DOA Estimation Method in Multipath Environment for Passive ...downloads.hindawi.com/journals/ijap/2019/7419156.pdfDOA Estimation Method in Multipath Environment for Passive Bistatic

Research ArticleDOA Estimation Method in Multipath Environment for PassiveBistatic Radar

Panhe Hu 1 Qinglong Bao1 and Zengping Chen12

1National Key Laboratory of Science and Technology on ATR National University of Defense Technology Changsha 410073 China2College of Information Engineering Shenzhen University Shenzhen 518060 China

Correspondence should be addressed to Panhe Hu hupanhe2009163com

Received 24 January 2019 Revised 6 May 2019 Accepted 13 May 2019 Published 20 June 2019

Academic Editor Ana Alejos

Copyright copy 2019 PanheHu et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Direction-of-arrival (DOA) estimation in multipath environment is an important issue for passive bistatic radar (PBR) usingfrequency agile phased array VHF radar as illuminator of opportunity Under such scenario the main focus of this paper is tocope with the closely spaced uncorrelated and coherent signals in low signal-to-noise ratio and limited snapshots Making fulluse of the characteristics of moduli of eigenvalues the DOAs of the uncorrelated signals are firstly estimated Afterwards theircontributions are eliminated by means of spatial difference technique Finally in order to improve resolution and accuracy DOAestimation of remaining coherent signals while avoiding the cross-terms effect a new beamforming solution based iterative adaptiveapproach (IAA) is proposed to deal with a reconstructed covariancematrixThe proposedmethod combines the advantages of bothspatial difference method and the IAA algorithm while avoiding their shortcomings Simulation results validate its effectivenessmeanwhile the good performances of the proposedmethod in terms of resolution probability detection probability and estimationaccuracy are demonstrated by comparison with the existing methods

1 Introduction

Passive bistatic radar (PBR) exploiting illuminators of oppor-tunity as uncooperative transmitters has become an emergingtechnology because it allows target detection and localizationwith advantages such as low cost covert detection and lowvulnerability to electronic jamming [1 2] However theseresearches mainly focus on FM radio broadcasting [3 4]mobile network base station [5] navigation satellites [6]digital video broadcasting and digital audio broadcasting[7ndash9] Compared with the aforementioned illuminators ofopportunity a dedicated radar usually has high power andan idea ambiguity function In recent years frequencyagility and phased array technology are widely used invery high frequency (VHF) radar systems for remote sens-ing and surveillance purpose because it provides flexiblebeam scanning improved detection probability and highvaluable counter to stealth technology [10 11] The useof frequency agile phased array VHF radar can extendthe range of available illuminators of opportunity while italso poses many challenges in PBR signal processing such

as low signal-to-noise ratio (SNR) limited snapshots etcThus some complex signal processing algorithms must beemployed which increase the complexity of the systemsDue to multipath propagation the direct signal of target andits multipath signal are coherent and closely spaced in themainlobe Meanwhile the transmitting signals emitted fromthe exploited illuminator are received simultaneously andthey may come from the same direction as the direct signalThese problems significantly degrade the direction-of-arrival(DOA) estimation performance and even result in a failure oftarget detection and localization

At present many high resolution algorithms have beendeveloped to cope with the scenario when uncorrelatedand coherent signals coexist The most popular solutionsare the forwardbackward spatial smoothing techniques [12ndash15] The problem however is that the decorrelation viaspatial smoothing technique is acquired at the cost of thereduction in array aperture which would further increase thewidth of mainlobe and make the resolution worse Besidesthese methods cannot resolve the uncorrelated and coherentsignals that are in the same direction Spatial difference

HindawiInternational Journal of Antennas and PropagationVolume 2019 Article ID 7419156 9 pageshttpsdoiorg10115520197419156

2 International Journal of Antennas and Propagation

Illuminator of opportunity

Target

Receiver

Passive bistatic radar

Surveillanceantenna

Referenceantenna

Multipath propagation

Mainlobesidelobe

Mainlobe

sidelo

be

Direct signal

Transmitting signalMultipath signal

Figure 1 Basic operating scenario of PBR system

methods are proposed to separately estimate the DOAsof the uncorrelated and coherent signals in the multipathenvironment [16ndash25] These methods firstly estimate theDOAs of the uncorrelated signals and then eliminate theircontribution using spatial difference technique After thatthe DOAs of the remaining coherent signals are estimatedHowever the method in [16] is difficult in realization becausethe covariance matrix of the uncorrelated signals should beknown as a priori The method in [17] suffers from rankdeficient problem The required number of snapshots is toohigh to put [18 19] into application Since [20ndash23] eliminatethe contributions of uncorrelated signals by constructing adifference matrix which encounter the problem of cross-terms effect and may yield wrong DOA estimates when thenumber of snapshots is small the method introduced in [2425]makes full use of the property of themoduli of eigenvaluesto resolve the uncorrelated signals from the coherent signalsHowever the main disadvantage is that the DOA estimationperformance degrades significantly in the low SNR andlimited snapshots Recently an iterative adaptive approach(IAA) has attracted much attention [26ndash28] A number ofvariant IAA-based beamforming solutions have shown asuperresolution performance in dealing with closely spacedsignals and work well with few numbers of snapshots [29]Unfortunately when uncorrelated and coherent signals arein the same direction these methods fail to work eitherCombining the above analysis we can get some conclusionsthat little literature has been published up to now in this fieldpartly because of the difficulties in PBR signal processing

Motivated by previous work we propose a new DOAestimation method in multipath environment for PBR Bymaking full use of the property of moduli of eigenvaluesthe DOAs of the uncorrelated signals are estimated Thenthe contribution of the uncorrelated signals is eliminatedby using spatial difference technique that is only coherentsignals remain in the spatial difference matrix To eliminate

the cross-terms effect while improving the resolution andaccuracy of DOA estimation of the remaining coherentsignals a new IAA-based beamforming solution is conductedon a reconstructed covariance matrix Simulation resultsvalidate the better performance of the proposed method bycomparison with the existing algorithms

2 Problems Statement

The basic operating scenario of PBR system is illustrated inFigure 1 As shown in the dashed line box the PBR systemconsists of a surveillance antenna a reference antenna anda receiver Meanwhile a frequency agile phased array VHFradar is used as the uncooperative illuminator of opportunityRefer to analysis in [10] the frequency agility technologycan increase the ability of antijamming and increase thedetection probability The phased array technology providesflexible beam pattern angular diversity and more degrees offreedom which makes it competent for the task of detectingand tracking different targets

However the types of illuminator are enriched whilethe difficulties for PBR signal processing also come alongIn case of PBR exploiting uncooperative illuminator ofopportunity as the transmitter there is a problem that theinstantaneous parameters of transmitting signal are unknownat the receiver thus a dedicated reference antenna needsto steer toward the uncooperative illuminator to receive thetransmitting signal Meanwhile the frequency agility tech-nology destroys the coherency between pulses and the abilityof rapidly changing beam scanning makes it impossible forPBR to predict the next beam position Therefore theseproblems make the SNR become low and the number ofsnapshots is finite

