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1254 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005

On the Virtual Array Concept forHigher Order Array Processing

Pascal Chevalier, Laurent Albera, Anne Ferréol, and Pierre Comon, Senior Member, IEEE

Abstract—For about two decades, many fourth order (FO) arrayprocessing methods have been developed for both direction findingand blind identification of non-Gaussian signals. One of the maininterests in using FO cumulants only instead of second-order (SO)ones in array processing applications relies on the increase of boththe effective aperture and the number of sensors of the consid-ered array, which eventually introduces the FO Vitual Array con-cept presented elsewhere and allows, in particular, a better reso-lution and the processing of more sources than sensors. To still in-crease the resolution and the number of sources to be processedfrom a given array of sensors, new families of blind identification,source separation, and direction finding methods, at an order =2 ( 2) only, have been developed recently. In this context, thepurpose of this paper is to provide some important insights into themechanisms and, more particularly, to both the resolution and themaximal processing capacity, of numerous 2 th order array pro-cessing methods, whose previous methods are part of, by extendingthe Virtual Array concept to an arbitrary even order for several ar-rangements of the data statistics and for arrays with space, angularand/or polarization diversity.

Index Terms—Blind source identification, higher order, HO di-rection finding, identifiability, space, angular, and polarization di-versities, 2 -MUSIC, virtual array.

I. INTRODUCTION

FOR about two decades, many fourth order (FO) array pro-cessing methods have been developed for both direction

finding [4], [6], [9], [21], [23] and blind identification [1], [5],[10], [12], [14], [17] of non-Gaussian signals. One of the maininterests in using FO cumulants only instead of second-order(SO) ones in array processing applications relies on the increaseof both the effective aperture and the number of sensors of theconsidered array, which eventually introduces the FO VirtualArray (VA) concept presented in [7], [15], and [16], allowing, inparticular, both the processing of more sources than sensors andan increase in the resolution power of array processing methods.

In order to still increase both the resolution power ofarray processing methods and the number of sources to beprocessed from a given array of sensors, new families ofblind identification, source separation, and direction finding

Manuscript received August 1, 2003; revised May 4, 2004. The associate ed-itor coordinating the review of this manuscript and approving it for publicationwas Prof. Abdelhak M. Zoubir.

P. Chevalier and A. Ferréol are with Thalès-Communications,EDS/SPM/SBP, 92704 Colombes Cédex, France (e-mail: [email protected]; [email protected]).

L. Albera is with the Laboratoire Traitement du Signal et de l’Image (LTSI),Rennes University (INSERM U642), 35042 Rennes Cédex, France (e-mail: [email protected]).

P. Comon is with the I3S, Algorithmes-Euclide-B, F-06903 Sophia-AntipolisCedex, France (E-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2005.843703

methods, exploiting the data statistics at an arbitrary even orderonly, have been developed recently in [3]

and [8], respectively. More precisely, [8] mainly extends thewell-known high resolution direction finding method calledMUSIC [24] to an arbitrary even order , giving rise to theso-called -MUSIC methods, whose interests, for ,are also shown in [8]. In particular, for operational contextscharacterized by a high source density, such as airborne surveil-lance over urban areas, the use of Higher Order (HO) MUSICmethods for direction finding allows us to reduce or even tominimize the number of sensors of the array and, thus, thenumber of reception chains, which finally drastically reducesthe overall cost. Besides, it is shown in [8] that, despite of theirhigher variance and contrary to some generally accepted ideas,

-MUSIC methods with may offer better performancesthan 2-MUSIC or 4-MUSIC methods when some resolution isrequired, i.e., in the presence of several sources, when the latterare poorly angularly separated or in the presence of modelingerrors inherent in operational contexts. In the same spirit, toprocess both over and underdetermined mixtures of statisticallyindependent non-Gaussian sources, [3] mainly extends therecently proposed FO blind source identification method calledIndependent Component Analysis using Redundancies (ICAR)[1] in the quadricovariance matrix to an arbitrary even-order

, giving rise to the so-called -Blind Identification ofOvercomplete MixturEs of sources (BIOME) methods, whoseinterests for are shown in [3]. Note that the -BIOMEmethod gives rise, for , to the sixth order method calledBlind Identification of mixtures using Redundancies in thedaTa Hexacovariance matrix (BIRTH) presented recently in[2]. In particular, it is shown in [2] and [3] that -BIOMEmethods, for , outperform all the existing Blind SourceIdentification (BSI) methods that are actually available, interms of processing power of underdetermined mixtures ofarbitrary statistically independent non-Gaussian sources.

Contrary to papers [3] and [8], the present paper does notfocus on particular HO array processing methods for particularapplications but rather aims at providing some important in-sights into the mechanisms of numerous HO methods and, thus,some explanations about their interests, through the extension ofthe VA concept introduced in [7], [15], and [16] for the FO arrayprocessing problems, to an arbitrary even-orderand for several arrangements of the th order data statistics forarrays with space, angular, and/or polarization diversity. ThisHO VA concept allows us, in particular, to show off both theincreasing resolution and the increasing processing capacity of

th order array processing methods as increases. It allowsus to solve not only the identifiability problem of HO methods

1053-587X/$20.00 © 2005 IEEE

CHEVALIER et al.: VIRTUAL ARRAY CONCEPT FOR HIGHER ORDER ARRAY PROCESSING 1255

presented in [3] and [8] in terms of maximal number of sourcesthat can be processed by these methods from an array of sen-sors but, in addition, that of all the array processing methodsexploiting the algebraic structure of the th order datastatistics matrix only for particular arrangements of the latter.As a consequence of this result, the HO VA concept shows offthe impact of the th order data statistics arrangement on the

th order array processing method performances and, thus, theexistence of an optimal arrangement of these statistics. This re-sult is completely unknown by most of the researchers. Finally,one may think that the HO VA concept will spawn much prac-tical research in array processing and will also be consideredas a powerful tool for performance evaluation of HO array pro-cessing methods.

After an introduction of some notations, hypotheses, and datastatistics in Section II, the VA concept is extended to even HOstatistics in Section III, where the questions of both the op-timal arrangement of the latter and the resolution of the VAis addressed. Some properties of the HO VA for arrays withspace, angular, and/or polarization diversity are then presentedin Section IV, where explicit upper bounds, that are reached formost array geometries, on the maximal number of independentnon-Gaussian sources that can be processed by a th ordermethod exploiting particular arrangements of the th order datastatistics, are computed for . Note that the restriction tovalues of lower than or equal to 8 is not very restrictive sinceit corresponds to order of statistics that have the highest prob-ability to be used for future applications. The results of Sec-tions III and IV are then illustrated in Section V through thepresentation of HO VA examples for both the Uniform LinearArray (ULA) and the Uniform Circular Arrays (UCA). Somepractical situations for which the HO VA concept leads to betterperformance than SO or FO ones are pointed out and illustratedin Section VI through a direction finding application. Finally,Section VII concludes this paper.

