Ambiguity Analysis and Resolution for Phase-Based 3D ...downloads.hindawi.com/journals/ijap/2019/4743829.pdf · ResearchArticle Ambiguity Analysis and Resolution for Phase-Based 3D

Research ArticleAmbiguity Analysis and Resolution for Phase-Based 3D SourceLocalization under Given UCA

Zhen Liu1 Xin Chen 1 ZhenhuaWei2 Tianpeng Liu1 Linlin Li2 and Bo Peng1

1College of Electronic Science National University of Defense Technology Changsha 410073 China2Rocket Force University of Engineering Xirsquoan 710025 China

Correspondence should be addressed to Xin Chen chenxin10nudteducn

Received 31 August 2018 Revised 21 March 2019 Accepted 7 April 2019 Published 18 April 2019

Academic Editor Herve Aubert

Copyright copy 2019 Zhen Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Under uniform circular array by employing some algebraic schemes to exploit the phase information of receiving data andfurther estimate the sourcersquos three-dimensional (3D) parameters (azimuth angle elevation angle and range) a series of novelphase-based algorithms with low computational complexity have been proposed recently However when the array diameter islarger than sourcersquos half-wavelength these algorithms would suffer from phase ambiguity problem Even so there always existcertain positions where the sourcersquos parameters can still be determined with nonambiguity Therefore this paper first investigatesthe zone of ambiguity-free source 3D localization using phase-based algorithms For the ambiguous zone a novel ambiguityresolution algorithm named ambiguity traversing and cosine matching (ATCM) is presented In ATCM the phase differences ofcentrosymmetric sensors under different ambiguities are utilized to match a cosine function with sensor number-varying and thesourcersquos unambiguous rough angles can be derived from amplitude and initial phase of the cosine functionThen the unambiguousangles are employed to resolve the phase ambiguity of the phase-based 3D parameter estimation algorithm and the sourcersquos range aswell as more precise angles can be achievedTheoretical analyses and numerical examples show that apart from array diameter andsourcersquos frequency the sensor number and spacing of employed sensors are two key factors determining the unambiguous zoneMoreover simulation results demonstrate the effectiveness and satisfactory performance of our proposed ambiguity resolutionalgorithm

1 Introduction

Passive source localization using an array of sensors hasnumerous key applications for wireless communication elec-tron reconnaissance astronomy smart antennas etc [1ndash5]A uniform circular array (UCA) has advantages over theother array geometries due to its 360∘ azimuth coverage analmost identical beamwidth and additional elevation angleinformation [6ndash14]

In the context of three-dimensional (3D) parameter esti-mation by employing some algebraic schemes to exploit thephase information of receiving data and further estimate thesourcersquos 3D parameters a series of novel phase-based algo-rithms using UCA have been proposed in [11ndash14] which arecomputationally simple and also provide acceptable estima-tion performance Comparing with traditional schemes suchas theMultiple Signal Classification (MUSIC) and Estimation

of Signal Parameters via Rotational Invariance Techniques(ESPRIT) algorithms the phase-based algorithms have obvi-ous advantage on the computational complexity Howeverwhen the array diameter of UCA exceeds the half-wavelengthof source periodical ambiguities would be introduced intophase difference measurement of these algorithms Even so itshould be noticed that there still exists some accurate estima-tion of sourcersquos location under ambiguous situations and thecorrect estimation zone probably accords with the demand ofpractical application Therefore under ambiguous situationsit is significant to analyze the zone of ambiguity-free beforeusing the phase-based algorithms As for ambiguity-freezone Tan et al [15] presented the ambiguity in the MUSICand ESPRIT algorithms for 2D direction of arrival estimationunder a uniform linear array and provided the ambiguity-freezones of these two algorithms So far aiming to the phase-based algorithms ambiguous analysis has not been found in

HindawiInternational Journal of Antennas and PropagationVolume 2019 Article ID 4743829 12 pageshttpsdoiorg10115520194743829

2 International Journal of Antennas and Propagation

1 2 34

x

y

1m +m

z

M

θr

mr

R

(r sin cos

r sin sin r cos )

Figure 1 Geometry of circular array and source

existing literatures Moreover when the source is located atambiguous zone effective methods of ambiguity resolutionare necessary to obtain accurate sourcersquos location In the caseof ambiguity resolution for the phase-based algorithm underUCA Xin et al [16] proposed two different algorithms forunambiguous localization of a single far-field source basedon two user-defined types of phase difference observationmodels This two methods can achieve unambiguous DOAestimation However they only consider the localization of asingle far-field source and near-filed situation is inapplicableto the proposed methods As for the near-filed situation ofunambiguous sourcersquos 3D localization we recently proposeda feasible scheme In [17] by referring to the idea of ambiguityresolution via rotating interferometer we employ adaptiverotation to make the sensors form virtual short baselineto resolve phase ambiguity of stationary monofrequencysources However moving or frequency-hopping sources arecommon in various environment this rotaryway is inapplica-ble as the matching of sourcersquos frequency and location beforeand after rotation are demanded and relevant methods needto be presented

Accordingly in this paper we first analyze the ambiguityproblem of phase-based algorithms and the correct estimatezone of sourcersquos 3D localization under ambiguous situationsis then investigated which could provide useful guidancein practical appliance Thereafter we present a novel algo-rithm of ambiguity resolution named ambiguity traversingand cosine matching (ATCM) which can achieve bothmonofrequency and frequency-hopping sourcesrsquo unambigu-ous localization Herein sourcersquos angles and range are firstdecoupled by using the centrosymmetry of UCA with evennumber of sensors Then by ambiguity traversing a matrixof phase difference under different ambiguities is developedAfter that based on the cosine property of unambiguousphase differences all elements of the phase difference matrixare utilized to match with a cosine function with variableamplitude and initial phase which correspond to sourcersquosangles By employing themethod of exhaustive searching theoptimal matching can be found and its corresponding rough

angle estimation with no ambiguity is obtained subsequentlyFinally to realize sourcersquos range as well as more precise angleestimation we proposed an effective method in [18] by utiliz-ing the obtained unambiguous angles to resolve ambiguity inthe phase-based algorithm via the approximation of steeringvectors between planar wavefront and curved wavefrontwhich can be also applied here

This paper contributes to the area of 3D source localiza-tion in the following three aspects

(1) We analyze the ambiguity problem of the phase-basedalgorithms and present the ambiguity-free estimation zoneunder ambiguous situations which could provide guidance inpractical appliance

(2) For sourcersquos 3D localization this paper provides anovel ambiguity resolution method to obtain actual sourcersquosparameters with satisfactory performance and acceptablecomputational complexity

(3) As the proposed algorithm does not need rotationit can achieve real-time sourcersquos localization by only onceimplementation and the applied scope is wider than rotaryway to resolve ambiguity

