Research Article Transmit/Receive Spatial Smoothing with …downloads.hindawi.com/journals/ijap/2016/6271648.pdf · 2019-07-30 · ESPRIT algorithm for joint angles estimation is

Research ArticleTransmitReceive Spatial Smoothing with Improved EffectiveArray Aperture for Angle and Mutual Coupling Estimation inBistatic MIMO Radar

Haomiao Liu Xiaojun Yang Rong Wang Wei Jin and Weimin Jia

Xirsquoan Research Institute of Hi-Tech Xirsquoan 710025 China

Correspondence should be addressed to Xiaojun Yang yxj029163com

Received 13 October 2015 Revised 16 December 2015 Accepted 21 December 2015

Academic Editor Wei Liu

Copyright copy 2016 Haomiao Liu 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

We proposed a transmitreceive spatial smoothing with improved effective aperture approach for angle and mutual couplingestimation in bistatic MIMO radar Firstly the noise in each channel is restrained by exploiting its independency in both thespatial domain and temporal domainThen the augmented transmit and receive spatial smoothingmatrices with improved effectiveaperture are obtained by exploiting the Vandermonde structure of steering vector with uniform linear array The DOD and DOAcan be estimated by utilizing the unitary ESPRIT algorithm Finally themutual coupling coefficients of both the transmitter and thereceiver can be figured outwith the estimated angles ofDODandDOANumerical examples are presented to verify the effectivenessof the proposed method

1 Introduction

A novel array radar named as multiple-input multiple-output(MIMO) radar has improved the progress of array signalprocessing [1 2] Because the MIMO radar is more flexiblethan the phased array radar [3] it has been researched inmany fields [4] In general the MIMO radar can be dividedinto two classes statistical MIMO radar and collocatedMIMO radar It is statistical MIMO radar where the transmitand receive antennas are widely distributed [5] which ownsthe spatial diversity The transmit and receive antennas areclosely located in collocated MIMO radar [6] Because theantennas transmit totally or partially noncoherent waveformsin the transmitter a virtual array with larger aperture in thereceiver can be formed and higher resolution can be obtainedfor the waveform diversity In this paper we focus on thecollocated MIMO radar

The direction of departure (DOD) and direction of arrival(DOA) estimation are one of the most important aspects inbistatic MIMO radar with collocated antennas And a lotof algorithms have been presented for this issue In [7] theCapon based algorithm for DOD and DOA estimation inbistatic MIMO radar is presented An estimation of signal

parameters via rotational invariance technique (ESPRIT)method [8] is proposed by exploiting the invariance propertyof the transmit array and the receive array The unitaryESPRIT algorithm for joint angles estimation is presented in[9] which has comparable angle estimation performance asESPRIT with lower computational complexity The multiplesignal classification (MUSIC) is utilized for estimation ofDOD and DOA in [10] As the tensor is used widely in mul-tidimensional signal processing a three-dimensional tensordecomposition method parallel factor (PARAFAC) analysisis used to estimate DOD and DOA [11] which has a betterperformance than the othermethodsThere are a lot of meth-ods for joint DOD and DOA estimation in bistatic MIMOradar and we have just presented representative ones

In recent years although many institutes and researchershave been studying this novel radar only a few of instituteshave built up physical systems (eg the ONERA in France)In [12] it points out that the mutual coupling is a major causeof descending the performance of radar by real data exper-iment In order to eliminate the effect of mutual couplinga MUSIC-Like method is proposed in [13] and the mutualcoupling coefficients can be estimated But this method losespartial effective aperture and it needs lots of snapshots for

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 6271648 10 pageshttpdxdoiorg10115520166271648

2 International Journal of Antennas and Propagation

performance guarantee In [14] an ESPRIT-Like methodperforms better than MUSIC-Like method However in thecase of small snapshots the accuracy of angle estimationusing both of the above approaches will degrade remarkablyBy exploiting the multidimensional structure of the receiveddata a three-order tensor is constructed [15] which areDODDOA and temporal dimensions respectively And a real-valued subspace approach is proposed it computes the sub-space utilizing the higher order singular value decomposition(HOSVD) Due to use of the forward-backward averagingtechnique this approach is suitable for coherent targets andsmall snapshots However the tensor-based real-valued sub-space approach just employs partial aperture it loses a lot ofinformation When there exists coherent targets the perfor-mance of that approach will degrade and it cannot deal withmore than two coherent targets Spatial smoothing techniqueis an effective approach to deal with the situation of smallsnapshots and coherent targets In [16] a spatial smooth-ing with improved aperture (SSIA) method is proposed toestimate DOA for the phased array radar it improves theeffective array aperture twice larger than the conventionalspatial smoothing approaches This technique also can beused to estimate DOD and DOA for bistatic MIMO radar

In this paper we proposed a transmitreceive spatialsmoothing with improved effective aperture (TRSSIA) meth-od for joint DOD and DOA estimation in bistatic MIMOradar with unknown mutual coupling Firstly the whiteGaussian noise is restrained using its dependency in boththe spatial and temporal domains Then the TRSSIA is usedto construct the transmit spatial smoothing matrix and thereceive spatial smoothing matrix Due to the Vandermondestructure of steering vectorwith uniform linear arrays (ULA)the transmit and receive augmented spatial smoothingmatri-ces are constructed and these two matrices can improve theeffective aperture two times larger than conventional onesThirdly by using the centro-Hermitian structure of aug-mented matrices the real-valued subspace methods (egunitary ESPRIT) can be used to estimate DOD and DOAFinally an additional DOD and DOA pairing technique isproposed and the mutual coupling coefficients are estimatedThe proposed approach restrains white Gaussian noise andtakes full advantage of the received data so it provides betterangle estimation performance And it can deal with morethan two coherent targets

The remainder of the paper is organized as follows InSection 2 the bistatic MIMO radar signal model is intro-duced The coupling calibration approach is demonstrated inSection 3 In Section 4 simulations are employed to verify theanalytical derivations Finally Section 5 gives the conclusion

Notation (sdot)H (sdot)T (sdot)minus1 and (sdot)lowast denote conjugate trans-

pose transpose inverse and conjugate respectively otimes and⊙ denote the Kronecker product operation and Khatri-Rao product operation Toeplitz(c) denotes the symmetricToeplitz matrix constructed by the vector c diag(sdot) denotesthe diagonalization operation mat(sdot) denotes the matrixingoperation Resdot and Imsdot denote abstracting the real partand imaginary part of complex number respectively emin(sdot)is an operator of getting theminimum eigenvector mean(sdot) is

used to compute themean of numbers I119872

denotes the119872timesMidentitymatrix and 0

119872times119873

is the119872times119873 zeromatrixA[119894 119895 ]denotes the 119894th to the 119895th rows of A

2 Bistatic MIMO Radar Signal Model

Consider a bistatic MIMO radar system equipped with 119872

transmit antennas and119873 receive antennas both of which areULA with half-wavelength spacing antennas The transmitantennas emit119872 orthogonal waveforms S = [s

1

s2

s119872

]T

and SSH119870 = I119872

where119870 is the number of samples per pulseperiod All the targets aremodeled as point-scatters in the far-field and we assume that there exists 119875 targets in the samerange bin Consider the effect of mutual coupling in both thetransmitter and receiver the received data of the 119897th pulse isshown as

X (119897) = [C119903

A119903

]Λ (119897) [C119905

A119905

]T S +W (119897)

119897 = 1 2 119871(1)

where X(119897) isin C119873times119870 is the received data during the 119897th pulseperiod and 119871 is the number of pulses C

119905

and C119903

are themutual coupling matrices of the transmit and receive arraysrespectively which can be expressed as banded symmetricToeplitz matrices [17]

C119905

= toeplitz [1 1198881199051

119888119905119901

119905

0 0] isin C119872times119872

(2a)

C119903

= toeplitz [1 1198881199031

119888119903119901

119903

0 0] isin C119873times119873

(2b)

where 119888119905119894

119888119903119895

are the transmit and receive mutual couplingcoefficients and there are 119901

119905

and 119901119903

nonzero mutual couplingcoefficients with 119872 ge 2119901

119905

+ 1 119873 ge 2119901119903

+ 1 respectivelyA119905

= [a119905

1

a119905

2

a119905

119875

] A119903

= [a119903

1

a119903

2

a119903

119875

] a119905

119901

=

[1 exp(j2120587119889119905

sin120601119901

120582) exp(j2120587(119872minus1)119889119905

sin120601119901

120582)]T and

a119903

119901

= [1 exp(j2120587119889119903

sin 120579119901

120582) exp(j2120587(119873minus1)119889119903

sin 120579119901

120582)]T

are the receive steering vector and the transmit steeringvector where 119889

119905

and 119889119903

are the adjacent antenna spacing oftransmit and receive arrays respectively 120601

119901

and 120579119901

are DODand DOA of the 119901th target and 120582 denoting the wavelengthOne has Λ(119897) = diag([119904

1

(119897) 1199042

(119897) 119904119875

(119897)]) where 119904119901

(119897) =

120573119901

exp(j2120587119891119901

119897) is the reflected signal of the 119901th target and 120573119901

and119891119901

are the amplitude andDoppler frequency respectivelyW(119897) isin C119873times119870 is the complex white Gaussian noise matrixwith the covariance 1205752

119908

I119873

[18 19] After matched filtering theoutput data at the receiver can be expressed as

x (119897) = CAs (119897) + w (119897) (3)

whereC = C119905

otimesC119903

A = A119905

⊙A119903

s(119897) = [1199041

(119897) 1199042

(119897) 119904119875

(119897)]T

w(119897) = vec(W(119897)SH119870)It assumes that the noise is iid complex white Gaus-

sian noise and we can obtain the equation written as

International Journal of Antennas and Propagation 3

119864[vec(W(119897))vecH(W(119897))] = I119870

otimes 1205752119908

I119873

Furthermore accord-ing to the properties of Kronecker product [20 21] we canobtain the following equation

119864 [w (119897)w (119897)H] =

119864 [vec (W (119897) SH) vecH (W (119897) SH)]

1198702

=119864 [(Slowast otimes I

119873

) (vec (W (119897)) vecH (W (119897))) (ST otimes I119873

)]

1198702

=(Slowast otimes I

119873

) (I119870

otimes 1205752119908

I119873

) (ST otimes I119873

)

1198702=I119872

otimes 1205752119908

I119873

119870

=1205752119908

I119872119873

119870

(4)

3 Proposed Algorithm

31 Restraining Noise According to (4) the noise of all119872119873

channels are independent Meanwhile the noise of eachchannel is also independent in the temporal domain Sowe can restrain the noise both in the spatial and temporaldomain For the ULA the received data of each channel isexpressed as

119909119898119899

(119897) =

119875

sum119901=1

119886119905

119901119898

119886119903

119901119899

119904119901

(119897) + 119908119898119899

(119897) (5)

where 119909119898119899

(119897)means the (119898minus1)119873+119899th row of x(119897)119908119898119899

(119897) isthe (119898 minus 1)119873 + 119899th row of w(119897)and 119886

119905

119901119898

and 119886119903

119901119899

are the119898thelement and the 119899th element of a

119905

119901

and a119903

119901

respectivelyWhen119898 = 1 119899 = 1 119909

11

(119897) = sum119875

119901=1

119904119901

(119897) + 11990811

(119897) the correlationcoefficient can be expressed as

119903(11)(11)

(Δ119897) = 119864 [11990911

(119897) 119909lowast

11

(119897 + Δ119897)]

=

119875

sum119901

1=1

119875

sum119901

2=1

1205902

(119901

1119901

2)

(6)

where 1205902(119901

1119901

2)

is the correlation coefficient between 119904119901

1

(119897) and119904119901

2

(119897+Δ119897) For the white Gaussian noise the correlation coef-ficient of two adjacent noises and the correlation coefficientof signal and noise are both zeros in the temporal domainAnd in the spatial domain we can get the correlation efficientbetween any channel and the first channel which can beshown as