Furthermore among the characteristics of the environ-ment the multipath propagation in VHF band is seriousAs shown in the solid line box the received signal is a

International Journal of Antennas and Propagation 3

mixture of transmitting signal of the exploited illuminatorthe direct signal of target and its multipath signal The directsignal andmultipath signal are coherent and closely spaced inthe mainlobe Meanwhile the transmitting signal is receivedsimultaneously andmay come from the same direction as thedirect signal

Therefore under such scenarios the problems of DOAestimation in multipath environment for PBR system aresummarized as how to deal with the closely spaced uncorre-lated and coherent signals with high resolution and accuracyin the low SNR with limited snapshots

3 Signal Model

Without loss of generality in PBR scenario a uniform lineararray (ULA) with 119873 isotropic sensors is considered as thesurveillance antenna Let the first sensor be the reference andtake 120579 as the DOA of an impinging signal Then the steeringvector can be expressed as

푎 (120579) = [1 119890minus(푗2휋푑 sin 휃)휆 sdot sdot sdot 119890minus(Nminus1)((푗2휋푑 sin 휃)휆)]푇 (1)

where 120582 is the wavelength of the signal 119889 is the distancebetween adjacent sensors and the superscript 119879 denotes thetranspose operator

Assume that 119870 far-field narrowband signals are receivedsimultaneously and are mixtures of 119870푈 uncorrelated signalsand 119870퐶 coherent signals with several groups The 119870푈 signalsare uncorrelated and the signal that comes from direction 120579푘corresponding to the signal 119904푘 with power 1205902푘 119896 = 1 sdot sdot sdot 119870푈The remaining119870퐶 coherent signals are divided into119862 groupswhich come from 119862 statistically independent signals withpower 1205902푘 119896 = 119870푈 + 1 119870푈 + 119862 and with 119871푘 multipathsignals for each signal That is the relationship betweenuncorrelated and coherent signals is

119870 = 119870푈 + 119870퐶 119870퐶 = 퐾119880+퐶sum푘=퐾119880+1

119871푘 (2)

In the 119896119905ℎ coherent group the multipath signal that comesfrom the direction 120579푘푙 119897 = 1 sdot sdot sdot 119871푘 corresponds to the119897119905ℎ multipath propagation of the independent signal and itscomplex reflection coefficient is 120588푘푙 It is assumed that theuncorrelated signals and coherent signals in different groupsare regarded as uncorrelated with each other Consequentlythe model of the received signals is given by

푥 (119905) = 퐾119880sum푘=1

푎 (120579푘) 119904푘 (119905)+ 퐾119880+퐶sum푘=퐾119880+1

119871119896sum119897=1푎 (120579푘푙) 120588푘푙119904푘 (119905) + 푛 (119905)

= 퐴푈푠푈 (119905) + 퐴퐶푠퐶 (119905) + 푛 (119905) = 퐴s (119905) + 푛 (119905)= 퐴Υ푠 (119905) + 푛 (119905)

(3)

where 퐴 = [퐴푈퐴퐶] 퐴푈 = [푎(1205791) sdot sdot sdot 푎(120579퐾119880)] and 퐴퐶 =[퐴퐾119880+1휌퐾119880+1 sdot sdot sdot 퐴퐾119880+1휌퐾119880+1] are the array flow pattern

퐴푘 = [푎(120579푘1) sdot sdot sdot 푎(120579푘퐿119896)] and 휌푘 = [120588푘1 sdot sdot sdot 120588푘퐿119896]푇contain the reflection coefficient of 119896119905ℎ multipathpropagation 퐴 = [퐴119880 퐴퐾119880+1 sdot sdot sdot 퐴퐾119880+퐶] Υ =blkdiag퐼퐾119880 휌퐾119880+1 sdot sdot sdot 휌퐾119880+퐶 푠(119905) = [푠푇푈(119905) 푠푇퐶(119905)]119879푠119880(119905) is the source vector of the uncorrelated signals and푠119862(119905) is the source vector of the coherent groups 푛(119905) is theadditive Gaussian white noise with mean zero and variance1205902푛 which is uncorrelated with each other and the signalsBesides the operator blkdiag represents a block diagonalmatrix

After that the array covariance matrix can be obtained

푅 = 119864 푥 (119905)푥119867 (119905) = 퐴푈푅푈퐴퐻푈 + 퐴퐶푅퐶퐴퐻퐶 + 휎2푛퐼푁 (4)

where 푅푈 = diag12059021 sdot sdot sdot 1205902퐾119880 is the covariance matrix ofuncorrelated signals and 푅퐶 = diag1205902퐾119880+1 sdot sdot sdot 1205902퐾119880+119862 isthe covariance matrix of coherent signals 퐼푁 stands for the119873times119873 identity matrix Besides the superscript119867 denotes thetranspose operator the operator diag1198911 sdot sdot sdot 119891푁 representsa diagonal matrix with the diagonal entry 119891푘 119896 = 1 sdot sdot sdot 119873

4 The Proposed DOA Estimation Method

41 DOA Estimation of Uncorrelated Signals For the DOAestimation of the uncorrelated signals the eigenvalue decom-position (EVD) of the covariance matrix 푅 is conducted andthe result is given

푅 = 푈푆Λ푆푈퐻푆 +푈푁Λ푁푈퐻푁 (5)where 푈푆 = [푢1 sdot sdot sdot 푢퐾119880+퐶] Λ푆 = diag1205821 sdot sdot sdot 120582퐾119880+퐶푈푁 = [푢퐾119880+퐶+1 sdot sdot sdot 푢푁] and Λ푁 = diag120582퐾119880+퐶+1 sdot sdot sdot 120582푁The 푈푆 is signal subspace matrix whose columns are theeigenvectors corresponds to 119870푈 + 119862 largest eigenvalues 푈푁is noise subspace composed of the eigenvectors correspondto 119873 minus (119870푈 + 119862) smallest eigenvalues Λ푆 and Λ푁 are thediagonal matrixes constituted by their eigenvalues

Moreover the푈푆 is also spanned by the 퐴Υ [24 25] thusit is written as

푈푆 = 퐴ΥT (6)where 푇 is a full-rank matrix The푈푆 can be partitioned intotwo overlapped submatrixes with the size of (119873minus1)times(119870푈+119862)which are expressed as

푈1 = 푈푆 (1 119873 minus 1 ) = 퐴1ΥT (7)

푈2 = 푈푆 (2 119873 ) = 퐴2ΥT = 퐴1ΦΥT (8)

where 퐴1 and 퐴2 are the first and last 119873 minus 1 rows of 퐴Φ = diag1206011 sdot sdot sdot 120601퐾 with 120601푘 = 119890minus(푗2휋푑 sin 휃119896)휆 119896 = 1 sdot sdot sdot 119870Afterwards a new matrix is constructed as

푈 = 푈dagger1푈2 = Tminus1ΥdaggerΦΥT = Tminus1Λ푇T (9)

where the superscript dagger denotes the matrix pseudo inverseoperator and the matrixes and vectors in (9) have thefollowing forms