The rest of the paper consists of six sections Sec-tion 2 reviews the model of a single narrowband sourcersquos3D parameter estimation Section 3 presents the ambiguityproblem of phase-based algorithms and gives the equation ofambiguity-free estimation zone under ambiguous situationsSection 4 introduces the method of ATCM to obtain sourcersquosunambiguous angle estimation Section 5 carries out simula-tions to demonstrate the rationality of theoretic analysis onunambiguous estimation zone and the effectiveness of theproposed ATCM algorithm Section 6 concludes the wholepaper

2 Signal Modeling

As shown in Figure 1 a fixed UCA with radius R and Midentical omnidirectional sensors is employed where the Msensors are uniformly spaced on the circumference in the

International Journal of Antennas and Propagation 3

xy-plane Assuming that a single narrowband source withcurved wavefront is located at (120601 120579 119903) where the azimuthangle 120601 isin [minus120587 120587] is measured counterclockwise from the x-axis the elevation angle 120579 isin [0 1205872] is measured downwardfrom the z-axis and the range 119903 is measured from the centerof the UCA

Without loss of generality we consider the sourcersquos rangeis beyond the Fresnel area but dissatisfies the condition of far-field Under these assumptions the119898th output of the sensorat time n can be written as

119909119898 (119899) = 119904 (119899) exp 1198952120587120582 (119903 minus 119903119898 (120601 120579 119903)) + 119908119898 (119899) (1)

for119898 = 1 119872 119899 = 1 119873 119904(119899) is the complex envelopewith power 1205902119904 and 119908119898(119899) is the white complex Gaussiannoise vector with power 1205902119899 which is independent of the 119904(119899)120582 is the wavelength of the source 119903119898(120601 120579 119903) is the rangebetween the mth sensor and the source which has the formof

119903119898 (120601 120579 119903) = 119903radic1 + (119877119903 )2 minus 2119877120577119898 (120601 120579)119903 (2)

where 120577119898(120601 120579) = cos(120574119898 minus 120601) sin 120579 with 120574119898 = 2120587(119898 minus 1)119872Consider a function defined as

119891 (120580119903) = 1 + 1205802119903 minus 2120580119903 sdot 120577119898 (120601 120579)12 (3)

where 120580119903 = 119877119903 According to a sufficiently large 119903 relative to119877 119891(120580119903) can be well approximated by a second-order Taylorseries expansion as

119891 (120580119903) asymp 119891 (0) + 1198911015840 (0) 120580119903 + 1211989110158401015840 (0) 1205802119903 (4)

Then 119903119898(120601 120579 119903) can be approximated as

119903119898 (120601 120579 119903) asymp 119903 minus 119877120577119898 (120601 120579) + 11987722119903 (1 minus 1205772119898 (120601 120579)) (5)

Substituting (5) into (1) yields the approximated signalmodel as

119909119898 (119899)= 119904 (119899) exp 1198952120587119877120582 (120577119898 (120601 120579) minus 1198772119903 (1 minus 1205772119898 (120601 120579)))+ 119908119898 (119899)

(6)

It can be seen from (6) that the sourcersquos angles and rangeare included in its exponent By employing some algebraicschemes to exploit the phase information of receiving dataand further estimate the parameters of angles as well as rangesourcersquos 3D localization can be well achieved and we definethis kind of technology as the phase-based algorithms

3 Ambiguity Analysis ofPhase-Based Algorithms

Recently for sourcersquos 3D localization phase-based algorithmsreceive growing attention as its low computational complexity

and high estimation precision However when the arraydiameter exceeds half of sourcersquos wavelength phase-basedalgorithms would suffer from ambiguity problem It shouldbe pointed out that not all sourcersquos location estimation wouldappear phase ambiguity Here we first introduce the phase-based algorithms After that we analyze the ambiguity prob-lem of phase-based algorithms and the still correct estimationzone under ambiguous situations is then investigated

31 Introducing of the Phase-Based Algorithms In phase-based algorithms a correlation function is always defined as

119877119898119889 = 119864 119909119898 (119899) 119909119889 (119899)lowast= 1205902119904 exp j2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119889 (120601 120579 119903)) + 1205902119899 (7)

where 119864 denotes the expected value

119889=

119872 for 119898 + 119897 = 119872 119897 = 1 2 119872 minus 1mod (119898 + 119897119872) otherwise

(8)

119897 represents the spacing between employed sensors 120595119898(120601120579 119903) = 120577119898(120601 120579) minus (1198772119903)(1 minus 1205772119898(120601 120579)) and (sdot)lowast denotesthe complex conjugate Assuming that the receiving data isnoiseless the phase of 119877119898119889 is

119906119898119889 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119889 (120601 120579 119903)) + 2120587119902 (9)

where 119902 is a certain integer Thereafter under unambiguoussituation the phase-based algorithms reformulate unambigu-ous phase angle as the form of matrix and utilize the leastsquare (LS) schemes to obtain sourcersquos 3D parameters [11ndash13]

32 Ambiguity Problem of Phase-Based Algorithms It shouldbe pointed out that the exploited phase of correlation func-tion belongs to [minus120587 120587) and the value is inconsistent to the realphase when 119902 = 0 which would cause incorrect estimationof sourcersquos location parameters Therefore the situation thatineffectiveness of sourcersquos localization caused by incorrectphase estimation as 119902 = 0 is called phase ambiguity Toguarantee no phase ambiguity in (9) the condition 119877 le 1205824is always necessary to ensure 119902 = 0 since |120595119898(120601 120579 119903) minus120595119889(120601 120579 119903)| le 2

It is worth noting that once the array has been designedthe limitation 119877 le 1205824 is useless for localization of over-frequency source However not all the estimated positions ofover-frequency sources would appear phase-ambiguous andit is significant for practical demand to investigate the correctestimation zone of sourcersquos locations under ambiguous situa-tions

33 Unambiguous Estimation Zone of Sourcersquos LocationsUnder ambiguity-free situation 119902 = 0

120588119898119889 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119889 (120601 120579 119903)) (10)


By employing trigonometric function the unambiguousphase difference 120588119898119889 can be easily simplified as

120595119898 (120601 120579 119903) minus 120595119889 (120601 120579 119903) = 120577119898 (120601 120579) minus 120577119889 (120601 120579)+ 1198772119903 (1205772119889 (120601 120579) minus 1205772119898 (120601 120579)) = sin 120579 (cos (120574119898 minus 120601)minus cos (120574119889 minus 120601)) + 1198772119903 sin2 120579 (cos2 (120574119889 minus 120601)minus cos2 (120574119889 minus 120601))

(11)

120588119898119889 = 4120587119877120582 sin (120594 (119898 + 119889 minus 2) minus 120601) sin (120594 (119889 minus 119898))sdot sin 120579 + 1198772119903 sdot sin (2120594 (119889 minus 119898))sdot sin (2120594 (119898 + 119889 minus 2) minus 2120601) sin2 120579

(12)

where 120594 = 120587119872 According to a sufficiently large value of 119903relative to 119877 (12) can be well approximated as

120588119898119889asymp 4120587119877120582 sin (120594 (119898 + 119889 minus 2) minus 120601) sin (120594 (119889 minus 119898)) sin 120579 (13)

