2 decades of array signal processing

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ecades of Array

Signal Processin Research

The ParametricApproach

HAMID KRlMand MATS VIBERG

OSteven Huntfrhe Image Bank

stimation problems in theoretical as well as appliedstatistics have long been of great research interestE iven their importance in a great variety of applica-

tions. Parameter estimation has particularly been an area offocus by applied statisticians and engineers as problemsrequired ever improving performance [7, 8, 91. Many tech-niques were the result of an attempt by researchers to gobeyond the classical Fourier-limit.

As applications expanded, the interest in accurately esti-mating relevant temporal as well as spatial parameters grew.Sensor array signal processing emerged as an active area ofresearch and was centered on the ability tofuse data collectedat several sensors in order to carryout a given estimation task(space-time processing). This framework, as will be de-scribed in more detail below, uses to advantage prior infor-mation on the data acquisition system (i.e., array geometry,sensor characteristics, etc.). The methods have proven usefulfor solving several real-world problems, perhaps most nota-bly source localization in radar and sonar. Other more recentapplications are discussed in later sections.

The first approach to carrying out space-time processingof data sampled at am array of sensors was spatial filtering or

beamforming. The conventional (Bartlett) beamformer datesback to the second world-war, and is a mere application ofFourier-based spectral analysis to spatio-temporally sampleddata. Later. adaptive bearmformers [6, 25,451 and classicaltime delay estimation techniques [81 were applied to enhanceone’s ability to resolve closely spaced signal sources. Thespatial filtering approach, however, suffers from fundamentallimitations: its performance, in particular, is directly depend-ent upon the physical size of the array (the aperture), regard-less of the available data collection time and signal-to-noiseratio (SNR). From a statistical point of view, the classicaltechniques can be seen as spatial extensions of spec tral Wie-ner filtering [1501 (or matchedfifiltering).

The extension of the time-delay estimation methods tomore than one signal (these techniques originally used onlytwo sensors), and the limited resolution of beamformingtogether with an increasing number of novel applications,renewed interest of researchers in statistical signal process-ing. We might add at this stage, that the word resolution is

used in a rather informal way. It generally refers to the abilityto distinguish closely sp,aced signal sources. One typicallyrefers to some spectral-like measure, which would exhibit

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peaks at the locations of the sources. Whenever there are twopeaks near two actual emitters, the latter are said to beresolved. However, for parametric techniques, the intuitivenotion of resolution is non-trivial to define in precise terms.This in turn, resulted in the emergence of the parameterestimation approach as an active research area. Importantinspirations for the subsequent effort include the MaximumEntropy (ME) spectral estimation method in geophysics by[23] and early applications of the maximum likelihood prin-ciple [8 1, 1061. The introduction of subspace-based estima-tion techniques [13, 1051 marked the beginning of a new erain the sensor array signal processing literature. Th e subspace-based approach relies on certain geometrical properties of theassumed data model, resulting in a resolution capabilitywhich (in theory) is not limited by the array aperture, pro-vided the data collection time and/or SNR are sufficientlylarge and assuming the data model accurately reflects theexperimental scenario.

The quintessential goal of sensor array signal processingis the estimation ofparametersby fusing temporal and spatial

information, captured via sampling a wavefield with a set ofjudiciously placed antenna sensors. The wavefield is as-sumed to be generated by a finite number of emitters, andcontains information about signal parameters characterizi ngthe emitters. Given the great number of existing applicationsfor which the above problem formulation is relevant, and thenumber of newly emerging ones, we feel that a review of thearea, with the hindsight and perspective provided by time, isin order. The focus is on parameter estimation methods, andmany relevant problems are only briefly mentioned. Themanuscript is clearly not meant to be exhaustive, but ratheras a broad review of the area, and more importantly as a guidefor a first time exposure to an interested reader. We deliber-ately emphasize the relatively more recent subspace-based

methods in relation to bearrzforming, for which the reader isreferred for more in depth treatment to the excellent, and insome sense complementary, review by Van Veen and Buck-ley [133]. For more extended presentations, the reader isreferred to textbooks such as [SO,52, 5 8 , 1021.

The balance of this article consists of the backgroundmaterial and of the basic problem formulation. Then weintroduce spectral-based algorithmic solutions to the signalparameter estimation problem. We contrast these suboptimalsolutions to parametric methods. Techniques derived frommaximum likelihood principles as well as geometric argu-ments are covered. Later, a number of more specializedresearch topics are briefly reviewed. Then, we look at anumber of real-world problems for which sensor array proc-essing methods have been applied. We also include an exam-ple with real experimental data involving closely spacedemitters and highly correlated signals, as well as a manufac-turing application example.

A studentlpractitioner who is somewhat familiar with thefield might read the various sections sequentially. For afirst-time exposure, however, it may be best to scan theapplications section before the description and somewhatmore mathematical treatment of the algorithms are discussed.

Background and For

In this section, we motivate the data model assumed through-out this paper, via its derivation from first principles inphysics. Statistical assumptions about data collection arestated and basic geometrical properties of the model arereviewed.

Wave Propagation

Many physical phenomena a re either a result of waves propa-gating through a medium (displacement of molecules) orexhibit a wave-like physical manifestation. A wave propaga-tion which may take various forms (with variations depend-ing on the phenomenon and on the medium, e.g. anelectro-magnetic (EM) wave in free space or an acousticwave in a pipe), generally follows from the homogeneoussolution of the wave equation.

The models of interest in this paper may equally apply to

an EM wave as well as to an acoustic wave (e.g., SONAR).Given that the propagation model is fundamentally the same,we will for analytical expediency, show that it can followfrom the solution of Maxwell’s equations, which, clearly areonly valid for EM waves. In empty space (no current orcharge) the following holds

aBV X E = - - at

(3)

(4)

where ., and x, respectively, denote the “divergence” and“curl.” Further, B is the magnetic induction, E is the electric

field, whereas po and EO are the magnetic and dielectricconstants. Invoking Eq. 1 the following curl property results,

( 5 )x (V x E) = V(V .E) - V*E = -VE .

Using Eqs. 3 and 4 leads to

a a2EV X ( V x E ) = - - ( V x B ) = - &t o /J - , t2

which, when combined with Eq. 5, yields the fundamentalwave equation

(7)

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The constant c is generally referred to as the speed ofpropagation, and for EM-waves in free space it follows from the

above derivation c = 1/ & 3 x 10’m / s . The homogene-

ous (no forcing function) wave equation (Eq. 7) constitutes thephysical motivation for our assumed data model. This is regard-less of the type of wave or medium (EM or acoustic). In someapplications, the underlying physics are irrelevant, it is merelythe mathematical structure of the data model that counts.

Though Eq. 7 is a vector equation, we only consider one ofits components, say E(r,t)where r is the radius vector. It willlater be assumed that the measured sensor outputs are propor-tional to E(r, t) . nterestingly enough, any field of the formE(r,t)= A d a )atisfies Eq. 7, , provided Ia1 = l/c, with “T’ denoting

transposition. Througlh its dependence on t-rTa only, the solu-

tion can be interpreted as a wave traveling in the direction a,with the speed of propagation l d a1 = e. For the latter reason, ais referred to as the slowness vector. The chief interest herein isin narrowband (This is not really a restriction, since any signalcan be expressed as a linear combination of narrowband com-ponents.) forcing functions. The details of generating such a

forcing function (i.e. radiation of an antenna) can be found inthe classic book by Jordan [59]. In complex notation (see e.g.[63, Section 15.31) and taking the origin as a reference, anarrowband transmitted waveform can be expressed as (upper-case and lowercase Greek letters are to be understood as vectorsor matrices within their context)

E(0,t)= s ( t ) i o t ,

where s( t ) s slowly time-varying compared to the carrier dot .For Irl<< c / B ,where B is the bandwidth of s ( t ) , we can write

In the last equa1it:y the so-called wave-vector k = a m wasintroduced, and its magnitude 1kl = k = o /c is the wave-number. One can also write k = 2z/h, where h is the wave-length. Note that k also points in the direction of propagation.For example, in the xy-plane we have

k = k(cos0 sine)‘ , (9)

where 0 is the direction of propagation, defined counter-clockwise relative the x-axis (Fig. 1).

It should be noted that Eq. 8 implicitly assumed far-fieldconditions, since an isotropic (Isotropic refers to uniform

propagation/transmission in all directions.) point sourcegives rise to a spherical traveling wave whose amplitude isinversely proportional to the distance to the source. All pointslying on the surface of a sphere of radius R will then share acommon phase andl are referred to as a wavefront. Thisindicates that the distance between the emitters and the re-ceiving antenna array determines whether the sphericity ofthe wave should be taken into account. The reader is referredto e.g., [lo, 241 for treatments of near field reception. Far-field receiving conditions imply that the radius of propaga-

tion is so large (compared to the physical size of the array)that a flat pilane of constant phase can be considered, thusresulting in a plane wave as indicated in Eq. 8. Though notnecessary, the latter will be our assumed working model forconvenience of exposition.

Note thalt a linear medium implies the validity of thesuperposition principle, and thus allows for more than onetraveling wave. Equation 8 carries both spatial and temporalinformation and represents an adequate model for distin-guishing signals with distinct spatio-temporal parameters.These may #come n various forms, such as DOA (in generalazimuth and elevation), signal polarization (if more than onecomponent of the wave is taken into account), transmittedwaveforms, temporal frequency etc. Each emitter is generallyassociated with a set of such characteristics. The interest inunfolding the signal parameters forms the essence of sensorarray signal processing as presented herein, and continues tobe an important and ac tive topic of research.

Parametr ic Data Model

Most modern approaches to signal processing are model-based, in the sense that they rely on certain assumptions madeon the observed data. In this section we describe the prevail-ing model used in the rernainder of this article. A sensor isrepresented as a point receiver at given spatial coordinates.

TIn the 2D-case and as sholwn in Fig. 1, we have ri = (xi yi) .Using Eqs, 8 and 9, the field measured at sensor 1 and due toa source at az imuthal DOA 8 is given by

If a flat Frequency response, say g@) , is assumed for thesensor 1 over the signal hnd wid th, its measured output will

be proportilonal to the field at r i. Dropping the carrier termdot or convenience (in practice, the s ignal is usually down-converted to baseband before sampling), the output is mod-eled by

Referring to Eq. 8, wiz see that Eq. 11 requires that thearray aperture (i.e. the physical size measured in wave-lengths) be much less than the inverse relative bandwidth

I y t

X

I . Two-dimensional array geometry

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. Uniform Linear Array geometry

(f /B). n the array processing literature, this is referred to asthe narrowband assumption. For an L-element antenna arrayof arbitrary geometry, the array output vector is obtained as

~ ( t ) a(B)s(t).

A single signal at the DOA 8, thus results in a scalarmultiple of the steering vector (Other popular names for a(0)include action vector, array propagation vector and signal

replica vector.) a(B)= U , ( € ) ) ...,U,(€))]' as the array output.

