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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 9, SEPTEMBER 2012 4683

Joint 2-D DOA Estimation and Phase Calibration forUniform Rectangular Arrays

Philipp Heidenreich, Student Member, IEEE, Abdelhak M. Zoubir, Fellow, IEEE, andMichael Rübsamen, Member, IEEE

Abstract—A precise model of the array response is required tomaintain the performance of direction-of-arrival (DOA) estima-tion. When modeling errors are present or the sensor environmentis time-varying, autocalibration becomes necessary. In this paper,the problem of phase autocalibration for uniform rectangulararray (URA) geometries is considered. For the case with a singlesource, a simple and robust least-squares algorithm for joint2-D DOA estimation and phase calibration is presented. Whenperforming phase autocalibration with a URA, the phase andDOA parameters cannot be identified together without ambiguity.This problem is discussed and a suitable remedy is suggested. Anapproximate Cramér-Rao bound and analytical expressions forthe mean squared error performance of the proposed estimatorare presented. The proposed algorithm for phase autocalibrationis extended for the case with multiple sources. The results areevaluated using a representative body of simulations.

Index Terms—2-D direction-of-arrival (DOA) estimation, auto-calibration, phase autocalibration, uniform rectangular array.

I. INTRODUCTION

N ARROWBAND 2-D direction-of-arrival (DOA) estima-tion plays an important role in many applications, such

as radar, sonar, or wireless communications [2]–[4]. Severalstudies have shown that the performance of DOA estimationalgorithms, especially subspace-based high-resolution algo-rithms, critically depends on a precise knowledge of the arrayresponse, e.g., [5] or [6]. In practice, a sensor array implicatesseveral systematic errors, e.g., gain and phase perturbations ofthe sensors, sensor position errors, or mutual coupling, whichcan lead to a substantial degradation in performance. A numberof off-line calibration techniques have been developed toacquire an accurate model of the array response. Such off-linecalibration techniques involve the measurement of the arrayresponse for a finite set of known directions. Subsequently, the

Manuscript received September 23, 2011; revised December 22, 2011 andMay 07, 2012; accepted May 20, 2012. Date of publication June 06, 2012; dateof current version August 07, 2012. The associate editor coordinating the reviewof this manuscript and approving it for publication was Prof. Arie Yeredor. Thiswork was supported in part by the German Research Foundation (DFG) projectCAMUS. Preliminary results of this work were presented at the ICASSP 2011,Prague, Czech RepublicMay2011P. Heidenreich and A. M. Zoubir are with the Signal Processing Group, Insti-

tute of Telecommunications, Technische Universität Darmstadt, 64283 Darm-stadt, Germany (e-mail: [email protected]; [email protected]).M. Rübsamen is with the Concept Engineering Team of Intel Mobile Com-

munications, 85579 Neubiberg, Germany (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2012.2203125

measured data is fitted to a presumed model, see, e.g., [7] or [8,Ch. 3]. However, due to unmodeled effects or a time-varyingsensor environment, the resulting model of the array responsemay still deviate from its true value. Multiple autocalibrationtechniques have been proposed to mitigate the effect of theremaining errors in the array response.Several autocalibration techniques have been developed to

correct gain and phase errors in the presumed array response[9]–[13]. Such errors may result from sensor imperfections orfrom coherent local scattering effects [14]. In other applica-tions, phase errors dominate gain errors. Therefore, the localiza-tion performance in the presence of phase errors has been ana-lyzed [15], and autocalibration techniques have been developed,which take into account only phase errors [11], [16]. Such errorsmay be caused, e.g., by wavefront distortions due to mediuminhomogeneities [16], [17], errors in the presumed array geom-etry, an imperfectly known carrier frequency, or the near-far spa-tial signature mismatch.Even though the deviation of the presumed array response

