Signal Integrity and Radiated Emission of High-Speed Digital Systems (ISBN 9780470511664)

8/8/2019 Signal Integrity and Radiated Emission of High-Speed Digital Systems (ISBN 9780470511664)

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588 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 2, FEBRUARY 2009

Direction-of-Arrival Estimation for NonuniformSensor Arrays: From Manifold Separation to

Fourier Domain MUSIC MethodsMichael Rbsamen, Student Member, IEEE, and Alex B. Gershman, Fellow, IEEE

AbstractIn this paper, the problem of spectral search-freedirection-of-arrival (DOA) estimation in arbitrary nonuniformsensor arrays is addressed. In the first part of the paper, wepresent a finite-sample performance analysis of the well-knownmanifold separation (MS) based root-MUSIC technique. Then,we propose a new class of search-free DOA estimation methodsapplicable to arrays of arbitrary geometry and establish theirrelationship to the MS approach. Our first technique is referredto as Fourier-domain (FD) root-MUSIC and is based on the factthat the spectral MUSIC function is periodic in angle. It usesthe Fourier series to expand this function and reformulate theunderlying DOA estimation problem as an equivalent polynomialrooting problem. Our second approach applies the zero-paddedinverse Fourier transform to the FD root-MUSIC polynomial toavoid the polynomial rooting step and replace it with a simple linesearch. Our third technique refines the FD root-MUSIC approachby using weighted least-squares approximation to compute thepolynomial coefficients. The proposed techniques are shown tooffer substantially improved performance-to-complexity tradeoffsas compared to the MS technique.

Index TermsDirection-of-arrival (DOA) estimation, nonuni-form sensor arrays, root-MUSIC.

I. INTRODUCTION

THE multiple signal classification (MUSIC) algorithm[1], [2] is one of the most popular subspace-based

techniques for estimating the directions-of-arrival (DOAs)of multiple signal sources. As the conventional (spectral)MUSIC algorithm involves a computationally demandingspectral search step, its use can be prohibitively expensive inscenarios where real-time processing is required. To reduce thecomputational complexity of spectral MUSIC, a numericallyefficient search-free modification of this approach has beenproposed [3]. The latter algorithm is commonly referred toas root-MUSIC because it makes use of polynomial rooting

instead of spectral search. Although the root-MUSIC techniqueenjoys a substantially improved computational complexity and

Manuscript received March 05, 2008; revised August 29, 2008; accepted Au-gust 29, 2008. First published October 31, 2008; current version published Jan-uary 30, 2009. The associate editor coordinating the review of this manuscriptand approving it for publication was Dr. Brian M. Sadler. This work was sup-ported in part by the European Research Council (ERC) Advanced InvestigatorGrants program. The results of the paper were presented in part at InternationalConference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas,NV, March 30April 4, 2008, and at the IEEE Sensor Array and MultichannelSignal Processing (SAM) Workshop, Darmstadt, Germany, July 2123, 2008.

The authors are with the Communication Systems Group, TechnischeUniversitt Darmstadt, 64283 Darmstadt, Germany (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TSP.2008.2008560

threshold performance as compared to spectral MUSIC [4], it isonly applicable to uniform linear arrays (ULAs) or nonuniformarrays (NUAs) whose sensors lie on a uniform grid. Anotherpopular search-free DOA estimation method is the ESPRIT(estimation of signal parameters via rotational invariance tech-niques) algorithm [5]. However, the array geometry is requiredin ESPRIT to be shift-invariant.

Several useful extensions of root-MUSIC and ESPRIT to cir-

cular arrays can be found in [6] and [7]. Several other exten-sions of root-MUSIC and ESPRIT to more general classes ofarray geometries have been developed in [8][10]. However, thearray geometries used in these papers are still rather specific,that is, the methods of [8][10] are not applicable to arbitraryarray configurations.

There have been several promising attempts to extend theconcept of root-MUSIC to arbitrary NUA geometries. For ex-ample, the array interpolation method [11][18] employs theidea of approximating any actual NUA by a virtual ULA and ap-plying the standard root-MUSIC technique to the virtual ULAobservations. The resulting method is commonly referred to as

interpolated root-MUSIC.Another approach to extend root-MUSIC to arbitrary NUAshas been reported in [19][21], and has been recently further ad-dressed in [22] where it has been termed the manifold separation(MS) technique. The essence of this technique is to model thereceived wavefield by means of an orthogonal expansion thatapproximates the true array steering vector as the product of amatrix that depends only on the array parameters and a Vander-monde vector depending only on the angle. The Vandermondestructure of the latter vector is then exploited to obtain a poly-nomial that can be rooted to estimate the source DOAs.

In this paper, we first present an asymptotic performanceanalysis of the MS technique using the general idea of the

performance analysis of the conventional root-MUSIC tech-nique [4]. A significant difference between our analysis andthat presented in [22] is that our analysis is valid for the finitesample case. Interestingly, its results can be used to characterizethe performance of interpolated root-MUSIC as well.