Common array geometries are depicted in Figs. 2 and 3. Forthe uniform linear array (ULA) we have ri = ( ( I - 1)d O)

and assuming that al l elements have the same directivity gl(0)

= , . = g ~ ( 0 )g(e) , the ULA steering vector takes the form(cf. (1 1))

where d denotes the inter-element distance. The radius vec-tors of the uniform circular array (UCA) have the form rz =

R(cos(2n(l- 1)lL) in(2n(l - l)/L))T, from which the form ofthe UCA steering vector can easily be derived. As previouslyalluded to, a signal source can be associated with a numberof characteristic parameters. For the sake of clarity and easeof presentation by referring to Figs. 2 and 3, we assume that8 is a real-valued scalar referred to as the DOA. For most ofthe discussed methods the extension to the multiple parame-ter per source case is straightforward.

As noted earlier, the superposition principle is applicableassuming a linear receiving system. If A4 signals impinge onan L-dimensional array from distinct DOAs 01, .., ,w, theoutput vector takes the form

where sm(t), m = 1, .., A4 denote the baseband signal wave-forms. The output equation can be put in a more compactform by defining a steering matrix and a vector of signalwaveforms as

UnqormCircular Array Geometry

In the presence of an additive noise n(t) we now get themodel commonly used in array processing

x(t) = A(B)s( t ) n(t) . (13)

The methods to be presented a ll require that M < L, whichis therefore assumed throughout the development. It is inter-esting to note that in the noiseless case, the array output isthen confined to an M-dimensional subspace of the complexL-space, which is spanned by the steering vectors. This is thesignal subspace, and this observation forms the basis ofsubspace-based methods .

The sensor outputs are appropriately pre-processed andsampled at arbitrary time instances, labeled t = 1,2,..,A7 forsimplicity. Clearly, the process x(t) can be viewed as amultichannel random process, whose characteristics can bewell understood from its first and second order statisticsdetermined by the underlying signals and noise. The pre-processing of the signal is often done in such a way that x(t)can be regarded as temporally white.

Assumptions

The signal parameters which are of interest in this article arespatial in nature, and thus require the cross-covariance infor-mation among the various sensors, i.e. the spatial covariancematrix

R = E{x(t)xH(t)} = AE(s(t)sH(t)}AH E{n(t)nH(t)) (14)

where E( } denotes statistical expectation,

E { s ( t ) s H ( t ) )P

is the source covariance matrix and

E{n(t)nH(t)} 02 1 (16)

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is the noise covariance matrix. The latter covariance structure

is a reflection of the noise having a common variance CJ atall sensors and being uncorrelated among all sensors. Suchnoise is usually termed spatially whit e, and is a reasonablemodel, for example receiver noise. However, other man-made noise sources need not result in spatial whiteness, inwhich case the noisle must be pre-whitened in many of the

methods to be described. More specifically, if the noisecovariance matrix is Q, he sensor outputs are multiplied by

(Q-"' denotes a Hermitian square-root factor of Q-')prior to further processing. The source covariance matrix, P,is often assumed to be nonsingular (a rank-deficient P, as inthe case of coherent signals, is discussed later) or near-singu-lar for highly correlated signals.

In the later development, the spectral factorization of Rwill be of central importance, and its positivity guarantees thefollowing representation,

2

R = A P A " + d I = U A U H , (17)

with U unitary and A.= diag{ hi , 12 , . ., h ~ )diagonal matrix

of real eigenvalues ordered such that hi 2 h2 2 2 h ~ 0.Observe that any vector orthogonal to A is an eigenvector of

R with the eigenvalue C J ~ . here are L - M linearly inde-pendent such vectors. Since the remaining eigenvalues are alllarger than 02, we can partition the eigenvaluehector pairsinto noise eigenvectors (corresponding to eigenvalues h ~ + i= . = 2 h~ = 02 ) nd signal eigenvectors (corresponding to

eigenvalues hi 2 2 M > CJ ). Hence, we can write

2where An = CJ I. Since all noise eigenvectors are orthogonalto A, the columns of Us must span the range space of Awhereas those of U, span its orthogonal complement (thenullspace of A H ) .Tlhe projection operators onto these s ignaland noise subspaces, are defined as

n u,u: = A ( A ~ A ) - ' A ~

nL unug= I - A ( A ~ A ) - ' A ~ ,

provided that the inverse in the expressions exists. It thenfollows

Problem Definition

The problem of central interest herein is that of estimating theDOAs of emitter signals impinging on a receiving array,when given a finite data set { ~ ( t ) ]bserved over t = 1,2, ...,N.As noted earlier, vie will primarily focus on reviewing anumber of techniques based on second-order statistics ofdata.

All of the earlier formulation assumed the existence ofexact quantities, i.e. infinite observation time. It is clear that

in practice only sample estimates which we denote by a hat ,i.e., *, are ;available. A natural estimate of R is the samplecovariance matrix

1

N * = I

R = Z x ( t ) x " ( t ) ,

for which (a spectral repiresentation similar to that of R isdefined as

A , . , . A A , .

R U ,A U: +U,AnUf.

This representation will1 be extensively used in the descrip-tion and implementation of the subspace-based estimationalgorithms. Indeed, if xit) is a stationary white Gaussianprocess wifh unknown structure (i.e., the data model (13) is

not used), then R and its eigen-elements are maximumlikelihood estimates of the corresponding exact quantities

Throughout the paper, the number of underlying signals,M , in the observed process is considered known. There are,

however, good and consiistent techniques for es timating theM signals present [31, 68, 108, 1441 in the event that suchinformation is not available (see also the "Additional Top-ics"). In the following two sec tions we discuss the best knownparameter estimation techniques, respectively classified asSpectral-Based and Parametric methods. Due to space limi-tations, a number of good variations which address specificaspects of the underlying problem and which have appearedin the literature are overlooked. Our reference list attempts tocover a portion of the gap which clearly can never be filled.

~41.

Summary of Estimators

Since the number of algorithms is extensive, it makes senseto give an overview of their properties already at this stage.The following "glossary" is useful for this purpose:

Coherent signals Two signals are coherent if one is a scaledand delayed version of the other.

Consistency An estimate is consis tent if it converges to thetrue value when the number of data tends to infinity.

Statistical efficiency A,n estimator is statistically efficientif it asyrnptotically attains the CramCr-Rao Bound (CRB),which is ; a lower bound on the covariance matrix of anyunbiased estimator (see e.g. [63]).

We will distinguish between methods that are applicableto arbitrary array geometries and those tha t require a uniformlinear array. Tables 1 and 2 summarize the message conveyedin the algorithm descriptions. The major computational re-quirements for each method are also included, assuming thatthe sample covariance matrix has already been acquired.Here, 1-D search means that the parameter estimates arecomputed from M one-dimensional searches over the pa-rameter space, whereas M-D search refers to a full M-dimen-

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Method Consistency Coherent ’ Statistical Computations1Signals Performance~ ~~

Bartlett M = 1 1 - 1-D Search

Capon No N o Poor 1-D Search

MUSIC Yes

,No Good EVD, 1-D Search

~~ ~

Min-Norm

DML

SML

WSF

1Method

IYes ~ No Good EVD, 1-D Search

Yes Yes , Good M-D Search

Yes Yes Efficient M-D Search

Yes Yes ~ Efficient EVD,M-D Search

~~

Consistency ~ Statistical Performance

~ Good

1CoherentISignalsComputations

EVD, polynomialRoot-Music

ESPRIT

IQML

Root-WSF

1Yes

Yes Yes I Good EV D

Yes Yes Good Iterative

Yes IYes , Efficient EVD,LS~

1Yes

sional numerical optimization. By a “good” statistical per-formance, we mean that the theoretical mean square error ofthe estimates is close to the CRB, typically within a few dBin practical scenarios.

Spectral-Based Algorithmic Solutions

As mentioned earlier, we classify the parameter estimationtechniques into two main categories, namely spectral-basedand parametric approaches. In the former, one forms somespectrum-like function of the parameter(s) of interest, e.g.,the DOA. The locations of the highest (separated) peaks ofthe function in question are recorded as the DOA estimates.Parametric techniques, on the other hand, require a simulta-neous search for all parameters of interest. The latter ap-proach often results in more accurate estimates, albeit at theexpense of an increased computational complexity.

Spectral-based methods which are discussed in this sec-tion, can be classified into beamforming techniques andsubspace-based methods.

Beamformi ng Techniques

The first attempt to automatically localize signal sourcesusing antenna arrays was through beamforming techniques.The idea is to “steer” the array in one direction at a time andmeasure the output power. The steering locations whichresult in maximum power yield the DOA estimates. The arrayresponse is steered by forming a linear combination of thesensor outputs

L

y ( t )= C w ; x , ( t ) W H X ( t ) ./= I

Given samples y (l ) ,y(2) ,..,y(A9, the output power is meas-ured by

where R is defined in (21). Different beamforming ap-proaches correspond to different choices of the weightingvector w. For an excellent review of beamforming methods ,we refer to [132].

Conventional BeamformerThe conventional (or Bartlett) beamformer is a natural exten-sion of classical Fourier-based spectral analysis [121 to sensorarray data. For an array of arbitrary geometry, this algorithmmaximizes the power of the beamforming output for a giveninput signal. Suppose we wish to maximize the output powerfrom a certain direction 8. Given a signal emanating fromdirection 8 , a measurement of the array output is corruptedby additive noise and written as

x ( t )= a(O)s(t)+n(t)

The problem of maximizing the output power is thenformulated as,

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m;xE{wHx(t)xH(t)w}maxwHE{x(t)xH(t)}w ( 25 )

where the assumptiton of spatially white noise is used. Toobtain a non-trivial solution, the norm of w is constrained toIwI = 1 when carrying out the above maximization. The re-

sulting solution is then

The above weight vector can be interpreted as a spatialfilter, which has been matched to the impinging signal. Intui-tively, the array weighting equalizes the delays (and possiblyattenuations) experienced by the signal on various sensors tomaximally combine their respective contribut ions.

Inserting the weighting vector Eq. 26 into Eq. 24, theclassical spatial spectrum is obtained

For a uniform linear array of isotropic sensors, the steering

vector a(0) takes the form (cf. (12))

where

is termed the electrical angle. By inserting Eq. 28 into Eq.

27, and noting that ja,,IA(0)12 = M , we obtain P B F ( ~ )s the

spatial analogue of the classical periodogram in temporaltime series analysis, see e.g. [96]. Unfortunately, the spatialspectrum shares the: same resolution limi tation as the perio-dogram. The standard beamwidth for a ULA is q B= 2x L ,

and sources whose electrical angles are closer than (I ~ ill

not be resolved by the conventional beamformer, regardlessof the available data quality. This point is illustrated in Figure4, where the beamforming spectrum is plotted versus theDOA in two different scenarios. A ULA of A4 = 10 sensorsof half-wavelength inter-element spacing (Such a UL A is

often termed a standard ULA, because d = n/k is the maxi-mum allowable eleiment separation to avoid ambiguities.) isused to separate two uncorrelated emitters, based on a batchof N = 100 data samples. The signal-to-noise ratio (SNR) forboth sources is 0 dlB. The beamwidth for such an array is2x 10 = 0.63, implying that sources need to be at least 12'apart in order to be separated by the beamformer. This is alsoverified in the figune, because for IO " separation, the sourcesare nearly (but not quite) resolved.

