2166 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 8, AUGUST 1998 Modeling nd Estimtion of Ambiguities in Liner Arrys Athnssios Mniks, Member, IEEE, nd Christos Proukkis Abstrct In this pper, the problem of mbiguities inherent in the mnifold of ny liner rry structure is investigted Ambiguities, which cn be clssified into trivil nd nontrivil, depending on the ese of their identifiction, rise when n rry cnnot distinguish between two different sets of directionl sources Initilly, the new concept of n mbiguous genertor set is introduced; it represents/genertes n infinite number of mbiguous sets of directions Then, by uniformly/nonuniformly prtitioning the rry mnifold curve of liner rry, different mbiguous genertor sets cn be clculted, nd s direct result, sufficient condition for the presence of mbiguities is obtined The theoreticl spects of the investigtion re followed by the proposl of n innovtive pproch tht clcultes not only ll such mbiguities existing in liner rry of rbitrry geometry but the rnk of mbiguity in ech cse s well The min results presented in the pper re supported by number of representtive exmples Index Terms Ambiguities, differentil geometry, liner rrys NOMENCLATURE, Sclr,, Vector,A Mtrix identity mtrix Trnspose, conjugte trnspose Absolute vlue of sclr Eucliden norm of vector Integer prt Sum of vector elements Element-by-element exponentil Mtrix exponentil th row of mtrix Element-by-element th power Difference between the th nd th sensor loctions Generl bering prmeter Azimuth bering Arc length prmeter Mnifold length Aperture of liner rry -dimensionl complex spce -dimensionl rel spce Mnuscript received July 30, 1997; revised Jnury 29, 1998 The ssocite editor coordinting the review of this pper nd pproving it for publiction ws Dr Jen-Jcques Fuchs The uthors re with the Digitl Communictions Reserch Section, Deprtment of Electricl nd Electronic Engineering, Imperil College of Science, Technology nd Medicine, London, UK Publisher Item Identifier S 1053-587X(98)05226-X Set of positive integers tr Trce of mtrix I INTRODUCTION WHEN finding the directions-of-rrivl (DOA s) of nrrowbnd signls using rry sensor mesurements, it is importnt to be certin tht the problem hs unique solution If the rry hs identicl responses to two different sets of DOA s, then the mbiguity problem is sid to rise The first ttempt to introduce mthemticl frmework to del with the mbiguity problem ws by Schmidt [1], who clssified the mbiguities ccording to their rnk bsed on the liner dependence between mnifold vectors, s rnk-1 or rnk greter thn one The former (rnk-1), which will lter be chrcterized s trivil, is comprtively esy to detect nd, for this reson, hs received much more ttention thn the relly importnt cse, which will be referred to s nontrivil mbiguities Any subsequent reserch to hndle the mbiguity problem ws minly concerned with either the performnce of specific rry geometries or with the identifiction of rry structures tht re free of mbiguities up to certin rnk of mbiguity Hence, in [2], it ws observed tht s the perture of circulr rry increses, so does the risk of trivil mbiguities occuring In [3], the mbiguity problem ws exmined for the cse of liner rrys, nd specil type of liner rry ws identified tht does not suffer from trivil mbiguities In [4], conjecture is mde providing reltively simple wy of identifying whether set of directions is rnk- mbiguous or not This would gretly ese the tsk of identifying mbiguities However, in [5], the conjecture ws shown, through counterexmple, to be, in generl, incorrect In ddition, in this pper, specific clss of plnr rry ws presented whose members were shown to be free of both trivil nd nontrivil mbiguities Finlly, in [6], specific clss of uniform circulr rry is shown to be free of rnk-2 mbiguities when the sources re coplnr with the rry In this pper, we ttempt to model nd clculte mbiguous sets of directions tht exist for ny liner rry geometry The orgniztion of the pper is s follows In Section II, the problem of mbiguities is formulted, nd useful clssifiction of mbiguities is given In Section III, prmetriztion of curves, such s the rry mnifold of liner rry, is presented In Section IV, the new notion of the mbiguous genertor set is introduced In Section V, technique is proposed for the identifiction nd estimtion of new clss of mbiguous genertor set bsed on uniform prtitioning of the rry mnifold curve This clss exists for ny rry geometry 1053 587X/98$1000 1998 IEEE
MANIKAS AND PROUKAKIS: MODELING AND ESTIMATION OF AMBIGUITIES IN LINEAR ARRAYS 2167 However, if the rry is symmetric liner, then second clss of mbiguity is identified bsed on nonuniform prtitioning of the mnifold curve Finlly, the pper is concluded in Section VI Note tht in compnion pper [7], the investigtion of mbiguities is extended to plnr rrys, nd n pproch is proposed tht provides mbiguous sets of DOA s tht re not coplnr with the rry II PROBLEM FORMULATION AND A CLASSIFICATION OF AMBIGUITIES It is well known tht in the bsence of clibrtion errors, the signl-vector received by n rry of sensors from nrrowbnd fr-field signl sources with directions, cn be modeled s where denotes the vector of the complex received bsebnd signls t the reference point, is the dditive noise, nd is the mtrix with columns the mnifold vectors, ie, (1) (2) The locus of ll mnifold vectors is continuum lying in -dimensionl complex spce nd is known s the rry mnifold