As the phase difference 1199061015840119898119889 is unambiguous it satisfies|1199061015840119898119889| le 120587 Substituting (13) into the condition yields

sin (120579)le 1205824119877 sdot 1003816100381610038161003816sin (120594 (119889 minus 119898)) sin (120594 (119898 + 119889 minus 2) minus 120601)1003816100381610038161003816

(14)

To further simplify the expression above we take noaccount of the influence of sourcersquos azimuth angle by settingsin(120594(119898 + 119889 minus 2) minus 120601) = 1 and (14) can be expressed as

sin (120579) le 1205824119877 sdot 1003816100381610038161003816sin (120594 (119889 minus 119898))1003816100381610038161003816 (15)

And then

120579max asymp arcsin 1205824119877 sdotmax119898 (1003816100381610038161003816sin (120594 (119889 minus 119898))1003816100381610038161003816) (16)

It can be found that the maximum elevation angle of sourceis related to the number of sensors the array diameter thechoice of spacing 119897 and the sourcersquos frequency Thereforeunder the maximum elevation angle of source determined bycertain factors in (16) the phase differences in phase-basedalgorithms would not introduce ambiguity and the parameterestimation of related sourcersquos locations is effective Howeverthe ambiguous zone is wide and methods of resolvingambiguity need to be proposed Thus in the next section wewould present a novel ambiguity resolution algorithm basedon the cosine property of phase differences whose part workhas been published in [19]

4 Ambiguity Resolution by Using the Methodof ATCM

It is clear that incorrect range estimation is caused bylarge angle estimation errors so ambiguity in 3D parameterestimation of source is mainly brought from angle estima-tion However looking at (6) ambiguity resolution is anintractable task as sourcersquos angles interweave with the rangeTherefore the troublesome separation of sourcersquos angles andrange is demanded to realize ambiguity resolution Here wefirst define two sensors which are symmetric about the centreof a circle as the centrosymmetric sensors (CCSs) so thenumber of sensors must be even to ensure this property Andsourcersquos angles and range are decoupled by computing thephase differences of CCSsThen based on the cosine propertyof unambiguous phase differences the method of ambiguitytraversing and actual-value matching are employed to obtainrough sourcersquos angle estimation with no ambiguity

41 Separation of Sourcersquos Angles and Range It can be noticedthat 120574119898+1198722 = 120574119898 + 120587 under a UCA with even sensors andthen

120577119898+1198722 (120601 120579) = minus120577119898 (120601 120579) (17)

for119898 = 1 2 1198722 According to (9) the phase differencesof CSSs can be expressed as

119906119898119898+1198722 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119898+1198722 (120601 120579 119903))+ 2120587119902 = 4120587119877120577119898 (120601 120579)120582 + 2120587119902

(18)

Note that 119906119898119898+1198722 is only represented by the 2D angleparameter 120577119898(120601 120579) of the sourcersquos locations and effectivemethods of resolving ambiguity can be utilized to obtainactual sourcersquos angles under ambiguous situation with 119902 = 042 Ambiguity Resolution of Sourcersquos Angles The unambigu-ous phase differences 120588119898119898+1198722 = 4120587119877 sin 120579 cos(2120587(119898 minus1)119872 minus 120601)120582 can be regarded as some sampling points of acosine function where the angular frequency is 2120587119872 initialphase is minus120601 and the amplitude is 4120587119877 sin 120579120582 Therefore wecan develop an optimal equation which is given by

min119860120593

1198722sum119898=1

10038161003816100381610038161003816100381610038161003816120588119898119898+1198722 minus 119860 cos (2120587 (119898 minus 1)119872 + 120593)10038161003816100381610038161003816100381610038161003816st 119860 gt 0 120593 isin [minus120587 120587)

(19)

where minsdot denotes the minimum value It is obvious thatthe actual sourcersquos angles can be obtained by searching theminimum in (19)

Before searching the optimal matching the probabletarget value 120588119898119898+1198722 should be firstly determined so themaximum ambiguity of phase differences is computed as

119863 = ceil (2119877120582 ) (20)


where ceil(sdot) denotes upper nearest integer By ambigu-ity traversing all probable unambiguous phase differences

are included in the following matrix which is givenby

U =[[[[[[[

11990611+1198722 minus 2120587119863 11990622+1198722 minus 2120587119863 sdot sdot sdot 1199061198722119872 minus 212058711986311990611+1198722 minus 2120587 (119863 minus 1) 11990622+1198722 minus 2120587 (119863 minus 1) sdot sdot sdot 1199061198722119872 minus 2120587 (119863 minus 1) sdot sdot sdot

11990611+1198722 + 2120587119863 11990622+1198722 + 2120587119863 sdot sdot sdot 1199061198722119872 + 2120587119863

]]]]]]]

(21)

Due to uncertain amplitude and initial phase of cosine func-tion the method of exhaustive searching cosine matching isemployed to obtain the phase differences with no ambiguityAssuming that the beginning amplitude is1198600 = 0 the searchstep is Δ119860 and the upper bound is ceil(4120587119877120582) Meanwhilethe beginning phase is 1205930 = minus120587 the search step is Δ120593 andthe upper bound is 120587 To ascertain the unambiguous phasedifferences a new optimal equation is developed which isgiven by

119866(119860 120593)= 1198722sum119898=1

min119860120593

10038161003816100381610038161003816100381610038161003816U (∙ 119898) minus 119860 cos ((2119898 minus 1) 120587119872 + 120593)10038161003816100381610038161003816100381610038161003816(22)

where U(∙119898) denotes the elements of the mth column in(21) By searching the minimum value of119866(119860 120593) we can findthe unambiguous phase differences and the actual sourcersquosazimuth angle and elevation angle can be obtained by theircorresponding amplitude and initial phase which can beexpressed as

120601 = minus120593 (23)

120579 = arcsin ( 1198601205824120587119877) (24)

The procedure of ATCM could be summarized as

Step 1 By computing the phase differences of CCSs sourcersquosangles and range can be decoupled

Step 2 To ascertain the scope of ambiguity the maximumambiguity 119863 of phase differences needs to be calculatedaccording to (18)

Step 3 After obtaining the matrix of phase differences wedevelop the optimal equation and find its minimum by usingthe method of exhaustive searching and cosine matching

Step 4 Theambiguity-free estimation of sourcersquos angles (120601 120579)is obtained by substituting the certain phase and amplitudeinto (21) and (22)

It should be pointed out that the effectiveness of resolvingambiguity and estimation precision depend on the step size insearchThe smaller the step themore accurate the estimationHowever by employing small step the computational cost

is too huge to implement real-time sourcersquos localization In[18] we have proposed an effective solution by utilizing theobtained rough sourcersquos angles to resolve ambiguity in thephase-based algorithm via the approximation of steering vec-tors between planar wavefront and curved wavefront whichcan be also applied here Therefore the ATCM algorithmfirst employs proper step size in search to obtain roughsourcersquos angles and more precise angle as well as rangeestimation could be realized by employing our proposedway in [18] which could alleviate the conflict broughtby computational complexity and estimation performanceMoreover the aforementioned method aims to single sourceAs for multiple incoherent sources we have also proposed afeasible scheme in [17] by computing the phase differencesof receiving datarsquos corresponding spectrum peak Thereafterby using the ATCM algorithm each sourcersquos unambiguous3D parameters could be estimated The flow chart of sourcersquosunambiguous 3D localization is shown in Figure 2

