superresolution source location with planar arrays

• HATKESuperresolution Source Location with Planar Arrays

VOLUME 10, NUMBER 2, 1997 THE LINCOLN LABORATORY JOURNAL 127

Superresolution Source Locationwith Planar ArraysGary F. Hatke

■ The challenge of precision source location with a radio-frequency antennaarray has existed from the beginnings of radiometry and has continued inmodern applications with planar antenna arrays. Early work in this field waslimited to estimating single source directions in one dimension with systemslike crossed-loop radiometers. Currently, more advanced systems attempt toestimate azimuth and elevation by using two-dimensional arrays. Monopulsetechniques have been extended to two-dimensional arrays to provide acomputationally efficient method for estimating the azimuth and elevation of asingle source from a planar array, but all monopulse techniques fail if there isappreciable interference close to the source. In this situation, adaptive array(superresolution) processing techniques are needed for direction estimation.

This article discusses the results of a study on the proper way to design anadaptive planar array with a constrained antenna aperture. We consider thesegmentation of the antenna aperture, the polarization of the antenna segments,and the algorithms used to process the signals received from the antenna. Inparticular, we concentrate on interference that is within one Rayleighbeamwidth of the source. The interference can be highly localized in space, as ina single direct-path interferer, or diffuse in space (possibly due to multipath).We present results of tests conducted with a segmented antenna array, alongwith simulations and analytical bounds, that guide us in designing a source-location system.

E of a source in thepresence of interference is a complex chal-lenge for a direction-finding system. Most tra-

ditional direction-finding systems are designed with amonopulse antenna-array configuration that seg-ments the antenna array into four symmetrical subap-ertures, as shown in Figure 1. This array configura-tion, or topology, allows estimating both azimuth andelevation of the source.

In nonadaptive direction-finding systems, the an-tenna patterns of the subapertures are designed tosuppress most interference from outside the mainbeam of the array. When the interference is inside themain beam of the array—by definition closer (in radi-ans) to the source than the inverse (in wavelengths) ofthe direction-finding antenna-array aperture—a non-

adaptive technique cannot suppress the interference.Traditionally, radar designers have solved this prob-lem by increasing the aperture of the antenna arrayuntil the interference is no longer inside the mainbeam. This solution can be impossible to implement,however, if the array is physically constrained and thewavelength of the radiation cannot be changed. Inthis case, adaptive array (superresolution) processingtechniques must be employed to accurately estimatethe source location.

Two approaches for source-location estimates withmain-beam interference can be used, depending onwhether the interference is pointlike or diffuse. Forsimplicity, we assume that there is only one interfererin the main beam. If the main-beam interference iscaused by a pointlike interferer, conventional mono-


128 THE LINCOLN LABORATORY JOURNAL VOLUME 10, NUMBER 2, 1997

pulse antenna topology and superresolution algo-rithms may be applied to the data from the foursubapertures of the array to suppress the interference.Problems arise when the main-beam interference isdiffuse as a result of incoherent multipath scatteringof the interference. In this case, applying adaptivetechniques to a conventional quadrant array may notsuppress the diffuse interference in the main beam ofthe array. This limitation also applies if there are mul-tiple main-beam interferers. Again, for simplicity, weconcentrate on the single interferer case.

To suppress diffuse main-beam interference effec-tively, we can divide the array into more subaperturesthan the conventional four. This additional segmen-tation gives more channels of information, hencemore degrees of freedom, to the adaptive-array pro-cessing algorithm, allowing improved interferencesuppression. We can also increase the degrees of free-dom by receiving two different polarizations fromsome or all of the array subapertures. This strategy isuseful when the interference and source have differ-ent polarization states. These approaches to suppressmain-beam interference come at a price—each addi-tional channel from the array requires an additionalreceiver, and the complexity of the adaptive-arrayprocessing algorithms that process the multichanneldata increases in proportion to the cube (or higher) ofthe number of data channels.

We conducted a two-part study to explore the bestcombinations of array topologies and superresolutionalgorithms that accurately estimate source location.

