oblique projection doa

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1374 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 4, APRIL 2

Oblique Projections for Direction-of-ArrivalEstimation With Prior Knowledge

Rmy Boyer and Guillaume Bouleux

Abstract Estimation of directions-of-arrival (DOA) is an im-portant problem in various applications and a priori knowledgeon the source location is sometimes available. To exploit this in-formation, standard methods are based on the orthogonal projec-tion of the steering manifold onto the noise subspace associatedwith the a priori known DOA. In this paper, we derive and analyzethe Cramr-Rao bound associated with this model and in partic-ular we point out the limitations of this approach when the knownand unknown DOA are closely spaced and the associated sourcesare uncorrelated (block-diagonal source covariance). To ll thisneed, we propose to integrate a priori known locations of severalsources into the MUSIC algorithm based on oblique projection of the steering manifold. Finally, we show that the proposed approachis able to almost completely cancel the inuence of the known DOAon the unknown ones for block-diagonal source covariance and forsufcient signal-to-noise ratio (SNR).

Index Terms Cramr-Rao bound, MUSIC algorithm, orthog-onal and oblique projectors, prior knowledge of DOA.

I. INTRODUCTION

D IRECTIONS-OF-ARRIVAL (DOA) of narrowbandsources estimation is one of the central problems inpassive radar, sensor sonar, radio-astronomy, and seismology.This problem has received considerable attention in the last30 years, and a variety of techniques for its solution havebeen proposed. In practical situations, we have sometimesthe knowledge of some a priori known subset of the DOA asfor instance in a radar application where the emitted signal isbackscattered by a number of stationary objects with knownpositions situated in the radars viewing eld [6], [11]. So,several methods have been proposed to incorporate this priorknowledge into an estimation algorithm. Prior knowledgeof DOA can be classied into two families depending if weassume soft or hard constraints [11]. Soft constraints mean thatwe known approximatively all the DOA. This class of methodis known under the name of beamspace methods [20] and hasreceived attention as data reduction methods. The second classof approach incorporates the exact knowledge of a subset of the DOA. This constraint is somewhat more restricting but

Manuscript received September 19, 2006; revised April 27, 2007. This workwas presented in part at the Fourth IEEE Workshop on Sensor Array and Multi-Channel Processing (SAM-2006). The associate editor coordinating the reviewof this manuscript and approving it for publication was Prof. Simon J. Godsill.

The authors are wiith the Universit Paris XI (UPS), CNRSLSS-Suplec,91190 Gif-sur-Yvette, France (e-mail: [email protected]; [email protected]).

Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identier 10.1109/TSP.2007.909348

more interesting gains can be expected. The exact knowledgof DOA allows the deation of the signal subspace and, thus, mitigate the inuence of the known DOA on the unknown one

It is well known that subspace-based methods are very sensitive to over or underestimation of the number of sources ana small error on this parameter leads to a poor accuracy. Othe other hand, estimate all the DOA and extract from this seonly theunknown DOA is noteasy, especially forclosely spaceDOA or/and for low SNRs. So, a specic strategy must be caried out to set up an estimation scheme which extracts onlthe unknown DOA without bias. In [6] and [11], a constrainedMUSIC algorithm has been presented. The key idea is to othogonally project the noisy array response onto the noise subspace spanned by thesteering vectors associatedwith theknownDOA. Note that, we cannd thesame principle in some sequential MUSIC algorithms as in [3], [14], and [13]. But, in contrawith our framework, the prior knowledge is uncertain since it constituted from the previously estimated DOA.

In this paper, we derive and analyze the Cramr-Rao boun(CRB), named Prior-CRB (P-CRB), associated with theorthogonal deation of the signal subspace. In particular, we showthat the prior knowledge of a subset of the DOA leads to

smaller variance for coherent (or highly correlated) sources asociated with closely spaced DOA. Another result lies in thfact that if the known and unknown DOA are closely spaceand the associated sources are uncorrelated, the orthogonal dation cannot help. To ll this need, we advocate that a bettescheme for the deation of the signal subspace is based not onon orthogonal projectors but also on oblique projectors. Baseon this principle, we propose to rewrite the MUSIC [16], [1Least-Squares (LS) criterion in the context of the oblique pro jector algebra. The resulting LS criterion can be decomposeinto thesum of two contributions: the rst term is a MUSIC-likcriterion and the second one is a corrective function which intgrates the prior knowledge.