119903(119898119899)(11)

(Δ119897) = 119864 [119909119898119899

(119897) 119909lowast

11

(119897 + Δ119897)]

=

119875

sum119901

1=1

119886119905

1199011119898

119886119903

1199011119899

119875

sum119901

2=1

1205902

(119901

1119901

2)

(7)

As we known from (4) the noise is independent in spatialdomain So the correlation coefficient of noise between anychannel and the first channel equals zero

Factually we can only get limited snapshots so we obtainthe asymptotic correlation coefficients

(11)(11)

(Δ119897) =

1015840

119897+119871

sum

119897=

1015840

119897

11990911

(119897) 119909lowast

11

(119897 + Δ119897)

119871

=

119875

sum119901

1=1

119875

sum119901

2=1

2

(119901

1119901

2)

+ 2

(11)

(8a)

(119898119899)(11)

(Δ119897) =

1015840

119897+119871

sum

119897=

1015840

119897

119909119898119899

(119897) 119909lowast

11

(119897 + Δ119897)

119871

=

119875

sum119901

1=1

119886119905

1199011119898

119886119903

1199011119899

119875

sum119901

2=1

2

(119901

1119901

2)

+ 2

(119898119899)

(8b)

where 2(11)

and 2(119898119899)

are the correlation coefficients of noisewhich nearly equal zeros

According to (8a)-(8b) we obtain a new vector which canbe written as

r (Δ119897)

= [(11)(11)

(Δ119897) (12)(11)

(Δ119897) (119872119873)(11)

(Δ119897)]T

= A119905

⊙ A119903

120588 + k

(9)

where 120588 = [sum119875

119901

2=1

2

(1119901

2)

sum119875

119901

2=1

2

(2119901

2)

sum119875

119901

2=1

2

(119875119901

2)

]T k =

[2

(11)

2

(12)

2

(119872119873)

]T

Consider the effect of mutual coupling in both the trans-mitter and receiver we construct two selection matrices

J119905

= J1

otimes I119873

J1

= [0(119872minus2119901

119905)times119901

119905

I119872minus2119901

119905

0(119872minus2119901

119905)times119901

119905

] (10a)

J119903

= I119872

otimes J2

J2

= [0(119873minus2119901

119903)times119901

119903

I119873minus2119901

119903

0(119873minus2119901

119903)times119901

119903

] (10b)

Thenweuse these two selectionmatrices on the received datait can get the selected data

x119905

(119897) = J119905

x (119897) = [J1

C119905

A119905

] ⊙ [C119903

A119903

] s (119897) + J119905

w (119897)

= A119905

⊙ A119903

s (119897) + w (119897) (11)

where A119905

is the first 119872 minus 2119901119905

rows of A119905

A119903

=

[C119903

A119903

]diag(C119903

(1 )A119903

)minus1 and s(119897) = [119904

1

(119897) 1199042

(119897) 119904119875

(119897)]T

119904119901

(119897) = 120591119901

C119903

(1 )a119903119901

119904119901

(119897) 120591119901

= sum119901

119905

119902=minus119901

119905

119888119905|119902|

119890119895120587(119902+119901119905)sin120601119901 Mean-while we can get another selected data vector which can beexpressed as

x119903

(119897) = J119903

x (119897) = [C119905

A119905

] ⊙ [J2

C119903

A119903

] s (119897) + J119903

w (119897)

= A119905

⊙ A119903

s (119897) + w (119897) (12)

where A119903


rows of A119903

A119905

=

[C119905

A119905

]diag(C119905

(1 )A119905

)minus1 and s(119897) = [

1

(119897) 2

(119897) 119875

(119897)]T

119901

(119897) = 120574119901

C119905

(1 )a119905119901

119904119901

(119897) 120574119901

= sum119901

119903

119902=minus119901

119903

119888119903|119902|

119890119895120587(119902+119901119903)sin120579119901 Theselected noise vectors w and w are both white Gaussian


Thenwe restrain the noise of selected data based on (8a)-(8b) It obtains two received vectors which are shown as

r119905

(Δ119897) = [119905(11)(11)

(Δ119897) 119905(12)(11)

(Δ119897)

119905(1198721119873)(11)

(Δ119897)]T= A119905

⊙ A119903

119905

+ k119905

(13a)

r119903

(Δ119897) = [119903(11)(11)

(Δ119897) 119903(12)(11)

(Δ119897)

119903(1198721198731)(11)

(Δ119897)]T= A119905

⊙ A119903

119903

+ k119903

(13b)

where 1198721 = 119872 minus 2119901119905

1198731 = 119873 minus 2119901119903

119905

=

[sum119875

119901

2=1

2

119905(1119901

2)

sum119875

119901

2=1

2

119905(2119901

2)

sum119875

119901

2=1

2

119905(119875119901

2)

]T

119903

=

[sum119875

119901

2=1

2

119903(1119901

2)

sum119875

119901

2=1

2

119903(2119901

2)

sum119875

119901

2=1

2

119903(119875119901

2)

]T and

2

119905(119894119901

2)

2

119903(119895119901

2)

are the correlation coefficients of 119904119894

(119897) and 119904119901

2

(119897) andthe correlation coefficient of

119895

(119897) and 119901

2

(119897) respectivelyk119905

= [2

119905(11)

2

119905(12)

2

119905(1198721119873)

]T and k

119903

= [2

119903(11)

2

119903(12)

2

119903(1198721198731)

]T where

2

119905(119898119899)

and 2

119903(119898119899)

are the correlationcoefficients of w

119905(11)

(119897) and w119905(119898119899)

(119897 + Δ119897) w119903(11)

(119897) andw119903(119898119899)

(119897 + Δ119897) respectively

32 TransmitReceive Spatial Smoothing with Improved Aper-ture After restraining the noise we obtain the new receiveddata vector r

119905

(Δ119897) r119903

(Δ119897) In the following we omit Δ119897 andwrite the received data as r

119905

r119903

In [16] a spatial smoothingwith improved aperture (SSIA) method with single snapshotis proposed It is suitable for coherent sources and improvesthe effective aperture This approach performs well for DOAestimation in the phased array radar In this paper it willprove that this technique can work well in the MIMO radar

Firstly we define a119898times119898 exchange matrixΠ119898

with oneson its antidiagonal and zeros elsewhere Then the left-Π-realmatrixQ can be expressed as [22]

Q2119899+1

=1

radic2

[[[

[

I119899

0 jI119899

0T radic2 0T

Π119899

0 minusjΠ119899

]]]

]

(14)

Equation (14) is a left-Π-real matrix of odd order The 2119899

order one is obtained fromQ2119899+1

by dropping the center rowand center column

For r119905

and1198721 = 119872sub + 119871119905

minus 1 we define

X119905

= [(H1199051

otimes IN) r119905 (H1199052 otimes I119873

) r119905

(H119905119871

119905

otimes I119873

) r119905

]

H119905119897

= [0119872subtimes(119897minus1)

I119872sub

0119872subtimes(119871119905minus119897)

] 1 le 119897 le 119871119905

(15)

And (15) can be expressed as

X119905

= [A119905

⊙ A119903

] Λ119905

AT119905

+ V119905

(16)

where Λ119905

= diag(119905

) A119905

and A119905

are the first 119872sub rows andthe first 119871

119905

rows of A119905

respectively

Then we obtain an augmented matrix

Xtaug =[

[

X119905

Π119872sub119873

Xlowast119905

Π119871

119905

]

]

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

AT119905

+ [

[

V119905

Π119872sub119873

Vlowast119905

Π119871

119905

]

]

= AtaugAT119905

+ Vtaug

isin C2119872sub119873times119871119905

(17)

where Λ1199051

= diag([119886119905

1(1minus119872sub)

119886119905

2(1minus119872sub)

119886119905

119875(1minus119872sub)

]) andΛ1199052

= diag([119886119905

1(1minus119871

119905)

119886119905

2(1minus119871

119905)

119886119905

119875(1minus119871

119905)

]) The steeringmatrix Ataug is twice larger than A

119905

⊙ A119903

Although the noisematrix V

119905

is not white Gaussian its effect on Xtaug is weakas every element nearly equals zero See Appendix for a briefillustration of the rotational invariance property of Xtaug Thesubspace approaches such as MUSIC and ESPRIT can beused to estimate DOD

We note that Xtaug is a centro-Hermitian matrix itsatisfies the following identity

Π2119872sub119873

XlowasttaugΠ119871119905 = Xtaug (18)

So the real-valued space methods for example unitaryMUSIC and unitary ESPRIT are suitable for Xtaug to estimateangles In this paper we estimate angles by utilizing theunitary ESPRIT algorithm Then the complex matrix can betransformed into real-valued matrix as follows

120593 (Xtaug) = QH2119872sub119873

XtaugQ119871119905

(19)

The real-valued signal subspace can be obtained by mak-ing SVD on 120593(Xtaug) The property of rotational invariance inreal-valued subspace is shown as

K(1)taugEtaugΥtaug asymp K(2)taugEtaug (20)

where Etaug contains the 119875 dominant left singular vectors of120593(Xtaug)MeanwhileK(1)taug andK

(2)

taug are the transformed selec-tion matrices they are both obtained from J(2)taug in the follow-ing way

K(1)taug = 2 sdot Re QH2(119872subminus1)119873

J(2)taugQ2119872sub119873 (21a)

K(2)taug = 2 sdot Im QH2(119872subminus1)119873

J(2)taugQ2119872sub119873 (21b)

where J(2)taug = I2

otimes J1199052

J1199052

= [0(119872subminus1)119873times119873

I(119872subminus1)119873

]A class of least squares (LS) approaches [20] for example

structured least squares (SLS) and total least squares (TLS)can be used to solve (20) for Υtaug Ultimately the angles ofDOD can be figured out by 120601

119894

119875

119894=1

= asin(2atan(eig(Υtaug))120587)


In the same way we estimate the angles of DOA Firstlyfor1198731 = 119873sub + 119871

119903

minus 1 we obtain the augmented matrix

Xraug = [X119903

Π119872119873sub

Xlowast119903

Π119871

119903

] = AraugAT119903

+ Vraug

isin C2119872119873subtimes119871119903

(22)

where X119903

= [(I119872

otimesH1199031

)r119903

(I119872

otimesH1199032

)r119903

(I119872

otimesH119903119871

119903

)r119903

] andH119903119897


I119873sub

0119873subtimes(119871119903minus119897)

] 1 le 119897 le 119871119903

Furthermorethe augmented matrix can be exchanged into real-valuedmatrix 120593(Xraug) = QH

2119872119873subXraugQ119871

119903

The real-valued signalsubspace Eraug can be got by utilizing SVD on 120593(Xraug) and itmeets the following relationship

K(1)raugEraugΥraug asymp K(2)raugEraug (23)

where the selection matrices K(1)raugK(2)

raug are constructedas K(1)raug = 2 sdot ReQH

2119872(119873subminus1)J(2)raugQ2119872119873sub K

(2)

raug = 2 sdot

ImQH2119872(119873subminus1)

J(2)raugQ2119872119873sub and J(2)raug = I2

otimes J1199032

J1199032

= I119872

otimes

[0(119873subminus1)times1

I119873subminus1

] Then the angles of DOA can be given by120579119895

119875

119895=1

= asin(2atan(eig(Υraug))120587)

33 Pairing DOD and DOA Because the proposed TRSSIAapproach tries tomake full use of the received data the angles

of DOD and DOA are figured out from transmit aug-mented matrix and receive augmented matrix respectivelyBy exploiting the relationship between the steering vectorsa119905

a119903

and the signal subspace EtaugEraug then 120601119894

119875

119894=1

can bepaired with 120579

119895

119875

119895=1

correctly Note that their relationship canbe expressed as

A119905

⊙ A119903

T1

= J(3)taug (Q2119872sub119873Etaug) = U

1

(24a)