BEAMFORMINGSPECTRUM

I

I

50 611 70 80 90 100 110 120 130 140

DOA (deg)

BEAIL4FORMING S PECTRU M- -

0 9

I

i

ji

I

i

ii,

50 EO 70 80 90 100 110 120 130 140DOA (deg)

Normaliaed beamforming spectra versuyDOAfor two differentscenarios. True DOAs indicated by dotted vertical lines

Capon 's BeamformerIn an attempt to alleviate the limitat ions of the above beam-former, such as its resollving power of two sources spacedcloser than a beamwidth, researchers have proposed numer-ou s modifications. A well-known method was proposed byCapon [ 2 5 ] , nd was later interpreted as a dual of the beam-former by Lacoss [74]. The optimization problem was posedas

m i n P ( w )

subject to w"a(e) = 1,w

where P(w) s as defined in Eq. 24. Hence, Capon's beam-former (also known as the Minimum Variance DistorsionlessResponse filter in the acoustics literature) attempts to mini-

mize the power cont ributed by noise and any signals coming

from other directions than 8, while maintaining a fixed gain

in the "look direction" 0. (This can be viewed as a sharpspatial bandpass filter.) The optimal w can be found using,e.g., the technique of Lagrange multipliers, resulting in

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Inserting the above weight into (24) leads to the following“spatial spectrum”

It is easy to see why Capon’s beamformer outperforms theclassical one given in Eq . 27, as the former uses everyavailable degree of freedom to concentrate the received en-ergy along one direction, namely the bearing of interest. Thisis reflected by the constraint given in Eq . 30 . The powerminimization can also be interpreted as sacrificing somenoise suppression capability for more focused “nulling” inthe directions where there are other sources present. Thespectral leakage from closely spaced sources is thereforereduced, though the resolution capability of the Capon beam-former is still dependent upon the array aperture and clearly

on the SNR.A number of alternative methods for beamform-ing have been proposed, addressing various issues such aspartial signal cancelling due to signal coherence [149]andbeam shaping and interference control [22,44, 1321.

Subspace-Based Methods

Many spectral methods in the past, have implicitly calledupon the spectral decomposition of a covariance matrix tocarry out the analysis (e.g., Karhunen-Lokve representation).One of the most significant contributions came about whenthe eigen-structure of the covariance matrix was explicitly

invoked, and its intrinsic properties were directly used toprovide a solution to an underlying estimation problem for agiven observed process. Early approaches involving invari-ant subspaces of observed covariance matrices include prin-cipal component factor analysis [SS] and errors-in-variablestime series analysis [6S]. In the engineering literature, Pis-arenko’s work [94] in harmonic retrieval was among the firstto be published. However, the tremendous interest in thesubspace approach is mainly due to the introduction of theMUSIC (Multiple SIgnal Classification) algorithm [13,1053.It is interesting to note that while earlier works were mostlyderived in the context of time series analysis and later appliedto the sensor array problem, MUSIC was indeed originallypresented as a DOA estimator. It has later been successfully

brought back to the spectral analysis/system identificationproblem with its later developments (see e.g. [118, 1351).

The MUSICAlgorithmAs noted previously, the structure of the exact covariancematrix with the spatial white noise assumption implies thatits spectral decomposition can be expressed as

R = APA +0’1 = U s A s U r+o 2 U n U ; , (33)

where, assuming APAH o be of fu ll rank, the diagonal matrix

As contains the M largest eigenvalues. Since the eigenvectorsin U, (the noise eigenvectors) are orthogonal to A, we have

uga(e)=o,e €{e l,...,e,,,} (34)

To allow for unique DOA estimates, the array is usuallyassumed to be unambiguous; that is, any collection of Lsteering vectors corresponding to distinct DOAs q k forms a

linearly independent set {a(ql), .., a ( q L ) }recall M < L) . If

a ( . ) atisfies these conditions and P has full rank, then APAHis also of full rank. It then follows that 01,. , 0 ~re the onlypossible solutions to the relation in Eq. 34, which couldtherefore be used to exactly locate the DOAs.

In practice, an estimate R of the covariance matrix isobtained, and its eigenvectors are separated into the signaland noise eigenvectors as in Eq. 22. The orthogonal projectoronto the noise subspace is estimated as

fiL= u,,u;.The MUSIC “spatial spectrum” is then defined as

(35)

Although P W ( Q is not a true spectrum in any sense (it is

merely the distance between two subspaces), it exhibits peaksin the vicinity of the true DOAs, as suggested by (34). Theperformance of the various spectral-based estimators is illus-trated in Fig. 5, where the scenario is identical to that of Fig. 4.

The performance improvement of the MUSIC estimatorwas so significant that it became an alternative to mostexisting methods. In particular, it follows from the abovereasoning that estimates of an arbitrary accuracy can be

-110 120 130 14C

DOA (deg)

5. Normalized spectra versus DOA.The dash-dotted curve repre-sents the conventional beamformer, the dashed curve is Capon’smethod and the solid curve is the “Music Spectrum.” True DOAsindicated by dotted vertical lines.

74 IEEE SIGNAL PROCESSING MAGA ZINE JULY 1996



obtained if the da ta clollection time is sufficiently long or theSNR isadequately high, and the signal model is sufficientlyaccurate. Thus, in contrast to the beamforming techniques,the MUSIC algorithm provides stutistically consistent esti-mates. Though the MUSIC functional (Eq. 36) does notrepresent a spectral estimate, its important limitation is stillthe failure to resolve closely spaced signals in small samplesand at low SN R scenarios. This loss of resolution is morepronounced for highly correlated signals. In the limiting caseof coherent signals, the property (Eq. 34) is violated and themethod fails to yield consistent estimates, see e.g. 1671. Themitigation of this limitation is an important issue and isseparately addressed at the end of this section.

Extensions to M U S I CThe MUSIC algorithm spawned a significant increase inresearch activity which led to a multitude of proposed modi-fications. These were attempts to improve/overcome some ofits shortcomings in various specific scenarios. The mostnotable was the unifying theme of weighted MUSICwhich,

for different W , particularized to various algorithms

(37)

The weighting matrix W is introduced to take into account(if desired) the influence of each of the eigenvectors. It isclear that a uniform weighting of the eigenvectors, i.e. W =I , results in the original MUSIC method. As shown in [114],this is indeed the optimal weighting in terms of yieldingestimates of minimal asymptotic variance. However, in dif-ficult scenarios involving small samples , low SNR and highlycorrelated signals, a carefully chosen non-uniform weightingmay still improve the resolution capability of the estimator

without seriously increasing the variance.One particularly useful choice of weighting is given by

I691

where el is the first column of the L x L identity matrix. Thiscorresponds to the Min-Norm algorithm, derived in [72,99]for ULAs and extended to arbitrary arrays in [75]. As shownin [61], the Min-Norm algorithm indeed exhibits a lower biasand hence a better resolution than the original MUSIC algo-rithm, at least when applied to ULAs.

Resolution-Enhunced Spectral Bused MethodsThe MUSIC algoritlhm is known to enjoy a property of highaccuracy in estimating the phases of the roots correspondingto DOA of sources. The bias in the estimates’ radii 1691,however, affects the resolution of closely spaced sourceswhen using the spatial spectrum.

A solution first proposed in 1141, and later in [21, 401, isto couple the MUSIC algorithm with some spatial prefilter-ing, to result in what is known as Beumspuce Processing. Thisis indeed equivalent to preprocessing the received data with

a predefined matrix T, whose columns can be chosen as thesteering vectors for a set cif chosen directions:

~ ( t ) T H x ( t ) (39)

Clearly, the steering vectors a(@) re then replaced by

T a(0) andl the noise culvariance for the beamspace data

becomes 02TTEIT.or the latter reason, T is often madeorthogonal ,and of norm 1 before application to x( t ) .

It is clear that if a certain spatial sector is selected to beswept (e.g. some prior knowledge about the broad directionof arrival of the sources may be available), one can potentiallyexperience some gain, the most obvious being computational,as a smaller dimensionaliuy of the problem usually results. Ithas, in addition, been shown that the bias of the estimates isdecreased when employing MUSIC in beamspace as opposedto the “elernent space” NIKJSIC [40, 1581. As expected, thevariance of the DOA estimates is not improved, but a certainrobustness to spatially conrelated noise has been noted in 1401.The latter fact can intuitively be understood when one recalls

that the spatia l pre-filter has a bandpass character, which willclearly tendl to whiten the noise.Other attempts to improving the resolution of the MUSIC

method are presented in 135, 62, 1561, based on differentmodifications of the criteirion function.

H

Coherent Signals

Though unlikely that on(: would deliberately transmit twocoherent signals from distinct directions, such a phenomenonis not uncommon as either a natural result of a multipathpropagation effect, or in1 entional unfriendly jamming. The

end result i:i a rank deficiency in the source covariance matrixP. This, in tuum, results in ii ivergence of a signal eigenvectorinto the noise subspace. Therefore, in general Ufa(0)# 0 for

any 8 and the MUSIC “spectrum” may fail to produce peaksat the DOA locations. Iin particular, the ability to resolveclosely spaced sources is dramatically reduced for highlycorrelated signals, e.g., [YrOI.

In the simple case of iwo coherent sources being presentand a uniform linear array, there is a fairly straightforwardway to “dc-correlate” the signals. The idea is to employ aforward-backward (FB) averaging as follows. Note that aULA steering vector (28 1 remains invariant, up to a scaling,if its elements are reversed and complex conjugated. More

precisely, let J be anL L exchange matrix, whose componentsare zero except for ones om the anti-diagonal. Then, for a ULAit holds that

The so-cal led backward array covariance matrix thereforetakes the form

JULY 1996 IEEE SIGNA L PROCESSING MAGA ZINE 75



where CD is a diagonal matrix with i4k, = 1,. ,M n the

diagonal, and $k as previously defined. By averaging theusual array covariance and RB, one obtains the FB arraycovariance

R, = - ( R + JR*J)

2 -= APA” +0’1,

where the new “source covar iance matr ix”

= (P +W(L-’)P@-(L-’))2 generally has full rank. The FB

version of any covariance-based algorithm simply consists ofreplacing R with k F B ,efined as in Eq. 42. Note that this

transformation has also been used in noncoherent scenarios,and in particular in time series analysis, for merely improvingthe variance of the estimates.

In a more general scenario where more than two coherentsources are present, forward-backward averaging cannot re-store the rank of the signal covariance matrix on its own. Aheuristic solution of this problem was first proposed in [ 1491for uniform linear arrays, and later formalized and extendedin [33, 43 , 1081. The idea of the so-called spatial smoothingtechnique is to split the ULA into a number of overlappingsubarrays, as illustrated in Figure 6. The steering vectors ofthe subarrays are assumed to be identical up to differentscalings, and the subarray covariance matrices can thereforebe averaged. Similar to (42), the spatial smoothing induces arandom phase modulation which in tum tends to decorrelatethe signals that caused the rank deficiency. A compact ex-

pression for this smoothed matrix R can be written in termsof selection matrices Fk as follows. Let p denote the numberof elements in the subarrays, implying that the number of

subarrays is K = L - p + 1. Then, the spatially smoothed array

covariance matrix can be expressed as

The rank of the averaged source covariance matrix R canbe shown to increase by 1 with probability 1 [29] for eachadditional subarray in the averaging, until it reaches its maxi-mum value IM.