Note that oblique projectors have received relatively little atention in the literature of signal processing. However, we cand in [9] a rst application of oblique projectors to sensor arays. In [1], Behrens and Scharf propose a very detailed revieon this topic andseveral signal processing oriented applicationsMore recently, these projection operators have been exploitein [24] and [28] in the context of blind channel identication, image restoration [27], in noise reduction [8] and in the conteof DOA estimation by McCloud and Scharf [12]. Remark ththeframework of this latterpublication is different to thecontexof this paper since the authors propose a scaled version of thMUSIC algorithm based on oblique projection but they do no

assume prior knowledge of DOA. Finally, our methodology 1053-587X/$25.00 2008 IEEE


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BOYER AND BOULEUX: OBLIQUE PROJECTIONS FOR DOA ESTIMATION 1375

also valid for all the methods belonging to the class of subspacetting techniques [26]. In addition, some potential applicationsof this algorithm can be considered in biomedical signal anal-ysis [5], [22] and harmonic retrieval [25].

After this brief introduction, Section II discusses the matrix-based model of the DOA problem and de nes in particular the

partitioned steering manifold. In Section III, we present ourderivation and the analysis of the P-CRB. In Section IV, we ex-plain how integrate into the MUSIC algorithm the prior knowl-edge of a subset of the DOA by means of oblique projectors.Section V is dedicated to the practical implementation of thespectral and the root versions of the Prior-MUSIC algorithm.Section VII presents some numerical simulations. Finally, werecall some important facts on oblique projectors and we presentthe demonstration of the theorems in the Appendix. We de-note by bold font vectors and matrices. In addition,

and mean, respectively, transpose, con- jugated-transpose, Moore-Penrose pseudoinverse, trace, deter-minant, Kronecker product, Hadamard product, and the directsum of two subspaces.

II. M ATRIX -BASED REPRESENTATION OF THE DOAESTIMATION PROBLEM

In this section, we introduce the standard matrix-based repre-sentation of the DOA estimation problem for an Uniform LinearArray (ULA) and in particular, we de ne the notion of parti-tioned steering manifold.

A. Parametric Multi-Input Multi-Output (MIMO) Model

Assume there are narrowband plane waves simultaneouslyincident on an ULA with sensors. The complex array responsefor the th snapshot is given by

(1)

whereis the noisy observation on the th sensor and is the

complex amplitude of the th source. The noise vector is de-noted by in which the noise on eachsensor, denoted by , is assumed to be additive, white, andGaussian of parameter and is a positive real param-eter. Matrix is the Vandermonde steering manifold

dened by(2)

where is the steering vector parameterized by DOA (inradian), given by

in which is the intersensor distance and is the wavelength.Parameter is assumed to be known or previously estimated([18, Appendix C]). So, the parametric MIMO model forsnapshots can be written according to

(3)

where withand is the noise matrix.

B. Partitioned Steering Manifold and Deated SignalSubspace

Assume that we know DOA among . Without loss of

generality, the steering manifold can be partitioned accordingto

(4)

where the submanifold is the matrix composed by thedesired DOA and submanifold collects the a priori

known DOA. We name the subspace of interest or thedeated signal subspace as its dimension is which issmaller than , the dimension of the signal subspace .

In addition, we assume that the DOA are all distinct (for ) or equivalently the rank of the steering manifold is(here, we assume ) then and intersect

trivially, i.e., . This implies that.

C. Structure of the Spatial Covariance

The spatial covariance matrix admitsa Vandermonde-type plus noise decomposition according to

(5)

where is the mathematical expectation and the noise-freespatial covariance is given by

(6)

If we assume that all the sources are correlated, the source co-variance is a full matrix, otherwise in the sequel we willsometimes assume that this matrix is block-diagonal, i.e., thesources associated with the known and unknown parts of thesteering manifold are uncorrelated. In that case, we have

(7)

with and where (re-spectively, ) is the covariance associated with the unknown(respectively, known) sources.