5 Numerical Examples

In this section we report on simulation experiments that havebeen performed to demonstrate the rationality of theoreticalanalysis on unambiguous estimation zone and the effective-ness of the proposed method for resolving ambiguity

Simulation experiments are divided into four parts InSection 51 the comparison of theoretical analysis on unam-biguous estimation zones and simulative results under three-dimension situation is presented Then to further demon-strate the rationality of (15) the similar comparison undertwo-dimension situation is illustrated In Section 52 wemainly validate the effectiveness of the resolving ambiguitymethod ATCM Considering the influence of step size insearch and signal-to-noise ratio (SNR) the performance ofATCM algorithm is evaluated in Section 53 Finally inSection 54 we employ the histogram distribution to demon-strate the satisfactory precision of sourcersquos 3D parameterestimation

51 Comparison of Unambiguous Parameter Estimation Zoneunder Three-Dimensional Situation Without loss of general-ity we consider a single source with frequency 119891 = 900MHzthe fixed radius of UCA 119877 = 05m sampling rate 119891119904 = 2GHzand snapshots N=2000 It should be pointed out that thearray diameter is larger than the sourcersquos half-wavelength andthe parameter estimation of phase-based algorithms probably


Phase differences ofCCSs with even sensors

Unambiguousangle parameters

Cosine property ofunambiguous phase differences

Ambiguity resolution forthe phase-based algorithm

Optimalequation

Sourcersquos 3-D parameterswith high precision

Ambiguity traversing

Phasedifference

matrix

exhaustive searching andcosine matching

Figure 2 Flowchart of sourcersquos unambiguous 3D localization

3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 44

22

0 0

z (m) x (m)

(a)

3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 4

4

22

00

z (m)x (m)

(b)

Figure 3 Comparison of unambiguous parameter estimation zones under three-dimensional situation (a)119872 = 6 119897 = 2 (b)119872 = 8 119897 = 3

introduces phase ambiguity To verify the validity of derivedanalytical expression we consider two different combinationsof sensorsrsquo number and spacing respectively

The comparison of unambiguous parameter estimationzones computed by analytical expression and performed bysimulation experiments under noise-free circumstance areshown in Figure 3 Note that in Figure 3(a) six sensorsdisplayed as black dots are uniformly spaced on the UCAand the red dot represents the central location of the UCAWe assume that the searching region of source location is119909 isin [minus4 4]m 119910 isin [minus3 3]m and 119911 isin [0 6]m and employphase differences of sensors whose spacing is 119897 = 2 to obtainsourcersquos 3D parameter estimation The zone covered by green

lines is computed by the formula (16) and the zone coveredby blue dots representing the correct estimation of currentlocation based on the phase-based algorithm is performedby simulation experiments It can be noticed that these twozones are almost overlapped and the simulation results accordwith analytical expression It should be pointed out thatthere exists a threshold of sourcersquos range for accurate 3Dlocalization since a sufficiently large value of sourcersquos range isrelative to array radius to ensure the feasible approximationin (5) and (13) In Figure 3(b) the number of sensors is119872 = 8the employed space between sensors is 119897 = 3 and other typicalscenario settings keep unchanged It can be found that thepresented phenomena are similar to Figure 3(a) Accordingly


minus2 minus1 0 1 2X (m)

minus2

minus1

0

1

2Y

(m)

M=6l=2

Analytic Unambiguous ZoneSimulative Unambiguous Zone

(a)


minus2

minus1

0

1

2

Y (m

)

M=6l=3


(b)

M=8l=2



minus2

minus1

0

1

2

Y (m

)

(c)


M=8l=3


minus2

minus1

0

1

2Y

(m)

(d)

Figure 4 Comparison of unambiguous parameter estimation zones in the Z-plane (a) Unambiguous zone under 119872 = 6 119897 = 2 (b)Unambiguous zone under119872 = 6 119897 = 3 (c) Unambiguous zone under119872 = 8 119897 = 2 (d) Unambiguous zone under119872 = 8 119897 = 3

we can reach the conclusion that the analytical expressioncan provide useful guidance to determine the unambiguousestimation zone

Actually we can notice that unambiguous localizationzones obtained by analytical expression and simulationexperiments in Figure 3 are not entirely overlapped There-fore we further investigate the ambiguity-free estimationzones in the Z-plane by employing 119911 = 3m As shownin Figure 4 the unambiguous estimation zones obtainedby analytical expression are encircled by blue circles andthe red dots representing the estimation locations withnonambiguity are performed by simulation experiments Itcan be seen that under fixed array diameter and sourcersquosfrequency the ambiguity-free estimation zone is wider withincreasing sensors but is narrower against the longer spacingof employed sensors Further it can be found that some

unambiguous estimation positions obtained by simulationsoversteps the area of analytical expression which are causedby the approximation from (14) to (15) where the trivialinfluence of sourcersquos azimuth angle is neglected

52 The Effectiveness of ATCM We take two incoherentsources with identical amplitude as an example The numberof sensors is set as119872 = 8 SNR is 10dB the sampling rate is119891119904 = 4GHz snapshots 119873 = 4000 and other experimentalparameters keep unchanged A source with 119891 = 900MHzis located at (201∘ 105∘ 45m) and the other one with 119891 =15GHz is located at (1205∘ 302∘ 8m) As shown in Figure 5the blue dot is the source and the zone covered by green linesis also computed by the formula (16) where the spacing 119897 = 2It is obvious that the location of source with 119891 = 900MHz is


42minus3

6

minus2

0x (m)

minus1

0

1

y (m

)

2

3

4z (m)

Target source 1

Theoretic localizationscope with no ambiguity

minus22 0minus4

(a)

42minus3

minus2

8

minus1

0

1

y (m

)

2

3

0x (m)

6 4z (m)

minus22 0 minus4

Target source 2


(b)

Figure 5 Sourcersquos 3D location (a) Source 1 (119891 = 900MHz) (b) Source 2 (119891 = 15GHz)

Unambiguous cosine curve

minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)

2 3 4 5 6 7 81Sensor number

Phase differences underdifferent ambiguities

(a)


minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(b)

Figure 6 Phase differences under different ambiguities (a) Source 1 (b) Source 2

located at the unambiguous zone while the location of sourcewith 119891 = 15GHz is located outside the unambiguous zone

After computing the phase differences of CSSs phasedifference matrix could be developed by ambiguity travers-ing It can be seen from the Figure 6 that blue dotsrepresent the phase differences under different ambiguitiesand the red curve represents the target cosine functioncorresponding to unambiguous phase differences Note thatfour phase difference dots are just located at the cosinecurve It should be pointed out that there exist multiplecosine curve lines in the phase difference matrix but onlyone normal cosine curve corresponds to unambiguous phasedifferences