II. HYPOTHESES, NOTATIONS, AND STATISTICS OF THE DATA

A. Hypotheses and Notations

We consider an array of narrowband (NB) sensors, and wecall the vector of complex amplitudes of the signals at theoutput of these sensors. Each sensor is assumed to receive thecontribution of zero-mean stationary and stastistically inde-pendent NB sources corrupted by a noise. Under these assump-tions, the observation vector can approximately be written asfollows:

(1)

Fig. 1. Incoming signal in three dimensions.

where is the noise vector that is assumed zero-mean,is the vector whose components are the complex ampli-tudes of the sources, and are the azimuth and the elevationangles of source (Fig. 1), and is the matrix of thesource steering vectors , which contains, in particular,the information about the direction of arrival of the sources. Inparticular, in the absence of coupling between sensors, compo-nent of vector , which is denoted as , can bewritten, in the general case of an array with space, angular, andpolarization diversity, as (2), shown at the bottom of the page,[11] where is the wavelength, are the coordinatesof sensor of the array, and is a complex numbercorresponding to the response of sensor to a unit electric fieldcoming from the direction and having the state of polar-ization (characterized by two angles in the wave plane) [11].Let us recall that an array of sensors has space diversity if thesensors do not all have the same phase center. The array has an-gular and/or polarization diversity if the sensors do not have allthe same radiating pattern and/or the same polarization, respec-tively.

B. Statistics of the Data

1) Presentation: The th order array processingmethods currently available exploit the information containedin the th order circular covariance matrix,

, whose entries are the th order circular cumulantsof the data, Cum

, where corresponds to thecomplex conjugation. However, the latter entries can be ar-ranged in the matrix in different ways, and it is shown inthe next section that the way these entries are arranged in the

matrix determines in particular the maximal processingpower of the th order methods exploiting the algebraicstructure of , such as the -MUSIC [8] or the -BIOME[3] methods. This result is new and seems to be completelyunknown by most of the researchers.

In order to prove this important result in the next section,let us introduce an arbitrary integer such that ,

(2)


and let us arrange the -upletof indices into two -uplets indexedby and defined by and

, respectively. As the indicesvary from 1 to , the two latter -uplets take

values. Numbering, in a natural way, the values of eachof two latter -uplets by the integers and , respectively,such that , , we obtain

(3a)

(3b)

Using the permutation invariance property of the cumulants, wededuce that CumCum

and assuming that the latterquantity is the element of the matrix, thus noted

, it is easy to verify, from the Kronecker product defini-tion, the hypotheses of Section II-A and under a Gaussian noiseassumption that the matrix can be writtenas

(4)

whereCum ,

with , is the th order circularautocumulant of , corresponds to the conjugatetransposition, is the mean power of the noise per sensor,is the spatial coherence matrix of the noise such thatTr , Tr[.] means Trace, is the Kronecker symbol,

is the Kronecker product, and is the vector

defined by with a number ofKronecker product equal to .

In particular, for and , thematrix corresponds to the well-known data covariance matrix(since the observations are zero-mean) defined by

(5)

For and , the matrix correspondsto the classical expression of the data quadricovariance matrix

used in [7] and [15] and in most of the papers dealing with FOarray processing problems and is defined by

(6)

whereas for and , the matrixcorresponds to an alternative expression of the data quadrico-variance matrix that is not often used and is defined by

(7)

2) Estimation: In situations of practical interest, the thorder statistics of the data Cum

are not known a priori and have to be estimated

from samples of data , , whereis the sample period.

For zero-mean stationary observations, using the ergodicityproperty, an empirical estimator of Cum

that is asymptotically unbiased andconsistent may be built from the well-known Leonov–Shiryaevformula [22], giving the expression of a th order cumulantof as a function of its th order moments ,by replacing in the latter all the moments by their empiricalestimate. More precisely, the Leonov–Shiryaev formula isgiven by

Cum

(8)

where describes all the partitions in setsof , with the conven-tion , and and an empirical estimateof (8) is obtained by replacing in (8) in all the moments

by their empiricalestimate, which is given by

(9)

Explicit expressions of (8) for with are givenin Appendix A.

However, in radiocommunications contexts, most of thesources are no longer stationary but become cyclostationary


(digital modulations). For zero-mean cyclostationary obser-vations, the statistical matrix defined by (4) becomes timedependent, noted , and the theory developed in thepaper can be extended without any difficulties by consideringthat is, in this case, the temporal mean, ,over an infinite interval duration, of the instantaneous statistics,

. In these conditions, using a cyclo-ergodicity prop-erty, the matrix has to be estimated from the sampleddata by a non empirical estimator such as that presented in [18]for . Note finally that this extension can also be appliedto non zero-mean cyclostationary sources, such as some nonlinearly digitally modulated sources [20], provided that a nonempirical statistic estimator, such as that presented in [20] for

and in [19] for , is used.

C. Related th order Array Processing Problems

A first family of th order array processing methods thatare concerned with the theory developed in the next sectionscorresponds to the family of th order Blind Identificationmethods, which aim at blindly identifying the steering vectorsof the sources from the exploitation ofthe algebraic structure of an estimate of the matrixfor a particular choice of . Such methods are described in [2]and [3]. A second family of methods concerned with the resultsof the paper corresponds to the th order subspace-baseddirection finding methods such as the -MUSIC method,presented in [8], which aims at estimating the angles of arrivalof the sources from the exploitation of thealgebraic structure of an estimate of the matrix for aparticular choice of .

III. HIGHER ORDER VIRTUAL ARRAY CONCEPT

A. General Presentation

The VA concept has been introduced in [7], [15], and [16]for the classical FO array processing problem exploiting (6)only. In this section, we extend this concept to an arbitraryeven order , for an arbitrary arrangement

, of the data th order circular cumulantsCumin the matrix and for a general array with space, angular,and polarization diversities. This HO VA concept is presentedin this section in the case of statistically independentnon-Gaussian sources.

Assuming no noise, we note that the matrices and, which are defined by (4) and (5), respectively, have the

same algebraic structure, where the auto-cumulant andthe vector play, for ,the rule played for by the power and the steeringvector , respectively. Thus, for the th order arrayprocessing methods exploiting (4), the vector

can be considered as thevirtual steering vector of the source for the true array ofsensors with coordinates and amplitude pattern

, . The components of the vectorcorrespond to the quanti-

ties, where

is the component of vector . Using(2) in the latter components and numbering, in a natural way,the values of the -uplet byassociating with the latter the integer defined by

(10)

we find that the component of the vector, noted ,

takes the form (11), shown at the bottom of the page. Com-paring (11) to (2), we deduce that the vector

can also be considered as the true steeringvector of the source for the VA of Virtual Sensors (VSs)with coordinates and com-plex amplitude patterns , for

, which are given by

(12)

(13)

(11)


which introduces in a very simple, direct, and short way theVA concept for the th order array processing problem for thearrangement and whatever the kind of diversity. Notethat (13) shows that the complex amplitude response of a VSfor given direction of arrival and polarization corresponds to aproduct of complex amplitude responses of true sensors and

conjugate ones for the considered direction of arrivaland polarization.