Then, we propose an alternative spectral search-free DOA es-timation technique for arbitrary NUAs. Our approach is referredto as Fourier-domain (FD) root-MUSIC because it exploits thefact that the null-spectrum MUSIC function is periodic in angle.More specifically, it uses the truncated Fourier series expan-sion of this periodic function to reformulate the DOA estima-tion problem in terms of polynomial rooting rather than spectral

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RBSAMEN AND GERSHMAN: DOA ESTIMATION FOR NONUNIFORM SENSOR ARRAYS 589

As the order of the FD root-MUSIC polynomial is entirely de-termined by the number of terms used in the truncated Fourierseries and the resulting DOA estimation performance can sufferfrom truncation errors, rather high orders of the FD root-MUSICpolynomial have to be chosen. Hence, the resulting computa-tional cost of the polynomial rooting step may be substantial. To

avoid this potentially costly polynomial rooting step, we use theidea of [23] and apply the inverse Fourier transform to the FDroot-MUSIC polynomial to compute the null-spectrum. Then,the source DOA estimates can be obtained by means of a simpleline search.

Our third algorithm further refines the FD root-MUSICtechnique using a weighted least-squares approximation ofthe MUSIC null-spectrum to compute improved values of theFourier series coefficients. This improves the DOA estimationperformance as compared to the FD root-MUSIC technique.

It is demonstrated that the proposed methods offer substan-tially improved performance-to-complexity tradeoffs as com-pared to the MS technique.

II. BACKGROUND

Let an array of omnidirectional sensors receive signalsfrom narrowband far-field sources with the un-known DOAs . The array snapshot vector attime can be modeled as [1][4]

(1)

where is the vector of signal DOAs,

(2)

is the signal steering matrix, is the vector of signal waveforms, is the vector of sensor noise, and

stands for the transpose.Assuming an array of arbitrary geometry, the steering

vector can be expressed as

(3)

where is the signal wavelength, , are the

coordinates of the th array sensor, and it will be hereafter as-sumed that the array manifold is known exactly.

The array covariance matrix can be written as [1][4]

(4)

where is the source covariance matrix,is the sensor noise variance, is the identity matrix, is thestatistical expectation, and is the Hermitian transpose.

The eigendecomposition of the exact covariance matrix canbe written as [1][4]

where and are the eigenvalues and cor-responding eigenvectors. Let the eigenvalues be sorted innonascending order. Then, the matrices

(5)

contain the signal- and noise-subspace eigenvectors,respectively.In practical situations, the exact array covariance matrix

is unavailable and its sample estimate

(6)

is used, where is the number of snapshots.The eigendecomposition of the sample covariance matrix (6)

yields

(7)

where the sample eigenvalues are again sorted in nonascendingorder and the matrices

and contain intheir columns the signal- and noise-subspace eigenvectorsof , respectively. Correspondingly, the diagonal matrices

and arebuilt from the signal- and noise-subspace eigenvalues of ,respectively.

A. Spectral MUSIC

The conventional MUSIC null-spectrum function can be ex-pressed as [1]

(8)

where is the vector 2-norm. The spectral MUSIC techniqueestimates the signal DOAs from the minima of this function bysearching over with a fine grid. The computational complexityof this spectral search step is typically substantially higher thanthat of the eigendecomposition step because, as a rule,where is the total number of spectral points. Note that for each

spectral point, the product of and (or, alternatively of

and ) has to be computed.

B. Array Interpolation

Array interpolation is a popular approach to array processingthat enables to reduce DOA estimation problems in NUAs tomuch simpler virtual ULA problems [11][18]. Its essence is toapproximate the steering vector of the actual sensor array as

(9)

where is the steering vector of a virtual ULA,is the number of virtual sensors, and is the array

interpolation matrix that is designed so that the interpolationerror is minimized.



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For example, if the virtual sensors are aligned along the axisof the coordinate system and centered with respect to its origin,the virtual steering vector takes the following form:

(10)

where denotes the inter-element spacing for the virtual ULA.

Consequently, the MUSIC null-spectrum function (8) can beapproximated as

This function can be expressed in terms of as [11]

(11)

This is a polynomial of degree whose roots appearin conjugate reciprocal pairs, that is, if is a root of ,then is its root as well, where stands for the complex

conjugate. The signal DOAs can be estimated from the largest-magnitude roots located inside the unit circle.The approximation (9) is typically inaccurate for the whole

array angular field-of-view. Therefore, angular sectors have tobe defined and such an approximation has to be used separatelyfor each sector [11]. The number of sectors, the number of vir-tual sensors and their locations are the design parameters of thearray interpolation method.