First, we studied the minimum required segmenta-tion to allow an adaptive-array processing algorithmto accurately estimate the direction of a source in thepresence of either diffuse or pointlike main-beam in-terference. Results of this study indicate that for pla-nar arrays, more than four subapertures are desirablewhen diffuse main-beam interference is present. Wealso found that using nonidentically polarized subap-ertures can be beneficial when the interference has arelatively pure polarization that differs from thesource.

We then developed computationally efficient algo-rithms that could be coupled with a wide range of ar-ray-segmentation topologies to give nearly optimalestimates of source direction. The new algorithms arecalled PRIME (polynomial root intersection for mul-tidimensional estimation) and GAMMA (generalizedadaptive multidimensional monopulse algorithm).These algorithms have been shown through simula-tion and analytical derivations to have estimate vari-ances that equal previous state-of-the-art algorithmsat a fraction of the computational cost. We useGAMMA when we have an estimate of the spatial co-variance matrix (the spatial power distribution) of theinterference signal in the absence of the desired sourcesignal. We use PRIME when we cannot get desiredsource-free observations of the spatial power distribu-tion. Both algorithms are multidimensional generali-zations of previously proposed one-dimensional root-ing algorithms that convert the direction-estimationtask to one of finding roots to a polynomial equation.

We verified the performance of the selected an-tenna-array topology designs and the adaptive-arrayprocessing algorithms by using a segmented antennaarray. Source directions were successfully estimatedby using both the PRIME and GAMMA algorithmsin the presence of either diffuse main-beam interfer-ence or pointlike main-beam interference.

Planar-Array Primer

Because estimating the direction to a source is a two-dimensional problem, two angles must be estimatedper source, which requires at least a two-dimensionalarray. Figure 2 shows the simplest array topology thatmeets this requirement—a three-subaperture planararray. While this array topology allows the simulta-

FIGURE 1. Four-subaperture circular array. This configura-tion allows estimation of source azimuth and elevation.



FIGURE 2. Three-subaperture circular array. This configura-tion is the simplest array topology that allows estimation ofsource elevation and azimuth angles.

neous estimation of elevation and azimuth angles, anarray topology with four subapertures arranged as inFigure 1 can be transformed into a three-channel ar-ray via a unitary beamforming matrix to provide esti-mates of elevation and azimuth angles for a sourcesignal. The three-subaperture array has no suchsimple method for producing source-angle estimates,and therefore is rarely used. The three-channel (fromfour subapertures) system is often called a monopulsesystem, because it was originally designed to estimatea desired source’s azimuth and elevation by using asingle pulse from a pulsed radar.

To see how this transformation occurs, let us lookat the output of the four-subaperture array in the ab-sence of noise, with a desired signal waveform s(t) atan azimuth θ and an elevation φ. The array outputvector x(t) has the form

x( ) ( ) .

( )

( – )

– ( )

– ( – )

t s t

e

e

e

e

jk u u

jk u u

jk u u

jk u u

=

+

+

1 2

1 2

1 2

1 2

(1)

Here we define an array constant k as 2πd /λ, where dis the distance from the center of the array to the ef-fective phase center of one of the quadrants, and λ isthe wavelength of the center frequency of the radio-frequency (RF) signal. The array constant k, whichvaries inversely with the array beamwidth, can be

thought of as a scaled distance between the arrayphase reference (here taken to be the center of the ar-ray) and the phase center of a given subaperture. Theterms u1 and u2 are functions of the azimuth and el-evation of the desired source:

u

u1

2

==

sin

cos sin .

θ

θ φ

The monopulse beamforming network can be math-ematically represented by a 4 × 4 matrix M:

M =− −

− −− −

14

1 1 1 1

1 1 1 1

j j j j

j j j j.

The output y(t), after applying the beamforming net-work, is a vector of four beams with the form

a y Mxs t t t

ku ku

ku ku

ku ku

ku ku

s t( ) ( ) ( )

cos( ) cos( )

sin( ) cos( )

cos( ) sin( )

sin( ) sin( )

( ) .= = =

1 2

1 2

1 2

1 2

An estimate for u1 is the arctangent of the voltageratio of the second beam [ sin( ) cos( ) ( )ku ku s t1 2 ] tothe first beam [ cos( ) cos( ) ( )ku ku s t1 2 ] divided by thearray constant k , and an estimate of u2 is the similarlyscaled arctangent of the voltage ratio of the thirdbeam [cos( ) sin( ) ( )ku ku s t1 2 ] to the first beam. Thearray-response vector a after the monopulse transfor-mation is real for all values of u1 and u2 because ofthe centrosymmetric nature of the four-subaperturesquare antenna array. A centrosymmetric array is onewith a phase center such that all subapertures in thearray appear in pairs around the phase center. Cen-trosymmetric arrays are useful topologies becausethere is always a beamforming matrix that producesarray-response vectors that are entirely real.