from its true value often depends on the direction, a numberof autocalibration techniques assume that the gain and/or phaseerror are the same for several sources. The latter assumption isvalid if the sources are closely spaced so that the directional de-pendency of the error in the presumed array response can beneglected. This is practically relevant, e.g., for active radar sys-tems, if narrow transmit beams are used, or in radio astronomy,if the individual antennas have narrow fields-of-view [17]. Weremark that the method proposed in this paper is conceptuallyrelated to the redundancy averaging used in radio astronomy[18].Multiple autocalibration techniques have been developed,

which are applicable in the case of arbitrary array geometries[19]. For instance, a maximum a posteriori (MAP) approach isdescribed in [20] and [21], where the joint optimization w.r.t.DOA and array error parameters can be regularized by consid-ering the array errors as random nuisance parameters, leadingto a DOA estimator which is robust against array perturbations[20]. For arbitrary array geometries and multiple sources, theMAP approach requires a computationally demanding non-convex optimization procedure. The MAP approach is onlyable to correct small deviations from the nominal model as itis based on a local first-order approximation of the error in thepresumed array response.Uniform linear arrays (ULAs) are a popular and well-known

class of arrays. Several researches have devised autocalibrationtechniques, which exploit the particular properties of such ar-rays. For example, the MAP approach can be simplified in the

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4684 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 9, SEPTEMBER 2012

case of ULAs [22]. Furthermore, in [9], a least-squares (LS) ap-proach has been proposed to estimate gain and phase error pa-rameters, which exploits the Toeplitz structure of the spatial co-variance matrix of the unperturbed array. More specifically, theequality of the elements on the diagonals of the spatial covari-ance matrix is exploited to construct a set of equations, which isthen solved by means of the LS principle. The described prin-ciple of gain and phase autocalibration for a ULA can also beapplied by only utilizing the information on the main diagonalfor gain estimation, and on the first superdiagonal for phase es-timation [12]. This simplification has been analyzed in [13],where it was found that interestingly the best simulative resultsare achieved when only the first superdiagonal is considered forphase estimation.In the case of isotropic sensors, there is an unidentifiable rota-

tion factor for phase autocalibration. Using a known phase errordifference at two reference sensors, or the DOA of a referencesource, this ambiguity can be resolved. Since this additional in-formation is rarely available in practice, the authors of [11] haveconstrained the phase errors to have zero mean. An alternativeapproach to resolve the rotational ambiguity is to determine thephase errors with minimum norm [13].All references mentioned above have considered 1-D DOA

estimation. In this paper, we present an extension of the workin [11] and [13] to autocalibration and 2-D DOA estimationwith a uniform rectangular array (URA). Note that the case of asingle source is particularly relevant in many applications (e.g.,in many radar systems, it is rather unlikely that there are sev-eral sources in one range-Doppler cell). For this reason, we firstpropose a simple and robust LS algorithm for joint 2-D DOA es-timation and phase calibration for the single source case. To re-solve the rotational ambiguity, we develop a suitable constraintfor the phase error parameters, based on the idea in [11]. Anapproximate Cramér-Rao bound (CRB) is presented, and an an-alytical expression for the mean squared error (MSE) perfor-mance of the proposed estimator is derived. The performanceis analyzed in a representative body of simulations. Later, wedescribe an extension to the case with multiple sources, similarto [1], in which we exploit the Toeplitz-block Toeplitz struc-ture of the unperturbed spatial covariance matrix. In this case,the phase calibration is decoupled from the DOA estimation andcan be applied as a preprocessing step, to enhance the resolutionof 2-D DOA estimation.The paper is structured as follows. In Section II, the signal

model is introduced together with the problem formulation andan approximation of the CRB. Also, the problem of rotationalambiguity and a possible remedy is discussed. In Section III,the proposed algorithm for DOA estimation and phase calibra-tion is presented, and an analytical expression for the perfor-mance is given. An extension for multiple sources is consideredin Section IV. In Section V, a performance analysis by meansof simulations is presented. Finally, conclusions are drawn inSection VI.