C. Manifold Separation

Another elegant root-MUSIC-type method for arbitraryNUAs has been proposed in [19][21] and further developed in[22]. In [19], it has been shown that for any arbitrary array, the

steering vector can be approximated as

(12)

where is an matrix that depends only on the arrayparameters and

(13)

is an Vandermonde vector which depends only on and. The parameter characterizes the accuracy of the ap-

proximation in (12). Specifically, (12) becomes exact for, and the accuracy of the approximation in (12) improves

when increasing .Based on the results of [19], it has been proposed in [20] and[22] to use (12) with some finite to approximate the MUSICnull-spectrum function as

(14)

where, in contrast to (11), and the degree of the poly-nomial is . It has been suggested in [20] to obtain thesignal DOAs from the largest-magnitude roots of locatedinside the unit circle, in the way similar to that used in the con-ventional root-MUSIC algorithm.

Several methods to compute the matrix have been dis-cussed in [20] and [22] including the least-squares (LS) tech-

nique and another approach that determines each element ofvia the inverse discrete Fourier transform (IDFT) of differentcomponents of the steering vector taken at different an-gles. The LS technique is optimal in the sense that it minimizesthe manifold approximation error.

The parameter should be taken large enough to obtain an

acceptable DOA estimation performance. It has been suggestedin [19] to use as a rule of thumb

(15)

where is the signal wavenumber and is the largest dis-tance between the array sensors and the origin of the coordinatesystem.

III. PERFORMANCE ANALYSIS OF THE MANIFOLDSEPARATION TECHNIQUE

In this section, we present the results of our asymptotic per-formance analysis of the MS technique. It is different from theearlier analysis presented in [22] in that it takes into account

the finite sample and the array manifold approximation erroreffects.1 Our results are asymptotic because they are based onthe assumptions of high SNR, large number of snapshots, andsufficiently small manifold approximation errors.

It can be shown (see the Appendix) that under the assumptionthat the signal-subspace eigenvalues are pairwise distinct fromeach other, the first-order approximation of the mean squareerror (MSE) of the estimate of the signal DOA is given by

(16)

where

(17)

and denotes the real part of a complex value.Interestingly, changing the definitions of the vectors andas

(18)

it can be readily shown (see the Appendix) that (16) can bealso used to characterize the performance of interpolated root-MUSIC. This fact discovers an interesting relationship betweenthe MS and array interpolation methods.

1

Note that the analysis in [22] does not take into account the finite sampleeffects, but instead it focusses on the manifold separation and array calibrationerrors.



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Note that if there is no truncation error (in the case of man-ifold separation) or interpolation error (in the case of array in-terpolation), then and . In the lattercase, (16) can be simplified to

(19)

Equation (19) means that if the approximation errors in (9)and (12) are negligible, the array interpolation and MS tech-niques lead to the same DOA estimation performance. More-over, (19) is equivalent to the expression derived in [4] for theconventional root-MUSIC estimator in the ULA case. In partic-ular, this means that if the conventional root-MUSIC and MStechniques are applied to a ULA, and if the approximation errorin (12) is negligible, both algorithms have the same asymptoticperformance. As the degrees of the root-MUSIC polynomial (orinterpolated root-MUSIC polynomial) and the MS polynomial

may be quite different and has to be chosen to guar-antee that the approximation error is negligible, these are quiteunexpected results.

Note that the results of our performance analysis can be usedas an alternative [and more judicious than in (15)] way to deter-mine the parameter in the MS technique.

IV. FOURIER-DOMAIN ROOT-MUSIC

In this section, we will develop an alternative root-MUSIC-type approach to DOA estimation in arbitrary NUAs. We willuse the fact that, according to (3), the MUSIC null-spectrumfunction (8) is periodic in with the period . Therefore, it

can be expressed using its Fourier series expansion as

(20)

where the Fourier coefficients are given by

(21)

Truncating the Fourier series in (20) to points, the

function can be approximated as

(22)

where the notation is used. It should be stressed herethat the definition of in (22) is the same as used in (14), but isdifferent from that used in (11).

Interestingly, the expression (22) for the polynomial is

related to that for the MS polynomial in (14). Indeed, thelatter polynomial can be rewritten in the form similar to that of

(22) with replaced by and with the following specificvalues of :

(23)The following lemma holds for the function in (22).

Lemma 1: Let

(24)

where are arbitrary coefficients. Then,

with the equality if and only if

where the coefficients are defined by (21).Proof: See [27] where this lemma is proven for a more

general case.From Lemma 1, it follows that, if the degrees and

of the polynomials and , respectively, are equal toeach other, then the polynomial coefficients (21) provide betterapproximation of the original MUSIC null-spectrum than theMS technique. Therefore, it may be expected that the FD root-MUSIC technique will provide an improved DOA estimationperformance as compared to the MS technique.