Three-channel systems do well in estimating singlesource directions when the array receives no interfer-ence. However, if one or more of the three beamsused for direction estimation in a monopulse system



receive an interference signal, the resulting angle esti-mates are corrupted. Because of the limited degrees offreedom in a three-channel monopulse system, even ifthe outputs are combined adaptively it is impossibleto eliminate the effects of the interference from thedesired source-angle estimates. For this reason, allcentrosymmetric adaptive planar-array processingsystems have four or more antennas with at least fourindependent channels at the output of anybeamformer. When an interference environmentconsists of more than a single point source in themain beam, such as diffuse multipath or multiplepoint-source interference, then even a four-channelsystem is unable to cope with the interference envi-ronment. This limitation implies that an antenna-segmentation topology with more than four subaper-tures will be beneficial.

Array-Topology Study

Lincoln Laboratory studied the benefits of alternateantenna segmentation topologies while attempting totrack a source with diffuse interference within one ar-ray beamwidth of the source. A multi-aperture array,known as the sampled aperture sensor (SAS), wasconstructed with fourteen dual-polarized subaper-tures, giving twenty-eight possible outputs. The SASwas designed with an eight-subaperture sixteen-chan-nel large outer array, which gave us a larger array aper-

ture and allowed a larger number of array configura-tions, and a smaller, six-subaperture twelve-channelinner array, as shown in Figure 3. The outer eight-subaperture array has an effective beamwidth of sixdegrees; the inner six-subaperture array has an effec-tive beamwidth of sixteen degrees. Because of limitedreceiver resources, only eight subapertures could beused simultaneously. Thus we constructed a switch-ing system that allowed the array operator to choosefrom five configurations of eight subapertures. Figure4 shows the dual-polarized patch SAS array. Eachsubaperture has roughly 25 to 30 dB of polarizationisolation between the vertical and horizontal outputs.

To achieve desired source-angle estimates in thepresence of main-beam interference, the SAS must bewell calibrated [1]. We performed a pattern responseequalization for spatial similarity (PRESS) array cali-

FIGURE 3. Sampled aperture sensor (SAS) subapertureconfigurations. The SAS was designed with an eight-sub-aperture sixteen-channel large outer array, which allows awide range of aperture and array configuration trade-offs,and a smaller, six-subaperture twelve-channel inner array.

FIGURE 4. Sampled aperture sensor. The outer eight-aper-ture array has an effective beamwidth of six degrees; the in-ner six-aperture array has an effective beamwidth of sixteendegrees. Both arrays are dual-polarized with patch-antennatechnology.



bration of the SAS mounted with its receivers. Thecalibration-error residual is defined as the averagelength of the difference vector between the assumedarray response (normalized to length of one) and theproperly normalized array response from the PRESSalgorithm. The average calibration-error residual istaken over all azimuths and elevations for which thecalibration is expected to be used. The SAS has an av-erage calibration-error residual of –27 dB. At thislevel, our experience indicates that a point-source in-terferer cannot be tolerated within less than one-eighth of a beamwidth from the desired source.

To simulate the effect of diffuse main-beammultipath interference on direction-finding a source,we conducted an experiment with two horn antennaselevated above the earth. The first antenna was angleddownward such that its radiation pattern illuminatedthe earth, but did not directly radiate toward the re-ceive array. The second horn provided a direct-pathsignal to simulate the source of interest. The signalfed to the downward-pointing antenna had 30-dBhigher gain than that fed to the direct-path antenna,creating a 20-dB interference-to-signal ratio at theSAS. The SAS was positioned at varying distancesfrom the transmitting antenna complex to give a vari-ety of desired source-interference separations.