Therefore, the rry mnifold, where is the prmeter spce, is essentilly mpping from to the complex -dimensionl spce (3) Thus, if the system under considertion is n zimuth-only system, ie,, then is the intervl on the rel line, nd the rry mnifold is curve in Note tht the rry mnifold vector is function of the loctions nd chrcteristics of the rry elements nd represents the rry complex response to unity power signl impinging on the rry from direction For liner rry of omnidirectionl sensors, the only prmeter of interest is the zimuth, ie,, nd the rry mnifold vector cn be written s (4) where is the vector of sensor loctions in hlf wvelengths The rry mnifold plys very importnt role in the signl subspce lgorithms tht re pplied in direction-finding systems Signl subspce lgorithms serch the rry mnifold to identify the true DOA s s those tht stisfy specified criterion For instnce, MUSIC [1] serches the mnifold for those tht re closest to the signl subspce in the Eucliden sense In this study, we will consider the problem tht rises when the mpping is not one to one In this cse, the rry cnnot distinguish between two (or more) different signl environments For instnce, two different sets of signls impinging on the rry cn provide identicl responses t the rry output, ie, the sme mesurements Under such conditions, it is sid tht the mbiguity problem rises In such cse, ny direction-finding lgorithm my be unble to resolve the true directions from the flse ones At this point, it is importnt to note tht mbiguities rise only s result of the rry geometry, nd thus, different rry geometries hve different sets of mbiguous directions This implies tht n rry tht is unmbiguous for given set of directions might become mbiguous for the sme set if we chnge its sensor loctions even slightly so tht new rry geometry is obtined A typicl exmple of mbiguities ssocited with liner rry operting in the presence of two sources t 5 nd 35 is illustrted in Fig 1 When, for instnce, MUSIC is pplied on the rry of Fig 1(), then only two nulls corresponding to the DOA s of the true sources will rise If the fourth sensor locted t 1 hlf wvelengths is now moved to the loction 15 hlf wvelengths so tht the uniform liner rry of Fig 1(c) is obtined, then four nulls pper t directions 5, 35, 10970, 12094, which thus form n mbiguous set of directions In generl, we cn sy tht every rry suffers from mbiguities in some wy or nother, nd sometimes, some mbiguous sets of directions cn be esily identified For instnce, it is impossible to distinguish whether signl is impinging on liner rry from direction-of-rrivl or from the direction, ie, the mirror imge of with respect to 180 Therefore, if direction-finding lgorithm is pplied in the rnge, then two directions will be estimted for every true source Note tht in this cse (5) nd for this reson, the prmeter spce in the cse of liner rrys is confined to Eqution (5) is specil cse of where (6) for Eqution (6) sys tht there exists t lest one mnifold vector, which cn be written s sclr multiple of nother mnifold vector Then, the two wves with berings re indistinguishble by the rry even if This is the simplest type of mbiguity, which is known s trivil mbiguity A more complicted type of mbiguity known s nontrivil mbiguity rises when mnifold vector cn be written s liner combintion of two or more different mnifold vectors For instnce, let be liner combintion of with so tht In such cse, the rry will hve identicl responses for the sets of berings,,, etc For exmple, consider the nonuniform liner rry with sensor loctions mesured in hlf wvelengths given by Let the sources present in the environment be For this specific rry, rnk, nd hence, these four mnifold vectors re linerly dependent As result, signl subspce direction-finding lgorithm will provide spurious source-direction t
2168 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 8, AUGUST 1998 () (b) (c) (d) Fig 1 Exmple of mbiguities in liner rrys () Liner rry configurtion (in hlf-wvelengths) (b) True directions: 5 nd 35 ; directions estimted from MUSIC: 5 nd 35 (c) Liner rry configurtion (in hlf-wvelengths) (d) True directions: 5 nd 35 ; directions estimted from MUSIC: 5, 35, 10970, nd 12094 This type of mbiguity is much more difficult to identify since, unlike trivil mbiguity, it cnnot be detected by simple serch of the mnifold Note tht in this study, the rry centroid will be tken s the reference point, implying tht sum However, it cn be proved tht the mbiguities re independent of the choice of the reference point III THE GEOMETRY OF THE CURVES OF THE ARRAY MANIFOLD Since curves will be extensively used in this pper, it is necessry to provide n pproprite prmetriztion of curves This prmetriztion cn be obtined from n re of mthemtics clled differentil geometry; see [8] or [9] Differentil geometry is specificlly concerned with the ppliction of clculus to the investigtion of the geometric properties of curves imbedded principlly in the three-dimensionl (3-D) rel spce However, notions from differentil geometry cn be extended (see [10]) to include curves embedded in complex -dimensionl spce such s the mnifold curves, which re of interest here The most bsic feture of curve, ccording to differentil geometry, is the rc length, which is formlly defined s where (7) must be differentible t ll points, nd therefore (8) In the cse of the mnifold of liner rry of omnidirectionl sensors, the reltionship between the rc length nd the zimuth, ie, given by (8) is simplified to where corresponds to the direction tht is prllel to the rry line, ie, to Furthermore, since the prmeter spce is, it results tht the totl length of the rry mnifold is (9) mnifold length (10)