II. SIGNAL MODEL

Consider a URA with sensors, where andare the number of elements in -direction and -direction,

respectively, as depicted in Fig. 1. The corresponding element

Fig. 1. Array geometry of the URA.

spacings are and . The azimuth and elevation angles, and, are also shown in Fig. 1.In the single source case, the common baseband model for

the array output vectors is given by

(1)

where is the number of available snapshots, representsmeasurement noise, is the source waveform, and andare the DOA parameters in electrical angles in -direction and-direction, respectively. Steering vector representsthe nominal array response; following the notation in [3], it canbe written as

(2)

where denotes the Kronecker product, and the elements ofvectors and are given by

(3)

Electrical angles and are related to physical azimuth andelevation angles by

(4)

where , and is the wavelength. We note that the corre-sponding inverse relations for obtaining azimuth and elevationangles are

(5)

respectively, where is the four-quadrant inverse tan-gent. is modeled as a circular complex Gaussian randomprocess, temporally white, with zero mean and variance .Likewise, is modeled as a multivariate circular complexGaussian random process, temporally white, with zero meanand covariance , where denotes an identity matrixof size . and are assumed to be uncorrelated. Thenominal spatial covariance matrix is

(6)

HEIDENREICH et al.: JOINT 2-D DOA ESTIMATION 4685

where denotes the expectation operator. To describe theelement structure of a URA covariance matrix, we employ in-dexing variables

(7)

for and , so that theelements of can be expressed as

(8)

where is the Kronecker delta function, which is unity forand zero otherwise.

A. Phase Errors

When the phase responses of the sensors are corrupted byerrors, the corresponding perturbed array output model is

(9)

where , are the phaseerror parameters, and is the diagonal matrix operator.Based on (6) and (9), the perturbed spatial covariance matrix is

(10)

Its elements can be expressed as

(11)

B. Problem Formulation

Based on measurements , corresponding tomodel (9), we aim at jointly estimating parameters , and

. In the following, we present a simple and robustleast-squares (LS) estimator which employs the phase relationsof the perturbed spatial covariance matrix. For the estimation,we only consider elements on the superdiagonals, i.e.,with , since the subdiagonals contain the same informa-tion due to the Hermitian symmetry, and the main diagonal isreal-valued and contains no phase information.We remark that the proposed estimator can be generalized

straightforwardly to array geometries, which satisfy the outerproduct structure from (2).

C. Ambiguity

Without the knowledge of , it is only possible to deter-mine the phase response up to a complex scalar. Consequently,we set , and search for the remaining parameters

(12)

with .Another identification problem occurs when phase error pa-

rameter exhibits a structure which resembles a steering vector, as defined in (2). Note that the phases of the nom-

inal steering vector can be interpreted as a 2-D linear function of

the sensor positions. If the phase errors have a sim-ilar structure, they cannot be distinguished from the phases ofa steering vector, which leads to a pointing ambiguity. In otherwords, we can rewrite the perturbed steering vector as

(13)

with and

(14)

where

(15)

Consequently, for all , parameters

(16)

with

(17)

lead to the same perturbed steering vector. In other words, isnot uniquely identifiable. We define and corresponding phaseerror parameter , with selection matrix

(18)

such that .We assume the phase errors to be i.i.d. such that asymptot-

ically, when and are large, there is no linear structureacross the -direction and -direction of the URA and we have

. To obtain a unique solution, we therefore proposeto choose such that this condition is fulfilled.

D. Cramér-Rao Bound

We assume to be Gaussian distributed with zero mean andknown covariance . The full parameter set, denotedwith , contains deterministic parameters , , , and ,and stochastic parameter . Therefore, the Fisher informationmatrix (FIM) is of the form [23]

(19)

where represents information obtained from the data, i.e.,, and represents a priori information.

Its respective elements are given by

(20)

where is the likelihood function and is theassumed prior distribution.