To examine the effects of spatial antenna segmen-tation, we initially connected eight receivers to theeight copolarized outer-array subapertures on theSAS. Because of antenna-pattern limitations, themaximum angular separation between the hot spot of

the multipath interference and the desired source waslimited to less than one full-array beamwidth. Withthe separation between the multipath interferenceand the desired source solely in the vertical dimen-sion, the important topology variable became thenumber of different vertical subapertures (number ofdistinct phase centers in the vertical direction) in use.

We analyzed the measured data by using differentsubsets of the available eight channels to simulate ar-rays with three, four, and five distinct verticalsubapertures, as shown in Figure 5. Figure 6 showsthe results of applying a rooting variant of the maxi-

FIGURE 5. Examples of SAS array configurations. Arrays of three, four, and five distinct vertical subapertures were simulatedby analyzing data from different subsets of the available eight channels.

FIGURE 6. Performance of three, four, and five vertical (el-evation) subapertures in distinguishing desired signal fromdiffuse interference. A four- and five-subaperture array per-forms well but the three-subaperture array fails to find thedesired signal.

Three elevation subaperturesFour elevation subaperturesFive elevation subaperturesInterference elevation

0

1/6

1/3

1/2

2/3

–1/6

–1/3

–1/2

–2/3

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(bea

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Source-interference separation (beamwidths)



mum-likelihood method (MLM) algorithm [2] to thethree different SAS array configurations. The resultsshow that the estimates closest to the desired source ofthe three-subaperture array tend to track the interfer-ence signal rather than the source, while the four- andfive-subaperture arrays more or less track the desiredsource. This behavior indicates that three subaper-tures in elevation cannot resolve the desired sourcefrom the interference, while four or five subaperturesin elevation can resolve and track the desired source.

In addition to resolution, another factor that af-fects a direction-finding system is the variance of thesource-direction parameters produced by the estimat-ing algorithm. Unlike resolution, which can be highlydependent on the algorithm used, estimator variancecan be bounded from below by a number of algo-rithm-independent expressions. One expression, theCramér-Rao bound, is commonly used to determinelimits on estimates generated by arrays [3]. Thebound is a function of the subaperture topology. Tocompute the bound, we calculate the spatial covari-ance of the interference source. For diffuse multipathsources, the calculation requires knowledge of the re-flection coefficients, in this case those of the earth’ssurface.

By using models such as the Beckmann-Spizzi-chino model [4], we can calculate the spatial covari-ance matrix for a diffuse multipath source and aCramér-Rao bound on the variance of the directionparameters to the desired source. Figure 7 shows thatusing a six-subaperture array instead of a four-subaperture array decreases the bound on elevation-angle noise for the geometries tested with the SAS byover a factor of two. This improvement can be attrib-uted to the additional vertical phase center in the six-subaperture array. Figure 8 shows that the six-sub-aperture array has four distinct phase centers inelevation, while the four-subaperture array has onlythree. This improvement further confirms our desireto have at least four distinct phase centers in thesource-interference dimension.

Note that the bound on elevation-angle noise fromthe four-subaperture array is a minimum for this par-ticular orientation; if the array is rotated such thatthere are four distinct phase centers in elevation, thebounds on elevation-angle noise actually increase.

This increase in the elevation-angle noise bound im-plies that the number of degrees of freedom is as im-portant as the number of phase centers.

To understand why the elevation-angle noisebounds from the four-subaperture array are so large,we consider the maximum number of point sourcesfor which unambiguous direction estimates can befound, given a planar array of N sensors. A well-known result in the literature indicates that the maxi-mum number of point sources should be N – 1 for anarbitrary planar-array geometry [5]. However, whenthe array is centrosymmetric, this result is no longer

FIGURE 7. Cramér-Rao bounds on elevation-angle noise forfour- and six-subaperture arrays. A four-subaperture arraydoes not have enough degrees of freedom to handle the dif-fuse interference, and thus performs poorly. A six-subaper-ture array decreases the bound on elevation-angle noise bymore than a factor of two for the geometries tested with theSAS.

FIGURE 8. Six-subaperture and four-subaperture arrays.The six-subaperture array has four phase centers in eleva-tion, while the four-subaperture array has three.