MANIKAS AND PROUKAKIS: MODELING AND ESTIMATION OF AMBIGUITIES IN LINEAR ARRAYS 2169 Note tht the mnifold curve of liner rry with omnidirectionl sensors ws described in gret detil in [10] nd ws found to be hyperhelix Such curve is very ttrctive since it is uniquely described, except for its position in spce, by set of constnt curvtures The exct vlues of these curvtures cn be computed nlyticlly using the expressions given in [10] nd [11], nd they form skew symmetric mtrix known s the Crtn mtrix s C where if there is no sensor t the rry centroid otherwise with denoting the number of sensors in symmetricl pirs nd representing the th curvture The dimensionlity lso represents the number of mnifold coordinte vectors t ny point, which form the coordinte mtrix The mnifold coordinte vectors t re trnsformed to the coordinte vectors t point by continuous differentible rel trnsformtion mtrix (which is known s the Frme mtrix) s with (11) Note tht the Crtn mtrix nd the Frme mtrix lwys stisfy the differentil eqution C (12) which implies tht the Crtn mtrix cn be expressed s C (13) The rry mnifold vector t point cn be expressed s function of the coordinte mtrix of the mnifold nd its curvtures s (14) where the vector (which is known s the rdii vector of the mnifold) is given s if if (15) with, nd, even In this study, the mnifold curves (hyperhelices) will be used to nlyticlly determine nd formlly define mbiguities in liner rrys IV THE CONCEPT OF AN AMBIGUOUS GENERATOR SET Before continuing, it is necessry to present the following definitions which will be extensively used Definition 1: An ordered set of rc lengths, where, is sid to be n mbiguous set of rc lengths if the mtrix with columns the mnifold vectors hs rnk less thn, ie, Definition 2: If set of rc lengths, where, is mbiguous, then its rnk of mbiguity 1 is defined s the integer It must be noted tht the bove definitions need to be extended for since there exist cses where set of mnifold vectors intersect the rry mnifold t points Therefore, set of rc lengths, where, is n mbiguous set of rc lengths if ll subsets of tht contin exctly elements re themselves mbiguous sets of rc lengths In this cse, the rnk of mbiguity of is defined s the integer rnk Note tht the previously introduced definitions cn be directly pplied to sets of DOA s by simply substituting [using (9)] the rc length with the zimuth The theorem tht follows is the first result of this pper It essentilly sttes tht if ll the elements of n mbiguous set of rc lengths re rotted on the rry mnifold by the sme vlue, then the resulting set is lso n mbiguous set of rc lengths Theorem 1: If, for liner rry of sensors,, with, is n mbiguous set of rc lengths with rnk of mbiguity, then ny set, with nd, is lso n mbiguous set of rc lengths with the sme rnk of mbiguity The proof of this theorem cn be found in Appendix A, wheres its essentil fetures re illustrted in the following exmple Exmple 1: Consider the rry of Fig 2(), which hs sensor loctions given by nd mnifold length Let four sources be present in the environment with DOA s 5313, 7846, 10154, nd 12687 If, eg, MUSIC, is pplied, then n extr null will pper t 0, which implies tht the set is n mbiguous set of directions Using (9), the corresponding mbiguous set of rc lengths is computed to be If this set is rotted by, then new set is obtined, which corresponds to set of directions If only four sources with directions equl to the first four elements of re present in the environment, then pplying MUSIC will result in n extr null t 13778, which implies tht equivlently,, is lso n mbiguous set 1 Note tht in [12], the rnk of mbiguity of mbiguous set of DOA s is defined s m 0 rnk( (s)) or,
2170 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 8, AUGUST 1998 () (b) (c) (d) Fig 2 Effects of rotting n mbiguous set () Liner rry configurtion (in hlf-wvelengths) (b) True directions: 5313, 7846, 10154, nd 12687 ; directions estimted from MUSIC: 0, 5313, 7846, 10154, nd 12687 (c) True directions: 3075, 6265, 8659, nd 10991 ; directions estimted from MUSIC: 3075, 6265, 8659, 10991, nd 13778 (d) True directions: 7313, 9846, 12154, nd 14687 ; directions estimted from MUSIC: 7313, 9846, 12154, nd 14687 Note tht the rottion should be crried out in the rclength domin nd not in the directions-of-rrivl domin For instnce, if the directions of the originl sources re rotted by so tht the new set of directions is obtined, then s cn be seen from Fig 2(d), this set is not mbiguous Note tht Theorem 1 cn esily be generlized to incorporte ny rnk of mbiguity of the set s well s ny number of elements It becomes cler tht if one mbiguous set is identified, then by simple rottion, n infinite number of mbiguous sets cn be generted, nd therefore, two different mbiguous sets my, in fct, be just rottion of ech other Since ll these sets cn be generted from single set, the ide of the mbiguous genertor set rises nd is defined s follows Definition 3: An ordered set of rc lengths, where, is sid to be n mbiguous genertor set of rc lengths if nd only if we hve the following ) All the elements of the set except the first element re nonzero b) The rnk of the mtrix, with columns the mnifold vectors