III. D ERIVATION AND ANALYSIS OF THE PRIOR -CRB

A. Incorporate Prior Knowledge

In [6] and [11], a prior knowledge MUSIC algorithm hasbeen introduced and analyzed. This algorithm, called con-strained MUSIC, is based on the projection of the noisy arrayresponse onto . In a view to derive the Prior-CRB, wevectorize model (3) according to

(8)


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in which the noise is denoted by vector . Conse-quently, is an additive white Gaussian process of parameters

and

(9)

and .To incorporate prior knowledge of the known DOA set:

collected in submanifold , model (8) ismodi ed according to

(10)

where and the notation prior indicates thatwe have projected signal onto the noise subspace associatedwith the a priori known DOA. Following the same notation, we

dene the noise-free signal according to

(11)

where with.

B. Prior-CRB Based on Model (10) (P-CRB)

There are several ways to derive the P-CRB. A rst approachcould be to derive this bound in the framework of Gaussian col-ored processes where the inverse of the noise error covariance,involved in the general CRB formula [21], is computed from

a truncated pseudoinverse of projector . An equivalent andmore handy way is based on the following proposition.

Proposition 1: The compressed 1 signal

(12)

where is a unitary basis of , followsa Gaussian distribution of parameterswhere .

Proof: See Appendix B.Let de ne the signal plus nuisance model parameter vector

by where

and . A standard re-sult ([18], Appendix B) is that the Mean Squares Error (MSE)for any unbiased estimate, , of the parameter vector satis es

with

(13)

where is the likelihood function. Consequently, theP-CRB is a lower bound on the minimal achievable variance.

According to proposition 1, (13) can be rewritten for Gaussian1In the sense that is a fat matrix (more columns than rows).

distribution according to

(14)

where denotes the th block of vector for .Finally, we can formulate the following result.

Theorem 1: The P-CRB with respect to the signal param-eter is given by

(15)

where with and.

Proof: See Appendix C.

C. CRB Without Prior Knowledge Based on Model (8)

The derivation of these bounds is well known [18, p. 392,(B.6.32)]. Then, we have

(16)

(17)

where , and. Expressions (16) and (17) are, respectively,

the CRB over the subspace of interest and over the wholespace . Remark that the P-CRB mixes the and the

in the sense that only the unknown derivative steeringvectors are projected onto the orthogonal complement of thewhole space.

D. Comparison of the Bounds

The following theorem discusses some properties linking thederived bounds.

Theorem 2: For any unbiased estimator of , we have the

following relations. For full :

(i)(ii)

for . If is block-diagonal:

(iii)(iv) for closelyspaced DOA(v) for closely spacedDOA and de cient.(vi) for widely spacedDOA.

Proof : See Appendix D.


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Note that closely spaced DOA means that at least one knownDOA is close to at least one unknown DOA. 2

If the sources are correlated and according to property (i), wecannot expect perform better than the CRB over the subspaceof interest, . This is a fundamental limit. Regarding prop-erty (ii), prior knowledge cannot help for a large number of sen-

sors. If the known and unknown sources are uncorrelated, theknown sources are not coherent and the DOA are widely spaced,then properties (iii) and (vi) mean that all the CRB are merged.Inversely [ cf . property (v)], if the known sources are coherent orhighly correlated, prior knowledge can be exploited. In conclu-sion, the use of orthogonal projector to introduce prior knowl-edge into an estimation algorithm is recommended for somelimit situations as for closely spaced DOA associated with co-herent (or possibly highly correlated) sources with small/mod-erate number of sensors. Some of these conclusions have beenalready obtained in [6] and [11] in the context of the constrainedMUSIC algorithm but here we present a more general argumen-tation since we base our analysis on a statistical quantity whichis independent of thespeci c choice of the estimation algorithm.An important point is that, for nite , property (iv) suggeststhat the orthogonal projector does not completely cancel the in-uence of the known DOA on the estimation of the unknownones and does not reach the CRB over for block-diag-onal source covariance associated with closely spaced DOA. InSection IV, we propose an estimation scheme which solves thisproblem.

IV. P RIOR KNOWLEDGE -BASED MUSIC A LGORITHMS

To be in line with property (iv), we assume that the sourcesassociated with the known and with the unknown parts of thesteering manifold are uncorrelated. Consequently, the spatialcovariance matrix of the sources is block-diagonal and is de-ned in (7). In addition, we denote unknown quantities by thehat symbol. According to this notation, we have .