Υdagger = blkdiag 퐼퐾119880 휂퐾119880+1 sdot sdot sdot 휂퐾119880+퐶

휂푘 = [120578푘1 sdot sdot sdot 120578푘퐿119896] (10)


and

Λ푇 = diag 1205921 sdot sdot sdot 120592퐾119880 120592퐾119880+1 sdot sdot sdot 120592퐾119880+퐶 (11)

120592푘=119890minus(푗2휋푑 sin 휃119896)휆 119896 = 1 sdot sdot sdot 119870푈120578푘1119890minus(푗2휋푑 sin 휃1198961)휆 + 120578푘1120588푘1119890minus(푗2휋푑 sin 휃1198962)휆 + sdot sdot sdot + 120578푘퐿119896120588푘퐿119896minus1119890minus(푗2휋푑 sin 휃119896119871119896 )휆 119896 = 119870푈 + 1 sdot sdot sdot 119870푈 + 119862 (12)

Based on the property of the diagonal elements which is welldocumented in [24] and we can obtain the following results

10038161003816100381610038161003816120592퐾119880+퐶10038161003816100381610038161003816 le 10038161003816100381610038161003816120592퐾119880+퐶minus110038161003816100381610038161003816 le sdot sdot sdot le 10038161003816100381610038161003816120592퐾119880+110038161003816100381610038161003816 lt 10038161003816100381610038161003816120592퐾119880 10038161003816100381610038161003816 = sdot sdot sdot= 100381610038161003816100381612059221003816100381610038161003816 = 100381610038161003816100381612059211003816100381610038161003816 = 1 (13)

Clearly by performing EVD of the 푈 the moduli values ofthe eigenvalues corresponding to the uncorrelated signalsare equal to 1 and the moduli values of the eigenvaluescorresponding to the coherent signals are all less than 1Therefore we can choose a threshold 120585 to resolve the uncor-related signals from the coherent signals In the meanwhilethe number of uncorrelated signals is estimated by sorting the|120592푘| 119896 = 1 sdot sdot sdot 119870119880 + 119862 in descending order and substitutingthem into the following equation

120575푘 = 10038161003816100381610038161003816100381610038161003816120592푘1003816100381610038161003816 minus 11003816100381610038161003816 (14)

Thus if the |120592푘| is the first to satisfy 120575푘 gt 120585 the number of theuncorrelated signals is 119896 minus 1 and the corresponding DOA isobtained as follows

120579푘 = sinminus1 (minus 1205821198952120587119889angle (120592푘)) (15)

42 DOA Estimation of Coherent Signals The DOAs of theremaining signals are then estimated by using a novel IAA-based beamforming solution Herein the equation in (4) isrewritten as

푅 = 119864 푥 (119905)푥퐻 (119905) = 푅푈 + 푅퐶 + 휎2푛퐼푁 (16)

in (16) the (119894 119895) element in the covariance matrix of uncorre-lated signals can be expressed as

푅푈 (119894 119895) = 퐾119880sum푘=1

1205902푘120601푖minus푗푘 (17)

Apparently푅푈 is a Toeplitzmatrix according to the propertythat any Toeplitz matrix Ξ will satisfy the fact that

퐽Ξ퐽 = Ξ (18)

where 퐽 is the exchange matrix which has unity elementsalong the counter-diagonal and zeros elsewhere Thus the

uncorrelated signals can be eliminated by means of spatialdifference technique [17] which can be expressed as

푅푑 = 푅 minus 퐽푅푇퐽= (푅푈 + 푅퐶 + 휎2푛퐼푁) minus 퐽 (푅푈 + 푅퐶 + 휎2푛퐼푁)푇 퐽= 푅퐶 minus 퐽푅푇퐶퐽

(19)

Theoretically only coherent signals remain in the spatialdifferencing matrix in this step However in the practicalPBR application scenario the noise cannot be eliminatedcompletely Furthermore the EVD of 푅푑 would generatecross-terms effect which would result in a high probability offailure in DOA estimation of closely spaced coherent signals

To solve the aforementioned problems we propose anew IAA-based beamforming solution with respect to animproved covariance matrix Then we rewrite (4) as

푅 = 119864 푥 (119905)푥퐻 (119905) = 퐴푃퐴 (20)

where 퐴 and 푃 denote the reconstructed array flow patternand signal power matrix which can be expressed as

퐴 = [퐴푈퐴퐶 퐼119873] = [푎 (1205791) sdot sdot sdot 푎 (120579퐾119880) 푎 (120579퐾119880+1)sdot 휌퐾119880+1 sdot sdot sdot 푎 (120579퐾119880+퐶)휌퐾119880+퐶 퐼119873]= [푎 (1205791) sdot sdot sdot 푎 (120579퐾) sdot sdot sdot 푎 (120579퐾+푁)]

(21)

and

푃 = blkdiag 푃푈푃퐶휎2푛퐼푁= diag 1198751 sdot sdot sdot 119875퐾 sdot sdot sdot 119875퐾+푁 (22)

where the diagonal value 119875푘 represents the correspondingsignal power at each angle

Afterwards according the IAA algorithm [27] (seeAppendix) the IAA-based beamformer for each DOA of theremaining coherent signals is computed as

푤푘 = 푅minus1푎 (120579푘)푎퐻 (120579푘)푅minus1푎 (120579푘) (23)

Therefore the estimated signal power of correspondingdirection can be obtained

푘 = 푤퐻푘 푅푑푤푘 (24)


In order to eliminate the cross-terms effect while improvingthe resolution and accuracy of DOA estimation a newcovariance matrix is constructed by squaring the spatialdifference matrix [21]

푅퐷 = 푅푑푅퐻푑 (25)

Then the spatial spectrum of each direction is obtained byreplacing 푅푑 with 푅퐷

푘 = 푤퐻푘 푅퐷푤푘 (26)

As a result by performing the IAA algorithm with respectto the reconstructed covariance matrix the DOAs of theremaining coherent signals are estimated

43 Summary and Discussion of the Proposed Method Inthis subsection the proposed method is summarized anddiscussed as follows

(i) Since the pervious analysis of practical PBR applica-tion scenario the covariance matrix of the received signalis unavailable thus it is replaced by the sample-averageestimated array autocovariance matrix

푅 = 1119873푃푁119875sum푛=1

푥 (n)푥퐻 (119899) (27)

where 푥(119899) is the received signal of the 119899119905ℎ snapshot 119873푃 isthe number of snapshot

(ii) Then by performing the EVD of the matrix 푅 thesignal subspace 푈119878 is obtained and the 푈1 and 푈2 arecomputed to construct푈 Thus the moduli of eigenvalues of푈 are obtained

(iii) Based on the property of moduli of eigenvalues theuncorrelated signals are resolved from coherent signals whilethe DOAs of the uncorrelated signals are estimated

(iv) To eliminate the contribution of uncorrelated signalsand improve the resolution and accuracy DOA estimation ofcoherent signals the spatial differencematrix푅푑 is computedand a new covariance matrix 푅퐷 is constructed by squaringthe spatial difference matrix