ssocited with the elements of the set, is less thn, ie, c) For ny subset of elements of with, the rnk of is equl to According to the previous definition, set of rc lengths with, nd is not n mbiguous genertor set This is becuse, but (ie, third condition is not stisfied) On the other hnd, the set is n mbiguous genertor set since it stisfies ll three conditions of Definition 3 From the bove definition, it is obvious tht when considering n rry, it is n impossible tsk to try to identify ll the mbiguous sets Therefore, the objective of this pper is to identify mbiguous genertor sets existing in the mnifold of liner rry of rbitrry geometry In prticulr, two generl clsses of mbiguous genertor sets will be identified, bsed on whether the mnifold curve is prtitioned, ccording to some specific rules, into equl or unequl segments,
MANIKAS AND PROUKAKIS: MODELING AND ESTIMATION OF AMBIGUITIES IN LINEAR ARRAYS 2171 ie, uniform or nonuniform prtitioning, respectively, of the mnifold curve Although ll uniform mbiguities cn be found by using the frmework presented in the following section, there is no nswer t this moment s to how ll nonuniform mbiguities cn be estimted Therefore, in the following section, new clss of nonuniform mbiguity tht exists in symmetric liner rrys will be identified, modeled, nd estimted V IDENTIFYING AMBIGUOUS GENERATOR SETS IN MANIFOLD CURVES A Bsed on Uniform Prtitions of Hyperhelices In this section, technique is proposed for the clcultion of mbiguous genertor sets of directions existing in liner rrys, s well s their ssocited rnk of mbiguity The technique is bsed on the uniform prtitions of the rry mnifold, which re obtined by dividing the mnifold length by the difference between ny two rry sensor loctions In the cse of liner rrys, for which the direction-of-rrivl consists of only the zimuth, the rry mnifold is curve of specified length with well-known properties If this curve is divided into equl segments, ccording to the following theorem, then the end points of these segments form n mbiguous set Theorem 2: If is the mnifold length of liner rry of sensors with loctions in hlf wvelengths, then ny subset of elements of the set of rc lengths, where with (16) is n mbiguous set if i) the lst element of is greter thn 0 nd smller thn, nd ii) the number of nonzero entries is greter thn or equl to The proof of this theorem cn be found in Appendix B The requirement tht the lst element of the set is smller thn the mnifold length, combined with the fct tht, implies tht if if (17) where denotes the integer prt of number By setting the vlue of in the bove eqution to be equl to the biggest possible difference between two rry sensor loctions, which is obviously the perture, nd by combining the bove with the second requirement of the theorem, the following sufficient condition (which hs lso been rigorously proven in [3]) cn be obtined A SUFFICIENT condition for the presence of mbiguities in ALL liner rrys is: where It should be mde cler tht the bove provides sufficient but by no mens necessry condition for the presence of mbiguities This mens tht liner rry cn possibly suffer from mbiguities even if Only for the specific cse of uniform liner rrys hs it been proven tht no mbiguities exist if or, equivlently, if the intersensor spcing is not greter thn hlf wvelength Theorem 2 hs n impliction tht needs to be stressed It is well known [11] tht n increse in the perture results in better resolution cpbilities However, from the point of view of mbiguities, incresing the perture my not be very good ide, t lest for signl subspce-type techniques By incresing the perture, the number of elements in the set of (16) increses, nd once this number becomes greter thn, the set becomes mbiguous Furthermore, n increse of ll the intersensor spcings my be considered n even worse ide since n rry with sensor loctions given by, with, might hve mny differences between sensor loctions tht result in mbiguous sets of the form of (16) However, it is importnt to emphsize tht there re some cses where, by incresing the perture nd then using some ugmenttion lgorithm [13], [14], mbiguities my be resolved nd even, in some situtions, my result in n unmbiguous DF system According to Theorem 2, ll subsets of elements from set of the form of (16) re mbiguous sets but not necessrily mbiguous genertor sets In order for such subset to be n mbiguous genertor set, it must hve its first element equl to zero s well s stisfy the third condition of the definition of the mbiguous genertor set Note tht from now on, the vector, with elements in the sensor loctions (in hlf wvelengths), will be ssumed to be ordered in the sense tht Furthermore, the difference between the th nd th sensor loctions, mesured in hlf wvelengths, will be denoted by, nd hence, the perture of the rry will be Exmple 2: Consider n rry with sensor loctions given by The mnifold length of this rry is clculted to be 203016 Furthermore, the only difference between sensor loctions tht stisfies the two conditions of Theorem 2 is the perture In this cse, by evluting (16), the set of rc lengths is obtined, which corresponds to the set of DOA s According to Theorem 2, ny subset of four elements from the bove set will be n mbiguous set, nd the mtrix, which hs columns, nd the mnifold vectors corresponding to such subset will be singulr This mens tht if, for instnce, MUSIC is employed in signl environment of three sources with source directions in ny three elements of the bove set, then five nulls (one for ech element of ) will pper Furthermore, it cn be shown tht no mtrix corresponding to ny three elements of is rnk deficient This mens tht four different mbiguous genertor sets cn be identified from, which re ll the subsets of with their first element zero nd three nonzero elements of