However, is not easily evaluated because of the expec-tation w.r.t. , and because of nuisance parameters and .It has been proposed in [20] and [24] to obtain an approximateFIM by evaluating at . By following similar steps as in[20], the block of this approximate CRB matrix correspondingto parameters , , and is given by

(21)

with elements

(22)

where denotes the element-wise matrix product,

(23)

is the orthogonal projection matrix of perturbed steering vector, , and

(24)

are partial derivatives of w.r.t. , , and , respectively., , and have been defined in Section II-C.Let be the block of the approximate CRB ma-

trix in (21) regarding parameters and . The correspondingapproximate CRB for azimuth and elevation angles, and , canbe obtained as [3, Sec. 8.2.3.1]

(25)

where

(26)

can be easily derived from (4).

III. AUTOCALIBRATION ALGORITHM

In the following, we present the proposed algorithm for jointDOA estimation and phase calibration. An analytical expressionfor the MSE performance is also given. For estimating , weemploy the well-known sample covariance matrix, defined by

(27)

A. DOA Estimation and Phase Calibration

For constructing equations from the phase measurements ofthe elements of in (27), we follow the idea in [11]. Usingindexing variables and , as defined in (7), we have

(28)

for and .In Appendix A, it is shown that, for sufficiently high SNR

and/or a large number of snapshots, phase error term haszero mean and covariance

if

if

if

if

if

ifotherwise.

(29)

Based on (28), we form two groups of equations, for whicheither or equals zero, and the other equalsone. Note that these elements are least likely to require phaseunwrapping [11], and hence lead to an estimator which is morerobust.1) Group 1: Phase measurements from elements on the first

superdiagonal, for and

(30)

2) Group 2: Phase measurements from elements on super-diagonal , for and

(31)

Generally more elements of can be considered. These havelarger lag values or , so that a phase unwrap-ping procedure, depending on , may be necessary.A possible approach is to use the proposed method to deter-

mine in a first stage. In a second stage, is used for phaseunwrapping of the covariance matrix elements, which have notbeen used in the first stage, and a refined estimate is obtained.Note that when the proposed method fails in the first stage, then


a subsequent phase unwrapping is likely to fail as well. There-fore, this approach can only improve the result in the asymptoticregion, and is not addressed here. In total, (30) and (31) contain

(32)

equations. These can be summarized by

(33)

where, according to Groups 1 and 2, consists of thephase measurements from the sample covariance matrix,

is the corresponding systemmatrix, and is a vectorof measurement errors.Proposition: The rank of is .Proof: Let us consider the case without measurement errors,

i.e., we have , which corresponds to equations andunknowns. Due to the nonidentifiability of , as described

in Section II-C, we have .Now assume and are given. In this case, (30) and (31)

allow to recursively compute by

and, for

otherwise

Therefore, we have , which concludes theproof.As described in Section II-C, we have , which is

due to the nonidentifiability of . Therefore, the optimum pointsof the LS optimization problem

(34)

can be expressed as

(35)

where

(36)

is the minimum norm solution of (34) [25], and is theMoore-Penrose pseudoinverse of . We note that, given atruncated singular value decomposition , where

is a positive definite diagonal matrix of dimension ,we have .As discussed in Section II-C, to obtain a unique solution, we

constrain , or more specifically , such that .Thus, we have

(37)

which can be solved for

(38)

Insertion into (35) yields

(39)

with

(40)

where matrices , , and have been defined inSection II-C. Note that the latter solution can also be inter-preted as the vector of (35) with the minimum norm of .Note that the proposed estimator in (39) is computationally

simple. When measurement vector is available and is com-puted off-line, is determined by a real-valued matrix vectorproduct with operations. The construction of in-volves the computation of , and the phase calculation of se-lected elements.

B. Analytical Performance

The bias of the proposed estimator is evaluated as

(41)

with , and so depends on the parameter itselfand how much differs from an identity matrix. can beinterpreted as a projection-like operator onto the unidentifiablecomponents of .Likewise, the MSE is

(42)

where can be evaluated using (29). We observe that theMSE consists of a squared bias term and a variance term due tothe additive phase errors.We note that the MSE in terms of physical azimuth and ele-

vation angles can be obtained according to (25) from the corre-sponding block of (42).