The drawback with spatial smoothing is that the effectiveaperture of the array is reduced, since the subarrays aresmaller than the original array. However, despite this loss of

aperture, the spatial smoothing transformation mitigates thelimitation of all subspace-based estimation techniques whileretaining the computational efficiency of the one-dimen-sional spectral searches. As discussed in the next section, theparametric techniques generally do not experience such prob-lems when faced with coherent signals. They require, on theother hand, a more complicated multidimensional search. Weagain stress the fact that spatial smoothing is limited toregular arrays with a translational invariance property, andFB averaging requires a ULA. When using more general

c I

Subarray 14 z

Subarray 2< >

Subarray 3~

i Spatial smoothing means that the array is split into identicalsubarrays, the covariances of which are averaged

arrays (e.g. circular), some sort of transformation of thereceived data must precede the smoothing transformation.Such a transformation is conceptually possible, but generallyrequires some a priori knowledge of the source locations.

Parametric Methods

While the spectral-based methods presented in the previoussection are computationally attractive, they do not alwaysyield sufficient accuracy. In particular, for scenarios involv-ing highly correlated (or even coherent) signals, the perform-ance of spectral-based methods may be insufficient. Analternative is to more fully exploit the underlying data model,leading to so-calledpa ramet ric array processing methods. Aswe shall see, coherent signals impose no conceptual difficul-ties for such methods. The price to pay for this increasedefficiency and robustness is that the algorithms typicallyrequire a multidimensional search to find the estimates. Foruniform linear arrays (ULAs), the search can, however, beavoided with little (if any) loss of performance.

Perhaps the most well known and frequently used model-based approach in signal processing is the maximum likeli-hood (ML) technique. This methodology requires a statisticalframework for the data generation process. Two differentassumptions about the emitter signals have led to correspond-ing ML approaches in the array processing literature. In thissection we will briefly review both of these approaches,discuss their relative merits, and present subspace-based MLapproximations. Parametric DO A estimation methods are ingeneral computationally quite complex. However, for ULAsa number of less demanding algorithms are known, as pre-sented shortly.

Deterministic Maximum Likelihood

While the background and receiver noise in the assumed datamodel can be thought of as emanating from a large numberof independent noise sources, the same is usually not the casefor the emitter signals. It therefore appears natural to modelthe noise as a stationary Gaussian white random processwhereas the signal waveforms are deterministic (arbitrary)and unknown. (The carrier frequencies are assumed to beknown.) Assuming spatially white and circularly symmetric(A complex random process i s circularly symmetric if its real

76 IEEE SIGNAL PROCESSING MAGAZINE JULY1996



and imaginary parts are identically distributed and have askew-symmetric cross-covariance, i.e., E[Re(n(t))Im(n ( t ) ) )= -EIIm(n(t))Re(nT(t))]] noise, the second-order moments ofthe noise term take the form

T,. I NR = -CX(~)X"(~ )

N i = l

E{n(t)nH(s)} = 0*1[6,,~ (43)

E{n(t)n'(s)} = 0 . (44)

As a consequence of the statistical assumptions, the obser-vation vector x(t) is also a circularly symmetric and tempo-rally white Gaussian random process, with mean A(8)s(t)andcovariance matrix 0'1. The likelihood function is the prob-ability density function (PDF) of all observations given theunknown parameters. The PDF of one measurement vectorx(t) is the complex L,-variate Gaussian:

where 11.(1denotes thle Euclidean norm, and the argument of

A(0)has been droppled for notational convenience. Since themeasurements are independent, the likelihood function isobtained as

As indicated above, the unknown parameters in the like-lihood function are the signal parameters @,the signal wave-forms s ( t ) and the noise variance c2. he ML estimates ofthese unknowns are calculated as the maximizing argumentsof L(B,s ( t ) ,02), he rationale being that these values make

the probability of the observations as large as possible. Forconvenience, the ML estimates are alternatively defined asthe minimizing arguments of the negative log-likelihood

function -logL(0, ~ ( t ) ,I ). Normalizing by N and ignoring

the parameter-independent L log n-term, we get

2

l D M L ( 0 , ~ ( t ) , d )L l o g d + T C I l x ( t ) - A s ( t ) l (N , (47)CJ N ,=I

whose minimizing arguments are the deterministic maximumlikelihood (DML) estimates.

As is well-known [15, 1421, explicit minima with respect

to o2 and s ( t )are given by

6' = -Tr{IIiR}L

where R is the sample covariance matrix, A' is the Moore-Penrose pseudo-inverse of A and IIi is the orthogonal pro-

jector onto the nullspace of A , .e.,.

Substituting Eqs. 48 and 49 into Eq. 46 shows that theDML signal parameter estimates are obtained by solv ing thefollowing minimization problem:

(54)

The interpretation is that the tpeasurements x(t) are pro-jected onto a model subspace orthogonal to all anticipateds ignal components , and a power measurement

~ $ l l r ~ ~ ~ ( t ) \ l ~ =r{niR}. is evaluated. The energy shouldN i = ~

clearly be smallest when the projector indeed removes all thetrue signal components, i.e., when 8 = 80. Since only a finitenumber of noisy samples is available, the energy is not

perfectly measured and e,,, will deviate from 80. However,

if the scenario is stationary, the error will converge to zero asthe number of samples is increased to infinity . This remainsvalid for correlated or even coherent signals, although theaccuracy in finite samples is somewhat dependent upon sig-nal correlations. Notice also that Eq. 54 reduces to the Bartlettbeamfomer in the case of a single source ( M = 1).

To calculate the DML est imates, the non-linear M-dimen-sional optimization problem (Eq. 54) must be solved numeri-cally. Finding the signal waveform and noise varianceestimates (if desired) is then straightforward, by inserting

e,,, into Eqs. 48-49. Given a good initial guess, a Gauss-

Newton technique (see e.g. [24, 1371) usually convergesrapidly to the minimum of Eq. 47. Obtaining sufficientlyaccurate initial estimates, however, is generally a computa-tionally expensive task. ILf hese are poor, the search proce-dure may converge to a local minimum, and never reach thedesired global minimum. A spectral-based method is a natu-ral choice for an initial estimator, provided all sources can beresolved. Another possibility is to apply the alternating pro-jection technique of [ 1621. However, convergence to the

global minimum can still not be guaranteed. Some additionalresults on the global properties of the criteria can be found in

1901.

Stochastic Ma xim um Likelihood

The other M L technique reported in the literature is termedthe stochastic maximum likelihood (SML) method. Thismethod is obtained by modeling the signal waveforms as




Gaussian random processes. This model is reasonable, forinstance, if the measurements are obtained by filtering wide-band signals using a narrow bandpass filter. It is, however,important to point out that the method is applicable even ifthe data is not Gaussian. In fact, the asymptotic (for largesamples) accuracy of the signal parameter estimates can beshown to depend only on the second-order properties (powers

and correlations) of the signal waveforms [8 9,11 5]. With thisin mind, the Gaussian signal assumption is merely a way toobtain a tractable ML method. Let the signal waveforms bezero-mean with second-order properties

E{~(t)s‘(s)) = 0,

leading to the observation vector x(t) be a white, zero-meanand circularly symmetric Gaussian random vector with co-variance matrix

(57 )= A(0)PA (0) +0’1.

The set of unknown parameters is, in this case, differentfrom that in the deterministic signal model. The likelihood

function now, depends on 8, P and CJ . The negative log-like-lihood function (ignoring constant terms) is in this cas e easilyshown to be proportional to

2

Although this is a highly non-linear function, this criterionallows explicit separation of some of the parameters. For

fixed 0, the minimum with respect to CJ and P can be shown

to be [16, 571

2

1L - M

i,(O) = ---Tr{ni}(59)

With these estimates substituted into (58) , he followingcompact form is obtained

O = arg RinlogAPs,(8)AN +6tML(0)Il]

In addition, this criterion has a nice interpretation, namelythat the determinant, termed the generalized variance in thestatistical literature, measures the volume of a confidenceinterval for the data vector. Consequently, we are looking forthe model of the observations with the “lowest cost” and inharmony with the ML principle.

The criterion function in Eq. 61 is also a highly non-linearfunction of its argument 0. A Newton-type technique imple-mentation of the numerical search is reported in [90] and an

excellent statistical accuracy results when the global mini-mum is attained. Indeed, the SML signal parameter estimateshave been shown to have a better large sample accuracy thanthe corresponding DML estimates [89, 1151, with the dif fer-ence being significant only for small numbers of sensors, lowSN R and highly correlated signals. This property holds re-gardless of the actual distribution of the signal waveforms; in

particular they need not be Gaussian. For Gaussian signals,the SML estimates attain the Cram&-Rao lower bound(CRB) on the estimation error variance, derived under thestochastic signal model. This follows from the general theoryof ML estimation (see e.g. [1301), since all unknowns in thestochastic model are estimated consistently. Thi s in contrastwith the deterministic model for which the number of signalwaveform parameters ~ ( t )rows without bound, as thenumber of samples increases, implying that they cannot beconsistently estimated. Hence, the general ML theory doesnot apply and the DML estimates do not attain the corre-sponding (“deterministic”) CRB.

Su bspace-Based Appr oxim atio ns

As noted previously, subspace-based methods offer signifi-cant performance improvements in comparison to conven-tional beamforming methods. In fact, the MUSIC method hasbeen shown to yield estimates with a large-sample accuracyidentical to that of the DML method, provided the emittersignals are uncorrelated [1121. However, the spectral-basedmethods usually exhibit a large bias in finite samples, leadingto resolution problems. This problem is especially notable forhigh source correlations. Recently, parametric subspace-based methods that have the same statistical performance(both theoretically and practically) as the ML methods havebeen developed [116,1 17,136,137 ]. Thecomputational costfor these so-called Subspace Fitting methods is, however,less than for the ML ditto. As will be seen later, a computa-tionally attractive implementation for th e ubiquitous case ofa uniform linear array, is known.

Recall the structure of the eigendecomposition of the arraycovariance matrix (Eq.33),

R = A P A ~ 0‘1 (62)

As previously noted, the matrices A and Us span the samerange space whenever P has full rank. In the general case, the

number of signal eigenvectors in Us quals M’, the rank of P.The matrix Us will then span an M’-dimensional subspace ofA. This can easily be seen by first expressing the identity in(62) as I = U 5 U p+UnU,v.Cancelling the 02U,,Uf term in

(63) then yields

(64)PAH + oZUqU:= U,AsUf .