(v) Finally the DOAs of remaining coherent signals areestimated by performing IAA algorithm on the reconstructedcovariance matrix 푅퐷

It is noteworthy that our proposed method combinesthe advantages of both spatial difference method and theIAA algorithm while avoiding their disadvantages Whenthe uncorrelated and coherent signals coexist in multipathenvironment especially the uncorrelated signal coming fromthe same direction as the coherent signals do themethod canstill resolve them whereas the IAA algorithm fails to workThe performance of the spatial difference method degradessignificantly in the low SNR with limited snapshots but ourproposed method can still work well Besides by exploitingour method the DOAs of uncorrelated and coherent signalsare estimated separately which can be parallelized easily tospeed up the processing in practical application

minus100 minus50 0 50 100DOA (deg)

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

Spat

ial s

pect

rum

(dB)

Uncorrelated signalsCoherent signals

2 coherent signals space closely in mainlobe

1 uncorrelated and coherent signals come from

2

1

the same direction

Figure 2 Spatial spectrums of the uncorrelated and coherentsignals

5 Simulation ResultsIn this section the performance of the proposed method isinvestigated by simulation experiments For PBR system aULAof119873 = 16 omnidirectional antenna sensors spaced half-wavelength interspacing is used as the surveillance antennaand the half power width of mainlobe can be computed as12057905 asymp 64∘ In the simulations the threshold is set as 120585 = 005to resolve uncorrelated signals from coherent signals and thescanning size is uniform in the range from minus90∘ to 90∘ with01∘ increment

In the first simulation we consider five uncorrelatedsignals from [minus30∘ minus10∘ 0∘ 35∘ 50∘] and two groups ofcoherent signals from [0∘ 4∘] and [minus45∘ minus30∘ 15∘ 28∘] Thereflection coefficient of the first group of coherent signals is[1 03837 + 09017119895] and the other group is [1 minus07513 +00579119895 minus03427 minus 03155119895 05873 minus 07385119895] For simplicityall signals are of equal power It is worth pointing out that thetwo uncorrelated signals are set as 120579 = minus30∘ and = 0∘ whichcome from the same direction with the two groups of coher-ent signals in the first group of coherent signals the two sig-nals are closely spaced in themainlobeThe simulation is car-ried out in the SNR of 10 dB and the number of snapshots of100 Therefore the spatial spectrum of the uncorrelated andcoherent signals is shown in Figure 2 As is apparent the pro-posedmethod can easily resolve the uncorrelated signals fromthe coherent signal in same direction and it also has satisfac-tory performance in resolving closely spaced coherent signals

The second simulation studies the superresolution per-formance of two closely spaced signals with the proposedmethod In the simulation we define the probability ofresolution as

119875푟 = Ω푟Ω (28)


SNR = 0dB Np=10SNR = 10dB Np=10


30 1 52 4Δ (deg)

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f Res

olut

ion

Figure 3 The resolution probability versus the angular gap

where Ω푟 and Ω represent the times of successful resolutionresult and the total trials respectively

The angular gap between two closely spaced coherentsignals is defined as 120579 = |1205792 minus 1205791| and when the conditionof (119875푟(1205792) + 119875푟(1205791))2 gt 119875푟((1205791 + 1205792)2) is satisfied the resultis considered successful Herein we set the DOA of the firstsignal as 0∘ and the other one vary from 0∘ to 5∘ Undersuch simulation scenario the coherent signals are always inthe mainlobe Figure 3 shows the curves of probability ofresolution with respect to the SNR of 0 dB and 10 dB thenumber of snapshots of 10 and 100 It is shown that ourproposed method has a little performance degradation inthe low SNR but the overall performance is satisfactory asthe SNR is larger Especially when the angular gap is 3∘ theprobability of resolution of is almost equal to 1 in SNR = 10dBand119873푃 = 10Therefore the superresolutionDOA estimationperformance of the proposed method is verified

In the third simulation we investigate the high accuracyperformance of DOA estimation with our proposed methodThe detection probability in the simulation is defined as

119875푑 = Ω푑Ω (29)

where Ω푑 and Ω stand for the times of successful detectionresult of119870 signals and the total tests respectively

The DOAs detection result is considered successful whenthe difference between the estimated DOA and the trueDOA is less than 2∘ ie 119875푑 = 1 subject to Δ120579 = |120579푘 minus120579푘| le 2o 119896 = 1 sdot sdot sdot 119870 The simulation considers the samescenario as the first one The DOA estimation results withrespect to the changes of SNR and the number of snapshotsare plotted in Figures 4 and 5 Apparently the curves ofthe detection probability indicate that our proposed methodhas good performance of DOA estimation when the SNRvaries from 0 dB and 10 dB and the number of snapshots ishigher than 30 This is because that our proposed method

092

093

094

095

096

097

098

099

1

Prob

abili

ty o

f Det

ectio

n

86 100 2 4SNR (dB)


Figure 4 The detection probability versus the SNR with119873푃 = 100


20 30 40 50 60 70 80 90 10010Number of snapshots

0

01

02

03

04

05

06

07

08

09

1Pr

obab

ility

of D

etec

tion

Figure 5The detection probability versus the number of snapshotswith SNR = 10dB

makes full use of the true signal power when dealing withuncorrelated and coherent signals which is different from theexisting subspace-based high resolution algorithms Thesemethods use the orthogonality of the signal subspace andthe noise subspace in a certain direction to calculate thespatial spectrum and the performance of DOA estimation ispoor when the SNR is low and the number of snapshots issmallTherefore similar to the previous results the proposedmethod has high accuracy DOA estimation performance indealing with uncorrelated and coherent signals

In the last simulation we further examine the perfor-mance of the proposed method by comparison We select themethod in [23] and the method in [24] as the comparative


02

04

06

08

1

12

14

16

18

2

RMSE

(deg

)

86 100 2 4SNR (dB)

Ref[23](Uncorrelated)Ref[23](Coherent)Ref[24](Uncorrelated)

Ref[24](Coherent)Ours (Uncorrelated)Ours (Coherent)

Figure 6 The RMSE of DOA estimation versus SNR

methods The performance is tested by the root mean squareerror (RMSE) which is defined as

RMSE = radic 1200 1119870200sum휏=1

퐾sum푘=1

(120579(휏)푘

minus 120579푘)2 (30)

In each simulation scenario 200 Monte Carlo runs areperformed Figures 6 and 7 show the RMSE curves of DOAestimation with respect to the SNR and the number ofsnapshots