It can be seen that the Constrained-MUSIC (CMUSIC) algo-rithm [6], [11] attempts to nd one component at a time whichis most orthogonal to the noise subspace of the partially knownsteering manifold, . The CMUSIC optimization problem canbe described according to where

(18)

Note that this problem is different to the standard un-constrained problem associated with the MUSIC algorithm:

with whereboth matrices and are unknown.

Following the formalism introduced in [7], [16] for theMUSIC algorithm, the constrained Prior-MUSIC criterion isgiven by

(19)

2More precisely, the inter-DOA distance must be under the Rayleigh resolu-tion [23], i.e., 0 0 .

where the cost function is de ned by

(20)

in which is a complex amplitude vector. Letthen the cost function can be rewritten according to

(21)

(22)

since and . Consequently ac-cording to (22), minimizing is equivalent to look forvectors in the subspace of interest and in the sametime to discard the DOA belonging to the known space .In contrast with orthogonal projectors, the steering manifoldis not affected by projector [cf. (11)].

A. Prior-MUSIC (P-MUSIC) Algorithm

A standard minimal-norm solution of with respect toparameter is given by

(23)

This solution satis es .Consequently, the cost function of the P-MUSIC algorithm

(24)

reaches its minimum value ( wrt . ) for the known and unknownDOA. However, criterion (19) must be minimal only for theknown DOA. So, we can expect that in some limit situationsas for low SNR or for closely spaced DOA, this approach willbe suboptimal. Consequently, in the next paragraph, we proposeanother approach which completely solves criterion (19).

B. Weighted Prior-MUSIC (WP-MUSIC) Algorithm

1) A Second Resolution Based on the Obliquely Weighted Pseudoinverse: To solve criterion (19), consider the followingminimal-norm solution:

(25)

where is the obliquely weighted pseudoinverse 3 dened by. Then, the cost function can be rewritten ac-

cording to

(26)

(27)

(28)

3We do not confuse the obliquely weighted pseudoinverse with the standardoblique pseudoinverse de ned in [1].


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The previous expressions are derived by using some basicproperties of oblique projectors de ned in (53) and (54) and thefact that projectors are idempotent. So, the nal WP-MUSICcriterion is given by . Note that

, we have . So,and thus is minimal.

, we have andand thus . So, is not minimal.To show that is a minimal-norm solution of criterion (19),

we consider the partial derivative wrt . of thecost functionaccording to

(29)

We have two cases, then . So, is a minimal-

norm solution of criterion (19) for the unknown DOA. then . Thus for the known

DOA, is not a minimal-norm solution or in other words,the cost function does not reached its minimumwrt . for the known DOA. This fact is very desirablesince criterion (19) must be minimum only for theunknownDOA. This property can be understood as a reinforcementof the rejection of the known DOA. In addition, for theknown DOA, it comes .

2) Link To the MUSIC-Like Criterion: One can easily verifythat and, therefore, the cost functioncan be decomposed into two contributions according to

(30)

where has been de ned in (18) and. As we can see, the above expression is a

CMUSIC criterion with an additional corrective term whichtakes into account the prior knowledge.

C. Large Number of Sensors

We begin by exposing an asymptotic result regarding obliqueprojectors.

1) Proposition 2: For large and if and in-

tersect trivially, we have and.

Proof: As , we havewhere and

then and are mutually orthogonal. This implies. Using these properties

together with the de nition of an oblique projector given in(52), it is easy to show the proposition.

It is straightforward to see that for a large number of sensors,we have

where (31)

According to (23) and the fact that , it is direct

to show . In addition, it comes

(32)

(33)

(34)

(35)

where we have used the results given in proposition 2. Conse-quently, the P-MUSIC and WP-MUSIC cost functions become

This result means that asymptotically, the criterions of theP-MUSIC and WP-MUSIC are in fact the criterion of the

MUSIC over the orthonormalized subspace of interest. Inaddition, the WP-MUSIC and P-MUSIC algorithms based,respectively, on criterion and are asymptoti-cally equivalent. However, for more realistic situations where

takes moderate values, we show in the simulation part thatthe two approaches are no longer equivalent, as expected inSections IV-A and IV-B.

V. IMPLEMENTATION OF THE WP-MUSIC C RITERION

As claimed at the end of Section IV-B, we focus our analysison the WP-MUSIC algorithm. To implement this algorithm, wehave two possibilities. We can use (27) or (28). The latter is

more readable since it explains how the WP-MUSIC algorithmworks. However, it is preferable to implement (27) for the twofollowing reasons.