JULY 19968 IEEE SlGNAL PROCESSING MAGA ZINE



Post-multiplying on the right by U, (note that UpUA = I )and re-arranging givles the relation

where T is the full-rink M M’ matrix

The relation in 13q. 65 forms the basis for the SignalSubspace Fitting (SSF) approach. Since 0 and T are un-known, it appears natural to search for the values that solve

Eq. 65. The resulting 0 will be the true DOAs (under generalidentifiability conditions [146]), whereas T is an uninterest-

ing “nuisance paranieter.” If an estimate U > of Us is used

instead, there will be no such solution. In this case, one

attempts to minimize some distance measure between U, and

AT. For this purpose, the Frobenius norm turns out to be auseful measure, andl when squared, it can be conveniently

expressed as the sunn of the squared Euclidean norms of therows or columns. In light of this formulation, the connectionto standard least-squares estimators is made clear. The SSFestimate is obtained by solving the following non-linearoptimization problem:

Similar to the DML criterion (Eq. 47), this is a separablenonlinear least squares problem [47]. The solution for thelinear parameter T (for fixed unknown A) is

which, when substitiuted into Eq.67, leads to the concentratedcriterion function

Since the eigenvectors are estimated with a quality, commen-surate with the closeness of the corresponding eigenvalues tothe noise variance, it is natural to introduce a weighting of theeigenvectors and aririve at

A natural questiion which arises is how to pick W tomaximize the accuracy, i.e., to minimize the estimation errorvariance. It can be shown that the projected eigenvectorsHk(Oo)uL,k 1, .., Ad‘ are asymptotically independent.

Hence, following from the theory of weighted least squares,[46], W should be a diagonal matrix containing the inverse

of the covariance matrix of ni €ln)ukk = 1,. M’ .This leads

to the choice [117, 1361

Wept = (A s- o ~ I ) ~ A :

Since WOp, epends on unknown quantities, we use in-

stead

w”p, A i s

where 6’ denotes a consistent estimate of the noise variance,for example the average of the L - M’ smallest eigenvalues.The estimator defined by (70) with weights given by (72) istermed the Weight ed Subspace Fitting (WSF) method. It hasbeen shown to theoretically yield the same large sampleaccuracy as the SML method, and at a lower computationalcost provided a fast method for computing the eigendecom-position is used. Practical evidence, e.g., [90], have given byhand that the WSF and SML methods also exhibit similarsmall sample (i.e. threshold) behaviour.

An alternative subspace fitting formulation is obtained byinstead starting from the “MUSIC relation”

AH(e)Un o i f e = e , , (73)

which holds for P having full rank. Given an estimate of Un,it is natural to look for the signal parameters which minimizethe following Noise Subspace Fitting (NSF) criterion,

6 = arg{mulnTr(AHU~CJ~AV)], (74)

where V is , some positiviz (semi-)definite weighting matrix.Interestingly enough, the estimates calculated by Eqs. 70 and

74 asymptotical ly coincide, if the weighting matrices are(asymptotically) related by [90]

V = At(€ln)UyWU~At*(O,). (75)

Note also that the NlSF method reduces to the MUSICalgorithm for V = I, provided 1a(€))\ s independent of 0.Thus, in a sense the general estimator in Eq. 74 unifies theparametric methods and also encompasses the spectral-basedsubspace methods.

The NSF method has the advantage that the criterionfunction in Eq . 74 is a quadratic function of the steeringmatrix A. This is useful if any of the parameters of A enterlinearly. An analytical solution with respect to these parame-ters is then readily available (see e.g., [138]). However, thisonly happens in very special situations, rendering this fore-mentioned advantage of limited importance. The NSF formu-lation i s also fraught with some drawbacks, namely that it

cannot produce reliable estimates for coherent signals, andthat the optimal weighting Vopt depends on 00, so that atwo-step procedure has to be adopted.

JULY 1996 IEEE SIGNAL PROCESSING MAG AZINE 79



Uniform Linear Arrays attributed to a larger bias of estimates for spectral-basedmethods, as compared to the parametric techniques [69,98].

The steering matrix for a uniform linear array has a veryspecial, and as it turns out, useful structure. From Eq. 28, theULA steering matrix takes the form

This structure is referred to as a Vandermonde matrix.(Each component of a column of a Vandermonde matrix isan integer power of the first component, and the integercoincides with its row number.) [47].

Root-MUSICThe Root-MUSIC method [ 1 ], as the name implies, is apolynomial-rooting version of the previously described MU-

SIC technique. The idea dates back to Pisarenko's method[94]. Let us define the polynomials

p , ( z )= uYp(z), I = M + l , M + 2,. . (L , (77)

where ui is the 1:th eigenvector of R and

p(z) =[ l , z . . . , zL-L]T

From our problem formulation, one makes the basic ob-servation that pi(zj has M of its zeros at i@", = l ,2, . M ,provided that P has full rank. To exploit the information fromall noise eigenvectors simultaneously, we want to find thezeros of the MUSIC-like function

However, the latter is not a polynomial in i (note thatpowers of z* are now present), which complicates the searchfor zeros. Since the values of z on the unit circle are of interest,we can use pT(z-') for pH(z>,which gives the Root-MUSICpolynomial

Note that p ( z ) s a polynomial of degree 2(L-1), whoseroots occur in mirrored pairs with respect to the unit circle.Of the ones inside , the phases of the M that have the largestmagnitude, say i, &,. ,?,,yield the DOA estimates, as

G m= arccos -arg{z } , m = 1,2 ..., M .C d 1

It has been shown [112, 1131 that MUSIC and Root-MU-SIC have identical asymptotic properties, although in smallsamples Root-MUSIC has empirically been found to performsignificantly better. As previously alluded to, this can be

ESPRITThe ESPRIT algorithm [93, 1011 uses the structure of theULA steering vectors in a slightly different way. The obser-vation here is that A has a so-called shift structure.Define thesub-matrices A1 and A2 by deleting the first and last rowsfrom A (defined in (13)) respectively, i.e.,

first row

'=[kis:iow]=[ A, ]By the structure of Eq. 76, A1 and A2 are related by the

formula

where CD is a diagonal matrix having the roots m =l ,2, .. , M , on the diagonal. Thus, the DOA estimation prob-

lem can be reduced to that of finding Q. Analogously to theother subspace-based algorithms, ESPRIT relies on proper-ties of the eigendecomposition of the array covariance ma-

trix. Applying this deletion transformation to Eq. (6Sj, we get

where Us has been partitioned conformably with A into thesub-matrices U1 and U2. Combining (83) and (84) yields

which, by defining '€ = T - b T , becomes

U, = U ] Y (86)

Note that '€ and CD are related by a similarity transforma-tion, and hence have the same eigenvalues. The latter is ofcourse given by e@", m = 1,2,.. , M , and are related to theDOAs as in Eq . 81. The ESPRIT algorithm is now stated:

I . Compute the eigendecomposition of the array covariance

matrix R

2. Form U, and Uz rom the M principal eigenvectors

3. Solve the approximate relation Eq. 86 in either a Least-Squares sense (LS-ESPRIT) or a Total-Least-Squares [46,471 sense (TLS-ESPRIT)

4. The DOA estimates are obtained by applying the inversionformula Eq. 81 to the eigenvalues of \i.

It has been shown that LS-ESPRIT and TLS-ESPRITyield identical asymptotic estimation accuracy [97],althoughin small samples TLS-ESPRIT usually has an edge. In addi-tion, unlike LS-ESPRIT, TLS-ESPRIT accounts for the noisy

nature of both U, and U, which is more intuitively appealing.

Note also that the ESPRIT algorithm al lows more flexibility

80 IEEE SIGNAL PROCESSING MAGAZIN E JULY 1996



in partitioning of the array (or of the A matrix), so long as ashift structure can be obtained and subsequently estimated.

I Q M Land Root- W S FAnother interesting exploitation of the ULA structure waspresented in [ 191, although the idea can be traced back to theSteiglitz and McBride algorithm for system identification

[1111. The IQML (Iterative Quadratic Maximum Likelihood)algorithm is an iterative procedure for minimizing the DMLcriterion

V ( 0 )= Tr(nhR}. (87)

The idea is to re-parameterize the projection matrix IIi

using a basis for thc nullspace of AH . Towards this end, onedefines a polynomial b(z) to have its M roots at e‘4m, m =

1,2 ..., M , i.e.,

Then, by construction the following relation holds true

L

U BHA = 0.

Since B has a full rank of L - M , its columns do in factform a basis for the nullspace of AH. Clearly the orthogonalprojections onto these subspaces must coincide, implying

n; = H(H ~ B ) - ’ I z~ . ( 90 )

Now the DML criterion function can be parameterized by

the polynomial coefficients b k in lieu of the DOAs Ok . TheDML estimate (54) can be calculated by solving

b = arg min Tr(Bl(BHH)-’BHR)

and then applying Eq. 81 to the roots of the estimated poly-nomial. Unfortunately, Eq. 91 is still a difficult non-linearoptimization problem. However, [191suggested an iterativesolution as follows

1. Set U = I2. Solve the quajdratic problem

b = argmjnTr(BUHHR} (92)

3. Form B and put U = (B”B)-’

4. Check [or convergence. If not, goto Step 2

5 . Apply (81) to the roots of b ( z )

Since the roots of b(z)should be on the unit circle, [20]suggested iusing the constraint

b,,, = bz ~, = 1,2,. ,i M (93)

when solving Eq. 92. Now, Eq. 93 does not guarantee unitymagnitude roots, but it can be shown that the accuracy loss

due to this fact is negligible. While the above describedIQML algorithm cannot be guaranteed to converge, it hasindeed been found to perform well in simulations.

An improvement over IQMLwas introduced in [ 1161. Theidea is sirnply to apply the IQML iterations to the WSFcriterion Eq. 70. Since tlhe criteria have the same form, themodification is straightforward. However, there is a veryimportant advantage of using the rank-truncated form

UsWUy rather than R in Eq. 92, with W as defined in Eq.

72. That is , after the second pass of the iterative scheme, theestimates already have the asymptotic accuracy of the trueoptimum! Hence, the resulting Root-WSF algorithm is nolonger an iterative procedure:

1. Solve the quadratic problem

b = argm$Tr(HH”U,WUy}

2. Solve the quadratic problem

(94)

(95)

3. Apply (81) to the roots of i ( z ) .

Note that this algorithm is essentially in closed form (asidefrom the eigendecomposition and polynomial rooting), and

that the resulting estimates have the best possible asymptoticaccuracy. Thus, the Root-WSF algorithm is a strong candi-date for the “best” method for ULAs.

Additional Topics

The emphasis in this paper is on parameter estimation meth-ods in sensor array processing. Because of space limitations,we have clearly omitted1 several interesting issues from themain discussion. To partially fill the gap, a very brief over-view of these additional topics is given in the following. Westructure this section in the form of a commented enumera-tion of refierences to selected specialized papers.

Number of Signals Estimation

In applications of model-based methods, an important prob-lem is the determination of M , the number of signals. In thecase of non-coherent signals, the number of signals is equalto the number of “large” eigenvalues of the array covariancematrix. This fact is used to obtain relatively simple non-para-metric algorithms for diztermining M . The most frequentlyused approach emanate:s from the factor analysis literature




[4]. The idea is to determine the multiplicity of t he smallest

eigenvalue, which theoretically equals L - M . A statisticalhypothesis test is proposed in [105], whereas [144, 1611 arebased on information theoretic criteria, such as Akaike’s AIC(an information theoretic criterion) and Rissanen’s MDL(minimum description length). Unfortunately, the aforemen-tioned approach is very sensitive to the assumption of a

spatially white noise field [1571. An alte rnative idea based onusing the eigenvectors rather than the e igenvalues is pursuedin the referenced paper. Another non-parametric method ispresented in [31].