The Fourier series coefficientscan be approximately obtained by means

of the discrete Fourier transform (DFT) as

(25)

where .Although in this case a close approximation of the original

Fourier series coefficients can be achieved in a computa-tionally efficient way, the resulting DFT coefficients will be

different from due to aliasing effects introduced by samplingthe null-spectrum function . Using such a DFT approxima-tion of , the final expression for the FD root-MUSIC poly-nomial can be written as

(26)

According to (26), the degree of is . Next, let

us prove that the roots of (26) satisfy the so-called conjugatereciprocity property, that is, if is a root of , then



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is also a root of this polynomial. Assuming that is a rootof and taking into account that , we obtain

(27)

that is, we have proved that is also a root of .As the structures of the manifold separation and FD root-

MUSIC polynomials are related, we may expect that the re-quired value of is approximately equal to that of and,therefore, the results of our asymptotic performance analysis ofthe MS technique can be used to determine as well.

The functions and are non-negative by definition,which is not the case for the function , that is, maytake values that are slightly below zero in some of its minima.Therefore, a different procedure has to be used to estimate thesignal DOAs from the roots of the polynomial . The signalroots of can be divided into two separate groups. The firstgroup contains the root pairs such that the roots of each pair lieexactly on the unit circle close to each other. These root pairs are

caused by the two sign changes of whenever takesa negative value.2 The second group contains the roots that donot belong to the unit circle. These roots appear in conjugatereciprocal pairs and have the same nature as the roots of theconventional root-MUSIC polynomial.

In Fig. 1, we schematically depict the root locations for thesetwo different groups of roots. The root pairs of the first group ofroots (i.e, roots located exactly on the unit circle) can be usedfor estimating the corresponding source DOAs by averaging theroots in each root pair. The rest of the source DOAs can be esti-mated in the standard way (similar to that used in conventionalroot-MUSIC) from the roots of the second group.

Summarizing, the source DOAs can be estimated from the

roots of using the following procedure: Step 1: Take the root closest to the unit circle. Identify

whether it belongs to the first or second group by checkingwhether its conjugate reciprocal value is another root.

Step 2: If this root belongs to the first group, then estimatethe source DOA from the average of this root and its closestneighbor, and drop both these roots. Go to Step 4.

Step 3: If this root belongs to the second group, then useit to estimate the source DOA and then drop both this rootand its conjugate reciprocal pair.

Step 4: If less than DOAs have been estimated, then goto Step 1. Otherwise, stop.

2Note that these roots do not appear in conjugate reciprocal pairs as they lieexactly on the unit circle.

Fig. 1. Signal root locations in the case when the MUSIC null-spectrum func-tion has both a negative minimum value (right part of the figure) and positiveminimumvalue(left part of thefigure). The insetsdepictcorresponding MUSICnull-spectrum functions.

In [20], an alternative variant of the MS technique has beenproposed whose essence is to use a large matrix but then totruncate the resulting polynomial. Note that the similar approachcan be directly applied to FD root-MUSIC as well, that is, a largenumber of null-spectrum samples can be used in the DFT, andthen the resulting polynomial can be truncated. In both cases,the truncated polynomials may have negative values at the unitcircle and our root selection algorithm should be used to esti-mate the source DOAs from the two groups of the signal roots.

V. FOURIER-DOMAIN LINE-SEARCH MUSICThe FD root-MUSIC polynomial has the order of

. As the value of has to be sufficiently high towarrant that the truncation errors are small, the computationalcomplexity of finding the roots of may be rather high. Toreduce the computational complexity of the FD root-MUSICapproach in such cases, let us consider a modification of thistechnique that avoids the polynomial rooting step by replacingit with a computationally simple line search procedure. The keyidea of this modification follows the work of [23].

Let us use in(26) zero-padding to values whereis the required number of points of the null-spectrum (note that,

as a rule, ). Then, the FD root-MUSIC polynomial canbe rewritten as

(28)

where

ifif .

(29)

Hence, a total of uniform in the intervalsamples of the null-spectrum can be obtained by directlyapplying IDFT to the zero-padded sequence of the DFT coeffi-cients , . As a result, no polynomial

rooting is needed anymore as the polynomial rooting step is re-placed by a line search over .



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We stress that there is a substantial computational differencebetween the conventional spectral MUSIC algorithm and FDline-search MUSIC. In particular, the spectral search in (8) re-

quires to evaluate the matrix-vector product (or, alter-

natively, ) for each value of involved in this search,while the proposed approach uses IDFT (that can be computed

in a numerically efficient way using FFT) to obtain the null-spectrum for all values of . As a result, the FD line-searchMUSIC approach allows to compute with a substantiallyreduced complexity as compared to the conventional computa-tion of the MUSIC null-spectrum function.

Note that FD line-search MUSIC can be also interpretedas a computationally efficient method for interpolating theMUSIC null-spectrum function which is optimal in the fol-lowing sense: If the aliasing that results from the originalsampling of the MUSIC null-spectrum is negligible, then theresulting null-spectrum function optimally approximatesthe original MUSIC null-spectrum in the sense of Lemma 1.