IV. EXTENSION FOR MULTIPLE SOURCES

The perturbed array output of a superposition of uncorre-lated sources, where we assume to be known, is modeled by

(43)

where and are the DOA parameters, and ,are the corresponding source waveforms.

The covariance matrix is

(44)


with . As in Section II, weuse indexing variables and , which allows expressing theelements as

(45)

Clearly, this constitutes a Toeplitz-block Toeplitz matrix, sincethe elements are only a function of and . Inthe unperturbed case with , the covariance matrix in(44) is a scaled sum of Toeplitz-block Toeplitz matrices andan identity matrix, and therefore also satisfies a Toeplitz-blockToeplitz structure.This structure can be exploited for phase calibration [1]. In

particular, we use the fact that elements of the unperturbed co-variance matrix, which are on the same superdiagonal withinblocks, and at the same position from block to block, are equal.So by selecting corresponding elements, the phases differenceswill only depend on the phase error parameters. We follow theidea in [12] and [13], and select only elements from , forwhich either or equals zero, and the otherequals one. Four groups of relations have been identified, andare described below. , denote the additive phaseerrors of respective groups.1) Group 1: Relations within blocks, on first superdiagonal,

for and

(46)

2) Group 2: Relations from block to block, on first super-diagonal, for and

(47)

3) Group 3: Relations within blocks, on superdiagonal ,for and

(48)

4) Group 4: Relations from block to block, on superdiagonal, for and

(49)

In total, (46)–(49) contain

(50)

equations. These can be summarized by

(51)

where, according to Groups 1 to 4, consists of thedifferences of phase measurements from the sample covariancematrix, is the corresponding systemmatrix, andis a vector of measurement errors.

The number of independent relations, and so the rank of ,is at most , which results from the unidentifiable linearcomponents of , as indicated in Section II-C. Hence, the LSapproach

(52)

leads to a nonunique solution. The minimum norm solution, i.e.,with the smallest , is given by

(53)

where is the Moore-Penrose pseudoinverse of , obtainedsimilarly as in Section III.

V. SIMULATION RESULTS

We first present simulation results using the proposed algo-rithm for joint DOA estimation and phase calibration in thesingle source case. If not stated otherwise, we consider a squareURA with 16 elements and an element spacing of . We gen-erate snapshots according to model (9), where theDOA parameters are and . The SNR is de-fined as . Phase errors are generated from a Gaussian dis-tribution with zero mean and variance .We compare the performance of the proposed algorithm with

the analytically foundMSE from (42) and the approximate CRBfrom (21). As performance metrics, we consider the root meansquared error (RMSE) for DOA estimation of azimuth and ele-vation angles, and an averaged RMSE for phase parameter es-timation, defined by

(54)

where are the phase error estimates in run , andis the number of Monte Carlo runs.Later, we present simulation results using the proposed algo-

rithm for multiple sources.

A. Variance of the Phase

Since the result for the phase error covariance in (29) differsfrom the results obtained in [11] or [13], we have used a simula-tion example for validation. For several scenarios of and ,and varying , a comparison between the square root of (29) forthe case and empirical results for the standarddeviation of the phase errors of is shown in Fig. 2.It can be observed that our derivation in Appendix A rep-

resents a good match with empirical data. For small values of, the scenarios with the same product are asymptoticallyequivalent, which is reasonable in view of (29). For large valuesof , the phase measurements become arbitrary and we observea limit from above at , which corresponds to the standarddeviation of a uniform distribution on .

B. Effect of SNR and Array Size

We simulate both DOA estimation and phase calibration per-formance of the proposed algorithm versus SNR. We consider


Fig. 2. Empirical results for the standard deviation (std) for the complex phaseof the elements of the sample covariance matrix, compared to (29); varyingand several scenarios.

Fig. 3. Simulation results: RMSE of azimuth and elevation DOA estimationversus SNR for various URA sizes, with .