A119905

⊙ A119903

T2

= J(3)raug (Q2119872119873subEraug) = U2

(24b)

where A119905

A119903

are the first1198721 and1198731 rows of A119905

and A119903

andT1

and T2

are both 119875times119875 nonsingular matricesThe selectionmatrices abstract parts of signal subspace are shown as

J(3)taug = [1 0] otimes J1199053

J1199053

= I119872sub

otimes [01198731times119901

119903

I1198731

01198731times119901

119903]

(25a)

J(3)raug = [1 0] otimes J1199033

J1199033

= [01198721times119901

119905

I1198721

01198721times119901

119905] otimes I119873sub

(25b)

Then the pairing matrix can be written as follows

P (119894 119895)

=1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I119872sub1198731

minus U1

UH1

) (a119905

(120601119894

) otimes a119903

(120579119895

))∙

1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I1198721119873sub

minus U2

UH2

) (a119905

(120601119894

) otimes a119903

(120579119895

))(26)

where P is a 119875 times 119875matrix a119905

(120601119894

) a119905

(120601119894

) are the first119872sub1198721

rows of a119905

(120601119894

) respectively a119903

(120579119895

) a119903

(120579119895

) are the first119873sub 1198731

rows of a119903

(120579119895

) respectively If P(119894 119895) is the largest in the 119894throw then we pair 120601

119894

with 120579119895

34 Mutual Coupling Estimation In order to calibrate theantennas with unknown mutual coupling the mutual cou-pling coefficients need to be estimated

Lemma 1 For any 119872 times 1 complex vector x and any 119872 times

119872 banded complex symmetric Toeplitz matrix A we have(Lemma 3 in [17])

A sdot x = Q (x) sdot a (27)

where the 119871 times 1 vector a is given by

119886 (119897) = A (1 119897) 119897 = 1 2 119871 (28)

and 119871 is the highest superdiagonal that is different from zero

The119872 times 119871 matrix Q(x) is given by the sum of the two119872 times 119871

following matrices

[W1

]119901119902

=

x119901+119902minus1

119901 + 119902 le 119872 + 1

0 119900119905ℎ119890119903119908119894119904119890

[W2

]119901119902

=

x119901minus119902+1

119901 le 119902 le 2

0 119900119905ℎ119890119903119908119894119904119890

(29)

By Lemma 1 we can get

C119905

a119905

(120601) = T119905

(120601) c119905

c119905

= [1 1198881199051

1198881199052

119888119905119901

119905

]T (30a)

C119903

a119903

(120579) = T119903

(120579) c119903

c119903

= [1 1198881199031

1198881199032

119888119903119901

119903

]T (30b)

The steeringmatrix and the signal subspace own the relation-ship as follows

([C119905

A119905

] ⊙ A119903

)T3


J(4)raug = [I119872119873sub

0119872119873sub

]

(31)


minus10 minus5 0 5 10 15 20 25 30SNR (dB)

RMSE

(deg

)

MUSIC-LikeESPRIT-LikeHOSVD-MUSIC

HOSVD-ESPRITTRSSIACRB

102

100

10minus2

(a)


RMSE

(deg

)



102

100

10minus2

10minus4

(b)

Figure 1 RMSE of angle estimation versus SNR (a) 119875 = 3 119901119905

= 119901119903

= 1119872sub = 4119873sub = 4 119871 = 64 (b) 119875 = 3 119901119905

= 119901119903

= 2119872sub = 3119873sub = 3and 119871 = 128

where T3

is a 119875 times 119875 nonsingular matrix In theory([C119905

a119905

(120601119894

)] otimes a119903

(120579119894

))HUperp3

2

= 0 and it also can be writtenas ((T

119905

(120601119894

)c119905

) otimes (diag(a119903

(120579119894

))i119903

)HUperp3

2

= (c119905

otimes i119903

)H(T119905

(120601119894

) otimes

diag(a119903

(120579119894

)))HUperp3

2

= 0 where i119903

is a119873subtimes1 vector whose allelements are onesUperp

3

is the orthogonal complement space ofU3

Then the mutual coupling coefficients in transmitter canbe obtained as

ctaug = c119905

otimes i119903

= emin (1

119875

119875

sum119894=1

Q119905119894

) (32)

whereQ119905119894

= (T119905

(120601119894

) otimesdiag(a119903

(120579119894

)))H(I119872119873sub

minusUH3

U3

)(T119905

(120601119894

) otimes

diag(a119903

(120579119894

))) By exploiting the structure of ctaug we get amatrix C

119905

= mat(ctaug) = c119905

iT119903

Then the estimated mutualcoupling coefficient can be obtained with the operations

C119905

(1 ) = iT119903

119905119894

119901

119905

119894=1

= mean (C119905

(119894 + 1 )) (33)

The mutual coupling coefficients in receiver also canbe figured out in the same way By exploiting the relation-ship between steering matrix and signal subspace (A

119905

⊙

[C119903

A119903

])T4

= J(4)taug(Q2119872sub119873Etaug) = U

4

where J(4)taug = [I119872sub119873

0119872sub119873

] The vector craug can be obtained and the estimatedmutual coupling coefficients can be obtained by

119903119895

119901

119903

119895=1

=

mean(C119903

(119895 + 1 ))

4 Simulation Results

In the following simulations we assume that both thetransmit and receive arrays are ULAs with119872 = 10 and119873 =

10 and 119889119905

and 119889119903

are both half wavelength The transmitter

emits the orthogonal waveforms S = (1 + 119895)Hradic2 whereH isin C119872times119870 is constructed with119872 rows of a119870times119870Hadamardmatrix and 119870 = 256 There are three targets at the samerange binwithRCS120573 = [1 1 1]

T and theDoppler frequenciesf119889

= [100 255 500]T Hz And these targets locate at (120601

1

1205791

) =

(minus15∘ 40∘) (1206012

1205792

) = (35∘ minus20∘) and (1206013

1205793

) = (0∘ minus10∘)All the simulation results are carried out by 100 Monte Carlotrials in this paper

In the first simulation we investigate the angle estimationperformances of MUSIC-Like ESPRIT-Like and tensor-based real-valued subspace (in this paper we call it HOSVD)methods and our proposed TRSSIA method There are twocases (1) 119901

119905

= 119901119903

= 1 c119905

= [1 02 + 11989500061]T and c

119903

=

[1 015 + 11989500251]T the number of pulses is 119871 = 64 and

(2) 119901119905

= 119901119903

= 2 c119905

= [1 07 + 1198950002 02 + 11989500061]T and

c119903

= [1 06+11989500121 015+11989500251]T the number of pulses is

119871 = 128 In Figure 1(a) it shows that TRSSIA estimates angleswith more accuracy than the other approaches The MUSIC-Like and ESPRIT-Likemethods performwell in the high SNRregion for case (1) as shown in Figure 1(b) andMUSIC-Likefails to work in case (2) owing to angle ambiguity which isexplained in [13] At both cases the TRSSIA method is betterthan the other methods Although the HOSVD methodsmake use of the multidimensional structure of the receiveddata it just selects 119872 minus 2119901

119905

elements of transmit array and119873 minus 2119901

119903

elements of receive array for angles estimationThe TRSSIA algorithm restrains the noise of received dataimproves the effective aperture and uses all the elements oftransmitter and receiver

The second simulation is carried out to show the RMSE ofangle estimation versus number of pulses with SNR = 5 dBFigure 2(a) shows that the performances of MUSIC-Like andESPRIT-Like approaches highly depend on the number of


20 40 60 80 100 120Number of pulses

RMSE

(deg

)



100

101

102

10minus1

10minus2

(a)RM

SE (d

eg)


102

103

104

100

101

10minus1

1205902

diag1205902n

of conventional method1205902

off1205902n


diag1205902n

of proposed method

(b)

Figure 2 (a) RMSE of angle estimation versus number of pulses (b)The ratio of signal power to noise power versus number of pulses (119875 = 3119901119905

= 119901119903

= 1119872sub = 4119873sub = 4 and SNR = 5 dB)

pulses and the TRSSIA method can give out more preciseangle estimation even with small snapshots In Figure 2(b) itanalyses the ratio of the power of signal to the power of noise1205902diag 120590

2

off and 1205902119899

represent the sum of diagonal elements ofautocorrelation matrix of signal the sum of off-diagonal ele-ments of autocorrelation matrix of signal and the sum of allelements of autocorrelation matrix of noise respectively Inour proposed algorithm the newly autocorrelation matricesof signal are Λ

119905

and Λ119903

For the other methods we call themconventional methods it issum119871

119897=1

s(119897)s(119897)H119871 According to (1)the 1205902off of the TRSSIA method is zero Because the TRSSIAmethod restrains the noise the 1205902diag120590

2

119899

is tremendouslyhigher than conventional methods Though the 1205902diag120590

2

119899

ofconventional methods is constant the 1205902off120590

2

119899

gets lower asthe number of pulses increases which improves the accuracyof signal subspace estimation and angle estimation

In the third simulation it compares the angle estimationperformance of the TRSSIA method with the performanceof the HOSVD methods in the scenario of coherent targetsThere are three cases (1) f

119889

= [100 110 200]T Hz (2) f

119889

=

[100 100 150]T Hz and (3) f

119889

= [100 100 100]T Hz As

Figure 3 shows the angle estimation performance of theHOSVDmethods degrade when all the targetsmove in a nar-row speed zone This method even cannot work when morethan two coherent targets exist Equation (1) proves that therank is always119875whether the targets are coherent or not so theangle estimation performance of the TRSSIA method doesnot degrade even if there are more than two coherent targets

As we know the forward-backward (FB) smoothing is apreferable technique to cope with coherent signals [23] Itshould be notable that our proposed approach can performas well as the joint transmit and receive diversity smoothing


RMSE

(deg

)

HOSVD-MUSICHOSVD-ESPRIT

TRSSIACRB

Case 1Case 3

Case 210

0

102

101

10minus2

10minus3

10minus4

10minus1

Figure 3 RMSE of angle estimation versus SNR with coherenttargets (119875 = 3 119901

119905

= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)

(TRDS) algorithm with FB smoothing in [24] as it is shownin Figure 4(a) But the TRSSIA needs to compute covariancematrices only once We present an evaluation of computa-tional complexity using TIC and TOC instruction in MAT-LAB In Figure 4(b) it demonstrates that the runtime gapbetween two methods gets wider with the number of pulsesincreasing

The performance of mutual coupling estimation isdemonstrated in the fourth simulationTheRMSEs of the realpart and the imaginary part of mutual coupling are adapted



RMSE

(deg

)

TRDSTRSSIACRB

10minus13

10minus15

10minus17

10minus19

10minus11

(a)

20 40 60 80 100 1200

0005

001

0015

002

0025

003

0035

Number of pulses

Runt

ime (

seco

nd)

TRDSTRSSIA

(b)

Figure 4 (a) RMSE of angle estimation versus number of pulses with coherent targets (b) Runtime of both TRDS and TRSSIA algorithmsversus number of pulses (119875 = 3 119891

119889

= [100 100 100]Hz 119901119905

= 119901119903

= 1 SNR = 5 dB TRDS119872sub = 5119873sub = 5 TRSSIA119872sub = 4119873sub = 4)


RMSE

Real part of conventional methodImaginary part of conventional methodReal part of TRSSIAImaginary part of TRSSIAReal part of CRBImaginary part of CRB

102

100

10minus2

10minus4

Figure 5 RMSE of mutual coupling estimation versus SNR (119875 = 3119901119905

= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)

to measure the performance In [13 14] a technique of esti-mating mutual coupling is proposed we call it conventionalmethod in this paper Figure 5 shows that our proposedapproach can estimate mutual coupling more accuratelyThere are two reasons On one hand the angle estimation ofour proposed method is more accurate On the other handour proposed method computes the mean of every mutualcoupling coefficient