VI. FOURIER-DOMAIN WEIGHTED LEAST-SQUARESROOT-MUSIC

As follows from Lemma 1, the Fourier series coefficientsprovide the best (in the -norm sense) approximation of theMUSIC null-spectrum within the class of harmonic ex-pansion functions (24). However, to estimate the signal DOAsit is sufficient and most important to have a close approxima-tion of the MUSIC null-spectrum function in the vicinity of thetrue DOAs, i.e., in the areas where the null-spectrum has itsminima. Using this idea, let us take unequal positive weightsfor the samples of the function , so that largest weights are

assigned to smallest values of (that likely correspond tothe angular areas where the sources are located) and, vice versa,lowest weights are assigned to largest values of . This leadsto the following weighted least squares problem

(30)

where are the weight coefficients, and and are,respectively, the number of the MUSIC null-spectrum samplesand the number of the expansion coefficients used in (30).

It will be assumed that .A natural choice of the weightcoefficients to stress low valuesof is given by

(31)

The constraints in (30) are used to guarantee that the roots ofthe resulting FD weighted least squares (FDWLS) root-MUSICpolynomial

(32)

occur in conjugate reciprocal pairs, whereare the optimal coefficients that solve (30).

Taking into account the equality constraints in (30), thisproblem can be rewritten in the following equivalent uncon-strained form:

(33)

Let us use the notation

where denotes the imaginary part of a complex value. Also,let the matrix be defined as

where is a vector of ones and the entries of thematrix are defined as

(34)

Then, the problem (33) can be written as

(35)

where .The solution to (35) can be expressed as

(36)

Equation (36) determines the coefficients of the FDWLS root-MUSIC polynomial (32). The roots of this polynomial obey theconjugate reciprocity property, which can be proven in the sameway as in (27), taking into account that . Asthe function can take negative values, the same procedureas in FD root-MUSIC has to be used to estimate the signal DOAsfrom the roots of the FDWLS root-MUSIC polynomial.

VII. COMPUTATIONAL COMPLEXITY ANALYSIS

The orders of computational complexities of the proposedFD root-MUSIC, FD line-search MUSIC, and FDWLS root-MUSIC algorithms are compared in Table I with that of the con-

ventional spectral MUSIC, interpolated root-MUSIC, and MStechniques. In this table, we assume that the number of virtualsensors in the interpolated root-MUSIC technique is equal tothe number of the actual sensors , and that the FDroot-MUSIC, FDWLS root-MUSIC, and MS polynomials areof the same degree . The number of angular sectors in the interpolated root-MUSIC technique is de-noted as .

Note that all the methods in Table I include the eigendecom-positionstep whichis representedby the term . The com-putation of samples of the MUSIC null-spectrum function re-quires operations. The complexities to compute theMS and FD root-MUSIC polynomial coefficients are given by

and , respectively.As there is a variety of computationally efficient polynomial



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TABLE ITHE ORDERS OF COMPUTATIONAL COMPLEXITIES OF SPECTRAL

MUSIC, INTERPOLATED ROOT-MUSIC, THE MS TECHNIQUE, AND THEPROPOSED THREE TECHNIQUES

rooting algorithms with the cubic and lower complexities [24],

[25], the orders of complexity of the rooting steps are not ex-plicitly shown in the table, but are assumed to be not higher than

. Moreover, as the matrix inversion complexityin FDWLS root-MUSIC dominates the complexity of the poly-nomial rooting step, the latter step is not included in the table.

As can be seen from Table I, depending on the scenario andarray parameters, the proposed techniques offer different trade-offs in terms of computational complexity. For example, in thetypicalcasewhen , the computationalcomplexityof FDroot-MUSIC is comparable to that of the MS technique, and thecomplexity of FDWLS root-MUSIC is somewhat higher thanthat of the other root-MUSIC-type techniques. However, in thetypical for spectral search-based DOA estimation case of large

, the proposed techniques can offer a signif-icantly reduced complexity with respect to the spectral MUSICmethod.

VIII. SIMULATION RESULTS

Throughout our simulations, 1000 independent Monte Carloruns have been used in each example. As the LS technique hasthe lowest manifold approximation error as compared to theother techniquesto obtain , ithasbeen usedin the MSmethodin all simulation examples. The performance curves have beenalways averaged over the sources.

In our first example, we validate the results of our asymptoticperformance analysis of the MS technique. A randomly gener-ated (but fixed throughout all simulation runs) NUA ofsensors is used. The sensor locations are depicted in Fig. 2. Twoequally powered signal sources are assumed to impinge on thearray from the directions and . The numberofsnapshots toestimatethe array covariance matrixis .

Fig. 3 displays the DOA estimation root-mean-square-errors(RMSEs) of the MS approach versus the sensor signal-to-noise-ratio (SNR) for different values of . In this figure, the ana-lytical performance curves are computed by means of (16).