Thus, as a summary, we can consider that the th orderarray processing problem of statistically independent NBnon-Gaussian sources from a given array of sensors withcoordinates and complex amplitude patterns

, is, for the arrangement ,similar to an SO array processing problem for which thesestatistically independent NB sources impinge, with the virtualpowers , on a VA of VS havingthe coordinates and the com-plex amplitude patterns , for

, which are defined by (12) and (13) respectively.Thus, HO array processing may be used to replace sensorsand hardware and, thus, to decrease the overall cost of a givensystem.

From this interpretation based on the HO VA concept, wenaturally deduce that the 3-dB beamwidth of this VA controlsthe resolution power of th order array processing method fora finite observation duration and the considered arrangement,whereas the number of different sensors of this VA controlsthe maximal number of sources that can be processed by suchmethods for this arrangement. More precisely, as some of the

VS may coïncide, we note as the number of differentVSs of the VA associated with the th order array processingproblem for the arrangement . Then, the maximumnumber of independent sources that can be processed by a thorder BSI method exploiting the algebraic structure ofis , whereas the th order direction finding methods ex-ploiting the algebraic structure of , such as the familyof -MUSIC methods [8], are able to process up tonon-Gaussian sources.

Another important result shown by (12) and (13) is that, fora given array of sensors, the associated th orderVA depends on the parameter and thus on the arrangement ofthe th order circular cumulants of the data in the ma-trix. This new result not only shows off the importance of thechosen arrangement of the considered data th order cumu-lants on the processing capacity of the methods exploiting thealgebraic structure of but also raises the problem of theoptimal arrangement of these cumulants for a given even order.This question is addressed in the next section.

Finally, note that (4) holds only for sources that are NBfor the associated VA, i.e., for sources such that the vector

does not depend on the fre-quency parameter within the reception bandwidth, i.e., forsource in the reception bandwidth such that

(14)

where is the propagation velocity, is the aperture of theVA for the considered parameters and , is wave vector forthe source , and is the vector whose norm isand whose direction is the line formed by the two most spacedVSs and . As increases with , the accepted re-ception bandwidth ensuring the NB assumption for the HO VAdecreases with . In particular, for HF or GSM links, the nar-rowband assumption for the HO VA is generally verified up to

or 10, i.e., up to a statistical order equal to 16,18, or 20 from a classically used array of sensors for these ap-plications [13].

B. Optimal Arrangement

For a given value of and a given array of sensors,we define in this paper the optimal arrangement , whichis denoted , as the one that maximizes the numberof different VS of the associated VA, since the processingpower of a th order method exploiting the algebraic structureof is directly related to the number of different VS ofthe associated VA.

To get more insight into , let us analyze(12) and (13). These expressions show that the -up-lets and

, where andare arbitrary permutations of

and , respectively, give riseto the same VS (same coordinates and same radiation pat-tern) of the VA associated with , which are definedby (12) and (13). The number of permutations of a given setof indices depends on the number of indices with differentvalues in the set. For this reason, let us classify all the -uplets

in familiessuch that corresponds to the set of -uplets

with different elements . For example, we have (15) and(16), shown at the bottom of the page. For the general case ofan arbitrary array of sensors, both the number of -upletsof and the number of different VSs of the VA associatedwith for an arbitrary arrangement areproportional to . Indeed, among the differentelements of , once the value of one of them is chosenamong possibilities, there are still possibilities forthe second one and then possibilities for the third one,and so on, and, finally, possibilities for the th one,which finally corresponds to possible solutionsfor the different elements. Then, for each of the latter solutions,

such that for and (15)

such that for and (16)


the value of elements have to be chosen among theconsidered different elements, finally giving rise to a numberof -uplets of proportional to . This quantityis equal to zero for but becomes a polynomial functionof degree with respect to variable for . Thus, asbecomes large, provided that , the number of differentVS of the VA for a given arrangement , is mainlydominated by the number of different VSs associated withfor this arrangement . The number of -uplets of isexactly equal to , whereas the number of differentVSs associated with for the arrangement is, forthe general case of an arbitrary array of sensors with noparticular symmetries, equal to

(17)

In fact, when all the are different, the number of permuta-tions andof and are equal to and ,respectively.

As a summary, in the general case of an arbitrary array ofsensors with no particular symmetries, for a given value of

and for large values of , the optimal arrange-ment is such that maximizes definedby (17) and, thus, minimizes the quantity with respectto . We deduce from this result that the arrange-ments and give rise to thesame number of VS (in fact, the first arrangement is the con-jugate of the other, whatever the values of and ). It is thensufficient to limit the analysis to if is even and to

if is odd. We easily verify that

for if is even (18)

for if is odd (19)

which proves that if is even andif is odd. In other words, is, in all cases, the integer thatminimizes . It generates steering vectors

for which the number of conjugate vectors isthe least different from the number of nonconjugate vectors. Inparticular, for , it corresponds to (6). We verify in Sec-tion IV for , 3, and 4 that this result, which is shown forlarge values of , remains valid, whatever the value of .

C. Virtual Array Resolution

To get more insights into the gain in resolution obtainedwith HO VA, let us compute the spatial correlation coefficientof two sources, with directions and ,respectively, for the VA associated with statistical order

and arrangement indexed by . This coefficient, whichis noted such that , isdefined by the normalized inner product of the steeringvectors and

and can bewritten as

(20)

For an array with space diversity only, this coefficient is propor-tional to the value, for the direction , of the complex ampli-tude pattern of the conventional beamforming in the direction

from the considered VA. It is shown in Appendix B that thiscoefficient (20) can be written as

(21)which implys that

(22)

Expression (22) shows that despite the fact thatis a function of and , its modulus does not depend on butonly on and on the normalized amplitude patternof the considered array of sensors for the pointing direction

. Moreover, as , we deduce from (22)that is a decreasing function of , which proves theincreasing resolution of the HO VA as increases. In particular,if we note dB, the 3-dB beamwidth of the th order VAassociated with a given array of sensors, we find from (22)that dB can be easily deduced from the normalized amplitude

pattern of the latter and is such that for

dB, i.e., such that , 0.794, and 0.84for , 3 and 4, respectively. As increases, this generates

dB values corresponding to a decreasing fraction of the 3-dBbeamwidth dB of the considered array of sensors, and wewill verify in Section V that dB dB, dB, and

dB for , 3 and 4, respectively. Finally, (22) provesthat rank-1 ambiguities (or grating lobes [11]) of the true and VAcoincide, regardless of the values of and since the directions

giving rise to are exactly the ones thatgive rise to . A consequence of this result isthat a necessary and sufficient condition to obtain VA withoutany rank-1 ambiguities is that the considered array of sensorshave no rank-1 ambiguities.