URA sizes , 8, and 16, and employ .Figs. 3 and 4 show the obtained results.It can be observed, that the analytically found MSE from (42)

accurately predicts the performance of the proposed approach.For , the proposed algorithm achieves the approxi-mate CRB for both DOA estimation and phase calibration. Note

Fig. 4. Simulation results: RMSE of phase calibration versus SNR for variousURA sizes, with .

that this is even possible by considering only selected elementsfrom . From considering more elements, the threshold perfor-mance can be improved, but this would require a nontrivial tech-nique to deal with the phase unwrapping problem. Also, it canbe observed that the convergence RMSE level of DOA estima-tion and phase calibration is inversely proportional to the URAsize. This convergence level is due to the unidentifiable com-ponents of , and corresponds to a remaining uncertainty. Thisbehavior seems to be captured by the approximate CRB, so thatit can be considered as a valid performance bound.

C. Comparison With BF Matching and MAP-NSF

The proposed algorithm is compared with two other methods.First, an intuitive approach, based on the conventional beam-former (BF) is used, in which is obtained by matching theideal steering vector with an unstructured estimate of the arrayresponse [1]. Second, a maximum a posteriori (MAP) approachis used, in which distributional information of the random pa-rameter is employed. In particular, the implementation in [21]with noise subspace fitting (NSF) is applied. Both methods aredescribed briefly.1) BF Matching: First, estimate the source waveform using

(55)

where and are estimated by the location of the globalmaximum of the BF spectrum, given by

(56)

Second, find an unstructured estimate of the array response as

(57)

Third, estimate the desired phase error parameters as

(58)


for . We remark that this approach assumes thephase errors to be sufficiently small such that the BF spectrumis not perturbed significantly.2) MAP-NSF: This approach uses the fact that the maximum

likelihood criterion, which is required for MAP estimation, isasymptotically as equivalent to the NSF criterion [21].For a single source, the latter is given by

(59)

where is the estimated noise subspace eigenvector matrix,obtained from an eigendecomposition of , and

(60)

where is the average of the smallest eigenvalues,and are initial DOA estimates, e.g., obtained from the

global maximum of (56), is the estimated signal eigenvector,and is its respective eigenvector. In the considered case, theMAP-NSF cost function is given by

(61)

Assuming is small, a local first-order approximation of ispossible around ,

(62)

with nominal steering vector , and

(63)

This approximation allows to concentrate the MAP-NSF crite-rion w.r.t. [21]. The DOA estimates are obtained from theglobal minimum of

(64)

where , and

(65)

(66)

The phase estimates are obtained by .3) Simulation: To compare the proposed algorithm with the

described methods, we simulate DOA estimation and the phasecalibration performance versus , at . Figs. 5and 6 show the obtained results.Regarding DOA estimation performance, the MAP-NSF and

BF matching method prove to be robust against small phase er-rors. Both achieve the approximate CRB and should thereforebe preferred over the proposed algorithm. However, for largerphase errors, the proposed algorithm is able to achieve optimumphase calibration and results in improved DOA estimation. Thesame can be observed regarding phase calibration performance,

Fig. 5. Simulation results: RMSE of azimuth and elevation DOA estimationversus phase error magnitude for , at .

Fig. 6. Simulation results: RMSE of phase calibration versus phase error mag-nitude for , at .

except that the BF matching approach provides suboptimal re-sults. We note that all methods break down when the phase errormagnitude is larger than .Regarding computational cost, both the MAP-NSF and BF

matching method are based on a 2-D spectral search, and aretherefore more complex than the proposed algorithm.


Fig. 7. Simulation results: Probability of resolution of 2-D MUSIC with nocorrection, proposed phase calibration and ideal calibration, for a URA with

, at , and with .