Fig. 3 clearly demonstrates that our first-order performanceanalysis accurately predicts the performance of the MS tech-

nique in the asymptotic domain. However, as the first-order per-formance analysis is valid only for sufficiently small estimation

Fig. 2. Array geometry.

Fig. 3. DOA estimation RMSEs of the MS technique versus SNR forK =

1 0 0 , = 0 1 0 , and = 2 0 . The array of Fig. 2 is used.

errors (see the Appendix for details), our theoretical results be-come inaccurate in the threshold domain (for SNR dB). Ascan be observed from Fig. 3, the performance curves saturate athigh SNRs. This saturation is caused by the approximation errorin (12) and (14). As can be expected, the negative effect of theapproximation error reduces when increasing .

In our second example, we compare the DOA estimation

performances of FD root-MUSIC, FD line-search MUSIC,FDWLS root-MUSIC, and the MS technique in the case of(note that such a choice guar-

antees that the asymptotic complexity of FD root-MUSIC isnot higher than that of the MS technique). For FD line-searchMUSIC, has been chosen to make sure that the errorscaused by the angular grid step size are negligible.3 In theFDWLS root-MUSIC approach, we useuniform samples of the MUSIC null-spectrum function. Allother parameters are chosen as in the first example.

Fig. 4 displays the DOA estimation RMSEs of the methodstested versus the SNR. The stochastic Cramr-Rao bound

3

Note that even though such a choice ofJ

is impractical because of a highcomputational complexity, our objectivehere is to demonstrate the inherent per-formance of FD line-search MUSIC without the effect of limited step size.



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Fig. 4. DOA estimation RMSEs versus SNR forK = 1 0 0

, = 0 1 0

, = 2 0 , M = 1 9 , J = 1 0 , and Q = 7 4 . The array of Fig. 2 is used.

(CRB) is also shown. From this figure, it can be observed thatthe FD root-MUSIC, FDWLS root-MUSIC, and FD line-searchMUSIC methods substantially outperform the MS technique.As expected, the performance of the FDWLS root-MUSICapproach is the highest among the three proposed methods.The performances of the FD root-MUSIC and FD line-searchMUSIC techniques are very similar to each other. The fact thatthe proposed root-MUSIC-type algorithms outperform the MStechnique can be explained by their lower level of truncationerrors as compared to that of the approximation errors in theMS technique.

In our third example, we assume two closely spaced signalsthat impinge on the array of Fig. 2 from the directionsand . All other parameters are chosen as in the previousexample. Fig. 5 displays the DOA estimation RMSEs of themethods tested versus the SNR. This figure shows that, again,the FD root-MUSIC, FD line-search MUSIC, and FDWLS root-MUSIC techniques outperform the MS technique in the highSNR region where the FDWLS root-MUSIC approach has thebest performance among the methods tested.

In our fourth example, we examine the performance ofthe methods tested versus the angular separation between thesources. Fig. 6 shows the DOA estimation RMSEs of the

methods tested versus the source angular separation, wherethe DOA of the first source is varied while the DOA of thesecond source is fixed and equal to . In this figure,SNR dB and all other parameters are chosen as in theprevious example. As follows from Fig. 6, FD root-MUSIC andFDWLS root-MUSIC perform consistently better than the MStechnique, with most pronounced performance improvementsachieved at medium/large source angular spacings. Also, theFD line-search MUSIC outperforms the MS technique in thecase when the source angular spacing is larger than 7 . How-ever, the performance of FD line-search MUSIC gets worsethan that of the MS technique in the case of closely spacedsources because of the threshold effect.

Our fifth example studies the impact of the number of snap-shots on the methods tested. In this example, and


, = 1 5

, =

2 0

,M = 1 9

,J = 3 : 6 1 1 0

, andQ = 7 4

. The array of Fig. 2 is used.

Fig. 6. DOA estimation RMSEs versus 0

forK = 1 0 0

, SNR= 2 0

dB, = 2 0 , M = 1 9 , J = 3 : 6 1 1 0 , and Q = 7 4 . The array of Fig. 2 is used.

are taken, and all other parameters are chosen as in theprevious example. Fig. 7 displays the DOA estimation RMSEsversus . This figure shows that for high values of we obtain

rather similar results as for high SNR values, i.e., the proposedmethods outperform the MS technique.Note that Figs. 57 also demonstrate that the rooting-based

methods have a better resolution threshold than the spectralsearch based FD line-search MUSIC technique. This can beexplained by the fact that the rooting based methods are insen-sitive to the radial root errors [4].

In our sixth example, we generalize the above results in thesense that we now consider the case when different (randomlygenerated) array geometries are used in each simulation run. Inthis example, the locations of 8 array sensors have beendrawn uniformly from the interior of a circle of radius . Theperformances of the FD root-MUSIC and the MS techniques are

compared for SNR dB and snapshots. The an-gular spacing between the two sources has been assumed in this



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Fig. 7. DOA estimation RMSEs versus K for SNR = 2 0 dB, = 1 5 , = 2 0

,M = 1 9

,J = 3 : 6 1 1 0

, andQ = 7 4

. The array of Fig. 2 is used.