IV. PROPERTIES OF HIGHER ORDER VIRTUAL ARRAYS

A. Case of an Array With Space, Angular, and PolarizationDiversity

For an array with space, angular, and polarization diversities,the component of vector and the component of

are given by (2) and (11), re-spectively, which shows that the th order VA asso-ciated with such an array is also an array with space, angular,


TABLE ICOORDINATES, COMPLEX RESPONSES AND MULTIPLICITY ORDER OF VS FOR

SEVERAL VALUES OF l, FOR q = 2 AND FOR ARRAYS WITH SPACE,ANGULAR AND POLARIZATION DIVERSITIES

and polarization diversities, regardless of the arrangement of theth order circular cumulants of the data in the matrix.Ideally, it would have been interesting to obtain a general ex-

pression of the number of different VSs of the th orderVA for the arrangement and for arbitrary array geome-tries. However, it does not seem possible to obtain such a re-sult easily since the computation of the number of VSs that de-generate in a same one is very specific of the choice of andand of the array geometry. For this reason, we limit our subse-quent analysis, for arbitrary array geometries, to some values of

, which extends the results of [7] up to the eighthorder for arbitrary arrangements of the data cumulants, despitethe tedious character of the computations. Moreover, this anal-ysis is not so much restrictive since the considered th orderdata statistics correspond in fact to the statisticsthat have the most probability to be used for future applications.Note that the general analysis for arbitrary values of and ispossible for ULA and is presented in Section V.

To simplify the analysis, for each sensor , , wenote as its complex response ,

as the triplet of its coordinates, , and we define

and. Moreover, for a given value of , we define the order of

multiplicity of a given VS of the considered th order VAby the number of -uplets , giving rise to thisVS. When the order of multiplicity of a given VS is greaterthan 1, this VS can be considered to be weighted in amplitudeby a factor corresponding to the order of multiplicity, and theassociated VA then becomes an amplitude tappered array.

The coordinates, the complex responses, and the order ofmultiplicity of the VS of HO VA, which are deduced from (12)and (13), are presented in Tables I– III for , 3 and 4, respec-tively, and for several values of the parameter . In these tables,the integers take all the values between 1 and (for ) but under the constraint that if fora given line of the tables. A VS is completely characterized bya line of a table for given values of the .

The results of Tables I–III show that for arrays with sensorshaving different responses, the VA associated with the parame-ters is amplitude tappered for , (3, 3), (3, 2),(4, 4), (4, 3) and (4, 2), whereas it is not for . Inthis latter case, the order of multiplicity of each VS is 1. Then,the number of different VS of the associated VA may be max-imum for and equal to . It is, in particular, the case

TABLE IICOORDINATES, COMPLEX RESPONSES AND MULTIPLICITY ORDER OF VS FOR


TABLE IIICOORDINATES, COMPLEX RESPONSES AND MULTIPLICITY ORDER OF VS FOR


if the responses of all the VSs are different. However, for arbi-trary values of and , the maximum number of VSs of the as-sociated VA, noted , is generally strictly lower than


TABLE IVN [2q; l] AS A FUNCTION OF N FOR SEVERAL VALUES OF q AND l AND FOR

ARRAYS WITH SPACE, ANGULAR AND POLARIZATION DIVERSITIES

due to the amplitude tappering of the VA. Table IV showsprecisely, for arrays with different sensors, the expression of

, computed from the results from Tables I–III, as afunction of for and several values of . Notethat has already been obtained in [7]. Note also that

corresponds to in most cases of sensors havingdifferent responses. We verify in Table IV that for a given valueof and the considered values of , whatever thevalue of , small or large, is a decreasing functionof , which confirms the optimality of the arrangementfor the integer that minimizes , as discussed in Sec-tion III-B. Note, in addition, for a given value of , and foroptimal arrangements, the increasing values of asincreases.

In order to quantify the results of Table IV, Table V summa-rizes the maximal number of different VS of theassociated VA for several values of , , and . As and in-crease, note the increasing value of the loss in the processingpower associated with the use of a suboptimal arrangement in-stead of the optimal one. For a given value of , note the in-creasing value of as increases for optimal arrange-ments of the cumulants, whereas note the possible decreasingvalue of as increases when the arrangement movesfrom optimality to suboptimality (for example,

for ).

B. Case of an Array With Angular and Polarization DiversityOnly

For an array with angular and polarization diversities only, allthe sensors of the array have the same phase centerbut have different complex responses , .Such an array is usually referred to as an array with colocalizedsensors having different responses in angle and polarization. Forsuch an array, (12) shows that for given values of and , theVSs of the associated VA have all the same coordinates givenby , whereas their complex response is given by (13).This shows that the th order VA associated with anarray with angular and polarization diversities only is also an

array with angular and polarization diversities only, whateverthe arrangement of the th order circular cumulants of the datain the matrix.

The complex responses of the colocalized VS of the thorder VA for the arrangements are all presented in Ta-bles I–III for . In particular, all the upper bounds

presented in Tables I–III for an array with space,angular, and polarization diversities remain valid for an arrayof colocalized sensors with angular and polarization diversitiesonly. This shows that for sensors having different complex re-sponses, the geometry of the array does not generally play animportant role in the maximal power capacity of the th orderarray processing methods exploiting the algebraic structure of

in terms of number of sources to be processed.

C. Case of an Array With Space Diversity Only

Let us consider in this section the particular case of an arraywith space diversity only. In this case, all the sensors of thearray are identical, and the complex amplitude patterns of thelatter , may be chosen to be equal toone. Under these assumptions, we deduce from (13) that for agiven value of , whatever the-uplet and whatever the arrangement index .

This shows that the th order VA associated with anarray with space diversity only is also an array with space diver-sity only, whatever the arrangement of the th order circularcumulants of the data in the matrix.

For such an array, the th order VA are presented hereafterfor , which extends the results of [7] up to the eighthorder for arbitrary arrangements of the data cumulants. Moreprecisely, for arrays with space diversity only, the coordinatesand the order of multiplicity of the VSs of the HO VA, deducedfrom Tables I–III, are presented in Tables VI–VIII for ,3, and 4, respectively, and for several values of the parameter .Again, in these tables, the integers take all the values between1 and ( for ) but under the constraintthat if for a given line of the tables. A VS iscompletely characterized by a line of a table for given values ofthe .

The results of Tables VI–VIII show that for arrays with iden-tical sensors, the VA associated with the parameters is al-ways amplitude tappered, whatever the values of and , whichimplys in particular that . Table IXshows precisely, for arrays with identical sensors, the expres-sion of computed from results of Tables VI–VIII asa function of for and several values of . Notethat has already been obtained in [7]. Note also that

corresponds to in most cases of array geome-tries with no particular symmetries. We verify in Table IX thatfor a given value of and the considered valuesof , whatever the value of , small or large, is adecreasing function of , which confirms the optimality of thearrangement for the integer that minimizes , asdiscussed in Section III-B. Note, also, for a given value of andfor optimal arrangements, the increasing values ofas increases. A comparison of Tables IV and IX shows thatwhatever the values of and , can only remainconstant or increase when an array with space diversity only is


TABLE VN [2q; l] FOR SEVERAL VALUES OF N , q, AND l AND FOR ARRAYS WITH SPACE, ANGULAR, AND POLARIZATION DIVERSITIES

TABLE VICOORDINATES AND MULTIPLICITY ORDER OF VSS FOR SEVERAL VALUES OF l

FOR q = 2 AND FOR ARRAYS WITH SPACE DIVERSITY ONLY

TABLE VIICOORDINATES AND MULTIPLICITY ORDER OF VSS FOR SEVERAL VALUES OF l,

FOR q = 3 AND FOR ARRAYS WITH SPACE DIVERSITY ONLY

replaced by an array with space, angular, and polarization diver-sities.