Six-subaperture arrayFour-subaperture array

0.09

0.11

0.07

0.05

0.03

0.01

Elev

atio

n st

anda

rd d

evia

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(bea

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Phase center



valid. The appendix entitled “A Maximum Numberof Sources Uniquely Localizable with a Centrosym-metric Array” shows that, in this case, only N – 2sources can be unambiguously located. Thus a four-subaperture square array can handle at most twopointlike sources. A diffuse interference source can bethought of as two or more closely spaced pointsources. Thus, with respect to degrees of freedom, itseems unlikely that a four-subaperture square arraywould perform well against a diffuse interferencesource.

Polarization Diversity

The previous results show that to mitigate diffusemain-beam interference, an array should use a seg-mentation with more than three subapertures in theline defined by the source-interference pair, and morethan four channels. We can further eliminate main-beam interference effects if there are polarization-state differences between the desired signal and theinterference, and not all the subapertures of the arrayhave the same polarization response. This approachacknowledges that the difficulty in estimating the di-rection to a source with main-beam interference is afunction of the array-beamspace distance between thedesired source array-response vector and the interfer-ence array-response vectors. This distance depends onthe inner product of the two normalized array-re-sponse vectors [6].

For unipolar arrays experiencing main-beam inter-ference, the desired source-interference array-beam-space distance can be mapped monotonically to an-gular separation between the desired source andinterference. With diversely polarized subapertures inthe array, such a mapping is not possible because thedistance between the desired source array-responsevectors and the interference array-response vectors isnow a function of not only the angular locations ofthe desired source and interference, but also their po-larization states. Note that just as any polarization-state plane wave can be synthesized by some complexcombination of vertical and horizontal plane waves,any array response from a polarization-diverse arraycan be generated by a complex linear combination ofthe array response to a purely vertically polarizedplane wave av and the array response to a purely hori-

zontally polarized plane wave ah . Thus the array-re-sponse vector to a desired source from a direction uwith polarization state p can be written as

a u p a a p( , ) [ , ] ,= v h

where p is defined as a two-dimensional complex vec-tor of unit magnitude.

The inner product of the normalized array-re-sponse vectors from directions u1 and u2 with polar-ization states p1 and p2 seen by a polarization-diversearray is given by

pa u a u a u a u

a u a u a u a up

a u a u p a u a u p

11 2 1 2

1 2 1 22

1 1 1 2 2 2

H vH

v hH

v

vH

h hH

h

v h v h

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

[ ( ), ( )] [ ( ), ( )].

This equation shows that even if the directions u1 andu2 are identical, the polarization states p1 and p2 canhave values such that the inner product of the two-ar-ray-response vectors can range from zero to one. Thusit may be much easier to estimate the direction to asource with main-beam interference if the receivingarray has diversely polarized subapertures, rather thanall subapertures of the same polarization. This ar-rangement, however, comes with a computationalcost. Because the array is now sensitive to polariza-tion, the number of parameters that we must estimateto determine the desired source direction includes thepolarization state of the source. Consequently, thefast estimation algorithms require an additional de-gree of freedom, as we discuss later. Thus, instead ofneeding a four-channel array to estimate the directionparameters to a source in the presence of one point-like main-beam interferer, a diversely polarized arraywould require five channels.

The above representation is valid when the polar-ization states of the plane waves are constant withtime. When the polarization state changes during theobservation time, the energy that the array sees is notcontained in a one-dimensional subspace, as it is for asource of constant polarization, but occupies a two-dimensional subspace spanned by the vertical andhorizontal array-response vectors av(u) and ah(u). Ifthis condition occurs with the interference energy, itwill increase the number of degrees of freedom in thearray required to accurately estimate the desired



source direction parameters because the energy fromthe interference direction acts as if there were twosources. In addition, the distance between the desiredsource array-response vector and the interference ar-ray-response subspace is measured as the closest ap-proach between the interference subspace and the de-sired source array-response vector. This definition ofdistance implies that if the polarization state of the in-terference varies during the observation time, the dif-ficulty in estimating the desired source direction willbe determined by the inner product of the desiredsource array-response vector and the array-responsevector from the interference direction with the worstpossible polarization state, even if that state is neveractually observed.