1) More DOA can be estimated. Since the second expres-sion of the WP-MUSIC criterion involves projector ,we have to satisfy constraint to ensure that isa rank- matrix. For the rst expression, only matricesand are involved through projector . In that case,we have to satisfy the following constraints:

is of rank-

is of rank-The two last constraints can be reformulated as

which is less restrictive thansince we have . In fact,

combining the constraints on the rank of matrices and, we obtain which allows possible values for

greater than .2) In a computational point of view, (28) involves the estima-

tion of projectors and while (27) involves onlythe estimation of projector .

A. Estimation of Oblique Projectors

1) Invariant to Change of Basis: The oblique projectorsand are invariant to change of basis. Indeed


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Fig. 1. CRB versus SNR [dB]. (a) Widely spaced DOA .(b) Closely spaced DOA .

Note that due to the fact that is Hermitian,the coef cients of , noted , are conju-gate centrosymmetry, i.e.,

and therefore occur in pairs. Consequently, the rootWP-MUSIC is based on the following result.

Theorem 3: The roots of polynomialwhere and has

been de ned in (44) and (45), respectively, are the set of theDOA without the subset of the known DOA.

Proof: See Appendix E.

VI. N UMERICAL SIMULATIONS

A. Numerical Analysis of the Prior-CRB

The geometry of the array is Uniform and Linear (ULA) of sensors and snapshots. So, the Rayleigh res-

olution is about 0.49. To illustrate the comparison between the

Fig. 2. CRB versus SNR [dB], Closely spaced DOA, (top) , (middle) , (bottom) .

three derived bounds: the , the and the ,we consider two situations.

1) One Known DOA and One Unknown: In this situation,is known and have to be estimated. The covariance ma-

trix of the sources is given by with

and . The coherent scenariois considered for singular spatial covariance, i.e., for .In Fig. 1(a), we consider uncorrelated, correlated and coherentsources with widely spaced DOA. In that case, the Prior-CRB iscomparable with the CRB for a single or two DOA. So, in this

situation, the knowledge of cannot help the estimation of .This situation con rms property (vi) and (iii) even if the sourcesare correlated.

In Fig. 1(b), the DOA are closely spaced. In this case, the CRBfor one DOA is much lower than the P-CRB and the CRB fortwo DOA, as expected in (iv). In addition, for block-diagonalsource covariance, the P-CRB and the CRB for two DOA aremerged according to property (iii). This illustrates in particularproperty (iv). The important point is that even if the sourcesare highly correlated, the gain associated with the P-CRB withrespect to the CRB for two DOA is small. Inversely, for coherentsources with closely spaced DOA, the P-CRB is much lowerthan the CRBfor two DOA. This observation illustrates property(v). In that case, prior knowledge is bene cial.

Finally, in Fig. 2, we vary the number of sensors for closelyspaced DOA. We can see that the CRB are asymptoticallymerged which con rms property (ii).

2) Two Known DOA and One Unknown: Here, we knowand and we want to estimate . The spatial covariance isgiven by

(50)

where and

(51)


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Fig. 3. CRB versus SNR [dB]. (a) Widely spaced DOA .(b) Closely spaced DOA .

with , and varies according toin the previous section. So, the known DOA are highly corre-

lated and the correlation coef cient between the unknown DOAassociated with the rst and third sources varies until to the co-herent scenario. On Fig. 3, we have drawn the CRB for closelyspaced and for widely spaced DOA. Like in the previous situa-tion, the P-CRB indicates that the exploitation of prior knowl-edge is interesting only for coherent sources with closely spacedDOA.

B. Illustration of the WP-MUSIC Algorithm

Here, we consider a numerical example to illustrate theWP-MUSIC algorithm. On Fig. 4(a), we have drawn the pseu-dospectrums of the CMUSIC and the WP-MUSIC algorithmsfor three DOA where one is known and two others have to beestimated. First, note on the Prior-MUSIC pseudospectrum thatthe known DOA at 100 has been ef ciently cancelled fromthe CMUSIC pseudospectrum without altering the unknown

Fig. 4. (a) CMUSIC and WP-MUSIC pseudospectrums for three DOA(one known and two unknown). (b) and for sensors. (c) Zero location with respect to the unit circle.

one. In contrast with the CMUSIC algorithm, we can note onFig. 4(b) that has only two null values at 50 and 150 .