2172 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 8, AUGUST 1998 As expected from Theorem 2, the set, which is of the form of (16), is mbiguous since However, mbiguous sets might lso be defined from Theorem 2 when To see this, consider the set of the form of (16) for the difference, which is If the mtrix, with columns in the corresponding mnifold vectors, is obtined, then, nd hence, the bove set is n mbiguous set with rnk of mbiguity equl to 2 This mens tht when MUSIC is pplied on ny two elements of, three nulls will pper It should lso be noted tht not ll the sets of the form of (16) resulting from difference between two sensor loctions re mbiguous if To see this, consider the difference of the previous rry, which results in set, which is unmbiguous The fct tht cn be mbiguous even if is to be expected from the proof of Theorem 2 In tht proof, it ws stted tht the th nd the th rows of mtrix, with columns in the mnifold vectors corresponding to, re equl If, then the th row nd the th row of will be equl, s will the th nd the th rows As result, cn be rnk deficient even if it hs less thn rows In such cse, the mbiguous set will hve rnk of mbiguity less thn As direct result of the previous discussion, it cn be sid tht the uniform liner rry with intersensor spcing greter thn 1 suffers from trivil mbiguities This wellknown result is rediscovered by observing tht in such cse, the set will contin t lest two elements, nd the mtrix will be rnk deficient since ll its rows will be equl Thus fr, it hs been shown tht Theorem 2 cn be used in order to identify mbiguous sets inherent in liner rrys of ny geometry When this theorem is focused on specific rry geometries, it cn produce some more useful results s the following corollries indicte Corollry 1: All the mbiguous sets tht exist in twoelement rry cn be clculted from (16) The proof cn be found in Appendix C Corollry 2: The set of rc lengths is n mbiguous set for ll three-element symmetric liner rrys, s long s The proof cn be found in Appendix D Corollry 3: Let two rrys of sensors hve common difference between two sensor loctions, which results in set of rc lengths of the form of (16) Then, this set is different for the two rrys, but the corresponding mbiguous sets of DOA s re the sme The proof cn be found in Appendix E Bsed on Theorem 2 in conjunction with the previous discussion, new technique, followed by n indictive exmple, is presented in step-by-step form This technique, for given liner rry with sensor loctions given by, provides mtrix whose rows re ll the mbiguous genertor sets rising from uniform prtitions of the rry mnifold Furthermore, column vector is lso provided, with its th element representing the rnk of mbiguity ssocited with the th mbiguous genertor set (ie, the th row of ) VI PROPOSED TECHNIQUE 1) Clculte the mnifold length, nd then tke the Hdmrd difference between the vector (with elements the sensor loctions) nd itself, ie,, which results in -dimensionl rel vector 2) Crete new vector by eliminting ll the elements of tht re smller thn one Note tht the elements of tht re smller thn one do not stisfy Condition i) of Theorem 2 nd, therefore, cnnot possibly give rise to mbiguous genertor sets Let be the dimension of the new vector, ie, Due to the properties of the Hdmrd difference, elements of will be equl to zero, nd hlf of the remining elements will be negtive Therefore, 3) For ech element of, ie, for ech difference between two sensor loctions, construct the corresponding vector by using (16) Note tht there re different vectors nd tht ech corresponds to different uniform prtition of the rry mnifold 4) Identify ll the vectors tht do not stisfy Condition ii) of Theorem 2 (ie, the number of nonzero elements in the row is smller thn ) nd then eliminte those for which All the remining vectors produce mbiguous genertor sets 5) Ensure ll remining vectors of Step 4 re of the sme length, where is the length of the vector with the mximum number of elements, by ppending zeros where necessry It is obvious tht the set with the mximum number of elements is the set corresponding to the rry perture Then, crete the mtrix with rows the vectors 6) For ech, identify the mbiguous genertor sets bsed on the following rules: Rule ) If the nonzero elements of cnnot be found in other rows, then mbiguous genertor sets cn be produced by the elements of These mbiguous genertor sets re ll the possible subsets of elements of with their first element zero nd nonzero elements of All mbiguous genertor sets constructed in this wy hve rnk of mbiguity Note tht such rows must definitely hve [see Theorem 2(ii)]; otherwise, they would hve been eliminted in Step 4 Rule b) If the nonzero elements of cn be found in other rows, then mbiguous genertor sets with rnk of mbiguity less thn might be obtined This mens tht ll subsets of with their first element 0 nd with length must be considered These subsets re clssified s mbiguous genertor sets if the three conditions of the mbiguous