Thus we could employ the method of exhaust searchingand cosine matching to ascertain unambiguous phase differ-ences Herein in order to directly exhibit the rationality of

method we utilize elevation searching to substitute afore-mentioned amplitude searching and the step size in anglesearching is Δ120579 = 1∘ The sum of minimum differences ispresented in Figure 7 It is clear that the minimum of sumjust corresponds to the actual sourcersquos angles Therefore bysearching theminimum sourcersquos rough angle estimationwithno ambiguity can be obtained

Figure 8 shows the comparison of sourcersquos angle estima-tion by using our proposed ATCM method and the phase-based algorithm in [12] For angle estimation of source 1which is located at the unambiguous position both thetwo methods can achieve accurate localization However forangle estimation of source 2 which is located outside theunambiguous zone the phase-based algorithm is ineffectivewhile the ATCM method can resolve ambiguity and obtainactual sourcersquos angles


0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s

80 minus10060Azimuth (degree)Elevation (degree)

040 10020 0

(20deg 11deg)1deg)deg 1

(a)

(120deg 30deg)0minus100

5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060

Elevation (degree)40 10020 0

(120deg 30deg)0deg 30deg)(1220deg 30deg(12 1

(b)

Figure 7 Sum of minimum difference (a) Source 1 (b) Source 2

0

20

40

60

80

Elev

atio

n (d

egre

e)

minus100 minus50 0 50 100 150minus150Azimuth (degree)

Actual angle location of source 1

Angle estimation of the literature [12]

Angle estimation of the proposed algorithmMaximum unambiguous elevation

(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)

Actual angle location of source 2Angle estimation of the literature [12]

Angle estimation of the proposed algorithm

Maximum unambiguous elevation

(1205∘ 302∘)

(b)

Figure 8 Comparison of sourcersquos angle estimation (a) Source 1 (b) Source 2

53 The Performance Analysis of ATCM In this part wefirst define the precise estimation as the difference betweenparameter estimation and actual value is no more thanone degree The probability of precise estimation 119875119901119890 =(119873119901119890119873119898119900) sdot 100 representing the ratio between the countof precise estimation 119873119901119890 and Monte Carlo runs 119873119898119900 isthen utilized to present the performance of ATCM We havepointed out that the estimation precision mainly depends onthe step size in search Therefore considering three typicalsourcersquos frequencies (119891 = 1 GHz 2 GHz 3 GHz) threehundred Monte Carlo runs are performed to obtain theprobability of precise sourcersquos angle estimation Assumingthat the distribution of sourcersquos location is random thesampling rate is 119891119904 = 6 GHz snapshots 119873 = 6000 the

number of sensors is set as 119872 = 8 the spacing 119897 = 2 andother experimental parameters are kept unchanged It can beseen from Figure 9(a) that the probability decreases as thestep size increases and only when the step size is less than 02degree the probabilities under different sourcersquos frequenciesare near to a hundred percent However the smaller the stepsize the higher the computational cost The elapsed CPUtimes of the ATCM algorithm by employing the step sizesΔ120579 = 02∘ and Δ120579 = 1∘ for a single run are measured as057s and 1361s respectively Meanwhile we also study theinfluence of SNR on the ATCM algorithm by employing thestep size Δ120579 = 02∘ As shown in Figure 9(b) the probabilityof precise estimation obviously improves as SNR increasesand the amplitude of change is gentle when the SNR is larger


02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)

minus10 minus5 0 5 10

SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)

Figure 9 Probability of precise estimation versus step size and SNR (a) Step size in search (noiseless) (b) SNR (step size = 02)

than 0dB Moreover the probability of precise estimation isacceptable under different frequencies of estimated source athigh SNR

As a whole the proposed ATCM algorithm with smallstep size in search is more feasible for moving sourcesHowever for fixed sources it can always employ relativelylarge step size to obtain rough sourcersquos angles with noambiguity which has obvious advantage over small step sizein computational cost

54 The Estimation Performance of Sourcersquos 3D LocalizationThe rough sourcersquos angle estimation obtained by ATCM canbe utilized to resolve ambiguity in phase-based algorithmin [12] and sourcersquos range as well as more precise angleestimation can be realized To demonstrate the effectivenessof this scheme and the satisfactory performance of sourcersquos3D localization three hundred Monte Carlo runs are per-formed to obtain histogram distribution of sourcersquos param-eter estimation The experimental parameters are identical toSection 52 It should be noticed fromFigures 10(a) 10(b) and10(c) that the histogram distribution of sourcersquos 3D parameterestimation is near to the true value and the times locatedat true value are relatively larger than others The maximumestimation errors of azimuth angle elevation angle and rangeapproximate to 005∘ 001∘ 08m respectively In Figures10(d) 10(e) and 10(f) the situation is similar

As a whole our proposed method can achieve bothambiguous and unambiguous sourcersquos 3D localization withexcellent estimation performance

6 Conclusion

3D source localization of an over-frequency source alwayssuffers from phase ambiguity under UCA with a fixed

array diameter This paper first investigates the correctlocalization zone using phase-based algorithms and a novelambiguity resolution method based on the cosine propertyof ambiguity-free phase differences is then presented whichcan achieve multiple incoherent monofrequency as wellas frequency-hopping sourcesrsquo unambiguous 3D parameterestimation Simulation experiments demonstrate the validityof theoretical analysis on unambiguous estimation zone andthe effectiveness of ambiguity resolution algorithm Researchresults could provide useful guidance in ambiguity analysisand practical appliance if the phase-based algorithms are tobe implemented

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no conflicts of interest

Authorsrsquo Contributions

The main idea was proposed by Zhen Liu and Xin ChenXin Chen and Zhenhua Wei conceived and designed theexperiments Zhen Liu and Tianpeng Liu wrote the paperLinlin Li and Bo Peng revised the paper

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China (no 61401481) and in part by


2005 2007 2009 2011 2013 2015Estimation of azimuth (degree)

010203040506070

Tim

es

(a)

1049 10495 105 10505 1051Estimation of elevation (degree)

0

20

40

60

80

Tim

es

(b)

39 41 43 45 47 49 51 53Estimation of range (meter)

010203040506070

Tim

es

(c)

12049 120495 1205 120505 12051Estimation of azimuth (degree)

0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206

Estimation of elevation (degree)

0

10

20

30

40

50

60

Tim

es

(e)

78 785 79 795 8 805 81 815 82Estimation of range (meter)

0

20

40

60

80

Tim

es

(f)

Figure 10Histogramdistribution of sourcersquos 3Dparameter estimation (a) Azimuth angle (119891 = 900MHz) (b) Elevation angle (119891 = 900MHz)(c) Range (119891 = 900MHz) (d) Azimuth angle(119891 = 15GHz) (e) Elevation angle (119891 = 15GHz) (f) Range (119891 = 15GHz)

the Natural Science Foundation of Hunan Province China(no 2017JJ3368)