It can be seen from the RMSE curves of DOA estimationwith differentmethods that theDOAestimation performanceof our method is better than the method in [23] and themethod in [24] especially in the low SNR with small numberof snapshots For method in [23] the reason of performancedegradation is that it utilizes the whole array to estimatethe uncorrelated signals and then uses the reduced array toestimate the coherent signals in a subsequent stageWhen thecoherent signals are decorrelated by using forwardbackwardspatial smoothing technique the reduction in array aperturewould broaden the mainlobe which significantly degradesthe resolution and accuracy of DOA estimation For methodin [24] the performance of DOA estimation of uncorre-lated signals is similar to our method due to the sameprocess in dealing with uncorrelated signals However whendealing with the remaining coherent signals it also usesforwardbackward spatial smoothing technique and theperformance is not satisfactory either For our methodwhen dealing with uncorrelated and coherent signals wecombine the advantages of spatial difference method andIAA algorithm and propose a novel IAA-based beamformingsolution with respect to the improved covariance matrixwhich can effectively improve the performance of resolutionand accuracy in DOA estimation



0

1

2

3

4

5

6

7

8

RMSE

(deg

)


Figure 7 The RMSE of DOA estimation versus the number ofsnapshots

6 Conclusions

In this paper we propose a novel DOA estimation methodin the multipath environment for PBR using frequencyagile phased array VHF radar as illuminator of opportunityThe proposed method combines the advantages of spatialdifference method and the IAA algorithm while avoidingtheir shortcomings and then performs a new IAA-basedbeamforming solution on an improved covariance matrix todeal with closely spaced uncorrelated and coherent signalsThe simulation results show that it has satisfactory DOAestimation performance in the scenario of low SNR andsmall number of snapshots Furthermore in the proposedmethod the DOAs of uncorrelated and coherent signalsare estimated separately which can be parallelized easily tospeed up processing and can be a viable solution for PBRapplications Since the presented PBR system operates inthe VHF band multipath propagation in complex terrainis serious and unavoidable In order to further improvethe performance of target detection and localization thedynamic characteristics of electromagnetic and geographicalenvironment and antimultipath technology should be takeninto consideration in future works

Appendix

The IAA algorithm is a data-dependent nonparametricmethod which is based on a weighted least squares (WLS)approach Firstly we define the covariance matrix of noisebased on the above sparse signal model

Q푘 = 푅 minus 119875푘푎 (120579푘) 푎퐻 (120579푘) (A1)

Then the WLS cost function is given by

M푉 = 푁sum푛=1

1003817100381710038171003817푥 (119899) minus 푠119896 (119899) 푎 (120579푘)10038171003817100381710038172푉 (A2)


where 푦2푉 = 푦퐻푉푦 In (A2) we perform partial derivativeoperation ofM푉 with respect to 푠119896(119899) and make the result bezero to obtain that

푠119896 (119899) = 푎퐻 (120579푘)푉푥 (119899)푎퐻 (120579푘)푉푎 (120579푘) (A3)

Then we can compute the mean square error (MSE) as

MSE = 119864 [푠119896 (119899) minus 푠119896 (119899)] [푠119896 (119899) minus 푠119896 (119899 ]퐻= [푎퐻 (120579푘)푉푎 (120579푘)]minus1 푎퐻 (120579푘)푉119864 [푛 (119899) 푛퐻 (119899)]sdot푉퐻푎 (120579푘) [푎퐻 (120579푘)푉푎 (120579푘)]minus1= [푎퐻 (120579푘)푉푎 (120579푘)]minus1 푎퐻 (120579푘)푉Q푘푉퐻푎 (120579푘)sdot [푎퐻 (120579푘)푉푎 (120579푘)]minus1

(A4)

Suppose푍1푍2 are a column vector and a row vector and theinverse matrix of푍2푍

119867

2existsThen we can get the following

matrix inequality

푍퐻1 푍1 ge (푍2푍1)퐻 (푍2푍퐻2 )minus1 (푍2푍1) (A5)

Let 푍2 = 푎퐻(120579푘)푄minus12119896 푍1 = 푄minus12119896푍1198673 with 푍3 =[푎퐻(120579푘)푉푎(120579푘)]minus1푎퐻(120579푘)푉 then we can rewrite (A4) as

MSE = 푍퐻1 푍1 ge (푍2푍1)퐻 (푍2푍1198672 )minus1 (푍2푍1)= [푎퐻 (120579푘)푄minus1푘 푎 (120579푘)]minus1 (A6)

Clearly we can get the conclusion that when 푉 = 푄minus1푘 theresult of MSE is minimal

After that using the matrix inversion lemma to yield

푠119896 (119899) = 푎퐻 (120579푘)푄minus1119896

푎퐻 (120579푘)푄minus1119896 푎 (120579푘)푥 (119899)

= 푎퐻 (120579푘)푅minus1

푎퐻 (120579푘)푅minus1푎 (120579푘)푥 (119899) = 푤119867119896 푥 (119899)

(A7)

Therefore the beamformer for each direction of the corre-sponding signal is

푤푘 = 푅minus1푎 (120579푘)푎퐻 (120579푘)푅minus1푎 (120579푘) (A8)

Since the푅 in IAA algorithmdepends on the unknown signalpower it must be done in an iterative way The initializationis implemented by conventional beamforming It is worthnoting that thematrix푃 and푅 are updated from the previousiteration not the data snapshot

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China under Grant 61401489

References

[1] F Colone C Bongioanni and P Lombardo ldquoMultifrequencyintegration in FM radio-based passive bistatic radar Part I tar-get detectionrdquo IEEEAerospace and Electronic SystemsMagazinevol 28 no 4 pp 28ndash39 2013

[2] F Colone C Bongioanni and P Lombardo ldquoMultifrequencyintegration in FM radio-based passive bistatic radar Part IIdirection of arrival estimationrdquo IEEE Aerospace and ElectronicSystems Magazine vol 28 no 4 pp 40ndash47 2013

[3] C Shi FWangM Sellathurai and J Zhou ldquoTransmitter subsetselection in FM-based passive radar networks for joint targetparameter estimationrdquo IEEE Sensors Journal vol 16 no 15 pp6043ndash6052 2016

[4] J You X Wan Y Fu and G Fang ldquoExperimental study ofpolarisation technique on multi-FM-based passive radarrdquo IETRadar Sonar amp Navigation vol 9 no 7 pp 763ndash771 2015

[5] R Raja Abdullah N Abdul Aziz N Abdul Rashid A AhmadSalah and F Hashim ldquoAnalysis on target detection and classifi-cation in LTE based passive forward scattering radarrdquo Sensorsvol 16 no 10 p 1607 2016

[6] H Zeng J Chen P Wang W Yang and W Liu ldquo2-D coherentintegration processing and detecting of aircrafts using GNSS-based passive radarrdquo Remote Sensing vol 10 no 7 p 1164 2018

[7] L Chen P Thevenon G Seco-Granados O Julien and HKuusniemi ldquoAnalysis on the TOA tracking with DVB-T signalsfor positioningrdquo IEEE Transactions on Broadcasting vol 62 no4 pp 957ndash961 2016

[8] TMartelli F Colone E Tilli and A Di Lallo ldquoMulti-frequencytarget detection techniques for DVB-T based passive radarsensorsrdquo Sensors vol 16 no 10 p 1594 2016

[9] J E Palmer H Andrew Harms S J Searle and L M DavisldquoDVB-T passive radar signal processingrdquo IEEE Transactions onSignal Processing vol 61 no 8 pp 2116ndash2126 2013