MANIKAS AND PROUKAKIS: MODELING AND ESTIMATION OF AMBIGUITIES IN LINEAR ARRAYS 2173 genertor set definition re stisfied Furthermore, for ech mbiguous genertor set, rnk of mbiguity is estimted Note tht this step clrifies why, in Step 4, it is incorrect to eliminte two identicl rows of, lthough they will result in the sme mbiguous genertor sets 7) Crete the mtrix, whose rows re ll the different mbiguous genertor sets found in Step 6 If one mbiguous genertor set hs less thn elements, then complete the corresponding row of with zeros In ddition, eliminte duplicte rows of Finlly, form the vector, with elements of the rnk of mbiguity of ech mbiguous genertor set, s estimted in Step 6 The bove technique ws checked using mny simultions, nd n illustrtive exmple is presented below Exmple 3: The steps of the previously described technique for n rry with sensor positions re s follows: 1) The mnifold length of this rry is computed to be The Hdmrd difference between nd itself is The mtrix cn now be formed s 6) In this step, the mbiguous genertor sets rising from different uniform prtitions of the mnifold re clculted The nonzero elements of cn lso be found in, nd hence, is investigted bsed on Rule 6b The mbiguous genertor set definition is initilly checked for the set, which consists of the five first elements of This set is certinly mbiguous, nd the mtrix with columns the corresponding mnifold vectors hs However, ny one of the four subsets of with four elements of involving one zero nd three nonzero elements is lso n mbiguous set The mtrix with columns in the corresponding mnifold vectors hs Therefore, the set stisfies the third condition of the definition of the mbiguous genertor set Similrly, the four subsets re mbiguous genertor sets since ll their subsets with three elements re unmbiguous Hence, the following five mbiguous genertor sets cn be defined from, ll of which hve : 2) Eliminting those entries of tht re smller thn unity results in 3) The seven row vectors re 4) The row vectors tht hve less thn nonzero elements re exmined The number of nonzero elements in is Furthermore,, nd hence, this row is eliminted In ddition, is eliminted since nd Thus, the remining sets re,,,, nd 5) Since is the vector with the mximum number of elements (six in this cse), it mens tht, nd ll the vectors re mde to hve length equl to six The nonzero elements of cnnot be found in ny other row of, nd hence, is investigted bsed on Rule 6 Since the number of nonzero elements in is, we hve, which implies tht only one mbiguous genertor set (which is the sme s the set of Step 3) cn be identified, with Since is the sme s, it follows tht the mbiguous genertor sets obtined from this row re the ones tht were lredy obtined from The nonzero elements of cnnot be found in ny other row of, nd hence, is lso investigted bsed on Rule 6 This time,, nd therefore, mbiguous genertor sets cn be found by tking ll the subsets of elements of (with one zero nd nonzero elements) Ech of these sets hs
2174 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 8, AUGUST 1998 Finlly, is lso investigted bsed on Rule 6 Since, it genertes mbiguous genertor set This set is the sme s set of Step 3 nd hs 7) The mtrix with rows in the mbiguous genertor sets produced in Step 6 is now constructed, s well s the vector with elements in the rnk of mbiguity for the pproprite mbiguous genertor set theorem shows tht if the mnifold of symmetric liner rry hs chrcteristic points, where, then this set is mbiguous This is the strting point for the identifiction of new clss of mbiguous genertor set hving rnk of mbiguity equl to ( ) Theorem 3: If, (with ) re the chrcteristic rc lengths of the mnifold of symmetricl liner rry corresponding to the first winding (if even) or hlf winding (if odd) of the ssocited hyperhelix, then, where The proof cn be found in Appendix F We hve seen tht the Crtn mtrix nd the Frme mtrix lwys stisfy the differentil eqution C (19) which, with the initil condition, hs the solution C (20) Then, in the cse of symmetric liner rry, the chrcteristic points my be obtined (see Appendix G) s the roots of tr C C (21) where In ddition, the number of windings ( odd) or hlf windings ( even) of this hyperhelicl curve cn be estimted using the expression In conclusion, 12 mbiguous genertor sets cn be identified for this rry Five of these hve, wheres the remining hve B Bsed on Nonuniform Prtitions of Hyperhelices In this section, nother clss of mbiguous genertor set existing only in symmetric 2 liner rrys will be identified nd estimted bsed not on uniform but on nonuniform prtition of the rry mnifold This cn be chieved using the concept of chrcteristic points tht, s cn be shown, prtition the mnifold curve into unequl segments nd re defined s follows Definition 4: A point on the rry mnifold, with, is chrcteristic point if nd only if Re (18) where denotes the tngent vector t the mnifold point, ie, The bove definition indictes n importnt property of chrcteristic points, tht is, the tngent t ny chrcteristic point is orthogonl (wide sense orthogonlity since only the rel prt is zero) to the mnifold vector t The following 2 Ech sensor hs symmetric counterprt with respect to the rry reference point, ie, sum(r i )=0; 8i = odd (22) where indictes the length of one winding ( odd) or of one hlf winding ( even) nd is given s the ( )th positive root (see [9, Th 1]) of (21) Thus, sufficient condition for presence of nontrivil mbiguities in the mnifold of symmetric liner rry is ie, Corollry 4: If nd, then there exist berings in the region ( ) such tht the mnifold vectors corresponding to re linerly dependent It is cler from the bove discussion tht if the rry is symmetric, then more rows my be dded to the mtrix, which ws presented in the previous section, ie, more genertor sets cn be found using nonuniform prtition of the rry mnifold bsed on Theorem 3 For instnce, n rry with sensor positions hs the following mbiguous genertor sets mtrix, which is estimted bsed on the uniform