5 Conclusion

This paper has proposed an algorithm for angle estimationwith unknown mutual coupling in both the transmitterand receiver The preliminary work is to restrain the whiteGaussian noise of each channel by computing the correlationcoefficients because the noise is independent in both thespatial and temporal domains In order to use more informa-tion of the received data we do spatial smoothing in boththe transmit array and the receive array and construct theaugmented steering matrices with improved aperture TheTRSSIA algorithm adopts more elements from the transmit-ter and the receiver to estimate angles So more informa-tion improves the angle estimation For restraining noiseimproving aperture and the spatial smoothing technique theTRSSIAmethod proves better angle estimation thanMUSIC-Like ESPRIT-Like and tensor-based real-valued approachesat small number of pulses and low SNR cases and its angleestimation performance does not descend even formore thantwo coherent targets Based on the more accurately estimatedangles and computing the mean of every mutual couplingefficient the mutual coupling estimation is more accuratethan the other methods The simulation results verify theadvantage of the proposed method

Appendix

Rotational Invariance Property of Xtaug

Firstly we do SVD on Xtaug and get 119875major left eigenvectorswhich can be expressed as U

119879119878

isin C2119872sub119873times119875 And therelationship between U

119879119878

and Ataug is U119879119878 = AtaugT whereT is an nonsingular matrix of 119875 times 119875 dimensions We equallydivideU

119879119878

into two partsU119879119878

andU119879119878

mean the first and last


119872sub119873 rows ofU119879119878

respectivelyThen the above equation canbe written as

[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)

Then we explore the rotational invariance of A119905

A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)

and these matrices meet A119905

Φ119905

= A119905

where A119905

and A119905

are thefirst and last119872subminus1 rows of A119905Φ119905 = diag([exp(j2120587119889

119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901

120582)]) Thus the rotational invarianceof Ataug can be shown in the following equation

[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)

Then EVD can be employed to solve Φ119905

furthermore DODcan be got The above deducing procedure proves the rota-tional invariance property of Xtaug

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] DW Bliss and KW Forsythe ldquoMultiple-inputmultiple-output(MIMO) radar and imaging degrees of freedom and resolu-tionrdquo in Proceedings of the Conference Record of the 37th IEEEAsilomar Conference on Signals Systems andComputers (ACSSCrsquo03) vol 1 pp 54ndash59 PacificGrove Calif USANovember 2003

[2] E Fishler A Haimovich R Blum D Chizhik L Cimini andR Valenzuela ldquoMIMO radar an idea whose time has comerdquo inProceedings of the IEEE Radar Conference pp 71ndash78 April 2004

[3] E Fishler AHaimovich R Blum R Cimini D Chizhik andRValenzuela ldquoPerformance of MIMO radar systems advantagesof angular diversityrdquo in Proceedings of the 38th IEEE AsilomarConference on Signals Systems and Computers vol 1 pp 305ndash309 IEEE Pacific Grove Calif USA November 2004

[4] A Farina andM Lesturgation ldquoGuest editorial special issue onbistatic andMIMO radars and their applications in surveillanceand remote sensingrdquo IET Radar Sonar and Navigation vol 8no 2 pp 73ndash74 2014

[5] A M Haimovich R S Blum and L J Cimini ldquoMIMOradar with widely separated antennasrdquo IEEE Signal ProcessingMagazine vol 25 no 1 pp 116ndash129 2008

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

[7] H Yan J Li and G Liao ldquoMultitarget identification and local-ization using bistatic MIMO radar systemsrdquo EURASIP Journalon Advances in Signal Processing vol 2008 Article ID 283483 8pages 2008

[8] D Chen B Chen and G Qin ldquoAngle estimation using ESPRITinMIMO radarrdquo Electronics Letters vol 44 no 12 pp 770ndash7712008

[9] G Zheng B Chen and M Yang ldquoUnitary ESPRIT algorithmfor bistatic MIMO radarrdquo Electronics Letters vol 48 no 3 pp179ndash181 2012

[10] X Zhang L Xu L Xu and D Xu ldquoDirection of Departure(DOD) and Direction of Arrival (DOA) estimation in MIMOradar with reduced-dimensionMUSICrdquo IEEE CommunicationsLetters vol 14 no 12 pp 1161ndash1163 2010

[11] D Nion and N D Sidiropoulos ldquoTensor algebra and multidi-mensional harmonic retrieval in signal processing for MIMOradarrdquo IEEE Transactions on Signal Processing vol 58 no 11 pp5693ndash5705 2010

[12] L Constancias M Cattenoz P Brouard and A Brun ldquoCoher-ent collocated MIMO radar demonstration for air defenceapplicationsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo13) pp 1ndash6 Ottawa Canada April-May 2013

[13] X Liu and G Liao ldquoDirection finding and mutual couplingestimation for bistatic MIMO radarrdquo Signal Processing vol 92no 2 pp 517ndash522 2012

[14] Z Zheng J Zhang and J Zhang ldquoJoint DOD and DOAestimation of bistatic MIMO radar in the presence of unknownmutual couplingrdquo Signal Processing vol 92 no 12 pp 3039ndash3048 2012

[15] X Wang W Wang J Liu Q Liu and B Wang ldquoTensor-basedreal-valued subspace approach for angle estimation in bistaticMIMO radar with unknown mutual couplingrdquo Signal Process-ing vol 116 pp 152ndash158 2015

[16] AThakre M Haardt and K Giridhar ldquoSingle snapshot spatialsmoothing with improved effective array aperturerdquo IEEE SignalProcessing Letters vol 16 no 6 pp 505ndash508 2009

[17] B Friedlander and A J Weiss ldquoDirection finding in thepresence of mutual couplingrdquo IEEE Transactions on Antennasand Propagation pp 273ndash284 1991

[18] F Gao B Jiang X Gao and X-D Zhang ldquoSuperimposed train-ing based channel estimation for OFDM modulated amplify-and-forward relay networksrdquo IEEE Transactions on Communi-cations vol 59 no 7 pp 2029ndash2039 2011

[19] G Wang F Gao W Chen and C Tellambura ldquoChannelestimation and training design for two-way relay networksin time-selective fading environmentsrdquo IEEE Transactions onWireless Communications vol 10 no 8 pp 2681ndash2691 2011

[20] X Zhang Matrix Analysis and Applications Tsinghua Univer-sity Press Beijing China 2013


[21] F Roemer M Haardt and G Del Galdo ldquoAnalytical perfor-mance assessment of multi-dimensional matrix- and tensor-based ESPRIT-type algorithmsrdquo IEEE Transactions on SignalProcessing vol 62 no 10 pp 2611ndash2625 2014

[22] M Haardt and J A Nossek ldquoUnitary ESPRIT how to obtainincreased estimation accuracy with a reduced computationalburdenrdquo IEEE Transactions on Signal Processing vol 43 no 5pp 1232ndash1242 1995

[23] L Zhang W Liu and L Yu ldquoPerformance analysis for finitesample MVDR beamformer with forward backward process-ingrdquo IEEE Transactions on Signal Processing vol 59 no 5 pp2427ndash2431 2011

[24] WZhangW Liu JWang and S LWu ldquoJoint transmission andreception diversity smoothing for direction finding of coherenttargets in MIMO radarrdquo IEEE Journal on Selected Topics inSignal Processing vol 8 no 1 pp 115ndash124 2014

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Submit your manuscripts athttpwwwhindawicom

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



performance guarantee In [14] an ESPRIT-Like methodperforms better than MUSIC-Like method However in thecase of small snapshots the accuracy of angle estimationusing both of the above approaches will degrade remarkablyBy exploiting the multidimensional structure of the receiveddata a three-order tensor is constructed [15] which areDODDOA and temporal dimensions respectively And a real-valued subspace approach is proposed it computes the sub-space utilizing the higher order singular value decomposition(HOSVD) Due to use of the forward-backward averagingtechnique this approach is suitable for coherent targets andsmall snapshots However the tensor-based real-valued sub-space approach just employs partial aperture it loses a lot ofinformation When there exists coherent targets the perfor-mance of that approach will degrade and it cannot deal withmore than two coherent targets Spatial smoothing techniqueis an effective approach to deal with the situation of smallsnapshots and coherent targets In [16] a spatial smooth-ing with improved aperture (SSIA) method is proposed toestimate DOA for the phased array radar it improves theeffective array aperture twice larger than the conventionalspatial smoothing approaches This technique also can beused to estimate DOD and DOA for bistatic MIMO radar

In this paper we proposed a transmitreceive spatialsmoothing with improved effective aperture (TRSSIA) meth-od for joint DOD and DOA estimation in bistatic MIMOradar with unknown mutual coupling Firstly the whiteGaussian noise is restrained using its dependency in boththe spatial and temporal domains Then the TRSSIA is usedto construct the transmit spatial smoothing matrix and thereceive spatial smoothing matrix Due to the Vandermondestructure of steering vectorwith uniform linear arrays (ULA)the transmit and receive augmented spatial smoothingmatri-ces are constructed and these two matrices can improve theeffective aperture two times larger than conventional onesThirdly by using the centro-Hermitian structure of aug-mented matrices the real-valued subspace methods (egunitary ESPRIT) can be used to estimate DOD and DOAFinally an additional DOD and DOA pairing technique isproposed and the mutual coupling coefficients are estimatedThe proposed approach restrains white Gaussian noise andtakes full advantage of the received data so it provides betterangle estimation performance And it can deal with morethan two coherent targets

The remainder of the paper is organized as follows InSection 2 the bistatic MIMO radar signal model is intro-duced The coupling calibration approach is demonstrated inSection 3 In Section 4 simulations are employed to verify theanalytical derivations Finally Section 5 gives the conclusion

Notation (sdot)H (sdot)T (sdot)minus1 and (sdot)lowast denote conjugate trans-

pose transpose inverse and conjugate respectively otimes and⊙ denote the Kronecker product operation and Khatri-Rao product operation Toeplitz(c) denotes the symmetricToeplitz matrix constructed by the vector c diag(sdot) denotesthe diagonalization operation mat(sdot) denotes the matrixingoperation Resdot and Imsdot denote abstracting the real partand imaginary part of complex number respectively emin(sdot)is an operator of getting theminimum eigenvector mean(sdot) is

used to compute themean of numbers I119872

denotes the119872timesMidentitymatrix and 0

119872times119873

is the119872times119873 zeromatrixA[119894 119895 ]denotes the 119894th to the 119895th rows of A

2 Bistatic MIMO Radar Signal Model

Consider a bistatic MIMO radar system equipped with 119872

transmit antennas and119873 receive antennas both of which areULA with half-wavelength spacing antennas The transmitantennas emit119872 orthogonal waveforms S = [s

1

s2

s119872

]T

and SSH119870 = I119872

where119870 is the number of samples per pulseperiod All the targets aremodeled as point-scatters in the far-field and we assume that there exists 119875 targets in the samerange bin Consider the effect of mutual coupling in both thetransmitter and receiver the received data of the 119897th pulse isshown as

X (119897) = [C119903

A119903

]Λ (119897) [C119905

A119905

]T S +W (119897)

119897 = 1 2 119871(1)

where X(119897) isin C119873times119870 is the received data during the 119897th pulseperiod and 119871 is the number of pulses C

119905

and C119903

are themutual coupling matrices of the transmit and receive arraysrespectively which can be expressed as banded symmetricToeplitz matrices [17]

C119905

= toeplitz [1 1198881199051

119888119905119901

119905

0 0] isin C119872times119872

(2a)