References

[1] Y D Huang and M Barkat ldquoNear-field multiple source local-ization by passive sensor arrayrdquo IEEE Transactions on Antennasand Propagation vol 39 no 7 pp 968ndash975 1991

[2] S Zhou and P Willett ldquoSubmarine location estimation via anetwork of detection-only sensorsrdquo IEEETransactions on SignalProcessing vol 55 no 6 pp 3104ndash3115 2007

[3] M Morinaga H Shinoda and H Kondoh ldquoDOA estimationof coherent waves for 77 GHz automotive radar with threereceiving antennasrdquo in Proceedings of the 6th European RadarConference pp 145ndash148 Rome Italy 2009

[4] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[5] B-S Kwon T-J Jung C-H Shin andK-K Lee ldquoDecoupled 3-D near-field source localization with UCA via centrosymmetricsubarraysrdquo IEICE Transactions on Communications vol E94-Bno 11 pp 3143ndash3146 2011

[6] N Wu Z Qu W Si and S Jiao ldquoDOA and polarizationestimation using an electromagnetic vector sensor uniformcircular array based on the ESPRIT algorithmrdquo Sensors vol 16no 12 2016

[7] Y Wu and H C So ldquoSimple and accurate two-dimensionalangle estimation for a single sourcewith uniform circular arrayrdquoIEEE Antennas and Wireless Propagation Letters vol 7 pp 78ndash80 2008

[8] Y Wu H Wang Y Zhang and Y Wang ldquoMultiple near-field source localisation with uniform circular arrayrdquo IEEEElectronics Letters vol 49 no 24 pp 1509-1510 2013

[9] D Starer and A Nehorai ldquoPassive localization on near-fieldsources by path followingrdquo IEEE Transactions on Signal Process-ing vol 42 no 3 pp 677ndash680 1994

[10] J H Lee D H Park G T Park and K K Lee ldquoAlgebraicpath-following algorithm for localising 3-D near-field sourcesin uniform circular arrayrdquo IEEE Electronics Letters vol 39 no17 pp 1283ndash1285 2003

[11] E-H Bae and K-K Lee ldquoClosed-form 3-D localization forsingle source in uniform circular array with a center sensorrdquoIEICE Transactions on Communications vol E92-B no 3 pp1053ndash1056 2009

[12] T-J Jung and K Lee ldquoClosed-form algorithm for 3-D single-source localization with uniform circular arrayrdquo IEEE AntennasandWireless Propagation Letters vol 13 pp 1096ndash1099 2014

[13] Y Wu H Wang L Huang and H C So ldquoFast algorithmfor three-dimensional single near-field source localization withuniform circular arrayrdquo in Proceedings of the 6th InternationalConference on Radar RADAR 2011 pp 350ndash352 ChinaOctober2011

[14] L Zuo J Pan and Z Shen ldquoAnalytical Algorithm for 3-DLocalization of a Single Source with Uniform Circular ArrayrdquoIEEE Antennas and Wireless Propagation Letters vol 17 no 2pp 323ndash326 2018

[15] C M Tan S E Foo M A Beach and A R Nix ldquoAmbiguity inMUSIC and ESPRIT for direction of arrival estimationrdquo IEEEElectronics Letters vol 38 no 24 pp 1598ndash1600 2002

[16] J Xin G Liao Z Yang and H Shen ldquoAmbiguity resolution forpassive 2-D source localization with a uniform circular arrayrdquoSensors vol 18 no 8 2018


[17] X Chen Z Liu and X Wei ldquoUnambiguous parameter estima-tion of multiple near-field sources via rotating uniform circulararrayrdquo IEEE Antennas and Wireless Propagation Letters vol 16pp 872ndash875 2017

[18] X Chen Z Liu andX ZWei ldquoAmbiguity resolution for phase-based 3-d source localization under fixed uniform circulararrayrdquo Sensors vol 17 no 5 Article ID 1086 2017

[19] X Chen S H Wang Z Liu and X Z Wei ldquoAmbiguityresolving based on cosine property of phase differences for 3-Dsource localization with uniform circular arrayrdquo in Proceedingsof the 2017 International Conference On Digital Image Processing(ICDIP Hongkong) China May 2017

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wwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom


1 2 34

x

y

1m +m

z

M

θr

mr

R

(r sin cos

r sin sin r cos )

Figure 1 Geometry of circular array and source

existing literatures Moreover when the source is located atambiguous zone effective methods of ambiguity resolutionare necessary to obtain accurate sourcersquos location In the caseof ambiguity resolution for the phase-based algorithm underUCA Xin et al [16] proposed two different algorithms forunambiguous localization of a single far-field source basedon two user-defined types of phase difference observationmodels This two methods can achieve unambiguous DOAestimation However they only consider the localization of asingle far-field source and near-filed situation is inapplicableto the proposed methods As for the near-filed situation ofunambiguous sourcersquos 3D localization we recently proposeda feasible scheme In [17] by referring to the idea of ambiguityresolution via rotating interferometer we employ adaptiverotation to make the sensors form virtual short baselineto resolve phase ambiguity of stationary monofrequencysources However moving or frequency-hopping sources arecommon in various environment this rotaryway is inapplica-ble as the matching of sourcersquos frequency and location beforeand after rotation are demanded and relevant methods needto be presented

Accordingly in this paper we first analyze the ambiguityproblem of phase-based algorithms and the correct estimatezone of sourcersquos 3D localization under ambiguous situationsis then investigated which could provide useful guidancein practical appliance Thereafter we present a novel algo-rithm of ambiguity resolution named ambiguity traversingand cosine matching (ATCM) which can achieve bothmonofrequency and frequency-hopping sourcesrsquo unambigu-ous localization Herein sourcersquos angles and range are firstdecoupled by using the centrosymmetry of UCA with evennumber of sensors Then by ambiguity traversing a matrixof phase difference under different ambiguities is developedAfter that based on the cosine property of unambiguousphase differences all elements of the phase difference matrixare utilized to match with a cosine function with variableamplitude and initial phase which correspond to sourcersquosangles By employing themethod of exhaustive searching theoptimal matching can be found and its corresponding rough

angle estimation with no ambiguity is obtained subsequentlyFinally to realize sourcersquos range as well as more precise angleestimation we proposed an effective method in [18] by utiliz-ing the obtained unambiguous angles to resolve ambiguity inthe phase-based algorithm via the approximation of steeringvectors between planar wavefront and curved wavefrontwhich can be also applied here

This paper contributes to the area of 3D source localiza-tion in the following three aspects

(1) We analyze the ambiguity problem of the phase-basedalgorithms and present the ambiguity-free estimation zoneunder ambiguous situations which could provide guidance inpractical appliance

(2) For sourcersquos 3D localization this paper provides anovel ambiguity resolution method to obtain actual sourcersquosparameters with satisfactory performance and acceptablecomputational complexity

(3) As the proposed algorithm does not need rotationit can achieve real-time sourcersquos localization by only onceimplementation and the applied scope is wider than rotaryway to resolve ambiguity