Fig. 8. Scatter plot of the RMSEs of the MS and FD root-MUSIC techniquesfor N = 8 , K = 2 0 0 ,SNR = 2 0 dB, 0 = 3 0 ,and M = 1 5 .Randomlygenerated array geometry.

example to be equal to 30 . In Fig. 8, a scatter plot for the DOA

estimation RMSEs of the MS and FD root-MUSIC techniquesis shown for . Each dot of this plot corresponds toone particular realization of the array geometry. Fig. 8 demon-strates substantial performance improvements achieved by FDroot-MUSIC as compared to the MS technique. Note that thecases where FD root-MUSIC outperforms the MS techniquecorrespond to all the dots that are located above the diagonalline. The average RMSE improvement in this example is ap-proximately given by a factor of 4.4.

In the last example, we study the impact of the parameteron the performance of the methods

tested. In this example, the sources are assumed to be spaced 5apart from each other, the number of snapshots is ,

and the locations of array sensors have been drawnuniformly from the interior of a circle of radius . All other


, SNR= 2 0

dB, 0 = 5 , and Q = 2 ( 2 M 0 1 ) . Randomly generated array geometry.

parameters are chosen as in the previous example. In Fig. 9, theDOA estimation RMSEs are plotted versus the value of . Allcurves in this figure have been averaged over 1000 realizationsof the random array geometry. Fig. 9 clearly shows that FD root-MUSIC and FDWLS root-MUSIC require reduced degrees oftheir polynomials as compared to the MS technique, to achievethe same performance.

In summary, the proposed DOA estimation algorithms havebeen shown to be highly competitive alternatives to the MS tech-nique in that these algorithms offer substantially improved per-formance-to-complexity tradeoffs.

IX. CONCLUSION

In this paper, the problem of spectral-search-free direc-tion-of-arrival estimation in arbitrary nonuniform sensor arrayshas been addressed. First of all, an asymptotic performanceanalysis of the popular manifold separation technique of[19][22] has been presented.

Then, a novel root-MUSIC-type approach to DOA estimationin sensor arrays of arbitrary geometry has been developed. Thisapproach has been referred to as Fourier-domain root-MUSIC.It exploits the fact that the null-spectrum MUSIC function is pe-

riodic in angle, and uses the truncated Fourier series expansionof this function to reformulate the DOA estimation problem interms of polynomial rooting.

To avoid the polynomial rooting step and further reduce thecomputational complexity, a zero-padded inverse Fourier trans-form has been applied to the Fourier-domain root-MUSIC poly-nomial to compute the null-spectrum and estimate the sourcedirections-of-arrival by means of a simple line search.

To further improve the performance of the proposed Fourier-domain root-MUSIC technique, a weighted least squares ap-proach is used to refine the polynomial coefficients.

It has been demonstrated through simulations with differentarray configurations that the proposed techniques offer attrac-

tive alternatives to the current state-of-the-art DOA estimationmethods applicable to arrays with arbitrary geometries.



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Fig. 10. Root geometry.

APPENDIXPROOF OF EQUATION (16)

Let us now derive (16) for the MS technique. Using a closemathematical similarity between the MS and array interpola-tion techniques, the corresponding derivation for the interpo-lated root-MUSIC technique can be obtained by following the

same steps.Taking into account the conjugate reciprocity of the roots, thefunction can be written as

(37)

where and arethe roots of , and the scalar canbeobtainedfromequating(14) and (37). The hat sign stresses thatand are functions of the sample noise-subspace matrix .

If there are no manifold approximation and finite-sample er-rors, the function will be equal to zero for .

The manifold approximation and finite-sample errors, however,lead to the root (see Fig. 10)

(38)

From Fig. 10, we obtain

(39)

where and are the magnitude and phase of

(40)

respectively. If , (39) yields

(41)

From (14) and (37), we obtain

(42)

where . Using (38) and (40),for and its first and second derivatives we have

(43)

(44)(45)

Using (43) and (44), we obtain for the first derivative ofthat

(46)

Therefore, (41) can be reformulated as

(47)

Similarly, from (43)(45) we obtain that the second derivativeof at is given by

(48)

Inserting (48) into (47), we obtain

(49)

A first-order approximation of the second derivative of at

yields

(50)

Equations (49) and (50) lead to the following expression:

(51)

Let us introduce the eigenvector estimation error

(52)

The following relations will be used [26]:

(53)



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(54)

(55)

where denotes the Kronecker delta.From (14), we have

(56)

where

(57)

(58)

(59)

Hence, the numerator of (51) is given by

(60)

Neglecting all terms that contain higher than the second ordersof , we obtain

(61)

As is constant, (61) can be simplified as

(62)

Using (53)(55), we have

(63)

(64)

(65)

(66)

Inserting (63)(66) into (62) and then substituting (62) into (51),we prove (16).