C119903

= toeplitz [1 1198881199031

119888119903119901

119903

0 0] isin C119873times119873

(2b)

where 119888119905119894

119888119903119895

are the transmit and receive mutual couplingcoefficients and there are 119901

119905

and 119901119903

nonzero mutual couplingcoefficients with 119872 ge 2119901

119905

+ 1 119873 ge 2119901119903

+ 1 respectivelyA119905

= [a119905

1

a119905

2

a119905

119875

] A119903

= [a119903

1

a119903

2

a119903

119875

] a119905

119901

=

[1 exp(j2120587119889119905

sin120601119901

120582) exp(j2120587(119872minus1)119889119905

sin120601119901

120582)]T and

a119903

119901

= [1 exp(j2120587119889119903

sin 120579119901

120582) exp(j2120587(119873minus1)119889119903

sin 120579119901

120582)]T

are the receive steering vector and the transmit steeringvector where 119889

119905

and 119889119903

are the adjacent antenna spacing oftransmit and receive arrays respectively 120601

119901

and 120579119901

are DODand DOA of the 119901th target and 120582 denoting the wavelengthOne has Λ(119897) = diag([119904

1

(119897) 1199042

(119897) 119904119875

(119897)]) where 119904119901

(119897) =

120573119901

exp(j2120587119891119901

119897) is the reflected signal of the 119901th target and 120573119901

and119891119901

are the amplitude andDoppler frequency respectivelyW(119897) isin C119873times119870 is the complex white Gaussian noise matrixwith the covariance 1205752

119908

I119873

[18 19] After matched filtering theoutput data at the receiver can be expressed as

x (119897) = CAs (119897) + w (119897) (3)

whereC = C119905

otimesC119903

A = A119905

⊙A119903

s(119897) = [1199041

(119897) 1199042

(119897) 119904119875

(119897)]T

w(119897) = vec(W(119897)SH119870)It assumes that the noise is iid complex white Gaus-

sian noise and we can obtain the equation written as


119864[vec(W(119897))vecH(W(119897))] = I119870

otimes 1205752119908

I119873


119864 [w (119897)w (119897)H] =

119864 [vec (W (119897) SH) vecH (W (119897) SH)]

1198702


119873


)]

1198702

=(Slowast otimes I

119873

) (I119870

otimes 1205752119908

I119873


)

1198702=I119872

otimes 1205752119908

I119873

119870

=1205752119908

I119872119873

119870

(4)




119909119898119899

(119897) =

119875

sum119901=1

119886119905

119901119898

119886119903

119901119899

119904119901

(119897) + 119908119898119899

(119897) (5)

where 119909119898119899



119905

119901119898

and 119886119903

119901119899


119905

119901

and a119903

119901


11

(119897) = sum119875

119901=1

119904119901

(119897) + 11990811


119903(11)(11)

(Δ119897) = 119864 [11990911

(119897) 119909lowast

11

(119897 + Δ119897)]

=

119875

sum119901

1=1

119875

sum119901

2=1

1205902

(119901

1119901

2)

(6)

where 1205902(119901

1119901

2)


1

(119897) and119904119901

2


119903(119898119899)(11)

(Δ119897) = 119864 [119909119898119899

(119897) 119909lowast

11

(119897 + Δ119897)]

=

119875

sum119901

1=1

119886119905

1199011119898

119886119903

1199011119899

119875

sum119901

2=1

1205902

(119901

1119901

2)

(7)



(11)(11)

(Δ119897) =

1015840

119897+119871

sum

119897=

1015840

119897

11990911

(119897) 119909lowast

11

(119897 + Δ119897)

119871

=

119875

sum119901

1=1

119875

sum119901

2=1

2

(119901

1119901

2)

+ 2

(11)

(8a)

(119898119899)(11)

(Δ119897) =

1015840

119897+119871

sum

119897=

1015840

119897

119909119898119899

(119897) 119909lowast

11

(119897 + Δ119897)

119871

=

119875

sum119901

1=1

119886119905

1199011119898

119886119903

1199011119899

119875

sum119901

2=1

2

(119901

1119901

2)

+ 2

(119898119899)

(8b)

where 2(11)

and 2(119898119899)



r (Δ119897)

= [(11)(11)

(Δ119897) (12)(11)

(Δ119897) (119872119873)(11)

(Δ119897)]T

= A119905

⊙ A119903

120588 + k

(9)

where 120588 = [sum119875

119901

2=1

2

(1119901

2)

sum119875

119901

2=1

2

(2119901

2)

sum119875

119901

2=1

2

(119875119901

2)

]T k =

[2

(11)

2

(12)

2

(119872119873)

]T


J119905

= J1

otimes I119873

J1

= [0(119872minus2119901

119905)times119901

119905

I119872minus2119901

119905

0(119872minus2119901

119905)times119901

119905

] (10a)

J119903

= I119872

otimes J2

J2

= [0(119873minus2119901

119903)times119901

119903

I119873minus2119901

119903

0(119873minus2119901

119903)times119901

119903

] (10b)


x119905

(119897) = J119905

x (119897) = [J1

C119905

A119905

] ⊙ [C119903

A119903

] s (119897) + J119905

w (119897)

= A119905

⊙ A119903

s (119897) + w (119897) (11)

where A119905


rows of A119905

A119903

=

[C119903

A119903

]diag(C119903

(1 )A119903

)minus1 and s(119897) = [119904

1

(119897) 1199042

(119897) 119904119875

(119897)]T

119904119901

(119897) = 120591119901

C119903

(1 )a119903119901

119904119901

(119897) 120591119901

= sum119901

119905

119902=minus119901

119905

119888119905|119902|


x119903

(119897) = J119903

x (119897) = [C119905

A119905

] ⊙ [J2

C119903

A119903

] s (119897) + J119903

w (119897)

= A119905

⊙ A119903

s (119897) + w (119897) (12)

where A119903


rows of A119903

A119905

=

[C119905

A119905

]diag(C119905

(1 )A119905

)minus1 and s(119897) = [

1

(119897) 2

(119897) 119875

(119897)]T

119901

(119897) = 120574119901

C119905

(1 )a119905119901

119904119901

(119897) 120574119901

= sum119901

119903

119902=minus119901

119903

119888119903|119902|




r119905

(Δ119897) = [119905(11)(11)

(Δ119897) 119905(12)(11)

(Δ119897)

119905(1198721119873)(11)

(Δ119897)]T= A119905

⊙ A119903

119905

+ k119905

(13a)

r119903

(Δ119897) = [119903(11)(11)

(Δ119897) 119903(12)(11)

(Δ119897)

119903(1198721198731)(11)

(Δ119897)]T= A119905

⊙ A119903

119903

+ k119903

(13b)

where 1198721 = 119872 minus 2119901119905

1198731 = 119873 minus 2119901119903

119905

=

[sum119875

119901

2=1

2

119905(1119901

2)

sum119875

119901

2=1

2

119905(2119901

2)

sum119875

119901

2=1

2

119905(119875119901

2)

]T

119903

=

[sum119875

119901

2=1

2

119903(1119901

2)

sum119875

119901

2=1

2

119903(2119901

2)

sum119875

119901

2=1

2

119903(119875119901

2)

]T and

2

119905(119894119901

2)

2

119903(119895119901

2)


(119897) and 119904119901

2


119895

(119897) and 119901

2


= [2

119905(11)

2

119905(12)

2

119905(1198721119873)

]T and k

119903

= [2

119903(11)

2

119903(12)

2

119903(1198721198731)

]T where

2

119905(119898119899)

and 2

119903(119898119899)


119905(11)

(119897) and w119905(119898119899)

(119897 + Δ119897) w119903(11)

(119897) andw119903(119898119899)



119905

(Δ119897) r119903


119905

r119903




Q2119899+1

=1

radic2

[[[

[

I119899

0 jI119899

0T radic2 0T

Π119899

0 minusjΠ119899

]]]

]

(14)




For r119905

and1198721 = 119872sub + 119871119905

minus 1 we define

X119905

= [(H1199051


) r119905

(H119905119871

119905

otimes I119873

) r119905

]

H119905119897


I119872sub

0119872subtimes(119871119905minus119897)

] 1 le 119897 le 119871119905

(15)


X119905

= [A119905

⊙ A119903

] Λ119905

AT119905

+ V119905

(16)

where Λ119905

= diag(119905

) A119905

and A119905


119905

rows of A119905

respectively


Xtaug =[

[

X119905

Π119872sub119873

Xlowast119905

Π119871

119905

]

]

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

AT119905

+ [

[

V119905

Π119872sub119873

Vlowast119905

Π119871

119905

]

]

= AtaugAT119905

+ Vtaug

isin C2119872sub119873times119871119905

(17)

where Λ1199051

= diag([119886119905

1(1minus119872sub)

119886119905

2(1minus119872sub)

119886119905

119875(1minus119872sub)

]) andΛ1199052

= diag([119886119905

1(1minus119871

119905)

119886119905

2(1minus119871

119905)

119886119905

119875(1minus119871

119905)


119905

⊙ A119903


119905



Π2119872sub119873



120593 (Xtaug) = QH2119872sub119873

XtaugQ119871119905

(19)




(2)



J(2)taugQ2119872sub119873 (21a)


J(2)taugQ2119872sub119873 (21b)

where J(2)taug = I2

otimes J1199052

J1199052

= [0(119872subminus1)119873times119873

I(119872subminus1)119873



119894

119875

119894=1




119903


Xraug = [X119903

Π119872119873sub

Xlowast119903

Π119871

119903

] = AraugAT119903

+ Vraug

isin C2119872119873subtimes119871119903

(22)

where X119903

= [(I119872

otimesH1199031

)r119903

(I119872

otimesH1199032

)r119903

(I119872

otimesH119903119871

119903

)r119903

] andH119903119897


I119873sub

0119873subtimes(119871119903minus119897)

] 1 le 119897 le 119871119903


2119872119873subXraugQ119871

119903






(2)

raug = 2 sdot



otimes J1199032

J1199032

= I119872

otimes


I119873subminus1


119875

119895=1




a119903


119875

119894=1


119895

119875

119895=1


A119905

⊙ A119903

T1


1

(24a)

A119905

⊙ A119903

T2


(24b)

where A119905

A119903


and A119903

andT1

and T2


J(3)taug = [1 0] otimes J1199053

J1199053

= I119872sub


119903

I1198731

01198731times119901

119903]

(25a)

J(3)raug = [1 0] otimes J1199033

J1199033

= [01198721times119901

119905

I1198721

01198721times119901


(25b)


P (119894 119895)

=1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I119872sub1198731

minus U1

UH1

) (a119905

(120601119894

) otimes a119903

(120579119895

))∙

1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I1198721119873sub

minus U2

UH2

) (a119905

(120601119894

) otimes a119903

(120579119895

))(26)


(120601119894

) a119905

(120601119894


rows of a119905

(120601119894


(120579119895

) a119903

(120579119895


rows of a119903

(120579119895


119894

with 120579119895






119886 (119897) = A (1 119897) 119897 = 1 2 119871 (28)



following matrices

[W1

]119901119902

=

x119901+119902minus1

119901 + 119902 le 119872 + 1

0 119900119905ℎ119890119903119908119894119904119890

[W2

]119901119902

=

x119901minus119902+1

119901 le 119902 le 2

0 119900119905ℎ119890119903119908119894119904119890

(29)


C119905

a119905

(120601) = T119905

(120601) c119905

c119905

= [1 1198881199051

1198881199052

119888119905119901

119905

]T (30a)

C119903

a119903

(120579) = T119903

(120579) c119903

c119903

= [1 1198881199031

1198881199032

119888119903119901

119903

]T (30b)


([C119905

A119905

] ⊙ A119903

)T3


J(4)raug = [I119872119873sub

0119872119873sub

]

(31)



RMSE

(deg

)



102

100

10minus2

(a)


RMSE

(deg

)



102

100

10minus2

10minus4

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 119871 = 64 (b) 119875 = 3 119901119905