The rest of the paper consists of six sections Sec-tion 2 reviews the model of a single narrowband sourcersquos3D parameter estimation Section 3 presents the ambiguityproblem of phase-based algorithms and gives the equation ofambiguity-free estimation zone under ambiguous situationsSection 4 introduces the method of ATCM to obtain sourcersquosunambiguous angle estimation Section 5 carries out simula-tions to demonstrate the rationality of theoretic analysis onunambiguous estimation zone and the effectiveness of theproposed ATCM algorithm Section 6 concludes the wholepaper

2 Signal Modeling

As shown in Figure 1 a fixed UCA with radius R and Midentical omnidirectional sensors is employed where the Msensors are uniformly spaced on the circumference in the




119909119898 (119899) = 119904 (119899) exp 1198952120587120582 (119903 minus 119903119898 (120601 120579 119903)) + 119908119898 (119899) (1)


119903119898 (120601 120579 119903) = 119903radic1 + (119877119903 )2 minus 2119877120577119898 (120601 120579)119903 (2)


119891 (120580119903) = 1 + 1205802119903 minus 2120580119903 sdot 120577119898 (120601 120579)12 (3)


119891 (120580119903) asymp 119891 (0) + 1198911015840 (0) 120580119903 + 1211989110158401015840 (0) 1205802119903 (4)




119909119898 (119899)= 119904 (119899) exp 1198952120587119877120582 (120577119898 (120601 120579) minus 1198772119903 (1 minus 1205772119898 (120601 120579)))+ 119908119898 (119899)

(6)








119889=


(8)


119906119898119889 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119889 (120601 120579 119903)) + 2120587119902 (9)





120588119898119889 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119889 (120601 120579 119903)) (10)




(11)


(12)





(14)



And then






120577119898+1198722 (120601 120579) = minus120577119898 (120601 120579) (17)


119906119898119898+1198722 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119898+1198722 (120601 120579 119903))+ 2120587119902 = 4120587119877120577119898 (120601 120579)120582 + 2120587119902

(18)


min119860120593

1198722sum119898=1


(19)



119863 = ceil (2119877120582 ) (20)




U =[[[[[[[


11990611+1198722 + 2120587119863 11990622+1198722 + 2120587119863 sdot sdot sdot 1199061198722119872 + 2120587119863

]]]]]]]

(21)


119866(119860 120593)= 1198722sum119898=1

min119860120593

10038161003816100381610038161003816100381610038161003816U (∙ 119898) minus 119860 cos ((2119898 minus 1) 120587119872 + 120593)10038161003816100381610038161003816100381610038161003816(22)


120601 = minus120593 (23)

120579 = arcsin ( 1198601205824120587119877) (24)

















Optimalequation



Phasedifference

matrix



3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 44

22

0 0

z (m) x (m)

(a)

3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 4

4

22

00

z (m)x (m)

(b)







minus2

minus1

0

1

2Y

(m)

M=6l=2


(a)


minus2

minus1

0

1

2

Y (m

)

M=6l=3


(b)

M=8l=2



minus2

minus1

0

1

2

Y (m

)

(c)


M=8l=3


minus2

minus1

0

1

2Y

(m)

(d)







42minus3

6

minus2

0x (m)

minus1

0

1

y (m

)

2

3

4z (m)

Target source 1


minus22 0minus4

(a)

42minus3

minus2

8

minus1

0

1

y (m

)

2

3

0x (m)

6 4z (m)

minus22 0 minus4

Target source 2


(b)



minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(a)


minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(b)








0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s


040 10020 0


(a)


5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060



(b)


0

20

40

60

80

Elev

atio

n (d

egre

e)





(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)




(1205∘ 302∘)

(b)





02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119909119898 (119899) = 119904 (119899) exp 1198952120587120582 (119903 minus 119903119898 (120601 120579 119903)) + 119908119898 (119899) (1)


119903119898 (120601 120579 119903) = 119903radic1 + (119877119903 )2 minus 2119877120577119898 (120601 120579)119903 (2)


119891 (120580119903) = 1 + 1205802119903 minus 2120580119903 sdot 120577119898 (120601 120579)12 (3)


119891 (120580119903) asymp 119891 (0) + 1198911015840 (0) 120580119903 + 1211989110158401015840 (0) 1205802119903 (4)




119909119898 (119899)= 119904 (119899) exp 1198952120587119877120582 (120577119898 (120601 120579) minus 1198772119903 (1 minus 1205772119898 (120601 120579)))+ 119908119898 (119899)

(6)








119889=


(8)


119906119898119889 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119889 (120601 120579 119903)) + 2120587119902 (9)





120588119898119889 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119889 (120601 120579 119903)) (10)




(11)


(12)





(14)



And then






120577119898+1198722 (120601 120579) = minus120577119898 (120601 120579) (17)


119906119898119898+1198722 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119898+1198722 (120601 120579 119903))+ 2120587119902 = 4120587119877120577119898 (120601 120579)120582 + 2120587119902

(18)


min119860120593

1198722sum119898=1


(19)



119863 = ceil (2119877120582 ) (20)




U =[[[[[[[


11990611+1198722 + 2120587119863 11990622+1198722 + 2120587119863 sdot sdot sdot 1199061198722119872 + 2120587119863

]]]]]]]

(21)


119866(119860 120593)= 1198722sum119898=1

min119860120593

10038161003816100381610038161003816100381610038161003816U (∙ 119898) minus 119860 cos ((2119898 minus 1) 120587119872 + 120593)10038161003816100381610038161003816100381610038161003816(22)


120601 = minus120593 (23)

120579 = arcsin ( 1198601205824120587119877) (24)

















Optimalequation



Phasedifference

matrix



3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 44

22

0 0

z (m) x (m)

(a)

3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 4

4

22

00

z (m)x (m)

(b)







minus2

minus1

0

1

2Y

(m)

M=6l=2


(a)


minus2

minus1

0

1

2

Y (m

)

M=6l=3


(b)

M=8l=2



minus2

minus1

0

1

2

Y (m

)

(c)


M=8l=3


minus2

minus1

0

1

2Y

(m)

(d)







42minus3

6

minus2

0x (m)

minus1

0

1

y (m

)

2

3

4z (m)

Target source 1


minus22 0minus4

(a)

42minus3

minus2

8

minus1

0

1

y (m

)

2

3

0x (m)

6 4z (m)

minus22 0 minus4

Target source 2


(b)



minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(a)


minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(b)








0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s


040 10020 0


(a)


5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060



(b)


0

20

40

60

80

Elev

atio

n (d

egre

e)





(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)




(1205∘ 302∘)

(b)





02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(11)


(12)





(14)



And then






120577119898+1198722 (120601 120579) = minus120577119898 (120601 120579) (17)


119906119898119898+1198722 = 2120587119877120582 (120595119898 (120601 120579 119903) minus 120595119898+1198722 (120601 120579 119903))+ 2120587119902 = 4120587119877120577119898 (120601 120579)120582 + 2120587119902

(18)


min119860120593

1198722sum119898=1


(19)



119863 = ceil (2119877120582 ) (20)