In order to quantify the results of Table IX, Table X summa-rizes the maximal number of different VSs of theassociated VA for several values of , , and . Other results canbe found in Table XII for odd and higher values of . Again,the value of the loss in the processing power associated with theuse of a suboptimal arrangement also increases as and in-crease. For a given value of , we verify the increasing value

TABLE VIIICOORDINATES AND MULTIPLICITY ORDER OF VSS FOR SEVERAL VALUES OF l,

FOR q = 4, AND FOR ARRAYS WITH SPACE DIVERSITY ONLY

of as increases for optimal arrangements of thecumulants.

V. VA EXAMPLES

In this section, the th order VA associated with particulararrays of sensors is described in order to illustrate the resultsobtained so far.


TABLE IXN [2q; l] AS A FUNCTION OF N FOR SEVERAL VALUES OF q AND l AND FOR

ARRAYS WITH SPACE DIVERSITY ONLY

A. Linear Array of Identical Sensors

For a linear array, it is always possible to choose a coordinatesystem in which the sensor has the coordinates ( , 0, 0),

. As a consequence, the VSs of the th order VAfor the arrangement are, from (12), at coordinates

(23)for and . This shows that the th orderVA is also a linear array whatever the arrangement .

For a ULA, it is always possible to choose a coordinatesystem such that , where is the interelement spacing,and the VA is the linear array composed of the sensors whosefirst coordinate is given by

(24)

for and . This shows that the th orderVA is also a ULA with the same interelement spacing, whateverthe arrangement . Moreover, for given values of , , and

, the minimum and maximum values of (24), which are notedand , respectively, are given by

(25)

(26)

and the number of different VSs of the associated VA iseasily deduced from (25) and (26) and is given by

(27)

This is independent of and means that for given values of and, the number of VSs is independent of the chosen arrangement

. In other words, in terms of processing power, for agiven value of and due to the symmetries of the array, all thearrangements are equivalent for a ULA. Besides, wededuce from (24) that

(28)

(29)

which is enough to understand that for given values of and ,the th order VA associated with is just a translationof of the VA associated with . Indeed,when varies from 1 to , the quantity varies from

to and describes the sensors of the ULA. In the sametime, the quantity varys from to and describesthe initial ULA translated of . We then deduce from(28) and (29) that the coordinates and arebuilt in the same manner as two initial ULAs such that the firstone is in translation with respect to the other, which proves thatfor a ULA, the th order VA (i.e., both the number of differentVSs and the order of multiplicity of these VSs) is independentof the arrangement .

Table XI summarizes, for a ULA, the number of different VSsgiven by (27) of the associated VA for several values of

and . It is verified in [8] that the -MUSIC algorithm is ableto process up to statistically independentnon-Gaussian sources from an ULA of sensors.

Comparing (27), which is quantified in Table XI, to, which is computed in Table IX and quantified

in Table X, for and the associated values of , wededuce that

for (30)

since all the arrays with two sensors are ULA arrays, whereasfor . Finally, to complete these re-

sults, we compute below for the ULA the order of multiplicityof the associated VS for , and we illustrate

some VA pattern related to a ULA. After tedious algebraic ma-nipulations, indexing the VSs such that their first coordinate in-creases with their index, we obtain the following results.

1) Fourth Order VA : For , the order of multi-plicity of the VS is given by

(31)

This result has already been obtained in [7] for . Theseresults are illustrated in Fig. 2, which shows the FO VA of aULA of five sensors for which , together with the orderof multiplicity of the VSs, with the x and y axes normalized bythe wavelength .


TABLE XN [2q; l] FOR SEVERAL VALUES OF N , q AND l AND FOR ARRAYS WITH SPACE DIVERSITY ONLY

TABLE XIN FOR SEVERAL VALUES OF q AND N FOR A ULA

2) Sixth Order VA : For , the order of multi-plicity of the VS is given by

(32a)

(32b)

(32c)

(32d)

where if is odd and ifis even. These results are illustrated in Fig. 3, which shows thesixth order VA of a ULA of five sensors for which ,together with the order of multiplicity of the VS, with the x andy axes normalized by the wavelength .

3) Eighth Order VA : For , the order of multi-plicity of the VS is given by

(33a)

Fig. 2. Fourth order VA of a ULA of five sensors with the order ofmultiplicities of the VS.

Fig. 3. Sixth order VA of a ULA of five sensors with the order of multiplicitiesof the VS.


Fig. 4. Eighth order VA of a ULA of five sensors with the order ofmultiplicities of the VS.

(33b)

(33c)

(33d)

These results are illustrated in Fig. 4, which shows the eighthorder VA of a ULA of five sensors for which , togetherwith the order of multiplicity of the VSs, with the x and y axesnormalized by the wavelength .

4) VA Patterns: To complete these results and to illustratethe results of Section III-C related to the increasing resolutionof HO VA as increases, Fig. 5 shows the array pattern (thenormalized inner product of associated steering vectors) of anHO VA associated with a ULA of five sensors equispaced halfa wavelenght apart for , 2, 3, and 4 and for a pointingdirection equal to 0 . Note the decreasing 3 dB beamwidth andsidelobe level of the array pattern as increases in proportionsgiven in Section III-C.

B. Circular Array of Identical Sensors

For a UCA of sensors, it is always possible to choose acoordinate system in which the sensor has the coordinates

, where is the radius of

the array, and where . We now analyze theassociated th order VA for and for all the possiblearrangements .

Fig. 5. VA pattern for q = 1, 2, 3, and 4, ULA with five sensors, d = �=2,pointing direction: 0 .

1) Fourth order VA :a) : For and , the coordinates of the as-

sociated VSs are ,, , where

(34a)

(34b)

It is then easy to show that these VSs lie ondifferent circles if is odd, or different circles ifis even. Moreover, for odd values of , different VSs lieuniformly spaced on each circle of the VA. We deduce that theVA of a UCA of odd identical sensors has

(35)

different VSs, which corresponds with the associated upper-bound given in Table IX. The order of multiplicity of these sen-sors is given in Table VI. The previous results are illustrated inTable XII and Fig. 6. The latter shows the VA of a UCA of fivesensors for which , together with the order of multi-plicity of the VSs for and . Table XII reports boththe number of different sensors of the VA associated with aUCA of sensors and the upper-bound computedin Table IX for several values of and and for odd values of

.b) : For and , the coordinates of the as-

sociated VSs are ,, , where

(36a)

(36b)

It is then easy to show that the VSs that are not at coordinates(0, 0, 0) lie on different circles if is odd ordifferent circles if is even. Moreover, for odd values of ,


TABLE XIIN [2q; l] AND N ASSOCIATED WITH A UCA FOR SEVERAL VALUES OF N ,

q, AND l AND FOR ARRAYS WITH SPACE DIVERSITY ONLY

Fig. 6. Fourth order VA of a UCA of five sensors with the order ofmultiplicities of the VS for (q; l) = (2;2).

different VSs lie uniformly spaced on each circle of the VA. Wededuce from this result that the VA of a UCA of odd identicalsensors has

(37)

different VSs, which corresponds to the associated upper boundgiven in Table IX. This result has already been obtained in [7].The order of multiplicity of these sensors is given in Table VI.The previous results are illustrated in Fig. 7 and Table XII. InFig. 7, the VA of a UCA of five sensors, for which ,is shown together with the order of multiplicity of the VSs for

and .