= 119901119903

= 2119872sub = 3119873sub = 3and 119871 = 128

where T3


a119905

(120601119894

)] otimes a119903

(120579119894

))HUperp3

2


119905

(120601119894

)c119905


(120579119894

))i119903

)HUperp3

2

= (c119905

otimes i119903

)H(T119905

(120601119894

) otimes

diag(a119903

(120579119894

)))HUperp3

2

= 0 where i119903


3



ctaug = c119905

otimes i119903

= emin (1

119875

119875

sum119894=1

Q119905119894

) (32)

whereQ119905119894

= (T119905

(120601119894


(120579119894

)))H(I119872119873sub

minusUH3

U3

)(T119905

(120601119894

) otimes

diag(a119903

(120579119894


119905


iT119903


C119905

(1 ) = iT119903

119905119894

119901

119905

119894=1

= mean (C119905

(119894 + 1 )) (33)


119905

⊙

[C119903

A119903

])T4


4


0119872sub119873


119903119895

119901

119903

119895=1

=

mean(C119903

(119895 + 1 ))



10 and 119889119905

and 119889119903





1

1205791

) =

(minus15∘ 40∘) (1206012

1205792

) = (35∘ minus20∘) and (1206013

1205793



119905

= 119901119903

= 1 c119905

= [1 02 + 11989500061]T and c

119903

=


(2) 119901119905

= 119901119903

= 2 c119905

= [1 07 + 1198950002 02 + 11989500061]T and

c119903



119905


119903





RMSE

(deg

)



100

101

102

10minus1

10minus2

(a)RM

SE (d

eg)


102

103

104

100

101

10minus1

1205902

diag1205902n


off1205902n


diag1205902n

of proposed method

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 and SNR = 5 dB)


2

off and 1205902119899


119905

and Λ119903


119897=1


2

119899


2

119899


2

119899



119889

= [100 110 200]T Hz (2) f

119889

=

[100 100 150]T Hz and (3) f

119889

= [100 100 100]T Hz As




RMSE

(deg

)


TRSSIACRB

Case 1Case 3

Case 210

0

102

101

10minus2

10minus3

10minus4

10minus1


119905

= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)





RMSE

(deg

)

TRDSTRSSIACRB

10minus13

10minus15

10minus17

10minus19

10minus11

(a)

20 40 60 80 100 1200

0005

001

0015

002

0025

003

0035

Number of pulses

Runt

ime (

seco

nd)

TRDSTRSSIA

(b)


119889

= [100 100 100]Hz 119901119905

= 119901119903



RMSE


102

100

10minus2

10minus4


= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)


5 Conclusion


Appendix



119879119878


119879119878


119879119878


andU119879119878



119872sub119873 rows ofU119879119878


[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)


A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)


Φ119905

= A119905

where A119905

and A119905


119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901


[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)





References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119864[vec(W(119897))vecH(W(119897))] = I119870

otimes 1205752119908

I119873


119864 [w (119897)w (119897)H] =

119864 [vec (W (119897) SH) vecH (W (119897) SH)]

1198702


119873


)]

1198702

=(Slowast otimes I

119873

) (I119870

otimes 1205752119908

I119873


)

1198702=I119872

otimes 1205752119908

I119873

119870

=1205752119908

I119872119873

119870

(4)




119909119898119899

(119897) =

119875

sum119901=1

119886119905

119901119898

119886119903

119901119899

119904119901

(119897) + 119908119898119899

(119897) (5)

where 119909119898119899



119905

119901119898

and 119886119903

119901119899


119905

119901

and a119903

119901


11

(119897) = sum119875

119901=1

119904119901

(119897) + 11990811


119903(11)(11)

(Δ119897) = 119864 [11990911

(119897) 119909lowast

11

(119897 + Δ119897)]

=

119875

sum119901

1=1

119875

sum119901

2=1

1205902

(119901

1119901

2)

(6)

where 1205902(119901

1119901

2)


1

(119897) and119904119901

2


119903(119898119899)(11)

(Δ119897) = 119864 [119909119898119899

(119897) 119909lowast

11

(119897 + Δ119897)]

=

119875

sum119901

1=1

119886119905

1199011119898

119886119903

1199011119899

119875

sum119901

2=1

1205902

(119901

1119901

2)

(7)



(11)(11)

(Δ119897) =

1015840

119897+119871

sum

119897=

1015840

119897

11990911

(119897) 119909lowast

11

(119897 + Δ119897)

119871

=

119875

sum119901

1=1

119875

sum119901

2=1

2

(119901

1119901

2)

+ 2

(11)

(8a)

(119898119899)(11)

(Δ119897) =

1015840

119897+119871

sum

119897=

1015840

119897

119909119898119899

(119897) 119909lowast

11

(119897 + Δ119897)

119871

=

119875

sum119901

1=1

119886119905

1199011119898

119886119903

1199011119899

119875

sum119901

2=1

2

(119901

1119901

2)

+ 2

(119898119899)

(8b)

where 2(11)

and 2(119898119899)



r (Δ119897)

= [(11)(11)

(Δ119897) (12)(11)

(Δ119897) (119872119873)(11)

(Δ119897)]T

= A119905

⊙ A119903

120588 + k

(9)

where 120588 = [sum119875

119901

2=1

2

(1119901

2)

sum119875

119901

2=1

2

(2119901

2)

sum119875

119901

2=1

2

(119875119901

2)

]T k =

[2

(11)

2

(12)

2

(119872119873)

]T


J119905

= J1

otimes I119873

J1

= [0(119872minus2119901

119905)times119901

119905

I119872minus2119901

119905

0(119872minus2119901

119905)times119901

119905

] (10a)

J119903

= I119872

otimes J2

J2

= [0(119873minus2119901

119903)times119901

119903

I119873minus2119901

119903

0(119873minus2119901

119903)times119901

119903

] (10b)


x119905

(119897) = J119905

x (119897) = [J1

C119905

A119905

] ⊙ [C119903

A119903

] s (119897) + J119905

w (119897)

= A119905

⊙ A119903

s (119897) + w (119897) (11)

where A119905


rows of A119905

A119903

=

[C119903

A119903

]diag(C119903

(1 )A119903

)minus1 and s(119897) = [119904

1

(119897) 1199042

(119897) 119904119875

(119897)]T

119904119901

(119897) = 120591119901

C119903

(1 )a119903119901

119904119901

(119897) 120591119901

= sum119901

119905

119902=minus119901

119905

119888119905|119902|


x119903

(119897) = J119903

x (119897) = [C119905

A119905

] ⊙ [J2

C119903

A119903

] s (119897) + J119903

w (119897)

= A119905

⊙ A119903

s (119897) + w (119897) (12)

where A119903


rows of A119903

A119905

=

[C119905

A119905

]diag(C119905

(1 )A119905

)minus1 and s(119897) = [

1

(119897) 2

(119897) 119875

(119897)]T

119901

(119897) = 120574119901

C119905

(1 )a119905119901

119904119901

(119897) 120574119901

= sum119901

119903

119902=minus119901

119903

119888119903|119902|




r119905

(Δ119897) = [119905(11)(11)

(Δ119897) 119905(12)(11)

(Δ119897)

119905(1198721119873)(11)

(Δ119897)]T= A119905

⊙ A119903

119905

+ k119905

(13a)

r119903

(Δ119897) = [119903(11)(11)

(Δ119897) 119903(12)(11)

(Δ119897)

119903(1198721198731)(11)

(Δ119897)]T= A119905

⊙ A119903

119903

+ k119903

(13b)

where 1198721 = 119872 minus 2119901119905

1198731 = 119873 minus 2119901119903

119905

=

[sum119875

119901

2=1

2

119905(1119901

2)

sum119875

119901

2=1

2

119905(2119901

2)

sum119875

119901

2=1

2

119905(119875119901

2)

]T

119903

=

[sum119875

119901

2=1

2

119903(1119901

2)

sum119875

119901

2=1

2

119903(2119901

2)

sum119875

119901

2=1

2

119903(119875119901

2)

]T and

2

119905(119894119901

2)

2

119903(119895119901

2)


(119897) and 119904119901

2


119895

(119897) and 119901

2


= [2

119905(11)

2

119905(12)

2

119905(1198721119873)

]T and k

119903

= [2

119903(11)

2

119903(12)

2

119903(1198721198731)

]T where

2

119905(119898119899)

and 2

119903(119898119899)


119905(11)

(119897) and w119905(119898119899)

(119897 + Δ119897) w119903(11)

(119897) andw119903(119898119899)



119905

(Δ119897) r119903


119905

r119903




Q2119899+1

=1

radic2

[[[

[

I119899

0 jI119899

0T radic2 0T

Π119899

0 minusjΠ119899

]]]

]

(14)




For r119905

and1198721 = 119872sub + 119871119905

minus 1 we define

X119905

= [(H1199051


) r119905

(H119905119871

119905

otimes I119873

) r119905

]

H119905119897


I119872sub

0119872subtimes(119871119905minus119897)

] 1 le 119897 le 119871119905

(15)


X119905

= [A119905

⊙ A119903

] Λ119905

AT119905

+ V119905

(16)

where Λ119905

= diag(119905

) A119905

and A119905


119905

rows of A119905

respectively


Xtaug =[

[

X119905

Π119872sub119873

Xlowast119905

Π119871

119905

]

]

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

AT119905

+ [

[

V119905

Π119872sub119873

Vlowast119905

Π119871

119905

]

]

= AtaugAT119905

+ Vtaug

isin C2119872sub119873times119871119905

(17)

where Λ1199051

= diag([119886119905

1(1minus119872sub)

119886119905

2(1minus119872sub)

119886119905

119875(1minus119872sub)

]) andΛ1199052

= diag([119886119905

1(1minus119871

119905)

119886119905

2(1minus119871

119905)

119886119905

119875(1minus119871

119905)


119905

⊙ A119903


119905



Π2119872sub119873



120593 (Xtaug) = QH2119872sub119873

XtaugQ119871119905

(19)




(2)



J(2)taugQ2119872sub119873 (21a)


J(2)taugQ2119872sub119873 (21b)

where J(2)taug = I2

otimes J1199052

J1199052

= [0(119872subminus1)119873times119873

I(119872subminus1)119873



119894

119875

119894=1




119903


Xraug = [X119903

Π119872119873sub

Xlowast119903

Π119871

119903

] = AraugAT119903

+ Vraug

isin C2119872119873subtimes119871119903

(22)

where X119903

= [(I119872

otimesH1199031

)r119903

(I119872

otimesH1199032

)r119903

(I119872

otimesH119903119871

119903

)r119903

] andH119903119897


I119873sub

0119873subtimes(119871119903minus119897)

] 1 le 119897 le 119871119903


2119872119873subXraugQ119871

119903






(2)

raug = 2 sdot



otimes J1199032

J1199032

= I119872

otimes


I119873subminus1


119875

119895=1




a119903


119875

119894=1


119895

119875

119895=1


A119905

⊙ A119903

T1


1

(24a)

A119905

⊙ A119903

T2


(24b)

where A119905

A119903


and A119903

andT1

and T2


J(3)taug = [1 0] otimes J1199053

J1199053

= I119872sub


119903

I1198731

01198731times119901

119903]

(25a)

J(3)raug = [1 0] otimes J1199033

J1199033

= [01198721times119901

119905

I1198721

01198721times119901


(25b)


P (119894 119895)

=1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I119872sub1198731

minus U1

UH1

) (a119905

(120601119894

) otimes a119903

(120579119895

))∙

1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I1198721119873sub

minus U2

UH2

) (a119905

(120601119894

) otimes a119903

(120579119895

))(26)