[10] P Hu Q Bao and Z Chen ldquoWeak target detection method ofpassive bistatic radar based on probability histogramrdquo Mathe-matical Problems in Engineering vol 2018 Article ID 824368610 pages 2018

[11] Y Zheng and B Chen ldquoAltitude measurement of low-angletarget in complex terrain for very high-frequency radarrdquo IETRadar Sonar amp Navigation vol 9 no 8 pp 967ndash973 2015

[12] S U Pillai and B H Kwon ldquoForwardbackward spatialsmoothing techniques for coherent signal identificationrdquo IEEETransactions on Signal Processing vol 37 no 1 pp 8ndash15 1989

[13] B Cai Y M Li and H Y Wang ldquoForwardbackward spatialreconstruction method for directions of arrival estimation ofuncorrelated and coherent signalsrdquo IET Microwaves Antennasamp Propagation vol 6 no 13 pp 1498ndash1505 2012

[14] A Delis and G Papadopoulos ldquoEnhanced forwardbackwardspatial filtering method for DOA estimation of narrowbandcoherent sourcesrdquo IEEProceedings Radar Sonar andNavigationvol 143 no 1 pp 10ndash14 1996


[15] K Cui W Wu J Huang X Chen and N Yuan ldquoDOAestimation of LFM signals based on STFT and multiple invari-ance ESPRITrdquo AEU - International Journal of Electronics andCommunications vol 77 pp 10ndash17 2017

[16] E M Al-Ardi R M Shubair and M E Al-Mualla ldquoComputa-tionally efficient high-resolution DOA estimation in multipathenvironmentrdquo IEEE Electronics Letters vol 40 no 14 pp 908ndash910 2004

[17] C Qi Y Wang Y Zhang and Y Han ldquoSpatial differencesmoothing forDOAestimation of coherent signalsrdquo IEEE SignalProcessing Letters vol 12 no 11 pp 800ndash802 2005

[18] J Shi G Hu F Sun B Zong and X Wang ldquoImprovedspatial differencing scheme for 2-DDOAestimation of coherentsignals with uniform rectangular arraysrdquo Sensors vol 17 no 9p 1956 2017

[19] Y Wan B Xu S Tang et al ldquoDOA estimation for uncorrelatedand coherent signals based on Fourth-order Cumulantsrdquo inProceedings of the 2013 International Conference on Communi-cations Circuits and Systems ICCCAS 2013 vol 2 pp 247ndash250November 2013

[20] X Xu Z Ye and J Peng ldquoMethod of direction-of-arrivalestimation for uncorrelated partially correlated and coherentsourcesrdquo IETMicrowaves Antennas amp Propagation vol 1 no 4pp 949ndash954 2007

[21] Y F Zhang and Z F Ye ldquoEfficient method of DOA estimationfor uncorrected and coherent signalsrdquo IEEE Antennas andWireless Propagation Letters vol 7 pp 799ndash802 2008

[22] Z Ye Y Zhang and C Liu ldquoDirection-of-arrival estimationfor uncorrelated and coherent signals with fewer sensorsrdquo IETMicrowaves Antennas amp Propagation vol 3 no 3 pp 473ndash4822009

[23] F Liu J Wang C Sun and R Du ldquoSpatial differencing methodfor DOA estimation under the coexistence of both uncorrelatedand coherent signalsrdquo IEEE Transactions on Antennas andPropagation vol 60 no 4 pp 2052ndash2062 2012

[24] L Gan and X Luo ldquoDirection-of-arrival estimation for uncor-related and coherent signals in the presence of multipathpropagationrdquo IET Microwaves Antennas amp Propagation vol 7no 9 pp 746ndash753 2013

[25] H Shi W Leng A Wang and T Guo ldquoDOA estimation formixed uncorrelated and coherent sources inmultipath environ-mentrdquo International Journal of Antennas and Propagation vol2015 Article ID 636545 8 pages 2015

[26] T Yardibi J Li P Stoica M Xue and A B Baggeroer ldquoSourcelocalization and sensing a nonparametric iterative adaptiveapproach based on weighted least squaresrdquo IEEE Transactionson Aerospace and Electronic Systems vol 46 no 1 pp 425ndash4432010

[27] L Du T Yardibi J Li and P Stoica ldquoReview of user parameter-free robust adaptive beamforming algorithmsrdquo Digital SignalProcessing vol 19 no 4 pp 567ndash582 2009

[28] M J Jahromi and M H Kahaei ldquoTwo-dimensional iterativeadaptive approach for sparse matrix solutionrdquo IEEE ElectronicsLetters vol 50 no 1 pp 45ndash47 2014

[29] M Barcelo J Lopez Vicario and G Seco-Granados ldquoA reducedcomplexity approach to IAA beamforming for efficient DOAestimation of coherent sourcesrdquo EURASIP Journal on Advancesin Signal Processing vol 2011 no 1 pp 1ndash16 2010

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom


Illuminator of opportunity

Target

Receiver

Passive bistatic radar

Surveillanceantenna

Referenceantenna

Multipath propagation

Mainlobesidelobe

Mainlobe

sidelo

be

Direct signal

Transmitting signalMultipath signal

Figure 1 Basic operating scenario of PBR system

methods are proposed to separately estimate the DOAsof the uncorrelated and coherent signals in the multipathenvironment [16ndash25] These methods firstly estimate theDOAs of the uncorrelated signals and then eliminate theircontribution using spatial difference technique After thatthe DOAs of the remaining coherent signals are estimatedHowever the method in [16] is difficult in realization becausethe covariance matrix of the uncorrelated signals should beknown as a priori The method in [17] suffers from rankdeficient problem The required number of snapshots is toohigh to put [18 19] into application Since [20ndash23] eliminatethe contributions of uncorrelated signals by constructing adifference matrix which encounter the problem of cross-terms effect and may yield wrong DOA estimates when thenumber of snapshots is small the method introduced in [2425]makes full use of the property of themoduli of eigenvaluesto resolve the uncorrelated signals from the coherent signalsHowever the main disadvantage is that the DOA estimationperformance degrades significantly in the low SNR andlimited snapshots Recently an iterative adaptive approach(IAA) has attracted much attention [26ndash28] A number ofvariant IAA-based beamforming solutions have shown asuperresolution performance in dealing with closely spacedsignals and work well with few numbers of snapshots [29]Unfortunately when uncorrelated and coherent signals arein the same direction these methods fail to work eitherCombining the above analysis we can get some conclusionsthat little literature has been published up to now in this fieldpartly because of the difficulties in PBR signal processing

Motivated by previous work we propose a new DOAestimation method in multipath environment for PBR Bymaking full use of the property of moduli of eigenvaluesthe DOAs of the uncorrelated signals are estimated Thenthe contribution of the uncorrelated signals is eliminatedby using spatial difference technique that is only coherentsignals remain in the spatial difference matrix To eliminate

the cross-terms effect while improving the resolution andaccuracy of DOA estimation of the remaining coherentsignals a new IAA-based beamforming solution is conductedon a reconstructed covariance matrix Simulation resultsvalidate the better performance of the proposed method bycomparison with the existing algorithms