Data Examples of Polarization Diversity

We conducted an experiment to estimate the desiredsource directions by using both singly polarized anddiversely polarized arrays in the presence of a point-like interferer with a non-fluctuating polarizationstate. For this experiment, two horns were placed onthe ground. The source horn transmitted a continu-ous tone in a vertically polarized state, while the inter-ference horn radiated a noise waveform, and the hornwas tilted so that the polarization was linear at a 45°slant to vertical and had a 20-dB interference-to-sig-nal ratio in the processing band of interest. The singlypolarized array was a standard quadrant monopulseconfiguration. The diversely polarized array consistedof all four vertical monopulse channels plus a hori-zontally polarized sum and a horizontally polarizedazimuth difference beam. The results in Figure 9show that the addition of the two cross-polarizedchannels greatly decreases the variance of the sourcedirection estimates.

Estimation Algorithms

The results of our array-topology study indicate thattopologies more complicated than simple quadrantarrays, with perhaps some diversely polarized subap-ertures, are desirable to mitigate main-beam interfer-ence. Because of the computational stress that adap-tive-array processing entails, efficient algorithms forproducing desired source direction estimates are im-perative. This section of the article discusses two new

algorithms designed to efficiently estimate source di-rections from arbitrary multidimensional arrays. Thefirst algorithm, PRIME, estimates directions for notonly the desired source but also for all the interferencesources. This algorithm can be used on any type ofsignal and interference waveform. The second algo-rithm, GAMMA, provides estimates for only the de-sired source direction. This algorithm requires thatthe interference covariance matrix be estimated in theabsence of appreciable desired signal energy.

Both algorithms achieve their computational effi-ciency by posing the angle-estimation problem insuch a way that the estimates are generated from the

FIGURE 9. Azimuth errors as a function of source-interfer-ence separation for a vertically polarized source in the pres-ence of mixed-polarization interference. The improved per-formance of the diversely polarized array, evidenced by lessscattering of data points, can be attributed to its ability todifferentiate signal and interference both polarimetricallyand spatially.

Source direction-finding estimatesStandard monopulse estimatesInterferer location

0

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roots of a data-derived polynomial. To understandthe benefits of this technique, we consider a generaldirection-estimation algorithm. Most modern direc-tion-finding algorithms applied to unipolar arrayscan be posed as

ua u M a u

a u M a uu* arg max

( ) ( )

( ) ( ),=

H

H1

2 (2)

where u * is the source direction estimate, and M1 andM2 are arbitrary matrices whose exact values dependon the test statistic used.

The difficulty of calculating this test statistic is infinding the maximum over the direction parameters.Attempts at differentiation of the test statistic with re-spect to the parameters of the direction vector u typi-cally do not yield simple equations. Thus the extremalvalues of u are generally calculated by evaluating thetest statistic over a grid of potential values for u. Thisapproach is often referred to as searching over the ar-ray manifold, because the set of array-response vec-tors a(u) can be considered a submanifold in complexN-dimensional space (if there are N subapertures)with dimension equal to M, which is the number ofparameters in the vector u. The rough estimates forlocal maxima are then refined, often by some localgradient iteration algorithm.

When the number of parameters in the directionvector u becomes larger than one, this manifoldsearch technique can become computationally inten-sive. Much effort has gone into avoiding this searchtechnique for simple source and interference sce-narios with simple array topologies. In the case inwhich the spatial covariance of the interference is anidentity, Equation 2 simplifies to

u a u xu

* arg max ( ) .= H 2

For a quadrant array topology that estimates azimuthand elevation, this problem can be approximatelysolved by using the monopulse technique. Unfortu-nately, when main-beam interference is present, thiseasy approximation does not work.

In the case of a special array topology—a uni-formly spaced unipolar line array—another tech-nique has been proposed to avoid the manifold

search. In this case, the array manifold is only a func-tion of a single scalar u and has the form

a( ) .

( – )

u

e

e

e

jku

j ku

j N ku

=

1

2

1

M(3)

The phase center of the array is assumed to be the firstsubaperture. The constant k is the scaled distance be-tween subapertures, defined as in Equation 1, with dthe distance between adjacent subapertures. Refer-ence 7 notes that if a new complex variable, defined asz, is used to replace the term e jku in Equation 2, thena(z) has the form

a( ) .