(120601119894

) a119905

(120601119894


rows of a119905

(120601119894


(120579119895

) a119903

(120579119895


rows of a119903

(120579119895


119894

with 120579119895






119886 (119897) = A (1 119897) 119897 = 1 2 119871 (28)



following matrices

[W1

]119901119902

=

x119901+119902minus1

119901 + 119902 le 119872 + 1

0 119900119905ℎ119890119903119908119894119904119890

[W2

]119901119902

=

x119901minus119902+1

119901 le 119902 le 2

0 119900119905ℎ119890119903119908119894119904119890

(29)


C119905

a119905

(120601) = T119905

(120601) c119905

c119905

= [1 1198881199051

1198881199052

119888119905119901

119905

]T (30a)

C119903

a119903

(120579) = T119903

(120579) c119903

c119903

= [1 1198881199031

1198881199032

119888119903119901

119903

]T (30b)


([C119905

A119905

] ⊙ A119903

)T3


J(4)raug = [I119872119873sub

0119872119873sub

]

(31)



RMSE

(deg

)



102

100

10minus2

(a)


RMSE

(deg

)



102

100

10minus2

10minus4

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 119871 = 64 (b) 119875 = 3 119901119905

= 119901119903

= 2119872sub = 3119873sub = 3and 119871 = 128

where T3


a119905

(120601119894

)] otimes a119903

(120579119894

))HUperp3

2


119905

(120601119894

)c119905


(120579119894

))i119903

)HUperp3

2

= (c119905

otimes i119903

)H(T119905

(120601119894

) otimes

diag(a119903

(120579119894

)))HUperp3

2

= 0 where i119903


3



ctaug = c119905

otimes i119903

= emin (1

119875

119875

sum119894=1

Q119905119894

) (32)

whereQ119905119894

= (T119905

(120601119894


(120579119894

)))H(I119872119873sub

minusUH3

U3

)(T119905

(120601119894

) otimes

diag(a119903

(120579119894


119905


iT119903


C119905

(1 ) = iT119903

119905119894

119901

119905

119894=1

= mean (C119905

(119894 + 1 )) (33)


119905

⊙

[C119903

A119903

])T4


4


0119872sub119873


119903119895

119901

119903

119895=1

=

mean(C119903

(119895 + 1 ))



10 and 119889119905

and 119889119903





1

1205791

) =

(minus15∘ 40∘) (1206012

1205792

) = (35∘ minus20∘) and (1206013

1205793



119905

= 119901119903

= 1 c119905

= [1 02 + 11989500061]T and c

119903

=


(2) 119901119905

= 119901119903

= 2 c119905

= [1 07 + 1198950002 02 + 11989500061]T and

c119903



119905


119903





RMSE

(deg

)



100

101

102

10minus1

10minus2

(a)RM

SE (d

eg)


102

103

104

100

101

10minus1

1205902

diag1205902n


off1205902n


diag1205902n

of proposed method

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 and SNR = 5 dB)


2

off and 1205902119899


119905

and Λ119903


119897=1


2

119899


2

119899


2

119899



119889

= [100 110 200]T Hz (2) f

119889

=

[100 100 150]T Hz and (3) f

119889

= [100 100 100]T Hz As




RMSE

(deg

)


TRSSIACRB

Case 1Case 3

Case 210

0

102

101

10minus2

10minus3

10minus4

10minus1


119905

= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)





RMSE

(deg

)

TRDSTRSSIACRB

10minus13

10minus15

10minus17

10minus19

10minus11

(a)

20 40 60 80 100 1200

0005

001

0015

002

0025

003

0035

Number of pulses

Runt

ime (

seco

nd)

TRDSTRSSIA

(b)


119889

= [100 100 100]Hz 119901119905

= 119901119903



RMSE


102

100

10minus2

10minus4


= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)


5 Conclusion


Appendix



119879119878


119879119878


119879119878


andU119879119878



119872sub119873 rows ofU119879119878


[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)


A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)


Φ119905

= A119905

where A119905

and A119905


119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901


[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)





References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











r119905

(Δ119897) = [119905(11)(11)

(Δ119897) 119905(12)(11)

(Δ119897)

119905(1198721119873)(11)

(Δ119897)]T= A119905

⊙ A119903

119905

+ k119905

(13a)

r119903

(Δ119897) = [119903(11)(11)

(Δ119897) 119903(12)(11)

(Δ119897)

119903(1198721198731)(11)

(Δ119897)]T= A119905

⊙ A119903

119903

+ k119903

(13b)

where 1198721 = 119872 minus 2119901119905

1198731 = 119873 minus 2119901119903

119905

=

[sum119875

119901

2=1

2

119905(1119901

2)

sum119875

119901

2=1

2

119905(2119901

2)

sum119875

119901

2=1

2

119905(119875119901

2)

]T

119903

=

[sum119875

119901

2=1

2

119903(1119901

2)

sum119875

119901

2=1

2

119903(2119901

2)

sum119875

119901

2=1

2

119903(119875119901

2)

]T and

2

119905(119894119901

2)

2

119903(119895119901

2)


(119897) and 119904119901

2


119895

(119897) and 119901

2


= [2

119905(11)

2

119905(12)

2

119905(1198721119873)

]T and k

119903

= [2

119903(11)

2

119903(12)

2

119903(1198721198731)

]T where

2

119905(119898119899)

and 2

119903(119898119899)


119905(11)

(119897) and w119905(119898119899)

(119897 + Δ119897) w119903(11)

(119897) andw119903(119898119899)



119905

(Δ119897) r119903


119905

r119903




Q2119899+1

=1

radic2

[[[

[

I119899

0 jI119899

0T radic2 0T

Π119899

0 minusjΠ119899

]]]

]

(14)




For r119905

and1198721 = 119872sub + 119871119905

minus 1 we define

X119905

= [(H1199051


) r119905

(H119905119871

119905

otimes I119873

) r119905

]

H119905119897


I119872sub

0119872subtimes(119871119905minus119897)

] 1 le 119897 le 119871119905

(15)


X119905

= [A119905

⊙ A119903

] Λ119905

AT119905

+ V119905

(16)

where Λ119905

= diag(119905

) A119905

and A119905


119905

rows of A119905

respectively


Xtaug =[

[

X119905

Π119872sub119873

Xlowast119905

Π119871

119905

]

]

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

AT119905

+ [

[

V119905

Π119872sub119873

Vlowast119905

Π119871

119905

]

]

= AtaugAT119905

+ Vtaug

isin C2119872sub119873times119871119905

(17)

where Λ1199051

= diag([119886119905

1(1minus119872sub)

119886119905

2(1minus119872sub)

119886119905

119875(1minus119872sub)

]) andΛ1199052

= diag([119886119905

1(1minus119871

119905)

119886119905

2(1minus119871

119905)

119886119905

119875(1minus119871

119905)


119905

⊙ A119903


119905



Π2119872sub119873



120593 (Xtaug) = QH2119872sub119873

XtaugQ119871119905

(19)




(2)



J(2)taugQ2119872sub119873 (21a)


J(2)taugQ2119872sub119873 (21b)

where J(2)taug = I2

otimes J1199052

J1199052

= [0(119872subminus1)119873times119873

I(119872subminus1)119873



119894

119875

119894=1




119903


Xraug = [X119903

Π119872119873sub

Xlowast119903

Π119871

119903

] = AraugAT119903

+ Vraug

isin C2119872119873subtimes119871119903

(22)

where X119903

= [(I119872

otimesH1199031

)r119903

(I119872

otimesH1199032

)r119903

(I119872

otimesH119903119871

119903

)r119903

] andH119903119897


I119873sub

0119873subtimes(119871119903minus119897)

] 1 le 119897 le 119871119903


2119872119873subXraugQ119871

119903






(2)

raug = 2 sdot



otimes J1199032

J1199032

= I119872

otimes


I119873subminus1


119875

119895=1




a119903


119875

119894=1


119895

119875

119895=1


A119905

⊙ A119903

T1


1

(24a)

A119905

⊙ A119903

T2


(24b)

where A119905

A119903


and A119903

andT1

and T2


J(3)taug = [1 0] otimes J1199053

J1199053

= I119872sub


119903

I1198731

01198731times119901

119903]

(25a)

J(3)raug = [1 0] otimes J1199033

J1199033

= [01198721times119901

119905

I1198721

01198721times119901


(25b)


P (119894 119895)

=1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I119872sub1198731

minus U1

UH1

) (a119905

(120601119894

) otimes a119903

(120579119895

))∙

1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I1198721119873sub

minus U2

UH2

) (a119905

(120601119894

) otimes a119903

(120579119895

))(26)


(120601119894

) a119905

(120601119894


rows of a119905

(120601119894


(120579119895

) a119903

(120579119895


rows of a119903

(120579119895


119894

with 120579119895






119886 (119897) = A (1 119897) 119897 = 1 2 119871 (28)



following matrices

[W1

]119901119902

=

x119901+119902minus1

119901 + 119902 le 119872 + 1

0 119900119905ℎ119890119903119908119894119904119890

[W2

]119901119902

=

x119901minus119902+1

119901 le 119902 le 2

0 119900119905ℎ119890119903119908119894119904119890

(29)


C119905

a119905

(120601) = T119905

(120601) c119905

c119905

= [1 1198881199051

1198881199052

119888119905119901

119905

]T (30a)

C119903

a119903

(120579) = T119903

(120579) c119903

c119903

= [1 1198881199031

1198881199032

119888119903119901

119903

]T (30b)


([C119905

A119905

] ⊙ A119903

)T3


J(4)raug = [I119872119873sub

0119872119873sub

]

(31)



RMSE

(deg

)



102

100

10minus2

(a)


RMSE

(deg

)



102

100

10minus2

10minus4

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 119871 = 64 (b) 119875 = 3 119901119905

= 119901119903

= 2119872sub = 3119873sub = 3and 119871 = 128

where T3


a119905

(120601119894

)] otimes a119903

(120579119894

))HUperp3

2


119905

(120601119894

)c119905


(120579119894

))i119903

)HUperp3

2

= (c119905

otimes i119903

)H(T119905

(120601119894

) otimes

diag(a119903

(120579119894

)))HUperp3

2

= 0 where i119903


3



ctaug = c119905

otimes i119903

= emin (1

119875

119875

sum119894=1

Q119905119894

) (32)

whereQ119905119894

= (T119905

(120601119894


(120579119894

)))H(I119872119873sub

minusUH3

U3

)(T119905

(120601119894

) otimes

diag(a119903

(120579119894


119905


iT119903


C119905

(1 ) = iT119903

119905119894

119901

119905

119894=1

= mean (C119905

(119894 + 1 )) (33)


119905

⊙

[C119903

A119903

])T4


4


0119872sub119873


119903119895

119901

119903

119895=1

=

mean(C119903

(119895 + 1 ))



10 and 119889119905

and 119889119903





1

1205791

) =

(minus15∘ 40∘) (1206012

1205792

) = (35∘ minus20∘) and (1206013

1205793



119905

= 119901119903

= 1 c119905

= [1 02 + 11989500061]T and c

119903

=


(2) 119901119905

= 119901119903

= 2 c119905

= [1 07 + 1198950002 02 + 11989500061]T and

c119903



119905


119903





RMSE

(deg

)



100

101

102

10minus1

10minus2

(a)RM

SE (d

eg)


102

103

104

100

101

10minus1

1205902

diag1205902n


off1205902n


diag1205902n

of proposed method

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 and SNR = 5 dB)


2

off and 1205902119899


119905

and Λ119903


119897=1


2

119899


2

119899


2

119899



119889

= [100 110 200]T Hz (2) f

119889

=

[100 100 150]T Hz and (3) f

119889

= [100 100 100]T Hz As




RMSE

(deg

)


TRSSIACRB

Case 1Case 3

Case 210

0

102

101

10minus2

10minus3

10minus4

10minus1


119905

= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)





RMSE

(deg

)

TRDSTRSSIACRB

10minus13

10minus15

10minus17

10minus19

10minus11

(a)