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In Fig. 4(c), we have drawn the zero location with respect tothe unit circle for the root-CMUSIC and root WP-MUSIC al-gorithms. Note that the zeros occur in pairs, as expected. How-ever, in presence of noise, selecting the zeros (with unit mod-ulus constraint) based only on is a dif cult task due totheir proximity to the unit circle. So, a decision only based on

seems ineffective. Inversely, note that a decision oncriterion is a more practicable task.

C. Performances of the Algorithms

1) Accuracy of the Proposed Methods: We assume that thesource covariance is block-diagonal. The tested methods are

WP-MUSIC: The MUSIC algorithm with prior knowl-edge based on the obliquely weighted pseudoinverse ( cf .Section IV-B).

P-MUSIC: The MUSIC algorithm with prior knowledgebased on standard LS resolution ( cf . Section IV-A).

P-MUSIC (SI): The MUSIC algorithm with prior knowl-edge based on the implementation of the oblique projectorproposed by McCloud and Scharf [12].

MUSIC: The standard root version of the MUSIC algo-rithm.

CMUSIC: The root version of the constrained MUSIC al-gorithm presented by DeGroat et al. [6].

The accuracy of the DOA of interest estimation is measuredthrough the Standard Deviation (Std) which is de ned as theroot of the MSE. Each simulation is based on 1000 MonteCarlo trials. In Fig. 5(a), we consider widely spaced DOA,e.g., . In this situation, all the tested algorithms

are equivalent. On Fig. 5(b), we choose closely spaced DOA,e.g., which corresponds to a distance inter-DOAmuch lower than the Rayleigh resolution for 10 sensors. In thisscenario, the CRB for two DOA and the P-CRB for the DOA of interest are merged, as expected in property (iii) of Theorem 1(cf. Section III-D) in the context of more than two DOA. Wecan note that for suf cient SNR, the CMUSIC and the MUSICalgorithms reach these bounds but they cannot outperform it forclosely spaced DOA and for block-diagonal source covariance.The WP-MUSIC, P-MUSIC, and P-MUSIC (SI) show Stdclose to the CRB for only one DOA at high SNR. By the lightof this example, we can say that most of the in uence of the

known DOA has been ef ciently cancelled by the proposedalgorithms. This is not the case for the CMUSIC algorithm.According to Fig. 5(b) and (c), the WP-MUSIC algorithm isslightly more ef cient than the P-MUSIC algorithm at lowSNR ( dB) and for closely spaced DOA. This observationconrms the discussion in Sections IV-A and IV.B. Finally, weperform in Fig. 5(d), (e), and (f), some experiments with twoknown DOA and one unknown. The conclusions are similar tothe more simple case of one unknown and one known.

2) Robustness to a Small Error on the a Priori: The scenariois the same as for Fig. 5(a), i.e., the known DOA isand the unknown DOA is . We perturb according to

and we compute the standard deviation for the DOAof interest, . This scenario is repeated for different SNR andnumber of sensors. The number of snapshots is equal to 100.

Fig. 6(a) shows that without noise, the P-MUSIC (SI) and theCMUSIC are very sensitive to a small error on the known DOA.Inversely, the P-MUSIC and WP-MUSIC algorithms are morerobust. These observations are con rmed by Fig. 6(b) wherethe SNR is equal to 30 dB. Indeed, we can note the remarkablerobustness of the P-MUSIC and WP-MUSIC algorithms since

the Std associated with these methods follow a at curve. Thisis a clear advantage of these approaches. In Fig. 6(c), we can seethat all the tested methods have similar robustness at low SNR(0 dB). In this situation, the error induced by thenoise dominatesthe error associated with the error on the known DOA.

In Fig. 6(d), we have drawn the Std for the tested methodswithout noise and for a large number of sensor .As expected, in this asymptotic regime, the CMUSIC, theP-MUSIC and the WP-MUSIC have the same ef ciency androbustness. Note that in this case, the bad results for theP-MUSIC (SI) algorithm. These observations are con rmed inFig. 6(e) and (f) in the noisy situation.