In the presence of coherent signals, the methods as juststated will fail, since the dimension of the signal subspace isin this case smaller than M .However, for ULAs one can testthe eigenvalues of the spatially smoothed array covariancematrix to determine M as proposed in [1081 and improved by

[681.A more systematic approach to estimating the number of

signals is possible if the maximum likelihood estimator isemployed. A classical generalized likelihood ratio test(GLRT) is described in [90], whereas [143] presents aninformation theoretic approach. Another model-based detec-tion technique is presented in [90, 1371, based on theweighted subspace fitting method. For determining thenumber of signals the model-based methods require signalparameter estimates for an increasing hypothesized numberof signals, until some pre-specified criterion is met. Theapproach is thus inherently more computationally intensivethan the non-parametric tests. Its performance in difficultsignal scenarios is, however, improved and is in addition lesssensitive to small perturbations of the assumed noise covari-ance matrix.

Reduced Dimension Beamspace Processing

At the except ion of the beamforming-based methods, theestimation techniques discussed herein require that the out-puts of all elements of the sensor array be available in digitalform. In many applications, the required number of high-pre-cision receiver front-ends and AD converters may be pro-hibitive. Arrays of 10 elements are not uncommon, forexample in radar applications. Techniques for reducing thedimensionality of the observation vector with minimal effecton performance, are therefore of great interest. As alreadydiscussed previously, a useful idea is to employ a lineartransformation

4

~ ( t ) T*x(t), (96)

where T is L R , with (usually) R << L. The transformationis typically implemented in analog hardware, thus signifi-cantly reducing the number of required A/D onverters. Thereduced-dimension observation vector z ( t ) is usually referredto as the beamspace data, and T is a beamspace transforma-tion.

Naturally, processing the beamspace data significantlyreduces the computational load of the digital processor. How-

ever, reducing the dimension of the data also implies a lossof information. The beamspace transformation can bethought of as a multichannel beamformer. By designing thebeamformers (the columns of T) so that they focus on arelatively narrow DOA sector, the essential information inx(t) regarding sources in that sector can be retained in z( t ) .See e.g., [41, 134, 159, 1651 and the references therein. With

further a priori information on the locations of sources, thebeamspace transformation can i n fact be performed with noloss of information [5]. As previously alluded to, beamspaceprocessing can even improve the resolution (the bias) ofspectral-based methods.

Note that the beamspace transformation effectively

changes the array propagation vectors from a(0) nto T*a(0).It is possible to utilize this freedom to give the beamspacearray manifold a simpler form, such as that of a ULA, [43].Hence, the computationally efficient ULA techniques areapplicable in beamspace. In [82], a transformation that mapsa uniform circular array into a ULA is proposed and analyzed,enabling computationally efficient estimation of both azi-muth and elevation.

Estimation Under Model Uncertainty

As implied by the terminology, model-based signal process-ing relies on the availability of a precise mathematical de-scription of the measured data. When the model fails to reflectthe physical phenomena with a sufficient accuracy, the per-formance of the methods will clearly degrade. In particular,deviations from the assumed model will introduce bias in theestimates, which for spectral-based methods is manifested bya loss of resolving power and a presence of spurious peaks.In principle, the various sources of modeling errors can beclassified into noise covariance and array response perturba-tions.

In many applications of interest, such as communication,sonar and radar, the background noise is dominated by man-made noise. While the noise generated in th e receiving equip-ment is likely to fulfill the spatially white assumption, theman-made noise tends to be quite directional. The perform-ance degradation under noise modeling errors is studied in,e.g. 177,1401. One could in principle envision extending themodel-based estimation techniques to also include estimationof the noise covariance matrix. Such an approach has thepotential of improving the robustness to errors in the assumednoise model [143, 1521. However, for low SNR’s,this solu-tion is less than adequate. The simultaneous estimation of acompletely unknown noise covariance matrix and of thesignal parameters poses a problem unless some additionalinformation which enables us to separate signal from noise,is available. It is, for example, always possible to infer thatthe received da ta is nothing but noise, and that the (unknown)noise covariance matrix i s precisely the observed array sam-ple covariance matrix. Estimation of parametric (structured)noise models is considered in , e.g., [18,641. So-called instru-mental variable techniques are proposed, e.g. in [ I 191 (basedon assumptions on the temporal correlation of signals and

82 IEEE SIGNAL PROCESSING MAGAZ INE JULY 1996



noise) and [1531 (utilizing assumptions on the spatial corre-lation of the signals and noise). Methods based on otherassumptions appeared in [92, 13 11.

At high SNR, the modeling errors are usually dominatedby errors in the assuimed signal model. The signals have anon-zero bandwidth, the sensor positions may be uncertain,the receiving equipment may not be perfectly calibrated, etc.

The effects of such errors are studied e.g. in [42,76,125,126].In some cases it is possible to physically quantify the errorsources. A “self-calibrating’’ approach may then be applica-ble [loo, 138, 1471, albeit at quite a computational cost. In[ 125, 126, 1391, robust techniques for unstructured sensormodeling errors are considered.

Wideban d Data Processing

The methods presented herein are essentially limited to proc-essing narrowband data. In many applications (e.g. radar andcommunication), this is indeed a realistic assumption. How-ever, in other cases (e.g. sonar), the received signal may bebroadband. A natural extension of all these methods is toemploy narrowband filtering, for example using the FastFourier Transform (FFT). An optimal exploita tion of the dataentails combining information from different frequency bins1661, see also 11451. A simpler suboptimal approach is toprocess the different FFT channels separately using a stand-ard narrowband metlhod, whereafter the DOA estimates atdifferent bins must be combined in some appropriate way.

Another approach is to explicitly model the array outputas a multidimensional time series, using an autoregressivemoving average (ARMA) model. The poles of the system areestimated, e.g. using the overdetermined Yule-Walker equa-tions. A narrowband technique can subsequently be em-ployed using the estimated spectral density matrix, evaluatedat the system poles. In [1221, the MUSICalgorithmis applied,whereas [88] proposes to use the ESPRIT algorithm.

A wideband DOA estimation approach inspired by beam-space processing is the so-called coherently averaged signalsubspace method, originally introduced in [1411. The idea isto first estimate the signal subspace at a number of FFTchannels. The information from different frequency bins issubsequently merged by employing linear transformations.The objective is to make the transformed steering matrices Aat different frequencies as identical as possible, for exampleby focusing at the center frequency of the signals. See, e.g.[56, 71, 1091 for further details.

Fast Subspace Calculation and Tracking

The implementation of subspace-based methods in applica-tions with real-time operation usually experiences a bottle-neck in the calculation of the signal subspace. A scheme forfast computation of i.he signal subspace is proposed in [60],and methods for subspace tracking are considered in [30] .The idea is to exploit the “low-rank plus (5 I” structure of theideal array covarianc,e matrix. An alternative gradient-basedtechnique for subspace tracking is proposed in [160].

2

Therein, it is observed that the solution W , of the constrainedoptimization problem,

qxTr{W*RW)

subject to W‘W = I

spans the signal subspace.For some methods, estimates of the individual principal

eigenvectors are required, in addition to the subspace theyspan. A number of different methods for eigenvalue/vector(The referenced paper conisiders tracking of the principa l leftsingular values of the datal matrix. Mathematically, but per-haps not numerically, these are identical to the eigenvectorsof the sample covariance matrix.) tracking is given in [28]. Amore recent approach based on exploiting the structure of theideal covariance is proposed and analyzed in [155]. Theso-called fast subspace decomposition (FSD) technique isbased on a Lanczos method for calculating the eigendecom-position. It is observed that the Lanczos iterations can beprematurely terminated without any significant performanceloss.

Signal Striucture Methods

Most “standard” approaches to array signal processing makeno use of any available information about the signal structure.However, many man-made signals have a rich structure thatcan be used to improve the estimator performance. In digitalcommunicztion applications, the transmitted signals are oftencyclostationary, which innplies that their autocorrelationfunctions are periodic. This additional information is ex-ploited in e.g. [ l , 154, 1941, and algorithms for DOA estima-tion of cyclostationary signals are derived and analyzed. Inthis approach, the wideband signals are easily incorporated

into the framework, and the number of signals may exceedthe number of sensors with the provision that they do not allshare the same cyclic frequency (which refers to the fre-quency at which the spatial correlation function repeats it-self).

A different approach utilizing signal structure is based onhigh-order statistics. As is well-known, all information abouta Gaussian signal is conveyed in the first and second ordermoments. HLowever, for non-Gauss ian signals, there is poten-tially more to be gained by using higher moments. This isparticularly so if the noise can be regarded as Gaussian. Then,the high-order cumulants will be theoretically noise-free, asthey vanish for Gaussian signals. Methods based on fourth-order cumnlants are proposed in [26 , 951, whereas [lo71

proposes to use high-order cyclic spectra (for cyclostationarysignals).

A common criticism of both approaches (those based oncyclostationarity and those based on high-order statistics) isthat they require a considerable amount of data to yieldreliable results. This is due to the slower convergence of theestimated cyclic and high-or der moments to their theoreticalvalues as the number of dit a is increased compared to that ofsecond order moments.

JULY 1996 IEEE SIGNAL PROCESSING MAGAZIN E 83



Polarization Sensitivity Personal Commu nicati ons

Most antenna arrays used in electro-magnetic applicationsare sensitive to polarization differences among the receivedsignals. Recall that the wave equation (7) is in fact vector-valued, and that a similar equation holds for the magneticcomponents. Provided that each polarization component can

be calibrated separately, the signal polarization can be util-ized for improving the DOA estimation performance, even ifthe array only measures the sum of the polarization compo-nents. An extension of the MUSIC method to polarizationsensitive a rrays is proposed in [39], whereas [78, 127, 1631consider parametric methods. In [148], it is shown that PO-larization diversity can also significantly improve the signalwaveform estimates, a fact which has long been known inmicrowave communications and which could be of consid-erable interest in array processing applications, e.g. commu-nication applications.

A more complete signal model is considered in [54, 861.Therein, it is assumed that the receiving sensors measure allsix field components of the elect ric and magnetic fields. Thisallows source localization using only one so-called vectorsensor.

Time Series Analysis

The DOA estimation problem employing a ULA shares somecommon aspects with time series analysis. Indeed, an impor-tant pre-cursor of the MUSIC algorithm is Pisarenko’smethod [94] for estimation of the frequencies of damp edh n-damped sinusoids in noise, whose covariance function meas-urements are given. Similarly, Kung’s algorithm forstate-space realization of a measured impulse response [73]is an early version of the ESPRIT algorithm. An important

difference between the time series case and the (standard)DOAestimation problem is that the time series data is usuallyestimated using a sliding window. A “sensor array-like datavector” x(t) is formed from the scalar time series y ( t )by using

x(t)= [ y ( t ) , ( t+ l), y ( t+L - )17

The windowing transformation induces a temporal corre-lation in the L-dimensional time series x ( t ) ,even if the scalarprocess y ( l ) is temporally white. This fact complicates theanalysis, as in the case of spatial smoothing in DOA estima-tion, sec e.g. [27, 8, 1881. A further complication arises intime series analysis problems when there is a known inputsignal present. This problem has fortunately, also been suc-

cessfully tackled using ideas from sensor array signal proc-essing [ 1351.