D. Resolution

Finally, we demonstrate that the proposed algorithm for thecase with multiple sources, as described in Section IV, is able toimprove the resolution performance. Consider a scenario withtwo uncorrelated sources of equal power. The DOA parametersare and , such that theangular source separation is parameterized by

(67)

We use spectral 2-D MUSIC [3] for high-resolution DOAestimation; the cost function is given by

(68)

where is the noise subspace, estimated from in (27) andcan be estimated in a preprocessing step using the proposed

algorithm in Section IV. The two sources are considered as re-solved if there are two local maxima of closeto the true DOA parameters, i.e.

(69)

In Fig. 7, we show the empirical probability of resolution of2-D MUSIC with no correction, the proposed phase calibrationmethod, and the ideal phase calibration, versus angular separa-tion , at , and with .It can be observed that the proposedmethod for phase calibra-

tion is able to substantially enhance the resolution performancewhen compared with no correction.Note that the Friedlander-Weiss method [10] is not consid-

ered here, since it requires the two sources to be resolved in theuncalibrated state, and consequently cannot be used to enhancethe resolution ability.

VI. CONCLUSION

For the single source case, we have presented an algorithm forjoint 2-D DOA estimation and phase calibration for URAs. Theproposed algorithm results in a linear LS problem, which canbe solved efficiently. An accurate analytical expression for theasymptotic MSE performance has been determined. Our simu-lation results suggest that the proposed algorithm achieves theapproximate CRB under moderate conditions. We remark thatthe problem of phase autocalibration with URAs generally re-sults in unidentifiable estimation due to an ambiguity problem.This has been described adequately, and a remedy in terms of asuitable constraint on the phase error parameters has been pro-posed.Furthermore, an extension of the proposed algorithm to the

case with multiple sources has been presented. This algorithmexploits the Toeplitz-block Toeplitz structure of the unperturbedcovariance matrix. Note that the phase calibration method isdecoupled from the DOA estimation and can be applied as apre-processing step. The proposed algorithm allows to substan-tially enhance the resolution performance of high-resolution2-D DOA estimation.

APPENDIX

Statistical Properties of the Sample Covariance Matrix:The sample covariance has been defined in (27). Its mean and afirst-order approximation of the covariance between the phasesof matrix elements are determined next.In the following, the elements of vectors , , and are

denoted with , , and , respectively. Plugging inand , the elements of are

(70)

It can be easily seen that

(71)

which corresponds to (10), so is unbiased. Hence, also thephases of the elements are unbiased.When the SNR is high and/or the number of snapshots is large

such that the estimation error of is small [26], we can employa first-order approximation of

(72)


where and

(73)

represents the phase estimation error. It can be easily seen thatfor . Hence, we focus on the case in

the following. Using (70) and (71), we have

(74)

Let us define

(75)

Then, for and , we can calculate

(76)

where we have used that for white circular complex Gaussianrandom processes with zero mean [27], we have

(77)

any third-order terms are zero, and

(78)

Note that (76) is nonzero only for or .Likewise, we can calculate

(79)

which is nonzero only for or .

Note that (76) and (79) cannot be simultaneously nonzero forand . Moreover, since the expressions in (76) and

(79) are real-valued, it can be easily seen that

(80)

which, by subtracting the latter two equations, yields

(81)From (73) and (75), we have . Consequently,(81) approximately equals .In summary, since is unbiased, has zero mean. When

the SNR is high and/or the number of snapshots is large, thecovariance between the phases of matrix elements is

if

if

if

if

if

ifotherwise

(82)

To support this result, we have presented a simulation examplein Section V.

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Philipp Heidenreich (S’07) received the Dipl.-Ing.and Dr.-Ing. degrees, both in electrical engineering,from Technische Universität Darmstadt, Germany, in2006 and 2012, respectively.In March 2007, he joined the Signal Processing

Group, Technische Universität Darmstadt, where heis currently working as a Research Fellow. His re-search interests are focussed on statistical signal pro-cessing and array signal processing with emphasis ondirection-of-arrival estimation and applications to au-tomotive radar.