VII. C ONCLUSION OF THE SIMULATION PART

1) For a small number of sensors, the WP-MUSIC shows thebest accuracy and the best robustness to a small error on theknown DOA. In particular, the accuracy of this algorithmis near the CRB associated with the subspace of interest.Consequently, the in uence of the known DOA is almostcancelled.

2) As explained in Sections IV-A, IV-B and in the simula-tion part, the P-MUSIC algorithm is slightly less ef cientthan the WP-MUSIC algorithm but its robustness is com-parable. This is a valuable solution.

3) The P-MUSIC (SI) shows a comparable accuracy as theP-MUSIC and the WP-MUSIC algorithms at high SNR butthis algorithm is less ef cient in severely noisy situations.In addition, its robustness is weak. So, essentially for thelatest reason mentioned, we prefer the implementation of the oblique projector introduced in Section V-A rather thanthe one presented by McCloud and Scharf. 5

4) For a small number of sensors and for closely spacedDOA associated with block-diagonal source covariance,the P-MUSIC, the WP-MUSIC, and the P-MUSIC (SI)algorithms outperform the CMUSIC algorithms, in par-ticular at high SNR where the CMUSIC algorithm islower bounded by the CRB over the whole space. For a

large number of sensors and/or widely spaced sources,this algorithm is equivalent to the ones based on obliqueprojectors.

VIII. C ONCLUSION

In this paper, we have presented a subspace-based solutionto estimate DOA among using the knowledge of known DOA. In a rst part of this paper, we have derived andanalyzed the CRB associated with the orthogonal de ation of the signal subspace and we have shown several limitations of this approach. Consequently in the second part, we have pro-posed alternative solutions based on an oblique de ation of the

5Note that in this work, the proposed methodology also assumes the block-diagonal structure of the spatial covariance of the sources.


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Fig. 5. Std versus SNR for two sources. (a) Widely spaced DOA with sensors and snapshots. Closely spaced DOA

with

sensors. (b) With

snapshots. (c) With

snapshots. Std versus SNR for three sources. (d) Closely spaced DOA

with sensors and snapshots. (e) Closely spaced DOA sensors and snapshots. (f) Widely spaced DOA sensors and snapshots.


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signal subspace. We show that the proposed algorithm, calledPrior-MUSIC, mitigatesalmost the in uence of the known DOAon the DOA of interest in particular when the DOA are closelyspaced and the source covariance is block-diagonal. Finally, theoblique projector framework provides a suitable wayto integrateprior knowledge into subspace-based methods and more gener-

ally into subspace tting techniques.APPENDIX

A. Brief Discussion of Oblique Projectors

This appendix is dedicated to oblique projections [1]. In par-ticular, werecall that the onlyrequirementof a matrix tobe a projector is is idempotent, i.e., .Let and be subspaces of that intersect trivially,i.e., . Then, the projector on along

is the linear operator satisfying: ; ; .The geometric interpretation of the above properties is

where andthen . So, the complex Euclidean Space

is decomposed according to. Let be a complex matrix having full column rank,

obtaining by the concatenation of matrices and accordingto . The orthogonal projector onto is thendened as and

(52)

This property is important since it highlights the link betweenthe orthogonal projector and oblique projectors and

. In addition, the ranges for and areand , respectively, and the null spaces for

and are and , respec-tively. A useful rewritten of can be deduced from (VIII-A)according to

(53)

Finally, note that

(54)

B. Demonstration of Proposition 1First, note that an ordered eigendecomposition of any

rank- idempotent matrices is

(55)

where is the rst columns of the left eigen-factor. Thanks to the property that the noise is zero-mean and

, it is not dif cult to see that

(56)

Fig. 6. Std versus Error on the known DOA with sensors. (a) Withoutnoise. (b) With dB. (c) With dB. sensorsand snapshots. (d) Without noise. (e) With sensors and snapshots and dB. (f) With dB.