MANIKAS AND PROUKAKIS: MODELING AND ESTIMATION OF AMBIGUITIES IN LINEAR ARRAYS 2175 prtitioning of its mnifold s described in Section VI-A: (23) However, becuse the rry is symmetric, (21) cn be used to estimte dditionl mbiguous genertor sets (bsed on nonuniform prtitioning) Thus, the Crtn mtrix C of the rry is initilly formed, ie, C nd then, the roots of (21), ie, the set of chrcteristic points, re estimted Finlly, the following dditionl mbiguous genertor sets, nd their ssocited rnks of mbiguity, should be dded s extr rows to the mtrix nd vector in (23): is prtitioned, ccording to some specific rules, uniformly or nonuniformly APPENDIX A PROOF OF THEOREM 1 Since is n mbiguous set, the mtrix with columns in the corresponding mnifold vectors, ie, is rnk deficient This mens tht ny submtrix, which hs exctly rows, is singulr Tht is det Consider now the set, where is the -element column vector with ll its elements equl to unity The mnifold vector corresponding to the th element of is Therefore, the mtrix with columns in the mnifold vectors corresponding to the set is of with mbig gen sets The determinnt of ny submtrix of tht hs exctly rows is rnk of mbig (24) In the previous discussion, new clss of mbiguities hs been identified, bsed on the chrcteristic points However, there re lso other clsses of mbiguity bsed on nonuniform prtitions of the mnifold curve For instnce, for the rry, the set of rc lengths is n mbiguous set tht, lthough nonuniform, is not the set of chrcteristic points, ie, does not belong to the proposed clss, nd provides the mbiguous genertor sets mbig gen sets (25) There is no nswer t this moment s to how other nonuniform clsses of mbiguous genertor sets cn be identified VII CONCLUSIONS In this pper, the notion of the mbiguous genertor set, which represents/genertes n infinite number of mbiguous sets of directions, is introduced Then, completely innovtive pproch to clculte mbiguous genertor sets is constructed In prticulr, two generl clsses of mbiguous genertor sets of DOA s re clculted, bsed on whether the mnifold curve dig nd, hence, is singulr This implies tht is rnk deficient, nd therefore, the set is lso n mbiguous set Furthermore, the rnk of mbiguity of nd of is the sme This is becuse the submtrices of with less thn rows re not singulr, which implies tht the submtrices of with less thn rows re lso not singulr APPENDIX B PROOF OF THEOREM 2 Consider liner rry of sensors with loctions If the difference between the th nd th sensor loctions, ie,, stisfies the following two conditions i) ; ii) ; then we hve to prove tht ny subset of elements of the set of rc lengths, where is n mbiguous set
2176 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 8, AUGUST 1998 Let be the mtrix with columns in the mnifold vectors corresponding to the, shown in (B1) t the bottom of the pge Since, the number of columns of is equl to or greter thn Hence, by tking submtrix of with exctly columns, eg, the first columns of, squre submtrix of is obtined whose determinnt is in (B2), shown t the bottom of the pge Consider now the th nd th rows of (with ) By using the property tht for ny, the th row of cn be written s in (B3), shown t the bottom of the next pge, ie,, nd therefore, It is esy to see tht the fct tht ws chosen to consist of the first columns of nd not ny columns is not restrictive in the lest since in every cse, the th nd the th rows of ny submtrix of will be equl Since ll submtrices of re singulr, it results tht is rnk nd, by using, (B1) r (B2)
MANIKAS AND PROUKAKIS: MODELING AND ESTIMATION OF AMBIGUITIES IN LINEAR ARRAYS 2177 deficient, nd therefore, ny subset of elements is n mbiguous set with exctly (16) As result, necessry nd sufficient condition for the presence of mbiguities in two-element liner rry is APPENDIX C PROOF OF COROLLARY 1 Consider the two-element rry Let the set be n mbiguous set of rc lengths The mtrix with columns in the mnifold vectors corresponding to is APPENDIX D PROOF OF COROLLARY 2 For ll three-element symmetric liner rrys with sensor positions given by, the mnifold length is equl to The set of rc lengths of the form of (16) for the difference between the first nd the third sensors, ie, for, is Since it is ssumed tht is n mbiguous set, it results tht is singulr nd therefore nd therefore, ll three-element symmetric liner rrys suffer from mbiguities of the form of the set given bove, s long s with since APPENDIX E PROOF OF COROLLARY 3 Consider two different rrys, ech hving sensors nd sensor loctions nd, respectively Let there be common difference between two sensor loctions, which, for both rrys, stisfies the conditions of Theorem 2 In this cse, the two sets of rc lengths constructed from (16) will be First rry: with Therefore, the only mbiguous sets of rc lengths tht cn possibly exist for two-element rry re of the form of Second rry: (B3)