20 40 60 80 100 1200

0005

001

0015

002

0025

003

0035

Number of pulses

Runt

ime (

seco

nd)

TRDSTRSSIA

(b)


119889

= [100 100 100]Hz 119901119905

= 119901119903



RMSE


102

100

10minus2

10minus4


= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)


5 Conclusion


Appendix



119879119878


119879119878


119879119878


andU119879119878



119872sub119873 rows ofU119879119878


[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)


A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)


Φ119905

= A119905

where A119905

and A119905


119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901


[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)





References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119903


Xraug = [X119903

Π119872119873sub

Xlowast119903

Π119871

119903

] = AraugAT119903

+ Vraug

isin C2119872119873subtimes119871119903

(22)

where X119903

= [(I119872

otimesH1199031

)r119903

(I119872

otimesH1199032

)r119903

(I119872

otimesH119903119871

119903

)r119903

] andH119903119897


I119873sub

0119873subtimes(119871119903minus119897)

] 1 le 119897 le 119871119903


2119872119873subXraugQ119871

119903






(2)

raug = 2 sdot



otimes J1199032

J1199032

= I119872

otimes


I119873subminus1


119875

119895=1




a119903


119875

119894=1


119895

119875

119895=1


A119905

⊙ A119903

T1


1

(24a)

A119905

⊙ A119903

T2


(24b)

where A119905

A119903


and A119903

andT1

and T2


J(3)taug = [1 0] otimes J1199053

J1199053

= I119872sub


119903

I1198731

01198731times119901

119903]

(25a)

J(3)raug = [1 0] otimes J1199033

J1199033

= [01198721times119901

119905

I1198721

01198721times119901


(25b)


P (119894 119895)

=1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I119872sub1198731

minus U1

UH1

) (a119905

(120601119894

) otimes a119903

(120579119895

))∙

1

(a119905

(120601119894

) otimes a119903

(120579119895

))H(I1198721119873sub

minus U2

UH2

) (a119905

(120601119894

) otimes a119903

(120579119895

))(26)


(120601119894

) a119905

(120601119894


rows of a119905

(120601119894


(120579119895

) a119903

(120579119895


rows of a119903

(120579119895


119894

with 120579119895






119886 (119897) = A (1 119897) 119897 = 1 2 119871 (28)



following matrices

[W1

]119901119902

=

x119901+119902minus1

119901 + 119902 le 119872 + 1

0 119900119905ℎ119890119903119908119894119904119890

[W2

]119901119902

=

x119901minus119902+1

119901 le 119902 le 2

0 119900119905ℎ119890119903119908119894119904119890

(29)


C119905

a119905

(120601) = T119905

(120601) c119905

c119905

= [1 1198881199051

1198881199052

119888119905119901

119905

]T (30a)

C119903

a119903

(120579) = T119903

(120579) c119903

c119903

= [1 1198881199031

1198881199032

119888119903119901

119903

]T (30b)


([C119905

A119905

] ⊙ A119903

)T3


J(4)raug = [I119872119873sub

0119872119873sub

]

(31)



RMSE

(deg

)



102

100

10minus2

(a)


RMSE

(deg

)



102

100

10minus2

10minus4

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 119871 = 64 (b) 119875 = 3 119901119905

= 119901119903

= 2119872sub = 3119873sub = 3and 119871 = 128

where T3


a119905

(120601119894

)] otimes a119903

(120579119894

))HUperp3

2


119905

(120601119894

)c119905


(120579119894

))i119903

)HUperp3

2

= (c119905

otimes i119903

)H(T119905

(120601119894

) otimes

diag(a119903

(120579119894

)))HUperp3

2

= 0 where i119903


3



ctaug = c119905

otimes i119903

= emin (1

119875

119875

sum119894=1

Q119905119894

) (32)

whereQ119905119894

= (T119905

(120601119894


(120579119894

)))H(I119872119873sub

minusUH3

U3

)(T119905

(120601119894

) otimes

diag(a119903

(120579119894


119905


iT119903


C119905

(1 ) = iT119903

119905119894

119901

119905

119894=1

= mean (C119905

(119894 + 1 )) (33)


119905

⊙

[C119903

A119903

])T4


4


0119872sub119873


119903119895

119901

119903

119895=1

=

mean(C119903

(119895 + 1 ))



10 and 119889119905

and 119889119903





1

1205791

) =

(minus15∘ 40∘) (1206012

1205792

) = (35∘ minus20∘) and (1206013

1205793



119905

= 119901119903

= 1 c119905

= [1 02 + 11989500061]T and c

119903

=


(2) 119901119905

= 119901119903

= 2 c119905

= [1 07 + 1198950002 02 + 11989500061]T and

c119903



119905


119903





RMSE

(deg

)



100

101

102

10minus1

10minus2

(a)RM

SE (d

eg)


102

103

104

100

101

10minus1

1205902

diag1205902n


off1205902n


diag1205902n

of proposed method

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 and SNR = 5 dB)


2

off and 1205902119899


119905

and Λ119903


119897=1


2

119899


2

119899


2

119899



119889

= [100 110 200]T Hz (2) f

119889

=

[100 100 150]T Hz and (3) f

119889

= [100 100 100]T Hz As




RMSE

(deg

)


TRSSIACRB

Case 1Case 3

Case 210

0

102

101

10minus2

10minus3

10minus4

10minus1


119905

= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)





RMSE

(deg

)

TRDSTRSSIACRB

10minus13

10minus15

10minus17

10minus19

10minus11

(a)

20 40 60 80 100 1200

0005

001

0015

002

0025

003

0035

Number of pulses

Runt

ime (

seco

nd)

TRDSTRSSIA

(b)


119889

= [100 100 100]Hz 119901119905

= 119901119903



RMSE


102

100

10minus2

10minus4


= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)


5 Conclusion


Appendix



119879119878


119879119878


119879119878


andU119879119878



119872sub119873 rows ofU119879119878


[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)


A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)


Φ119905

= A119905

where A119905

and A119905


119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901


[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)





References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RMSE

(deg

)



102

100

10minus2

(a)


RMSE

(deg

)



102

100

10minus2

10minus4

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 119871 = 64 (b) 119875 = 3 119901119905

= 119901119903

= 2119872sub = 3119873sub = 3and 119871 = 128

where T3


a119905

(120601119894

)] otimes a119903

(120579119894

))HUperp3

2


119905

(120601119894

)c119905


(120579119894

))i119903

)HUperp3

2

= (c119905

otimes i119903

)H(T119905

(120601119894

) otimes

diag(a119903

(120579119894

)))HUperp3

2

= 0 where i119903


3



ctaug = c119905

otimes i119903

= emin (1

119875

119875

sum119894=1

Q119905119894

) (32)

whereQ119905119894

= (T119905

(120601119894


(120579119894

)))H(I119872119873sub

minusUH3

U3

)(T119905

(120601119894

) otimes

diag(a119903

(120579119894


119905


iT119903


C119905

(1 ) = iT119903

119905119894

119901

119905

119894=1

= mean (C119905

(119894 + 1 )) (33)


119905

⊙

[C119903

A119903

])T4


4


0119872sub119873


119903119895

119901

119903

119895=1

=

mean(C119903

(119895 + 1 ))



10 and 119889119905

and 119889119903





1

1205791

) =

(minus15∘ 40∘) (1206012

1205792

) = (35∘ minus20∘) and (1206013

1205793



119905

= 119901119903

= 1 c119905

= [1 02 + 11989500061]T and c

119903

=


(2) 119901119905

= 119901119903

= 2 c119905

= [1 07 + 1198950002 02 + 11989500061]T and

c119903



119905


119903





RMSE

(deg

)



100

101

102

10minus1

10minus2

(a)RM

SE (d

eg)


102

103

104

100

101

10minus1

1205902

diag1205902n


off1205902n


diag1205902n

of proposed method

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 and SNR = 5 dB)


2

off and 1205902119899


119905

and Λ119903


119897=1


2

119899


2

119899


2

119899



119889

= [100 110 200]T Hz (2) f

119889

=

[100 100 150]T Hz and (3) f

119889

= [100 100 100]T Hz As




RMSE

(deg

)


TRSSIACRB

Case 1Case 3

Case 210

0

102

101

10minus2

10minus3

10minus4

10minus1


119905

= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)





RMSE

(deg

)

TRDSTRSSIACRB

10minus13

10minus15

10minus17

10minus19

10minus11

(a)

20 40 60 80 100 1200

0005

001

0015

002

0025

003

0035

Number of pulses

Runt

ime (

seco

nd)

TRDSTRSSIA

(b)


119889

= [100 100 100]Hz 119901119905

= 119901119903



RMSE


102

100

10minus2

10minus4


= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)


5 Conclusion


Appendix



119879119878


119879119878


119879119878


andU119879119878



119872sub119873 rows ofU119879119878


[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)


A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)


Φ119905

= A119905

where A119905

and A119905


119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901


[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)





References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RMSE

(deg

)



100

101

102

10minus1

10minus2

(a)RM

SE (d

eg)


102

103

104

100

101

10minus1

1205902

diag1205902n


off1205902n


diag1205902n

of proposed method

(b)


= 119901119903

= 1119872sub = 4119873sub = 4 and SNR = 5 dB)


2

off and 1205902119899


119905

and Λ119903


119897=1


2

119899


2

119899


2

119899



119889

= [100 110 200]T Hz (2) f

119889

=

[100 100 150]T Hz and (3) f

119889

= [100 100 100]T Hz As




RMSE

(deg

)


TRSSIACRB

Case 1Case 3

Case 210

0

102

101

10minus2

10minus3

10minus4

10minus1


119905

= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)





RMSE

(deg

)

TRDSTRSSIACRB

10minus13

10minus15

10minus17

10minus19

10minus11

(a)

20 40 60 80 100 1200

0005

001

0015

002

0025

003

0035

Number of pulses

Runt

ime (

seco

nd)

TRDSTRSSIA

(b)


119889

= [100 100 100]Hz 119901119905

= 119901119903



RMSE


102

100

10minus2

10minus4


= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)


5 Conclusion


Appendix



119879119878


119879119878


119879119878


andU119879119878



119872sub119873 rows ofU119879119878


[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)


A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)


Φ119905

= A119905

where A119905

and A119905


119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901


[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)





References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RMSE

(deg

)

TRDSTRSSIACRB

10minus13

10minus15

10minus17

10minus19

10minus11

(a)

20 40 60 80 100 1200

0005

001

0015

002

0025

003

0035

Number of pulses

Runt

ime (

seco

nd)

TRDSTRSSIA

(b)


119889

= [100 100 100]Hz 119901119905

= 119901119903



RMSE


102

100

10minus2

10minus4


= 119901119903

= 1119872sub = 4119873sub = 4 and 119871 = 64)


5 Conclusion


Appendix



119879119878


119879119878


119879119878


andU119879119878



119872sub119873 rows ofU119879119878


[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)


A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)


Φ119905

= A119905

where A119905

and A119905


119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901


[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)





References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119872sub119873 rows ofU119879119878


[U119879119878

U119879119878

] = [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Π119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

T (A1)


A119905

⊙ A119903

Λ119905

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ A119903

Λ119905

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198781

= U119879119878

[1 (119872sub minus 1)119873 ]

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

T = U1198791198782

= U119879119878

[119873 + 1 119872sub119873 ]

(A2)


Φ119905

= A119905

where A119905

and A119905


119905

sin1206011

120582) exp(j2120587119889119905

sin120601119901


[

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

Φ119905

= [

[

A119905

⊙ A119903

Λ119905

A119905

⊙ (Φ119873

Alowast119903

)Λ1199051

Λlowast

119905

Λ1199052

]

]

997904rArr

[U1198791198781

U1198791198781

]

dagger

[U1198791198782

U1198791198782

] = Tminus1Φ119905

T

(A3)





References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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