2 Problems Statement

The basic operating scenario of PBR system is illustrated inFigure 1 As shown in the dashed line box the PBR systemconsists of a surveillance antenna a reference antenna anda receiver Meanwhile a frequency agile phased array VHFradar is used as the uncooperative illuminator of opportunityRefer to analysis in [10] the frequency agility technologycan increase the ability of antijamming and increase thedetection probability The phased array technology providesflexible beam pattern angular diversity and more degrees offreedom which makes it competent for the task of detectingand tracking different targets

However the types of illuminator are enriched whilethe difficulties for PBR signal processing also come alongIn case of PBR exploiting uncooperative illuminator ofopportunity as the transmitter there is a problem that theinstantaneous parameters of transmitting signal are unknownat the receiver thus a dedicated reference antenna needsto steer toward the uncooperative illuminator to receive thetransmitting signal Meanwhile the frequency agility tech-nology destroys the coherency between pulses and the abilityof rapidly changing beam scanning makes it impossible forPBR to predict the next beam position Therefore theseproblems make the SNR become low and the number ofsnapshots is finite

Furthermore among the characteristics of the environ-ment the multipath propagation in VHF band is seriousAs shown in the solid line box the received signal is a




3 Signal Model





119870 = 119870푈 + 119870퐶 119870퐶 = 퐾119880+퐶sum푘=퐾119880+1

119871푘 (2)


푥 (119905) = 퐾119880sum푘=1

푎 (120579푘) 119904푘 (119905)+ 퐾119880+퐶sum푘=퐾119880+1

119871119896sum119897=1푎 (120579푘푙) 120588푘푙119904푘 (119905) + 푛 (119905)


(3)











푈1 = 푈푆 (1 119873 minus 1 ) = 퐴1ΥT (7)

푈2 = 푈푆 (2 119873 ) = 퐴2ΥT = 퐴1ΦΥT (8)







and






120575푘 = 10038161003816100381610038161003816100381610038161003816120592푘1003816100381610038161003816 minus 11003816100381610038161003816 (14)




푅 = 119864 푥 (119905)푥퐻 (119905) = 푅푈 + 푅퐶 + 휎2푛퐼푁 (16)


푅푈 (119894 119895) = 퐾119880sum푘=1

1205902푘120601푖minus푗푘 (17)


퐽Ξ퐽 = Ξ (18)




(19)



푅 = 119864 푥 (119905)푥퐻 (119905) = 퐴푃퐴 (20)



(21)

and















푅 = 1119873푃푁119875sum푛=1

푥 (n)푥퐻 (119899) (27)








minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

Spat

ial s

pect

rum

(dB)




2

1

the same direction





119875푟 = Ω푟Ω (28)




30 1 52 4Δ (deg)

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f Res

olut

ion





119875푑 = Ω푑Ω (29)



092

093

094

095

096

097

098

099

1

Prob

abili

ty o

f Det

ectio

n

86 100 2 4SNR (dB)





0

01

02

03

04

05

06

07

08

09

1Pr

obab

ility

of D

etec

tion





02

04

06

08

1

12

14

16

18

2

RMSE

(deg

)

86 100 2 4SNR (dB)





RMSE = radic 1200 1119870200sum휏=1

퐾sum푘=1

(120579(휏)푘

minus 120579푘)2 (30)





0

1

2

3

4

5

6

7

8

RMSE

(deg

)



6 Conclusions


Appendix




M푉 = 푁sum푛=1

1003817100381710038171003817푥 (119899) minus 푠119896 (119899) 푎 (120579푘)10038171003817100381710038172푉 (A2)



푠119896 (119899) = 푎퐻 (120579푘)푉푥 (119899)푎퐻 (120579푘)푉푎 (120579푘) (A3)



(A4)


119867


matrix inequality






푠119896 (119899) = 푎퐻 (120579푘)푄minus1119896

푎퐻 (120579푘)푄minus1119896 푎 (120579푘)푥 (119899)


푎퐻 (120579푘)푅minus1푎 (120579푘)푥 (119899) = 푤119867119896 푥 (119899)

(A7)




Data Availability




Acknowledgments


References

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





3 Signal Model





119870 = 119870푈 + 119870퐶 119870퐶 = 퐾119880+퐶sum푘=퐾119880+1

119871푘 (2)


푥 (119905) = 퐾119880sum푘=1

푎 (120579푘) 119904푘 (119905)+ 퐾119880+퐶sum푘=퐾119880+1

119871119896sum119897=1푎 (120579푘푙) 120588푘푙119904푘 (119905) + 푛 (119905)


(3)











푈1 = 푈푆 (1 119873 minus 1 ) = 퐴1ΥT (7)

푈2 = 푈푆 (2 119873 ) = 퐴2ΥT = 퐴1ΦΥT (8)







and






120575푘 = 10038161003816100381610038161003816100381610038161003816120592푘1003816100381610038161003816 minus 11003816100381610038161003816 (14)




푅 = 119864 푥 (119905)푥퐻 (119905) = 푅푈 + 푅퐶 + 휎2푛퐼푁 (16)


푅푈 (119894 119895) = 퐾119880sum푘=1

1205902푘120601푖minus푗푘 (17)


퐽Ξ퐽 = Ξ (18)




(19)



푅 = 119864 푥 (119905)푥퐻 (119905) = 퐴푃퐴 (20)



(21)

and















푅 = 1119873푃푁119875sum푛=1

푥 (n)푥퐻 (119899) (27)








minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

Spat

ial s

pect

rum

(dB)




2

1

the same direction





119875푟 = Ω푟Ω (28)




30 1 52 4Δ (deg)

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f Res

olut

ion





119875푑 = Ω푑Ω (29)



092

093

094

095

096

097

098

099

1

Prob

abili

ty o

f Det

ectio

n

86 100 2 4SNR (dB)





0

01

02

03

04

05

06

07

08

09

1Pr

obab

ility

of D

etec

tion





02

04

06

08

1

12

14

16

18

2

RMSE

(deg

)

86 100 2 4SNR (dB)





RMSE = radic 1200 1119870200sum휏=1

퐾sum푘=1

(120579(휏)푘

minus 120579푘)2 (30)





0

1

2

3

4

5

6

7

8

RMSE

(deg

)



6 Conclusions


Appendix




M푉 = 푁sum푛=1

1003817100381710038171003817푥 (119899) minus 푠119896 (119899) 푎 (120579푘)10038171003817100381710038172푉 (A2)



푠119896 (119899) = 푎퐻 (120579푘)푉푥 (119899)푎퐻 (120579푘)푉푎 (120579푘) (A3)



(A4)


119867


matrix inequality






푠119896 (119899) = 푎퐻 (120579푘)푄minus1119896

푎퐻 (120579푘)푄minus1119896 푎 (120579푘)푥 (119899)