REFERENCES

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[2] G. Bienvenu and L. Kopp, Adaptivity to background noise spatial co-herence for high resolution passive methods,in Proc.Int. Conf. Acous-tics, Speech, Signal Processing (ICASSP), Denver, CO, Apr. 1980, pp.307310.

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[4] B. D. Rao and K. V. S. Hari, Performance analysis of root-MUSIC,IEEE Trans. Acoust., Speech, Signal Process., vol. 37, pp. 19391949,Dec. 1989.

[5] A. Paulraj, R. Roy, and T. Kailath, A subspace rotation approach tosignal parameter estimation, Proc. IEEE, vol. 74, pp. 10441046, Jul.1986.

[6] C. P. Mathews and M. D. Zoltowski, Eigenstructure techniques for2-D angle estimation with uniformcircular arrays,IEEE Trans. SignalProcess., vol. 42, pp. 23952407, Sep. 1994.

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[15] A. B. Gershman and J. F. Bhme, A note on most favorable arraygeometries for DOA estimation and array interpolation, IEEE Signal

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[22] F. Belloni, A. Richter, andV. Koivunen, DOA estimation viamanifoldseparation for arbitrary array structures, IEEE Trans. Signal Process.,vol. 55, no. 10, pp. 48004810, Oct. 2007.

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[27] W. Rudin , Principles of Mathematical Analysis. New York: Mc-Graw-Hill, 1964.

Michael Rbsamen (S05) was born in Germanyin March 1980. He received the Dipl.-Ing. degreein electrical engineering from Aachen University of

Technology, Germany, in 2006. His diploma thesiswas on spatial distributions of wireless networks.

From October 2004 until July 2005, he held a re-search scholarship from Bell Laboratories, CrawfordHill, NJ, where he was working on MLSE receiversfor high spectral efficiency of optical communicationsystems. Since June 2006, he hasbeen with theDarm-stadt Universityof Technology,Darmstadt, Germany,

as a Ph.D. student. His doctoral thesis will be on advanced array processingalgorithms.

Mr. Rbsamen was student contest finalist of the IEEE International Confer-ence on Acoustics, Speech, and Signal Processing (International Conference onAcoustics, Speech and Signal Processing (ICASSP)), Las Vegas, NV, in March2008.

Alex B. Gershman (M97SM98F06) receivedthe Diploma and Ph.D. degrees in radiophysicsand electronics from the Nizhny Novgorod StateUniversity, Russia, in 1984 and 1990, respectively.

From 1984 to 1999, he held several full-timeand visiting research appointments in Russia,Switzerland, and Germany. In 1999, he joined theDepartment of Electrical and Computer Engineering,

McMaster University, Hamilton, ON, Canada, wherehe became a Professor in 2002. Since April 2005,he has been with the Darmstadt University of Tech-

nology, Darmstadt, Germany, as a Professor of communication systems. Hisresearch interests are in the area of signal processing and communications withthe primary emphasis on array processing, beamforming, MIMO and multiusercommunications, and estimation and detection theory.

Dr. Gershman is the recipient of several awards, including the 2004 IEEESignal Processing Society (SPS) Best Paper Award; the IEEE Aerospace andElectronic Systems Society (AESS) Barry Carlton Award for the best paperpublished in 2004, the 2002 Young Explorers Prize from the Canadian Insti-tute for Advanced Research (CIAR); the 2001 Wolfgang Paul Award from theAlexander von Humboldt Foundation, Germany; and the 2000 Premiers Re-search Excellence Award, Ontario, Canada. He also coauthored the paper thatreceived the 2005 IEEE SPS Young Author Best Paper Award. He was Ed-itor-in-Chief for the IEEE SIGNAL PROCESSING LETTERS from 2006 to 2008,the Chair of the Sensor Array and Multichannel (SAM) Technical Committee of

the IEEE SPS from 2006 to 2008, and a Member of the IEEE SPS Publications,Conference, andTechnical DirectionsBoards from 2006to 2008. He is currentlya member of the IEEE SPS Board of Governors. He is on the Editorial Board ofthe IEEE Signal Processing Magazine and several other technical journals. Hewas Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from1999 to 2005; the Technical Co-Chair of the IEEE International Symposium onSignal Processing and Information Technology (ISSPIT), Darmstadt, Germany,December 2003; the Technical Co-Chair of the Fourth IEEE Sensor Array andMultichannel (SAM) Signal Processing Workshop, Waltham, MA, June 2006;the General Co-Chair of the First IEEE Workshop on Computational Advancesin Multi-Sensor Adaptive Processing (CAMSAP), Puerto Vallarta, Mexico, De-cember 2005; the Tutorial Chair of EUSIPCO, Florence, Italy, September 2006;and the General Co-Chair of the Fifth IEEE SAM Workshop, Darmstadt, Ger-many, July 2008.

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