2178 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 8, AUGUST 1998 with Since, the bove two sets re mbiguous Using (9), the corresponding set of DOA s for the first rry is However, the first column of the mtrix on the right-hnd side of the bove eqution is the vector which, in conjunction with the definition of the chrcteristic points (18), is the ll-zero vector tht implies tht wheres, for the second rry APPENDIX G RELATION BETWEEN (18) AND (21) For ny liner rry, we hve Re tr Re tr Re tr Re The two sets re obviously identicl APPENDIX F PROOF OF THEOREM 3 By using (14), the determinnt of the mtrix becomes tr (26) However, t the chrcteristic points Re Re tr [using (26)] The bove eqution, in the cse of symmetric rrys (using the fifth property of Appendix H), becomes tr tr which implies tht the chrcteristic points re given s the roots of the function tr C C where, in the lst equlity, the following properties hve been used: the symmetry of with respect to indices stisfying mod ; the ntisymmetry of with respect to indices stisfying mod ; for odd Furthermore, using the fourth property of Appendix H, it follows tht which, bsed on (11), becomes APPENDIX H ASUMMARY OF PROPERTIES OF SYMMETRIC LINEAR ARRAY PARAMETERS 1) odd 2) 3) 4) 5) tr tr 6) REFERENCES [1] R O Schmidt, A signl subspce pproch to multiple emitter loction nd spectrl estimtion, PhD disserttion, Stnford Univ, Stnford, CA, 1981 [2] B Bygün nd Y Tnik, Performnce nlysis of the music lgorithm in direction finding systems, in Proc IEEE ICASSP, 1989, vol IV, pp 2298 2301 [3] K C Tn, G L Oh, nd M H Er, A study of the uniqueness of steering vectors in rry processing, Signl Process, vol 34, pp 245 256, 1993 [4] J T H Lo nd S L Mrple, Observbility conditions for multiple signl direction finding nd rry sensor locliztion, IEEE Trns Signl Processing, vol 40, pp 2641 2650, Nov 1992
MANIKAS AND PROUKAKIS: MODELING AND ESTIMATION OF AMBIGUITIES IN LINEAR ARRAYS 2179 [5] K C Tn nd Z Goh, A detiled derivtion of rrys free of higher rnk mbiguities, IEEE Trns Signl Processing, vol 44, pp 351 359, Feb 1996 [6] K C Tn, S S Goh, nd E C Tn, A study of the rnk-mbiguity issues in direction-of-rrivl estimtion, IEEE Trns Signl Processing, vol 44, pp 880 887, Apr 1996 [7] A Mniks, C Proukkis, nd V Lefkditis, Investigtive study of plnr rry mbiguities bsed on hyperhelicl prmetriztion, Imperil Coll Intern Rep AM-97-1, London, UK, July 1997 [8] M Lipschutz, Differentil Geometry New York: McGrw-Hill, 1969 [9] H Guggenheimer, Differentil Geometry New York: McGrw-Hill, 1969 [10] I Dcos nd A Mniks, Estimting the mnifold prmeters of onedimensionl rrys of sensors, J Frnklin Inst, Eng Appl Mth, vol 332B, no 3, pp 307 332, July 1995 [11] A Mniks, H R Krimi, nd I Dcos, Study of the detection nd resolution cpbilities of one-dimensionl rry of sensors by using differentil geometry, in Proc Inst Elect Eng Rdr, Sonr, Nvigtion, vol 141, no 2, pp 83 92, 1994 [12] C Proukkis nd A Mniks, Study of mbiguities of liner rrys, in Proc IEEE ICASSP, 1994, vol IV, pp 549 552 [13] Y I Abrmovich, N K Spencer, nd A Y Gorokhov, Generlized ugmenttion pproch for rbitrry liner ntenn rrys, in Proc IEEE ICASSP, 1997, vol 5, pp 3757 3760 [14] Y I Abrmovich, A Y Gorokhov, nd N K Spencer, Asymptotic efficiency of mnifold mbiguity resolution for DOA estimtion in nonuniform liner ntenn rrys, in Proc IEEE SPAWC-97, Pris, Frnce, 1997, pp 173 176 Christos Proukkis received the Diplom in electricl engineering from the Ntionl Technicl University of Athens, Athens, Greece, in 1990 He received the MSc degree in communictions nd signl processing in 1991 nd the PhD nd DIC degrees in electricl engineering in 1997, ll from Imperil College of Science Technology nd Medicine, University of London, London, UK His reserch interests re in the res of sensor rry signl processing nd super-resolution sptiotemporl spectrl estimtion s well s the generl re of digitl communictions Dr Proukkis ws wrded the Frngoulis Memoril studentship in 1992 nd ws scholr of the Onssis Foundtion from 1993 to 1996 He is member of the Technicl Chmber of Greece Athnssios Mniks (M 88) received PhD nd DIC degrees from Imperil College, University of London, London, UK, in 1988 nd ws ppointed Lecturer t Imperil College the sme yer He is now Reder in digitl communictions in the Deprtment of Electricl nd Electronic Engineering, Imperil College of Science, Technology, nd Medicine, London He hs published extensive set of journl nd conference ppers relting to his reserch interests, which re in the generl re of communiction nd signl processing, where he hs developed wide nd deep interest in the topic of super-resolution rry processing nd the ppliction of rry theory to communiction systems He hs been involved in rry mnifold investigtions using differentil geometry, super-resolution bemformers, direction finding, ner nd fr field rry processing, performnce bounds in rry systems, rry uncertinties nd mbiguities, rry design, rry signl bnormlities nd qulity fctors, higher order signl subspce techniques, H-inf-type robust rry processing, robust rry receivers, nd integrted wireless rry communiction networks bsed on CDMA, GSM-like, etc, pltforms Furthermore, he is the Orgnizer of the Advnced Communictions nd Signl Processing Lbortory t Imperil College nd, over the lst eight yers, hs held number of importnt reserch consultncies for the EU, industry, nd government orgniztions Dr Mniks is member of IEICE, corporte member of IEE, nd Chrtered Electricl Engineer