푎퐻 (120579푘)푅minus1푎 (120579푘)푥 (119899) = 푤119867119896 푥 (119899)

(A7)




Data Availability




Acknowledgments


References

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



and






120575푘 = 10038161003816100381610038161003816100381610038161003816120592푘1003816100381610038161003816 minus 11003816100381610038161003816 (14)




푅 = 119864 푥 (119905)푥퐻 (119905) = 푅푈 + 푅퐶 + 휎2푛퐼푁 (16)


푅푈 (119894 119895) = 퐾119880sum푘=1

1205902푘120601푖minus푗푘 (17)


퐽Ξ퐽 = Ξ (18)




(19)



푅 = 119864 푥 (119905)푥퐻 (119905) = 퐴푃퐴 (20)



(21)

and















푅 = 1119873푃푁119875sum푛=1

푥 (n)푥퐻 (119899) (27)








minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

Spat

ial s

pect

rum

(dB)




2

1

the same direction





119875푟 = Ω푟Ω (28)




30 1 52 4Δ (deg)

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f Res

olut

ion





119875푑 = Ω푑Ω (29)



092

093

094

095

096

097

098

099

1

Prob

abili

ty o

f Det

ectio

n

86 100 2 4SNR (dB)





0

01

02

03

04

05

06

07

08

09

1Pr

obab

ility

of D

etec

tion





02

04

06

08

1

12

14

16

18

2

RMSE

(deg

)

86 100 2 4SNR (dB)





RMSE = radic 1200 1119870200sum휏=1

퐾sum푘=1

(120579(휏)푘

minus 120579푘)2 (30)





0

1

2

3

4

5

6

7

8

RMSE

(deg

)



6 Conclusions


Appendix




M푉 = 푁sum푛=1

1003817100381710038171003817푥 (119899) minus 푠119896 (119899) 푎 (120579푘)10038171003817100381710038172푉 (A2)



푠119896 (119899) = 푎퐻 (120579푘)푉푥 (119899)푎퐻 (120579푘)푉푎 (120579푘) (A3)



(A4)


119867


matrix inequality






푠119896 (119899) = 푎퐻 (120579푘)푄minus1119896

푎퐻 (120579푘)푄minus1119896 푎 (120579푘)푥 (119899)


푎퐻 (120579푘)푅minus1푎 (120579푘)푥 (119899) = 푤119867119896 푥 (119899)

(A7)




Data Availability




Acknowledgments


References

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










푅 = 1119873푃푁119875sum푛=1

푥 (n)푥퐻 (119899) (27)








minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

Spat

ial s

pect

rum

(dB)




2

1

the same direction





119875푟 = Ω푟Ω (28)




30 1 52 4Δ (deg)

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f Res

olut

ion





119875푑 = Ω푑Ω (29)



092

093

094

095

096

097

098

099

1

Prob

abili

ty o

f Det

ectio

n

86 100 2 4SNR (dB)





0

01

02

03

04

05

06

07

08

09

1Pr

obab

ility

of D

etec

tion





02

04

06

08

1

12

14

16

18

2

RMSE

(deg

)

86 100 2 4SNR (dB)





RMSE = radic 1200 1119870200sum휏=1

퐾sum푘=1

(120579(휏)푘

minus 120579푘)2 (30)





0

1

2

3

4

5

6

7

8

RMSE

(deg

)



6 Conclusions


Appendix




M푉 = 푁sum푛=1

1003817100381710038171003817푥 (119899) minus 푠119896 (119899) 푎 (120579푘)10038171003817100381710038172푉 (A2)



푠119896 (119899) = 푎퐻 (120579푘)푉푥 (119899)푎퐻 (120579푘)푉푎 (120579푘) (A3)



(A4)


119867


matrix inequality






푠119896 (119899) = 푎퐻 (120579푘)푄minus1119896

푎퐻 (120579푘)푄minus1119896 푎 (120579푘)푥 (119899)


푎퐻 (120579푘)푅minus1푎 (120579푘)푥 (119899) = 푤119867119896 푥 (119899)

(A7)




Data Availability




Acknowledgments


References

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





30 1 52 4Δ (deg)

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f Res

olut

ion





119875푑 = Ω푑Ω (29)



092

093

094

095

096

097

098

099

1

Prob

abili

ty o

f Det

ectio

n

86 100 2 4SNR (dB)





0

01

02

03

04

05

06

07

08

09

1Pr

obab

ility

of D

etec

tion





02

04

06

08

1

12

14

16

18

2

RMSE

(deg

)

86 100 2 4SNR (dB)





RMSE = radic 1200 1119870200sum휏=1

퐾sum푘=1

(120579(휏)푘

minus 120579푘)2 (30)





0

1

2

3

4

5

6

7

8

RMSE

(deg

)



6 Conclusions


Appendix




M푉 = 푁sum푛=1

1003817100381710038171003817푥 (119899) minus 푠119896 (119899) 푎 (120579푘)10038171003817100381710038172푉 (A2)



푠119896 (119899) = 푎퐻 (120579푘)푉푥 (119899)푎퐻 (120579푘)푉푎 (120579푘) (A3)



(A4)


119867


matrix inequality






푠119896 (119899) = 푎퐻 (120579푘)푄minus1119896

푎퐻 (120579푘)푄minus1119896 푎 (120579푘)푥 (119899)


푎퐻 (120579푘)푅minus1푎 (120579푘)푥 (119899) = 푤119867119896 푥 (119899)

(A7)




Data Availability




Acknowledgments


References

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



02

04

06

08

1

12

14

16

18

2

RMSE

(deg

)

86 100 2 4SNR (dB)





RMSE = radic 1200 1119870200sum휏=1

퐾sum푘=1

(120579(휏)푘

minus 120579푘)2 (30)





0

1

2

3

4

5

6

7

8

RMSE

(deg

)



6 Conclusions


Appendix




M푉 = 푁sum푛=1

1003817100381710038171003817푥 (119899) minus 푠119896 (119899) 푎 (120579푘)10038171003817100381710038172푉 (A2)



푠119896 (119899) = 푎퐻 (120579푘)푉푥 (119899)푎퐻 (120579푘)푉푎 (120579푘) (A3)



(A4)


119867


matrix inequality






푠119896 (119899) = 푎퐻 (120579푘)푄minus1119896

푎퐻 (120579푘)푄minus1119896 푎 (120579푘)푥 (119899)


푎퐻 (120579푘)푅minus1푎 (120579푘)푥 (119899) = 푤119867119896 푥 (119899)

(A7)




Data Availability




Acknowledgments


References

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




푠119896 (119899) = 푎퐻 (120579푘)푉푥 (119899)푎퐻 (120579푘)푉푎 (120579푘) (A3)



(A4)


119867


matrix inequality






푠119896 (119899) = 푎퐻 (120579푘)푄minus1119896

푎퐻 (120579푘)푄minus1119896 푎 (120579푘)푥 (119899)


푎퐻 (120579푘)푅minus1푎 (120579푘)푥 (119899) = 푤119867119896 푥 (119899)

(A7)




Data Availability




Acknowledgments


References

































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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