Fig. 7. Fourth order VA of a UCA of five sensors with the order ofmultiplicities of the VS for (q; l) = (2;1).

Fig. 8. Sixth order VA of a UCA of five sensors with the order of multiplicitiesof the VS for (q; l) = (3; 2).

2) th Order VA : For , the analytical com-putation of the VA is more difficult. However, the simulationsshow that for given values of and , the number of differentVSs of the VA corresponds to the upper boundwhen is a prime number. In this case, it is verified in [8]that the -MUSIC method is able to process up to

statistically independent non-Gaussian sourcesfrom a UCA of sensors. Otherwise, remains smaller than

. This result is illustrated in Table XII and Figs. 8 and9. Figs. 8 and 9 show the VA of a UCA of five sensors for which


Fig. 9. Eighth order VA of a UCA of five sensors with the order ofmultiplicities of the VS for (q; l) = (4; 2).

together with the order of multiplicity of the VSs forand , respectively.

VI. ILLUSTRATION OF THE HO VA INTEREST THROUGH A TH

ORDER DIRECTION FINDING APPLICATION

The strong potential of the HO VA concept is illustrated inthis section through a th order direction finding application.

A. -MUSIC Method

Among the existing SO direction finding methods, theso-called High Resolution (HR) methods, developed fromthe beginning of the 1980s, are currently the most powerfulin multisource contexts since they are characterized, in theabsence of modeling errors, by an asymptotic resolution thatbecomes infinite, whatever the source signal-to-noise ratio(SNR). Among these HR methods, subspace-based methodssuch as the MUSIC (or 2-MUSIC) method [24] are the mostpopular. However, a first drawback of SO subspace-basedmethods such as the MUSIC method is that they are not able toprocess more than sources from an array of sensors.A second drawback of these methods is that their performancemay be strongly affected in the presence of modeling errorsor when several poorly angularly separated sources have to beseparated from a limited number of snapshots.

Mainly to overcome these limitations, FO direction findingmethods [4], [6], [9], [21], [23] have been developed these two

last decades, among which the extension of the MUSIC methodto FO [23], called 4-MUSIC, is the most popular. FO directionfinding methods allow in particular both an increase in the res-olution power and the processing of more sources than sensors.In particular, it has been shown in [7] and Section IV of thispaper that from an array of sensors, the 4-MUSIC methodmay process up to sources when the sensors are iden-tical and up to sources for different sensors.

In order to still increase both the resolution power of HR di-rection finding methods and the number of sources to be pro-cessed from a given array of sensors, the MUSIC method hasbeen extended recently in [8] to an arbitrary even-order

, giving rise to the so-called -MUSIC methods. For a givenarrangement of the th order data statistics and aftera source number estimation , the -MUSIC method [8] con-sists of finding the couples minimizing the estimatedpseudo-spectrum defined by (38), shown at the bottom of thepage, where , where is the

identity matrix, and is the matrix of theorthonormalized eigenvectors of the estimated statistical ma-

trix associated with the strongest eigenvalues. Usingthe HO VA concept developed in the previous sections and towithin the background noise and the source’s SNR, the esti-mated pseudo-spectrum can also be consid-ered as the estimated pseudo-spectrum of the 2-MUSIC methodimplemented from the th order VA associated with the con-sidered array of sensors for the arrangement .

B. -MUSIC Performances

The performance of -MUSIC methods forand for arbitrary arrangements are analyzed in detailin [8] for both overdetermined and underdetermined

mixtures of sources, both with and without modelingerrors. In this context, the purpose of this section is not to presentthis performance analysis again but rather to illustrate the poten-tial of the HO VA concept through the performance evaluationof -MUSIC methods on a simple example. To do so, we in-troduce a performance criterion in Section VI-B1 and describethe example in Section VI-B2. We assume that the sources havea zero elevation angle .

1) Performance Criterion: For each of the consideredsources and for a given direction finding method, two criterionsare used in the following to quantify the quality of the associ-ated direction-of-arrival estimation. For a given source, the firstcriterion is a probability of aberrant results generated by a givenmethod for this source, and the second one is an averaged rootmean square error (RMSE), computed from the nonaberrantresults, which are generated by a given method for this source.

More precisely, for given values of and , a given numberof snapshots and a particular realization of the observationvectors , the estimation of the direction of

(38)


Fig. 10. RMS error of the source 1 and p(� � �) as a function ofL. (a) 2-MUSIC. (b) 4-MUSIC. (c) 6-MUSIC. P = 2,N = 3, ULA, SNR= 5 dB, � = 90 ,� = 82; 7 . No modeling errors.

Fig. 11. RMS error of the source 1 and p(� � �) as a function of L. (a) 2-MUSIC. (b) 4-MUSIC. (c) 6-MUSIC. P = 2, N = 3, ULA, SNR = 5 dB,� = 90 , � = 82;7 . With modeling errors.

arrival of the source from -MUSIC is definedby

(39)

where the quantities correspond to theminima of the pseudo-spectrum defined by (38)for . To each estimate , we associate thecorresponding value of the pseudo-spectrum, which is definedby . In this context, the estimate isconsidered to be aberrant if , where is a threshold to bedefined. In the following, .

Let us now consider realizations of the observation vec-tors . For a given method, the probability ofabberant results for a given source is defined by theratio between the number of realizations for which is aber-rant, and the number of realizations. From the nonaberrantrealizations for the source , we then define the averaged RMSerror for the source RMSE by the quantity

RMSE (40)

where is the number of nonaberrant realizations for thesource , and is the estimate of for the nonaberrant re-alization .

2) Performance Illustration: To illustrate the performanceof -MUSIC methods, we assume that two statistically inde-pendent quadrature phase shift keying (QPSK) sources with araise cosine pulse shape are received by a ULA of om-nidirectional sensors spaced half a wavelenght apart. The twoQPSK sources have the same symbol duration , where

is the sample period, the same roll-off , the sameinput SNR is equal to 5 dB, and the direction of arrival is equalto and , respectively. Note that the normal-ized autocumulant of the QPSK symbols is equal to 1 at theFO and 4 at the sixth order.