Applications

The research progress of parameter estimation and detectionin array processing has resulted in a great diversity of appli-cations, and continues to provide fertile ground for new ones.In this section we discuss three important application areas.

Receiving arrays and related estim atio ddete ction techniqueshave long been used in High Frequency communications.These applications have recently reemerged and received asignificant attention by researchers, as a potentially useful“panacea” for numerous problems in personal communica-tions (see e.g., [3, 120, 123, 1511). They are expected to playa key role in accommodating a multiuser communicationenvironment, subject to severe multipath.

One of the most important problems in a multiuser asyn-chronous environment is the inter-user interference, whichcan degrade the performance quite severely. This is also thecase in a practical Code-Division Multiple Access (CDMA)system, because the varying delays of different users inducenon-orthogonal codes. An interesting application of the MU-SIC algorithm for estimating these propagation delays ispresented in [121]. The base stations in mobile communica-tion systems have long been using spatial diversity for com-bating fading due to the severe multipath. However, using anantenna array of several elements introduces additional de-grees of freedom, which can be used to obtain higher selec-tivity. An adaptive receiving array can be steered in thedirection of one user at a time, while simultaneously nullinginterference from other users, much in the the same way asthe beamforming techniques described previously.

The multipath which may be caused by buildings reflec-tions or hills, etc. introduces difficulties for conventionaladaptive array processing. Undesired cancelation may occur[149], and spatial smoothing may be required to achieve aproper selectivity. In [ 3 ] , t is proposed to identify all signalpaths emanating from each user, whereafter an optimal signalcombination is performed. A configuration equivalent to thebeamspace array processing with a simple energy differenc-

ing scheme serves in localizing incoming users waveforms[1031. This beamspace strategy underlies an adaptive optimi-zation technique proposed in [120], which addresses theproblem of mitigating the effects of dispersive time varyingchannels.

Communication signals have a rich structure that can beexploited for signal separation using antenna arrays. Indeed,the DOAs need not be estimated. Instead, signal structuremethods such as the constant-modulus beamformer [48] havebeen proposed for directly estimating the steering vectors ofthe signals, thereby allowing for blind (i.e., not requiring atraining sequence) signal separation. Techniques based oncombining beamforming and demodulation of digitallymodulated signals have also recently been proposed (see [80,

100, 128,1291). It is important to note that the various optimalandor suboptimal proposed methods are of current researchinterest, and that in practice, many challe nging and interest-ing problems remain to be addressed.

Radar and Sonar

The classical application of array signal processing is in radarand sonar, and modern model-based techniques have also

84 IEEE SIGNAL PROCESSING MAGAZINE JULY1996



found their way to these areas [34, 51, 52, 79, 911. Theantenna array is, for example, used for source localization,interference cancelation a nd ground clutter suppression.

In radar applications, the mode of operation is referred toas active. This is on account of the role of the antenna arraybased system which radiates pulses of electro-magnetic en-ergy and listens for the return. The parameter space of interestmay vary according to the geometry and sophistication of theantenna array. The radar returns enable estimation of parame-ters such as velocity (Doppler frequency), range and DOAsof targets of interest [130]. Using passive far-field listeningarrays, only the DOAs can be estimated.

In sonar applications, on the other hand, the signal energyis usually acoustic, ,and measured using arrays of hydro-phones. The sonar can operate in an active as well as passivemode. In a passive mode, the receiving array has the capabil-ity of detecting and locating distant sources. Deformablearray models are often used in sonar, as the receiving antennais typically towed under water [79]. Techniques withpiecewise linear subarrays are used in the construction of the

overall solution. Recent experimental results using passivesonar and also seismic (seismometer) arrays a re presented in

In an active mode, a sonar system emits acoustic (electro-magnetic arrays are also used underwater) energy and moni-tors and retrieves any existing echo. This again can be usedfor parameter estimalion, such as bearings and velocity etc .,using the delay of the e cho. Despite its limitations due to thebending speed-of-propagation profiles and the high propaga-tion losses, sonar together with related estimation techniques,remains a reliable tool for range, bearing estimation and otherimaging tasks in underwater applications. However, the dif-ficult propagation conditions under water may call f or morecomplex signal modeling, such as in matchedfieldprocessing

(see e.g. [101).

~ 7 1 .

Industrial Applications

Sensor array signal processing techniques have drawn muchinterest from industriial applications, such as manufacturingand medical applications. In medical imaging and hyperther-mia treatment [37, 38] , circular arrays are commonly used asa means to focus eneirgy in both an injection mode as well asreception mode. It has also been used in treatment of tumors[36]. In electrocardiograms, planar arrays are used to trackthe evolution of wavefronts which in turn provide informa-tion about the condition of a patient’s heart. Array processing

methods have also been adopted to localize brain activityusing biomagnetic sensor arrays, so-called super-conductingquantum interference device (SQUID) magnetometers [85].It is expected that the medical applications will proliferate asmore resolution is sought, e.g. signals emanating from awomb may be of more interest than those of a mother.

Other applications in industry are almost exclusively inautomatic monitoring and fault detectionAocalization. In en-gines, sensors are placed in a judicious and convenient wayto detect and potentially localize faults such as knocks or

broken gears [49, 1661. Another emerging and importantapplication of array signal processing is in using notions fromoptics to cairry out tasks in product quality assurance in amanufacturing environment [ 2 ] , nd in object shape charac-terization in tomography [133,84].An example is provided in“Radar Example.”

Future Directions

In summary, and as demonstrated by the above applicationswhich are miore or less adrianced in their stage of investiga-tion, new connections of certain problems (new or old) keepbeing made to array processing methods whose resolvingpower, in turn, provides i n many cases a superb improvementover more conventional solutions. It is clear that the theoreti-cal interest in sensor array processing has somewhat dimin-ished in recent years, aiid is largely due to the lag ofapplications. However, we believe that the future holds nicesurprises as the applicati ons are catching up with the theoreti-cal foundations, and new perspectives from fields such as

geometry of polynomials :md other insights from communi-cations (e.g. equalization) start making inroads into the re-search community. It is our belief that we will b e witnessingan explosiv~e developmenit of array processing algorithmswithin personal communications, as the various networkoperators start demanding smart antennas (which alreadyhappened). ’We also look forward to more impact of model-based array signal processing in various imaging problems,where more traditional approaches dominate at present. Ex-amples include synthetic aperture radar (SAR) and underwa-ter acoustic imaging. Thiz interest in remote sensing andimaging is expected to grow, particularly on account of theapplications in environmental studies. Other “unexpected”environmental applications have also recently appeared, such

as chemical sensor arrays [87].

Measured Data Experiment

In this section, the viability of the various methods is testedon experimental data. In the first example, the results usingradar data are presented in some detail, whereas the secondexample uses image data.

Radar Example

The receiver is an L = 32-element uniform linear array, whichis mounted vertically so that the DOA is in this case the

elevation angle. With reference to Figure 1, we will assumethat the array is mounted along the y-axis, so that 8 = 0corresponds to the array broadside. The sensors have a hori-zontal polarization. The array is located on the shore of LakeHuron, whereas the transmitter is on the opposite side. Thus,the surface o f the lake will ideally generate a specular multi-path component. There will most likely also be a diffusemultipath. ‘The data collixtion system, referred to as theMulti-Parameter Adaptive Radar System (MARS), was de-veloped at the Communications Research Laboratory at

JULY 1996 IEEE SIGNAL PROCESSING MA GAZINE 85



McMaster University. More details of the MARSsystem andthe experimental conditions can be found in [ 3 2 ] .The par-ticular data sets used herein were also employed by Zoltowski[ 1641, who obtained good results with a maximum-likelihoodtechnique operating on beamspace data. The major goalherein is to investigate the achievable performance of “stand-ard” DOA estimation methods on experimental data. Abeamspace implementation is also considered.

Five data sets labeled 141through 145were available. Eachset consists of a total of 127snapshots, sampled at basebandwith a sampling rate of 62.5 Hz. Due to potentially non-sta-tionary conditions, the data set were divided into 8 batches,each comprising N = 16 snapshots, before further processing.The theoretical DOAs of the direct and reflected signal paths

are fixed (at 81=0.0835” and 82=-0.303” respectively), butthe carrier frequency varies between 8.62 GHz and 12.34GHz. This causes the electrical angle separation to vary

between A$ = BWl3 and A$ = BWl2, where BW = 2 d Lrepresents the standard beamwidth of the array (in radians).Note that the electrical angle for this setup is defined as

Q,= -kdsinO,

because the DOA is defined relative to the array broadside.The given f ive data sets were processed using the previouslydescribed methods assuming a perfect ULA array response(Eq. 28) ,given that the array is quite carefully calibrated. Asin any real data experiment, there are many possible sourcesof errors. The most obvious in this case, is the diffuse multi-path, which may be modeled as a spatially extended enlitter,the waveform of which probably has some correlation withthat of the more point-like signal paths. To illustrate theproblem, the eigenvalues of a typical sample covariance aredepicted in Fig. 7, along with a realization from the theoreti-

cal model with two point sources in spatially white noise.Other sources of errors include inaccurate calibration andinsufficient SN R for resolving within-beamwidth emitters.

Below, we display the results for the various methodsalong with the theoretical DOAs.However, one should bear

Eigenvalue distribution106,

o Simulated dataI lo5‘

io 3 6

x

1o2

5 10 15 20 25 30

7. Distribution of eigenvalues fo r 32-element un$orm linear ar-ray. Simulated and real data.

I BEAMFORMING1

0 9

0 8

0 7

0 3 ’ / x

-5 - 1 5 -1 -0.5 0 0 5 1 1.5 2Beamwidths

fa)

CAPON

L1 ‘2 1 5 1 --05 0 0 5 1 1 5 2

I Beamwidths

ib)

8. Spatial spectra for the beamforming (left) and Capon (right)methods. The results rom the 8 batches o the 12.33 GHzdataare overlayed. The horizontal axis contains the electrical angle infractions of the beamwidth. “True” DOAsindicated by verticaldotted lines.

in mind that in a real data experiment, there are actually notrue values available for the estimated parameters. Th e pos-tulated DOAs are calculated fro m the geometry of the experi-mental setup, but in practice the multipath component mayvery well be absent from some of the data sets, or it mayimpinge from an unexpected direction. We first apply beam-forming techniques to the “simplest” case, which is the 12.34GHz data. The angle separation is here about A$ = BWl2. InFigure 8 , the results from applying the traditional beamform-ing and the Capon methods to the 8 data batches are pre-sented. The “spatial spectra” are all normalized such that thepeak value is unity. The “true” DOAs (electrical angles,measured in fractions of the beamwidth) are indicated byvertical dotted lines in the plot. As expected, the Bartlettbeamformer cannot quite resolve the sources and neither canCapon’s method in this scenario. The high variance of the

86 IEEE SIGNA L PROCESSING MA GAZ INE JULY1996



latter is due to the short data record (in fact, the samplecovariance matrix is not invertible because N < L, so apseudo-inverse was used in Eq. 32).