–

z

z

z

z N

=

1

2

1

M

Reference 7 further notes that when the magnitudeof the variable z is unity, then the conjugate of a(z) isequal to a(z –1). By using these substitutions, we canexpress Equation 2 as

arg max( ) ( )

( ) ( ).

–

–z

T

T

z z

z z=1

11

12

a M a

a M a

Finally, if the condition z = 1 is relaxed, then theoptimization statistic can be driven to infinity for val-ues of z such that

z z z g zN T– –( ) ( ) ( ) .1 12 0a M a ≡ = (4)

The z N – 1 term is added to eliminate negative powersof z , allowing g(z ) to be a common polynomial.

This conversion from performing a manifoldsearch required by Equation 2 to solving for a set ofhomogeneous solutions of a complex polynomialproblem was applied to a number of superresolutiondirection-estimation algorithms, which yielded root-



ing variants of the multiple signal classification (MU-SIC) [8] and maximum entropy method (MEM) al-gorithms [2]. One benefit of the rooting techniques isthat they tend to have better resolution propertiesthan previous manifold-search techniques. Reference2 discusses how the root-MUSIC algorithm resolvestwo closely spaced sources with a 5-dB-lower signal-to-noise ratio than the spectral (manifold search)MUSIC algorithm. The rooting techniques havebeen generalized to handle nonuniform linear arrays[1], as well as polarization-diverse linear arrays [9].

By following the logic used to derive a polynomialrepresentation for uniform linear arrays, we can char-acterize the array manifold for a planar array withsubapertures centered on a regular grid in terms oftwo complex variables. To illustrate this concept, letus assume a rectangular gridding for the subaperturesof the array. Consider the subaperture in the upperleft corner as the phase-reference element (the phasereference can be arbitrarily chosen anywhere on thearray). The array-response vector element al ,p corre-sponding to the subaperture displaced to the right byl subapertures and down by p subapertures has theform

a e el pjk u l jk u p

,( ) ( ) .= 1 1 2 2

Here k1 is the scaled distance between subaperturephase centers in the vertical direction, and k2 is thescaled distance between subaperture phase centers inthe horizontal direction. By making the substitution

w e z ejk u jk u= =1 1 2 2and ,

we can represent the array-response vector a(u) asa(z, w). By using this representation, we can developa two-variable polynomial equation for a planar arraysimilar to the univariate polynomial in Equation 4.Using this representation does not give unique solu-tions for either z or w, however, because we have onlyone equation with two unknowns.

The PRIME algorithm was developed to solve thisproblem. It is often possible to generate two indepen-dent statistics that have local maxima correspondingto true source directions. The system of equations

g z w

g z w1

2

0

0

( , )

( , )

==

defines a finite set of (z, w) pairs that satisfy bothpolynomials. Some of these solutions correspond totrue source locations. Take, for example, the PRIMEversion of the MUSIC algorithm. As noted in Refer-ence 8, the estimation test statistic in this case is

ua u E E a uu

* arg max( ) ˆ ˆ ( )

.= 1H

n nH (5)

Here En are estimates of the eigenvectors of thesample covariance matrix (which is x xi i

Hi∑ by defi-

nition) that are orthogonal to all the source array-re-sponse vectors, including interference. If the rank ofthe matrix En, defined as q, is greater than or equal totwo, then we can construct two distinct subsets of theeigenvectors in En, defined as En1 and En2, where

rank rank

and

ˆ , ˆ ;

ˆ ˆ ;ˆ ˆ ˆ .

E E

E E

E E E

n n

n n

n n n

q q1 2

1 2

1 2

{ } < { } <

≠

∪ =

By using these two new matrices, we define the poly-nomials g z w1( , ) and g z w2( , ) as

g z w

z w z w z w

g z w

z w z w z w

N N Tn n

H

N N Tn n

H

1

1 1 1 11 1

2

1 1 1 12 2

1 2

1 2

( , )

( , ) ˆ ˆ ( , )

( , )

( , ) ˆ ˆ ( , ) .