Under these assumptions, Figs. 10 and 11 show the varia-tions, as a function of the number of snapshots , of the RMSerror for the source 1 RMSE and the associated probabilityof nonabberant results (we obtain similar resultsfor the source 2) estimated from realizations at theoutput of both 2-MUSIC, 4-MUSIC, and 6-MUSIC methods foroptimal arrangements of the considered statistics, without andwith modeling errors, respectively. In the latter case, the steeringvector of the source becomes an unknown function

of , where is a modeling error vector thatis assumed to be zero-mean, Gaussian, and circular with inde-pendent components such that . Notethat for omnidirectional sensors and small errors, is the sumof the phase and amplitude error variances per reception chain.For the simulations, is chosen to be equal to 0.0174, which


corresponds, for example, to a phase error with a standard devi-ation of 1 with no amplitude error.

Both in terms of probability of nonaberrant results and es-timation precision, Figs. 10 and 11 show, for poorly angularlyseparated sources, the best behavior of the 6-MUSIC methodwith respect to 2-MUSIC and 4-MUSIC as soon as becomesgreater than 400 snapshots without modeling errors and 500snapshots with modeling errors. For such values of , the reso-lution gain and the better robustness to modeling errors obtainedwith 6-MUSIC with respect to 2-MUSIC and 4-MUSIC, due tothe narrower 3 dB-beamwidth and the greater number of VSsof the associated sixth order VA, respectively, is higher than theloss due to a higher variance in the statistics estimates. A similaranalysis can be done for 4-MUSIC with respect to 2-MUSIC assoon as becomes greater than 2000 without modeling errorsand 1700 snapshots with modeling errors.

Thus, the previous results show that despite their highervariance and contrary to some generally accepted ideas,

-MUSIC methods with may offer better performancesthan 2-MUSIC or 4-MUSIC methods when some resolution isrequired, i.e., in the presence of several sources, when the latterare poorly angularly separated or in the presence of modelingerrors inherent in operational contexts, which definitely showsoff the great interest of HO VA.

VII. CONCLUSION

In this paper, the VA concept, which was initially introducedin [7], [15], and [16] for the FO array processing problem andfor a particular arrangement of the FO data statistics has been ex-tended to an arbitrary even-order for severalarrangements of the th order data statistics and for general ar-rays with space, angular, and polarization diversities. This HOVA concept allows us to provide some important insights into themechanisms of numerous HO methods and, thus, some explana-tions about their interests and performance. It allows us, in par-ticular, not only to show off both the increasing resolution and theincreasing processing capacity of the th order array processingmethodsas increasesbutalso tosolve the identifiabilityproblemofall theHOmethodsexploitingthealgebraicstructureofthe th

order data statistics matrix only for particular arrange-ments of the latter. The maximal number of sources that can beprocessed by such methods reached for most of sensors responsesand array geometries has been computed for andfor several arrangements of the data statistics in the matrix.For a given number of sensors, the array geometry together withthe number of sensors with different complex responses in thearray have been shown to be crucial parameters in the processingcapacity of these HO methods. Another important result of thepaper, which is completely unknown by most of the researchers,is that the way the th order data statistics are arranged gener-ally controls the geometry and the number of VSs of the VA and,thus, the number of sources that can be processed by a th ordermethod exploiting thealgebraic structure of . This gives riseto the problem of the optimal arrangement of the data statistics,which has also been solved in the paper. In the particular caseof a ULA of identical sensors, it has been shown that all theconsidered arrangements of the data statistics are equivalent and

give rise to VA with VSs, whereas whenis a prime number, the UCA of identical sensors seems to

generate VA with VSs, whatever the valuesof and . On the other hand, the HO VA concept allows us toexplain why, despite their higher variance, HO array processingmethods may offer better performances than SO or FO ones whensomeresolutionisrequired, i.e., in thepresenceofseveralsources,whenthe latterarepoorlyangularlyseparatedor in thepresenceofmodelingerrors inherentinoperationalcontexts.Finally,onemaythink that the HO VA concept will spawn much practical researchin array processing and will also be considered to be a powerfultool for performance evaluation of HO array processing methods.

APPENDIX A

We present in this Appendix explicit expressions of theLeonov–Shiryaev formula (8) for , 2 and 3, assumingzero-mean complex random vector .

Cum

(A.1)

Cum

(A.2)

Cum


(A.3)

APPENDIX B

We show in this Appendix that the spatial correlation coef-ficient defined by (20) can be written as (21). To this aim, theproperty (B.1), given for arbitrary complex vectors ,, , and , by

(B.1)

can easily be verified. Applying recurrently the property (B.1),we obtain

(B.2)Then, applying (B.2) to (20), we obtain (21).

REFERENCES

[1] L. Albera, A. Ferreol, P. Chevalier, and P. Comon, “ICAR, un algorithmed’ICA à convergence rapide robuste au bruit,” in Proc. GRETSI, Paris,France, Sep. 2003.

[2] L. Albera, A. Ferreol, P. Comon, and P. Chevalier, “Sixth order blindidentification of underdetermined mixtures (BIRTH) of sources,” inProc. ICA, Nara, Japan, Apr. 2003, pp. 909–914.

[3] , “Blind Identification of Overcomplete MixturEs of sources(BIOME),” Linear Algebra and Its Applications J., vol. 391, pp. 3–30,Nov. 2004.

[4] J. F. Cardoso, “Localization et Identification par la quadricovariance,”Traitement du Signal, vol. 7, no. 5, Jun. 1990.

[5] , “Super-symetric decomposition of the fourth-order cumulanttensor—blind identification of more sources than sensors,” in Proc.ICASSP, Toronto, ON, Canada, May 1991, pp. 3109–3112.

[6] J. F. Cardoso and E. Moulines, “Asymptotic performance analysis ofdirection finding algorithms based on fourth order cumulants,” IEEETrans. Signal Process., vol. 43, no. 1, pp. 214–224, Jan. 1995.

[7] P. Chevalier and A. Ferreol, “On the virtual array concept for the fourthorder direction finding problem,” IEEE Trans. Signal Process., vol. 47,no. 9, pp. 2592–2595, Sep. 1999.

[8] P. Chevalier, A. Ferreol, and L. Albera, “High resolution directionfinding from higher order statistics: The 2q-MUSIC algorithm,” IEEETrans. Signal Process., submitted for publication.

[9] H. H. Chiang and C. L. Nikias, “The ESPRIT algorithm with high orderstatistics,” in Proc. Workshop Higher Order Statistics, Vail, CO, Jun.1989, pp. 163–168.

[10] P. Comon, “Blind channel identification and extraction of more sourcesthan sensors,” in Proc. SPIE Conf., San Diego, CA, Jul. 1998, pp. 2–13.

[11] R. T. Compton Jr., Adaptive Antennas—Concepts and Perfor-mance. Englewood Cliffs, NJ: Prentice-Hall, 1988.

[12] L. De Lathauwer, B. De Moor, and J. Vandewalle, “ICA techniques formore sources than sensors,” in Proc. Workshop Higher Order Statistics,Caesara, Israel, Jun. 1999.