Dr. Heidenreich received the Best Student Paper Award at the Biennial Work-shop on Digital Signal Processing for In-Vehicle Systems in 2011, Kiel, Ger-many.

Abdelhak M. Zoubir (S’87–M’91–SM’97–F’08)received the Dr.-Ing. from Ruhr-UniversitätBochum, Germany, in 1992.He was with Queensland University of Tech-

nology, Australia, from 1992 to 1998, where he wasan Associate Professor. In 1999, he joined CurtinUniversity of Technology, Australia, as a Professorof Telecommunications and was Interim Head of theSchool of Electrical and Computer Engineering from2001 until 2003. In 2003, he moved to TechnischeUniversität Darmstadt, Germany, as Professor of

Signal Processing and Head of the Signal Processing Group. His researchinterest lies in statistical methods for signal processing with emphasis on boot-strap techniques, robust detection and estimation and array processing appliedto telecommunications, radar, sonar, car engine monitoring, and biomedicine.He published more than 300 journal and conference papers on these areas.Prof. Zoubir was Technical Chair of the 11th IEEE Workshop on Statistical

Signal Processing (SSP 2001), General Co-Chair of the 3rd IEEE InternationalSymposium on Signal Processing & Information Technology (ISSPIT 2003)and of the 5th IEEE Workshop on Sensor Array and Multi-channel Signal Pro-cessing (SAM 2008). He is the General Co-Chair of SPAWC 2013 to be heldin Darmstadt, Germany, General Co-Chair of EUSIPCO 2013 to be held inMarrakesh, Morocco, and Technical Co-Chair of ICASSP-14 to be held in Flo-rence, Italy. He was an Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING (1999–2005), a Member of the Senior Editorial Board of the IEEEJOURNAL ON SELECTED TOPICS IN SIGNAL PROCESSING (2009–2011). He cur-rently serves on the Editorial Boards of the EURASIP journals Signal Pro-cessing and the Journal on Advances in Signal Processing (JASP). He is theEditor-in-Chief of the IEEE Signal Processing Magazine (2012–2014). He isPast-Chair (2012) of the IEEE SPS Technical Committee Signal ProcessingTheory andMethods (SPTM) [Chair (2010–2011), Vice-Chair (2008–2009) andMember (2002–2007)] and a Member of the IEEE SPS Technical CommitteeSensor Array and Multi-channel Signal Processing (SAM) (2007–2012). Healso serves on the Board of Directors of the European Association of Signal Pro-cessing (EURASIP). He is a Fellow of the IEEE Distinguished Lecturer (Class2010–2011).

Michael Rübsamen (S’05–M’12) received theDipl.-Ing. degree from RWTH Aachen University,Germany, in 2006, and the Dr.-Ing degree fromTechnische Universität Darmstadt, Germany, in2011, both in electrical engineering.During 2004–2005, he held a research scholarship

from Bell-Labs, Crawford Hill, NJ, where he workedon signal processing methods for fiber optic commu-nication systems. From 2006 till 2012, he worked asa Research Scientist at the Communication SystemsGroup, Technische Universität Darmstadt. In 2009,

he visited Temasek Labs, Nanyang Technological University (NTU), Singapore.In June 2012, he joined the concept engineering team of Intel Mobile Com-munications, Neubiberg, Germany, where he is currently working as a Wire-less System Engineer. His research interests are focussed on array signal pro-cessing, signal processing for communication systems, beamforming, and di-rection-of-arrival estimation.Dr. Rübsamen received the Student Best Paper Award at the IEEE Sensor

Array and Multichannel Signal Processing (SAM) Workshop in 2010. He wasstudent contest finalist at the IEEE International Conference on Acoustics,Speech, and Signal Processing (ICASSP) in 2008. He was awarded a grantfrom the German National Academic Foundation in 2003. In 2002 and 2003,he received the Philips Student Award and the Henry-Ford-II Student Award,respectively. He served as local organization Co-Chair for the IEEE SAMWorkshop in Darmstadt in 2008.

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