U =[[[[[[[


11990611+1198722 + 2120587119863 11990622+1198722 + 2120587119863 sdot sdot sdot 1199061198722119872 + 2120587119863

]]]]]]]

(21)


119866(119860 120593)= 1198722sum119898=1

min119860120593

10038161003816100381610038161003816100381610038161003816U (∙ 119898) minus 119860 cos ((2119898 minus 1) 120587119872 + 120593)10038161003816100381610038161003816100381610038161003816(22)


120601 = minus120593 (23)

120579 = arcsin ( 1198601205824120587119877) (24)

















Optimalequation



Phasedifference

matrix



3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 44

22

0 0

z (m) x (m)

(a)

3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 4

4

22

00

z (m)x (m)

(b)







minus2

minus1

0

1

2Y

(m)

M=6l=2


(a)


minus2

minus1

0

1

2

Y (m

)

M=6l=3


(b)

M=8l=2



minus2

minus1

0

1

2

Y (m

)

(c)


M=8l=3


minus2

minus1

0

1

2Y

(m)

(d)







42minus3

6

minus2

0x (m)

minus1

0

1

y (m

)

2

3

4z (m)

Target source 1


minus22 0minus4

(a)

42minus3

minus2

8

minus1

0

1

y (m

)

2

3

0x (m)

6 4z (m)

minus22 0 minus4

Target source 2


(b)



minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(a)


minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(b)








0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s


040 10020 0


(a)


5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060



(b)


0

20

40

60

80

Elev

atio

n (d

egre

e)





(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)




(1205∘ 302∘)

(b)





02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





U =[[[[[[[


11990611+1198722 + 2120587119863 11990622+1198722 + 2120587119863 sdot sdot sdot 1199061198722119872 + 2120587119863

]]]]]]]

(21)


119866(119860 120593)= 1198722sum119898=1

min119860120593

10038161003816100381610038161003816100381610038161003816U (∙ 119898) minus 119860 cos ((2119898 minus 1) 120587119872 + 120593)10038161003816100381610038161003816100381610038161003816(22)


120601 = minus120593 (23)

120579 = arcsin ( 1198601205824120587119877) (24)

















Optimalequation



Phasedifference

matrix



3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 44

22

0 0

z (m) x (m)

(a)

3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 4

4

22

00

z (m)x (m)

(b)







minus2

minus1

0

1

2Y

(m)

M=6l=2


(a)


minus2

minus1

0

1

2

Y (m

)

M=6l=3


(b)

M=8l=2



minus2

minus1

0

1

2

Y (m

)

(c)


M=8l=3


minus2

minus1

0

1

2Y

(m)

(d)







42minus3

6

minus2

0x (m)

minus1

0

1

y (m

)

2

3

4z (m)

Target source 1


minus22 0minus4

(a)

42minus3

minus2

8

minus1

0

1

y (m

)

2

3

0x (m)

6 4z (m)

minus22 0 minus4

Target source 2


(b)



minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(a)


minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(b)








0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s


040 10020 0


(a)


5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060



(b)


0

20

40

60

80

Elev

atio

n (d

egre

e)





(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)




(1205∘ 302∘)

(b)





02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







Optimalequation



Phasedifference

matrix



3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 44

22

0 0

z (m) x (m)

(a)

3

2

1

0

minus1

minus2

minus3

minus4 minus2

y (m

)

6 4

4

22

00

z (m)x (m)

(b)







minus2

minus1

0

1

2Y

(m)

M=6l=2


(a)


minus2

minus1

0

1

2

Y (m

)

M=6l=3


(b)

M=8l=2



minus2

minus1

0

1

2

Y (m

)

(c)


M=8l=3


minus2

minus1

0

1

2Y

(m)

(d)







42minus3

6

minus2

0x (m)

minus1

0

1

y (m

)

2

3

4z (m)

Target source 1


minus22 0minus4

(a)

42minus3

minus2

8

minus1

0

1

y (m

)

2

3

0x (m)

6 4z (m)

minus22 0 minus4

Target source 2


(b)



minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(a)


minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(b)








0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s


040 10020 0


(a)


5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060



(b)


0

20

40

60

80

Elev

atio

n (d

egre

e)





(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)




(1205∘ 302∘)

(b)





02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




minus2

minus1

0

1

2Y

(m)

M=6l=2


(a)


minus2

minus1

0

1

2

Y (m

)

M=6l=3


(b)

M=8l=2



minus2

minus1

0

1

2

Y (m

)

(c)


M=8l=3


minus2

minus1

0

1

2Y

(m)

(d)







42minus3

6

minus2

0x (m)

minus1

0

1

y (m

)

2

3

4z (m)

Target source 1


minus22 0minus4

(a)

42minus3

minus2

8

minus1

0

1

y (m

)

2

3

0x (m)

6 4z (m)

minus22 0 minus4

Target source 2


(b)



minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(a)


minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(b)








0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s


040 10020 0


(a)


5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060



(b)


0

20

40

60

80

Elev

atio

n (d

egre

e)





(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)




(1205∘ 302∘)

(b)





02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



42minus3

6

minus2

0x (m)

minus1

0

1

y (m

)

2

3

4z (m)

Target source 1


minus22 0minus4

(a)

42minus3

minus2

8

minus1

0

1

y (m

)

2

3

0x (m)

6 4z (m)

minus22 0 minus4

Target source 2


(b)



minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(a)


minus40

minus20

0

20

40

Phas

e diff

eren

ce (r

adia

n)



(b)








0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s


040 10020 0


(a)


5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060



(b)


0

20

40

60

80

Elev

atio

n (d

egre

e)





(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)




(1205∘ 302∘)

(b)





02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 (20deg 11deg)

5

10

Sum

of m

inim

um d

iffer

ence

s


040 10020 0


(a)


5

Sum

of m

inim

um d

iffer

ence

s

10

80

Azimuth (degree)060



(b)


0

20

40

60

80

Elev

atio

n (d

egre

e)





(201∘ 105∘)

(a)


0

20

40

60

80El

evat

ion

(deg

ree)




(1205∘ 302∘)

(b)





02 04 06 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



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 08 120

40

60

80

100Pr

obab

ility

()

Step size (degrees)

f=1GHz

f=2GHz

f=3GHz

(a)


SNR (dB)

60

80

70

90

100

Prob

abili

ty (

)

f=1GHz

f=2GHz

f=3GHz

(b)






6 Conclusion



Data Availability






Acknowledgments




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




010203040506070

Tim

es

(a)


0

20

40

60

80

Tim

es

(b)


010203040506070

Tim

es

(c)


0

20

40

60

80

Tim

es

(d)

30194 30196 30198 302 30202 30204 30206


0

10

20

30

40

50

60

Tim

es

(e)


0

20

40

60

80

Tim

es

(f)



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


Documents

Ambiguity Analysis and Resolution for Phase-Based 3D ...downloads.hindawi.com/journals/ijap/2019/4743829.pdf · ResearchArticle Ambiguity Analysis and Resolution for Phase-Based 3D