(57)

(58)

The error covariance, noted , is given by the covariance of the following centered signalthen

(59)

(60)

(61)

C. Demonstration of Theorem 1

We recall that the signal plus nuisance model parametervector by where

and . The rstterm in (14) is associated with the noise and is given by

(62)

The second term which involved the partial derivatives of thenoise-free model with respect to the parameter vector can be


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expressed according to

(63)

(64)

(65)

where

...... (66)

with . So, we obtain

The P-CRB is given by the matrix at the bottom of the page.As the nuisance and signal parameters are decoupled, the

P-CRB has a block-diagonal structure. To obtain the P-CRB forsubvector , we follow the block-diagonalization method in-troduced in [18, p. 390]. Finally, we have

where

.After some straightforward algebraic derivations and for suf-

cient number of snapshots, it comes

Let (respectively, ) be the oblique projector on(respectively, ) along (respectively, )

dened in (52). Based on these operators, we have the followingproperty:

(67)

(68)

(69)

(70)

where to obtain (68) [respectively, (69)], we use (53) [respec-tively, (54)]. Consequently, the P-CRB can be simpli ed to (15).

D. Demonstration of Theorem 2

To show property (i), it is equivalent to prove thatis a positive semide nite

(psd) matrix. First, using , it isstraightforward to see that is idempotent. Therefore,the eigenvalues of is 1 or 0 and, thus, ispsd. Next, as is a nonde cient matrix,is also psd. Finally using that (1) the Hadamard product of twopsd matrices is also a psd matrix ( cf . result R19 in [18]) and (2)the real part of a psd matrix is psd itself, we prove (i).

Property (ii) can be proved in the following manner. In[21], it has been shown that

. For the P-CRB, we have

and then

(71)

Consequently, using the de nition of the P-CRB, it comes

which proves property (ii).Next, note that the CRB over the whole space can be rewritten

according to

(73)

where . In addition, suppose thatis block-diagonal in (73) then it is easy to deduce prop-

erty (iii) since

(74)

Note that this relation holds for widely or closely spacedDOA.

If the DOA are widely spaced, the in uence between the DOAis weak then it is well known that .Consequently, according to property (iii), thereaches its minimum near the by superiorvalues [ cf . relation (i)], which proves property (vi). In-versely, for closely spaced DOA, the is invariant


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while the inverse of is near the singularity(large condition number with respect to the inversion). Thus,

. This fact together with prop-erty (iii) give property (iv).

Now suppose that is de cient (coherent sources), thendue toits block structure, isalsode cient. 6 In addition, if we

consider closely spaced DOA then takes a largevalue as the inverse of the Hadamard product of two (near) sin-gular matrices. In the same time and through projector , theP-CRB remains sensitive to the known DOA but insensitive tothe correlation between the sources associated with the knownDOA, i.e., to the spatial covariance . Consequently, prop-erty (v) holds. For correlated sources, the problem is more com-plicated and we have deferred the discussion to the simulationpart.

E. Demonstration of Theorem 3

As we know occur in pairs, we can give the factor-

ized forms of polynomials and accordingto

(75)

and

(76)

where are the desired (known or unknown) zeros andand are the extraneous zeros. Based on (75) and

(76), admits the following factorization:

(77)

where

and

6We have since we assume that .

Clearly, has no trivial roots, i.e., any knownor unknownDOA are solution of . Inversely, we only have

for the unknown DOA. So, according to (77), zeros of are only the DOA which annulate , i.e., the unknown DOA.

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Rmy Boyer received the M.Sc. and Ph.D. degrees in the eld of signal pro-cessing and communications from cole Nationale Sup rieure des T lcom-munications (ENST), Paris, France, in 1999 and 2002, respectively.

From 2002 to 2003, he wasa Visiting Researcher with the University of Sher-brooke, QC, Canada. He is currently an Associate Professor with the Labora-toire des Signaux et Syst mes (L2S), Gif-sur-Yvette , with the CNRS-UPS-SU-PELEC (National Center of Scienti c Research and University of Paris-Sud).His research activities include multiway-arrays processing, sensor array, sinu-soidal modeling, and watermarking.

Guillaume Bouleux received the engineering degree in electronic and signalprocessing from the Ecole Nationale Sup rieure de l Electronique et de ses Ap-plications (ENSEA), Cergy, France, in 2004 and the Master degree from Uni-versit e de Cergy-Pontoise, France, in 2005.

He is currently working toward the Ph.D. degree in signal processing in col-laboration with both Universit e Jean Monnet (to LASPI laboratory), Saint Eti-enne, France and Universit e de Paris-Sud, Orsay, France (to LSS-Supelec lab-oratory). His research interest includes array signal processing, statistical anal-ysis, linear, and multilinear algebra.

oblique projection doa

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