[13] C. Demeure and P. Chevalier, “The smart antennas at Thomson-CSFCommunications: Concepts, implementations, performances, appli-cations,” Annales Télécommun., vol. 53, no. 11–12, pp. 466–482,Nov.–Dec. 1998.

[14] M. C. Dogan and J. M. Mendel, “Cumulant-based blind optimumbeamforming,” IEEE Trans. Aerosp. Electron. Syst., vol. 30, no. 3, pp.722–741, Jul. 1994.

[15] , “Applications of cumulants to array processing—part I: apertureextension and array calibration,” IEEE Trans. Signal Process., vol. 43,no. 5, pp. 1200–1216, May 1995.

[16] M. C. Dogan and J. M. Mendel, “Method and Apparatus for SignalAnalysis Employing a Virtual Cross Correlation Computer,” Patent no.5 459 668, Oct. 1995.

[17] A. Ferreol, L. Albera, and P. Chevalier, “Fourth Order Blind Identifi-cation of Underdetermined Mixtures of sources (FOBIUM),” in Proc.ICASSP, Hong Kong, Apr. 2003, pp. 41–44.

[18] A. Ferreol and P. Chevalier, “On the behavior of current second andhigher order blind source separation methods for cyclostationarysources,” IEEE Trans. Signal Process., vol. 48, no. 6, pp. 1712–1725,June 2000. Errata: vol. 50, no. 4, p 990, Apr. 2002.

[19] A. Ferreol, P. Chevalier, and L. Albera, “Higher order blind separation ofnon zero-mean cyclostationary sources,” in Proc. EUSIPCO, Toulouse,France, Sep. 2002, pp. 103–106.

[20] , “Second order blind separation of first and second order cyclo-stationary sources—application to AM, FSK, CPFSK and deterministicsources,” IEEE Trans. Signal Process., vol. 52, no. 4, pp. 845–861, Apr.2004.

[21] E. Gönen and J. M. Mendel, “Applications of cumulants to array pro-cessing—part VI: polarization and direction of arrival estimation withminimally constrained arrays,” IEEE Trans. Signal Process., vol. 47, no.9, pp. 2589–2592, Sep. 1999.

[22] P. Mc Cullagh, “Tensor methods in statistics,” in Monographs on Statis-tics and Applied Probability. London, U.K.: Chapman and Hall, 1987.

[23] B. Porat and B. Friedlander, “Direction finding algorithms based onhigher order statistics,” IEEE Trans. Signal Process., vol. 39, no. 9, pp.2016–2024, Sep. 1991.

[24] R. O. Schmidt, “Multiple emitter location and signal parameter estima-tion,” IEEE Trans. Antennas. Propag., vol. AP-34, no. 3, pp. 276–280,Mar. 1986.

Pascal Chevalier received the M.Sc. degreefrom Ecole Nationale Supérieure des TechniquesAvancées (ENSTA), Paris, France, and the Ph.D.degree from South-Paris University in 1985 and1991, respectively.

Since 1991, he has been with Thalés-Communi-cations, Colonbes, France, where he has shared in-dustrial activities (studies, experimentations, exper-tise, management), teaching activities both in Frenchengineering schools (Supelec, ENST, ENSTA), andFrench Universities (Cergy-Pontoise), and research

activities. Since 2000, he has also been acting as Technical Manager and Ar-chitect of the array processing subsystem as part of a national program of mili-tary satellite telecommunications. He has bee a Thalés Expert since 2003.Hispresent research interests are in array processing techniques, either blind orinformed, second order or higher order, spatial- or spatio-temporal, time-in-variant or time-varying, especially for cyclostationary signals, linear or non-linear, and particularly widely linear for noncircular signals, for applicationssuch as TDMA and CDMA radiocommunications networks, satellite telecom-munications, spectrum monitoring, and HF/VUHF passive listening. He is au-thor or co-author of about 100 papers, including journals, conferences, patents,and chapters of books.

Dr. Chevalier was a member of the THOMSON-CSF Technical and Scien-tifical Council from 1993 to 1998. He co-received the “2003 Science and De-fense” Award from the french Ministry of Defence for its work as a whole aboutarray processing for military radiocommunications. He is presently a EURASIPmember and an emeritus member of the Societé des Electriciens et des Electron-iciens (SEE).


Laurent Albera was born in Massy, France, in1976. He received the DESS degree in mathematics.In 2001, he received the DEA degree in authomaticand signal processing from the University of Science(Paris XI), Orsay, France, and the Ph.D. degree inscience from the University of Nice, Sophia-An-tipolis, France.

He is now Associate Professor with the Universityof Rennes I, Rennes, France, and is affiliated withthe Laboratoire Traitement du Signal et de l’Image(LTSI). His research interests include high order sta-

tistics, multidimensional algebra, blind deconvolution and equalization, digitalcommunications, statistical signal and array processing, and numerical analysis.More exactly, since 2000, he has been involved with blind source separation(BSS) and independent component analysis (ICA) by processing both the cyclo-stationary source case and the underdetermined mixture identification problem.

Anne Ferréol was born in 1964 in Lyon, France. Shereceived the M.Sc. degree from ICPI-Lyon, Lyon,France, and the Mastère degree from Ecole NationaleSupérieure des Télécommunications (ENST), Paris,France, in 1988 and 1989, respectively. She iscurrently pursuing the Ph.D. degree with the EcoleNormale Supérieure de Cachan, France, in collab-oration with both SATIE Laboratory and THALESCommunications.

Since 1989, she has been with Thomson-CSF-Communications, Gennevilliers, France, in the array

processing department.Ms. Ferréol co-received the 2003 “Science and Defense” Award from the

french Ministry of Defence for its work as a whole about array processing formilitary radiocomunications.

Pierre Comon (M’87–SM’95) graduated in 1982,and received the Doctorate degree in 1985, both fromthe University of Grenoble, Grenoble, France. Helater received the Habilitation to Lead Researches in1995 from the University of Nice, Sophia Antipolis,France.

For nearly 13 years, he has been in industry, firstwith Crouzet-Sextant, Valence, France, between1982 and 1985, and then with Thomson Marconi,Sophia-Antipolis, France, between 1988 and 1997.He spent 1987 with the ISL Laboratory, Stanford

University, Stanford, CA. He joined the Eurecom Institute, Sophia Antipolis, in1997 and left during the Fall of 1998. He was an Associate Research Directorwith CNRS, Sophia-Antipolis, from 1994 to 1998. He has been Research Di-rector at Laboratory I3S, CNRS, since 1998. His research interests include highorder statistics, blind deconvolution and equalization, digital communications,and statistical signal and array processing.

Dr. Comon was Associate Editor of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING from 1995 to 1998 and a member of the French National Com-mittee of Scientific Research from 1995 to 2000. He was the coordinator ofthe European Basic Research Working Group ATHOS from 1992 to 1995.Between 1992 and 1998, he was a member of the Technical and ScientificCouncil of the Thomson Group. Between July 2001 and January 2004, he actedas the launching Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS

AND SYSTEMS I in the area of Blind Techniques. He was IEEE DistinguishedLecturer from 2002 to 2003.

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