Next, the spectral-based subspace methods are applied.Since the signal comiponents are expected t o be highly corre-lated, these methods are likely to fail. However, the signalscan be “de-correlated’ by pre-processing the sample covari-ance matrix using forward-backward averaging (42). In Fig-ure 9, the results for the MUSIC method with and without FBaveraging are shown. It is clear that FB averaging improvesthe resolution capability and reduces the variance of theestimates in this case. The Min-Norm method was also appliedand the resulting normalized “spatial spectra” are depicted inFig. IO . The Min-Norm method is known to have better resolu-tion properties than MUSIC because of the inherent bias in theradii of the roots [61, 691. Indeed, the Min-Norm methodperforms slightly better than MUSIC in this case, and it nearlyresolves the sources (even without FB averaging.

Finally, we applied parametric methods to the same datasets. The “optimal” methods described earlier all produced

virtually identical estimates in all of the available data sets.Therefore, only those of the WSF method are displayed in the

In Fig. II1, we show the WSF estimates along with thoseof the ESPRIT method with FB averaging. For comparison,we also applied the W SF method to beamspace data. Thebeamspace dimension was chosen as R = 8, and the transfor-mation matrix was designed as follows: First, new “beam-space outputs” were obtained by summing the outputs ofsensors 1-25, 2-26, etc. up to 8-32. This corresponds to 8parallel Bartlett beamformers, all of them steered to the DOA0” (but with different phases). The SVD of the corresponding32 X 8 transformation matrix (which is a Toeplitz matrix with1/0entries was computed, and its left singular vectors weretaken as the beamspace transformation matrix T (cf. (Eq.96)). The !W D was applied because it is desirable to haveorthonormal columns in the transformation matrix, so as notto destroy the spatial color of the noise (if it was white in thefirst place) The beamspace data were processed using theWSF method similarly to the element-space version. The

plots.

1

0 7

::$ 0 5

E 0 4

N

m

MUSIC

~ ~~

-2 -1 5 -1 -05 0 0 5 1 1 5 2Beamwidths

id

FB MUSIC1,

091

0 8

0 7

0 3 li0 2

0 1

ibi

As in Figure 23, but o r the MUSICmethod, with (right ) andwithout ( left) orward-backward averaging.

MIN-NORM

’

0 9

Beamwidths

FB MIN-NORM

Beamwidths

1. As in Figure 23 , but or the Min-Norm method, with (right )and withoui+ left ) orward-backward averaging.




Parametric Methods1

0 8 WSF

BS WSF

0 6 F B E S P

0 4I

Frequency (GHz)

0 4 r

0 6

8.62 9.76 I 9.79 11.32 12.34

0 8

12 3 4 5 6 7 8

Batch No

1 Parameter ertimates or the W SFmethod in element-spaceand beamApace, and for the ESPRIT algorithm with onuard-back-ward averaging The methodr are applied to the 12 34 GHzdata

SNR speculartorrelation

results for the parametric methods are shown in Fig. 11. Theparametric methods also pe rf om satisfactorily on this dataset, and the estimation error is on the order of a 10:th of abeamwidth. This is clearly smaller than for any of the spec-tral-based methods.

Next, we applied the various algorithms to a more difficultcase, namely the 8.62 GHz data. The results for the Bartlettbeamformer and the FB Min-Norm method are shown inFig. 12. It is clear that the spectral-based methods as appliedhere fail to locate the multipath component. It is possible thatother modifications such as spatial smoothing or beamspaceprocessing can improve the situation, but such a test fallsbeyond the scope of this study.

The estimation results for the parametric methods, appliedto the remaining data sets, are shown in Figs. 13 and 14. Asseen in the figures, the WSF method performs reasonably

well in all data sets except for the one at 9.76 GHz where onlyone signal is detected, although there is a large bias in theestimate of the multipath component in the 8.62 GHz data.

13 dB 10 dB 10 dB 1 dB

0.985123" -J123" -J144"

1

0 9

0 8

0 7

1

>j 0 6

3 0 5

i: 0 42

0 3

:

0 2

BEAMFORMING

41

0 0 50/-2 -1 5 -1 -0 5

Beamwidths

\- I

1 5 2

FB MIN-NORM1

0 9

0 8

0 7

3 0 6

10.5

0 3

0 2

0.1

0-2

Beamwidths

fb)

2. Spatial spectra for the beamforming (left) and FB Min-NormThe FB ESPRIT method on the other hand, produces goodestimates in half of the batches at 8.62 GHz and fails in most

( l i g h t )methods applied to the 8.62 GH zdata.

1SNR direct 19dB 113dB 11 4 d B 110dB 116dB I

I i i I I I I

88 IEEE SIGNAL PROCESSING MAGAZINE JULY 1996



Parametric Methods1

0 8

0 6

0 4

-0 4

I

-0 6

-0 8

11 2 3 4 5

Batch No

(ai

WSF ~

ESWSF ,FEESP '

6

Parametric Methods1

0 8

0 6

0 4

-0 4

-0 6

-0 8

12 3 4 5 6

Batch No

fb)

7

WSF

BS WSF

FE ESP

7

3. As in Figure 11 , butfor the 8.62 GHz (left) and 11.32 GH z(right) datu.

GHz data. This is obviously intriguing, and perhaps it givesan indication of the expected accuracy of the SNR estimates.Another possibility is that more than one reflected componentconstructively interfered to giv e one diffuse specular compo-nent.

A Manufacturing Example

In the steady march towards a fully automated factory envi-ronment, different techniques are constantly being exploredfor various stages of a production line. Applications of sensorarray processing to quality assurance and fault detection haverecently been reported [ 2 , 5 3 ] .A linear array of photosensorsplaced along one side of a target is shown in Figure 15 anddata is accordingly collected. Assuming that the image ofinterest is quantized to 1 bit (i.e. 110 pixels), a simplified datamodel follows by assigning data z1 to a sensor in row 1as,

Parametric Methods1

0 8 W SF ~

ESWSF 1

0 6 FE ESP

0 4

-0

-0 6

-0 8

3 4 5Batch No

(4

Parametric Methods

6 7 I

1

0 6

0 6

04 '

-0.4

-0.6

-0.8

WSF

ES WSF

FE ESP

12 3 4 5 6 7 I

Batch No

fbl

4 .As in Figure 11 , but fo r the 9.76 GH z (le@)and 9.79 GHz(right) data.

where xii, ... XH represent the 1-pixels in the I: th row. If theimage contains a single straight line as in Fig. 15 ,there willbe only one non-zero pixel i n each row, and a little geometryshows that

where xo is the offset and 0 is the orientation of the line. Thevector of "sensor outputs" is modeled as a complex sinusoidwith the row number being the analog of sensor index in ourformulation previously. The line offset and orientation canbe easily computed from the amplitude and frequency of thesinusoid. The model is straightforwardly extended to themultiple-lines case in gray-scale images, and objects otherthan lines in the im age will appear as noise in the dat a model.Given the sum-of-sinusoids model, Aghajan and Kailath [2]

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ImagingSensors

4

-a4

2 4

4

4

4

4

, ines to be detected

Y

Target

xl=xO - (1-1)tan@)

5. Typical linear array setup o photosensors fo r detection oflines (defects) in a manufacturing environment, by merely using

suggested to apply the computationally efficient TLS ES-PRIT algorithm to spatially smoothed data to obtain estimates

of the frequencies and amplitudes.For illustration purposes, an image of a semiconductor

wafer is shown in Fig. 16. A linelscratch is manifested as abright line of pixels. Anomalous pixels (This may be a defectin the wafer or an inscription put in for orientation.) are on aline typically of brighter intensity and are to be detected andlocalized. The results of estimating the line orientation by asubspace-based detector (dashed) and by the more classicalHough transform (solid) a re displayed in Fig. 17. Figure 18shows the corresponding result for the line offset estimates.Though the two methods relatively perform equally well, thecomputational and storage efficiency are not an asset for theHough transform [2]. In light of the comparison of a sub-space-based technique to the more classical approach from

16. Chip with an inscribed line

1 1.2 1. 4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Angle (degrees)

7. Estimation of orientation o the line in Figure 16, using TL SESPRIT (vertical line ) and the Hough transform. The peak magni-tude of Hough Transfarm yields a similar orientation as TLS ES-PRIT

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

040 50 60 70 80I 2o 30

I

18. Estimation of the line offset using TLS ESPRIT (vertical lineand Hough Transfonn

tomography, namely the Hough transform, shown in thesefigures, there is reason to believe that sensor array processing

solutions need no longer be limited to the framework theywere developed in (i.e. signal processing in Radar, Sonar etc.)and potentially offer a wealth of efficient and highly perform-ing algorithms.

Conclusions

The depth acquired from the theoretical research in sensorarray processing reviewed in this paper has played a key rolein helping identify areas of applications. New real-world




problems which are solvable using array signal processingtechniques are regu1,arly introduced, and it is expected thattheir importance will only grow as automatization becomesmore widespread in iindustry, and as fas ter and cheaper digitalprocessing systems become available. This manuscript is notmeant to be exhaustive , but rather to serve as a broad reviewof the area, and more importantly as a guide for a first timeexposure to an interested reader. The focus is on algorithms,whereas deeper analyses and other more specialized researchtopics are only briefly touched upon.

In concluding, the recent advances of the field, we thinkit is safe to say that the spectral-based subspace methods and,in particular, the more advanced parametric methods haveclear performance improvements to offer as compared tobeamforming methods. Resolution of closely spaced sources(within fractions of a beamwidth) have been demonstratedand documented in :several experimental studies. However,high performance is not for free. The requirements on sensorand receiver calibration, mutual coupling, phase stability,dynamic range, etc. become increasingly more rigorous withhigher performance specifications. Thus, in some applica-tions the cost for achieving with in-beamwidth resolution maystill be prohibitive.

The quest for algorithms that perform well under idealconditions and the understanding of their properties has insome sense slowed down in light of the advances over thepast years. However, the remaining work is no less challeng-ing, namely to adapt the theoretica l methods to fit the particu-lar demands in specific applications. For example, real-timerequirements may place high demands on algorithmic sim-plicity, and special array and signal structures present incertain scenarios must be exploited. The future work, also inacademic research, will be much focused on bridging the gapthat still exists between theoretical methods and real-world

applications.

Acknowledgmeints

The authors are grateful to Professor Simon Haykin ofMcMaster University, for providing the data used in theMeasured Data Experiment section and to H. Aghajan ofStanford Univesity for providing figures in the Manufactur-ing Example section. The authors are also grateful to thereviewers who, through numerous comments, made this pa-per more readable. One of us is also grateful to Prof. A.S.Willsky of MIT for his constant encouragement and neverending enthusiasm.

Hamid Krim is with the Stocastic Systems Group, LIDS,Massachusetts Institute of Technology, Cambridge, MA 02139.

Mats Viberg is with the Department of Applied Electron-ics, Chalmers Univt:rsity of Technology, Gothenburg, Swe-den.

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