT( 1113857

ARD2 AT( 1113857

⋮

ARDN AT( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BT (3)

where B [b(1) b(2) b(L)]T isin CLtimesK the transmittingdirection matrix is defined as AT [at(θ1 r1)

at(θ2 r2) at(θk rk)] isin CMtimesK and the receiving direc-tion matrix is defined as AR [ar(θ1) ar(θ2)

ar(θk)] isin CNtimesKAT ⊙AR stands for KhatrindashRao product SoXn is expressed as

Xn ARDn AT( 1113857BT n 1 2 N (4)

Under the interference of noise the signal model isupdated as




xmnl 1113944K





R


Y

Y1

Y2

⋮

YM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

AR ⊙B1113858 1113859ATT (7)

Z

Z1

Z2

⋮ZL

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ B⊙AT1113858 1113859AT

R (8)






minATARB

1113957X minus AT ⊙AR1113858 1113859BT

F (9)


1113954BT

1113954AT ⊙ 1113954AR1113960 1113961+ 1113957X (10)


minATARB

1113957Y minus AR ⊙B1113858 1113859ATT

F

(11)


1113954AT

T 1113954AR ⊙ 1113954B1113960 1113961+ 1113957Y (12)


minATARB

1113957Z minus B⊙AT1113858 1113859ATR

F

(13)


θ θ

d d

f0 f1


Targets

fM



1113954AT

R 1113954B⊙ 1113954AT1113960 1113961+ 1113957Z (14)



T2F le 10minus 10






jπ(Nminus 1)sin θk( )1113876 1113877T

(15)



(16)


Gq 1113954ψ (17)


G

1 0

1 π

⋮ ⋮

1 π(N minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

isin CNtimes2 (18)


1113954qr GTG1113872 1113873minus 1GT 1113954ψr (19)


1113954qr(2)( 1113857 (20)



at θk rk( 1113857 at θk( 1113857 ∘ at rk( 1113857 (21)





(22)



ψt angle at θk rk( 1113857( 1113857



rk

c1113874 11138751113876 1113877

T

(23)


a1(3)

a1(1)

= + +a1

(2)

a2(1)

a2(3)

a2(2)

a3(3)

a3(1)

a3(2)

X




1113954qt GTG1113872 1113873minus 1GT 1113954ψt (24)



4Δf (25)












Sx ASfAH

+ σ2IMN (26)


CRB σ2

2LRe SfA


1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠ ∘ DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(27)



CRBACR σ2

2LRe Sf otimes

1 1

1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠deg DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(28)




RMSEθ

1K

1T

1113944

K

k11113944

T

t1

1113954θkt θk1113872 11138732

11139741113972

RMSEr

1K

1T

1113944

K

k11113944

T

t11113954rkt rk1113872 1113873

2

11139741113972

(29)




PSD V

Ttimes 100 (30)









ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)


Unitary ESPRITCRB


ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(km

)


Unitary ESPRITCRB



0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Rang

e (m

)


RMSE

(deg)


Unitary ESPRITCRB


10ndash2

10ndash1



6 Conclusions


Data Availability




Acknowledgments


References








Unitary ESPRIT


0

01

02

03

04

05

06

07

08

09

1

Ang

le p

roba

bilit

y of

succ

essfu

l det

ectio

n



Unitary ESPRIT


01

02

03

04

05

06

07

08

09

1

Rang

e pro

babi

lity

of su

cces

sful d

etec

tion

0


RMSE

(km

)


Unitary ESPRITCRB


10ndash2

10ndash1


































xmnl 1113944K





R


Y

Y1

Y2

⋮

YM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

AR ⊙B1113858 1113859ATT (7)

Z

Z1

Z2

⋮ZL

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ B⊙AT1113858 1113859AT

R (8)






minATARB

1113957X minus AT ⊙AR1113858 1113859BT

F (9)


1113954BT

1113954AT ⊙ 1113954AR1113960 1113961+ 1113957X (10)


minATARB

1113957Y minus AR ⊙B1113858 1113859ATT

F

(11)


1113954AT

T 1113954AR ⊙ 1113954B1113960 1113961+ 1113957Y (12)


minATARB

1113957Z minus B⊙AT1113858 1113859ATR

F

(13)


θ θ

d d

f0 f1


Targets

fM



1113954AT

R 1113954B⊙ 1113954AT1113960 1113961+ 1113957Z (14)



T2F le 10minus 10






jπ(Nminus 1)sin θk( )1113876 1113877T

(15)



(16)


Gq 1113954ψ (17)


G

1 0

1 π

⋮ ⋮

1 π(N minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

isin CNtimes2 (18)


1113954qr GTG1113872 1113873minus 1GT 1113954ψr (19)


1113954qr(2)( 1113857 (20)



at θk rk( 1113857 at θk( 1113857 ∘ at rk( 1113857 (21)





(22)



ψt angle at θk rk( 1113857( 1113857



rk

c1113874 11138751113876 1113877

T

(23)


a1(3)

a1(1)

= + +a1

(2)

a2(1)

a2(3)

a2(2)

a3(3)

a3(1)

a3(2)

X




1113954qt GTG1113872 1113873minus 1GT 1113954ψt (24)



4Δf (25)












Sx ASfAH

+ σ2IMN (26)


CRB σ2

2LRe SfA


1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠ ∘ DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(27)



CRBACR σ2

2LRe Sf otimes

1 1

1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠deg DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(28)




RMSEθ

1K

1T

1113944

K

k11113944

T

t1

1113954θkt θk1113872 11138732

11139741113972

RMSEr

1K

1T

1113944

K

k11113944

T

t11113954rkt rk1113872 1113873

2

11139741113972

(29)




PSD V

Ttimes 100 (30)









ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)


Unitary ESPRITCRB


ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(km

)


Unitary ESPRITCRB



0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Rang

e (m

)


RMSE

(deg)


Unitary ESPRITCRB


10ndash2

10ndash1



6 Conclusions


Data Availability




Acknowledgments


References








Unitary ESPRIT


0

01

02

03

04

05

06

07

08

09

1

Ang

le p

roba

bilit

y of

succ

essfu

l det

ectio

n



Unitary ESPRIT


01

02

03

04

05

06

07

08

09

1

Rang

e pro

babi

lity

of su

cces

sful d

etec

tion

0


RMSE

(km

)


Unitary ESPRITCRB


10ndash2

10ndash1
































1113954AT

R 1113954B⊙ 1113954AT1113960 1113961+ 1113957Z (14)



T2F le 10minus 10






jπ(Nminus 1)sin θk( )1113876 1113877T

(15)



(16)


Gq 1113954ψ (17)


G

1 0

1 π

⋮ ⋮

1 π(N minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

isin CNtimes2 (18)


1113954qr GTG1113872 1113873minus 1GT 1113954ψr (19)


1113954qr(2)( 1113857 (20)



at θk rk( 1113857 at θk( 1113857 ∘ at rk( 1113857 (21)





(22)



ψt angle at θk rk( 1113857( 1113857



rk

c1113874 11138751113876 1113877

T

(23)


a1(3)

a1(1)

= + +a1

(2)

a2(1)

a2(3)

a2(2)

a3(3)

a3(1)

a3(2)

X




1113954qt GTG1113872 1113873minus 1GT 1113954ψt (24)



4Δf (25)












Sx ASfAH

+ σ2IMN (26)


CRB σ2

2LRe SfA


1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠ ∘ DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(27)



CRBACR σ2

2LRe Sf otimes

1 1

1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠deg DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(28)




RMSEθ

1K

1T

1113944

K

k11113944

T

t1

1113954θkt θk1113872 11138732

11139741113972

RMSEr

1K

1T

1113944

K

k11113944

T

t11113954rkt rk1113872 1113873

2

11139741113972

(29)




PSD V

Ttimes 100 (30)









ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)


Unitary ESPRITCRB


ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(km

)


Unitary ESPRITCRB



0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Rang

e (m

)


RMSE

(deg)


Unitary ESPRITCRB


10ndash2

10ndash1



6 Conclusions


Data Availability




Acknowledgments


References








Unitary ESPRIT


0

01

02

03

04

05

06

07

08

09

1

Ang

le p

roba

bilit

y of

succ

essfu

l det

ectio

n



Unitary ESPRIT


01

02

03

04

05

06

07

08

09

1

Rang

e pro

babi

lity

of su

cces

sful d

etec

tion

0


RMSE

(km

)


Unitary ESPRITCRB


10ndash2

10ndash1

































1113954qt GTG1113872 1113873minus 1GT 1113954ψt (24)



4Δf (25)












Sx ASfAH

+ σ2IMN (26)


CRB σ2

2LRe SfA


1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠ ∘ DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(27)



CRBACR σ2

2LRe Sf otimes

1 1

1 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎛⎝ ⎞⎠deg DHPperpAD1113872 1113873

T⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus 1

(28)




RMSEθ

1K

1T

1113944

K

k11113944

T

t1

1113954θkt θk1113872 11138732

11139741113972

RMSEr

1K

1T

1113944

K

k11113944

T

t11113954rkt rk1113872 1113873

2

11139741113972

(29)




PSD V

Ttimes 100 (30)









ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)


Unitary ESPRITCRB


ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(km

)


Unitary ESPRITCRB



0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Rang

e (m

)


RMSE

(deg)


Unitary ESPRITCRB


10ndash2

10ndash1



6 Conclusions


Data Availability




Acknowledgments


References








Unitary ESPRIT


0

01

02

03

04

05

06

07

08

09

1

Ang

le p

roba

bilit

y of

succ

essfu

l det

ectio

n



Unitary ESPRIT


01

02

03

04

05

06

07

08

09

1

Rang

e pro

babi

lity

of su

cces

sful d

etec

tion

0


RMSE

(km

)


Unitary ESPRITCRB


10ndash2

10ndash1




































ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(deg)


Unitary ESPRITCRB


ndash5 0 5 10 15 20SNR (dB)

10ndash2

10ndash1

100

RMSE

(km

)


Unitary ESPRITCRB



0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Rang

e (m

)


RMSE

(deg)


Unitary ESPRITCRB


10ndash2

10ndash1



6 Conclusions


Data Availability




Acknowledgments


References








Unitary ESPRIT


0

01

02

03

04

05

06

07

08

09

1

Ang

le p

roba

bilit

y of

succ

essfu

l det

ectio

n



Unitary ESPRIT


01

02

03

04

05

06

07

08

09

1

Rang

e pro

babi

lity

of su

cces

sful d

etec

tion

0


RMSE

(km

)


Unitary ESPRITCRB


10ndash2

10ndash1
































6 Conclusions


Data Availability




Acknowledgments


References








Unitary ESPRIT


0

01

02

03

04

05

06

07

08

09

1

Ang

le p

roba

bilit

y of

succ

essfu

l det

ectio

n



Unitary ESPRIT


01

02

03

04

05

06

07

08

09

1

Rang

e pro

babi

lity

of su

cces

sful d

etec

tion

0


RMSE

(km

)


Unitary ESPRITCRB


10ndash2

10ndash1





























































Documents

Tensor …downloads.hindawi.com/journals/mpe/2020/5720189.pdftarget through spectral peak search. At present, there are some solutions to this problem. In [16], a nonlinear FDA radarisproposed,butthismethodneedstochangethearray