– – – –

=

=− − − −

a E E a

a E E a

The scalars N1 and N2 are the number of phase cen-ters in the elevation and azimuth planes of the array,respectively. Note that in the absence of estimationerror and array-calibration errors, the expressions

a E E a

a E E a

( , ) ˆ ˆ ( , )

( , ) ˆ ˆ ( , )

–

–

z w z w

z w z w

Tn n

H

Tn n

H

−

−

1 11 1

1 12 2

are zero for (z, w) pairs corresponding to true sourcedirections, just as the original term

a E E a( , ) ˆ ˆ ( , )–z w z wTn n

H−1 1

is zero. Thus some joint solutions to these two equa-tions correspond to directions having array-responsevectors orthogonal to En , maximizing Equation 5.

There are other methods for forming multiple in-



dependent polynomials. One method is to form inde-pendent statistics on two nonsimilar subarrays of theentire array aperture. This method allows us to incor-porate a larger number of subapertures into the direc-tion-estimation problem than the number of availablereceivers. Figure 10 provides an example of two dis-similar subarrays from a segmented array. A thirdmethod is to differentiate the original polynomialg(z,w) with respect to both z and w. This approachgenerates two new independent polynomials that re-tain the homogeneous solution properties of theoriginal statistic.

Polynomial Root Intersections

The heart of the PRIME algorithm is the formationof polynomial root intersections. For the cases treatedhere, root intersection means finding the common ze-ros of two polynomials in two unknowns. Several pro-cedures are relevant. First, numerical techniques canbe used if good initial guesses are available. A more so-phisticated global approach utilizes root tracking andhomotopy (e.g., linear interpolation) of the polyno-mial coefficients from a canonical, decoupled set ofequations to the equations of interest [10]. This ap-proach is most appropriate when the polynomialshave high degree. For lower-degree polynomials, asimpler procedure known as elimination theory pro-vides all intersection points. We now present a math-ematical explanation of elimination theory.

Consider two polynomials f (x) and g (x) of a single

variable with, for example, complex coefficients:

f x a x a x a

g x b x b x b

nn

mm

( )

( ) .

= + + +

= + + +

K

K

1 0

1 0

Let {γi } and {δi } denote the roots of f and g , respec-tively. The resultant R of the two polynomials f and gis given by Reference 11:

R f g a bnm

mn

iij

j( , ) ( ) .≡ −∏ γ δ

R( f , g ) can be shown to be a polynomial in the coeffi-cients {ai } and {bj } of the polynomials f and g thatvanishes if and only if f (x) and g (x ) have a commonroot, provided at least one of an and bm is nonzero.There are several different but equivalent expressionsfor the resultant. One of the simplest, from Sylvester[12], expresses R( f , g ) as the determinant of the(m + n) × (m + n) matrix

a a a

a a a

a a a

b b b

b b b

b b b

n n

n n

n

m m

m m

m

–

–

.

1 0

1 0

1 0

1 0

1 0

1 0

0 0

0 0

0 0

0 0

0 0

0 0

L L

L L

M M M M M M M

L L

L L

L L

M M M M M M M

L L

−

−

FIGURE 10. Two dissimilar five-subaperture subarrays from a six-subaperturearray. These configurations allow us to incorporate a larger number of subaper-tures into the direction-estimation problem than the number of available receivers.



PRIME must find the root intersections of two-variable polynomials g1(z, w) and g2(z, w). If we viewg1 and g2 as polynomials in w with coefficients that arepolynomials in z, the resultant, denoted Rw(z), is apolynomial in z whose roots are the z-coordinates ofthe common solutions of gi(z , w) = 0. Similarly, theresultant Rz(w) is a polynomial in w expressing the w-coordinates. For small degrees, we can determine theroots of the resultants Rz and Rw and pair the resultingz-w coordinates in all possible ways to form a list ofcandidate intersections that we can substitute in theoriginal equations. Alternatively, we can calculate thehomogeneous solutions for Rz and substitute them inthe original polynomials g1 or g2.

Not all common roots of gi(z, w) correspond tophysical angles. We can select physically relevantroots by evaluating a direction-finding statistic suchas spectral MUSIC for each common root or by uti-lizing prior information about signal angles of arrival.If Rw(z), or Rz(w), vanishes identically, the polynomi-als gi(z , w) = 0 have a common nontrivial factor (as-suming, for example, that the coefficient of the high-est w-order term of g1(z , w) or g2(z, w) is a nonzeropolynomial in z). In this case, the solution sets inter-sect along the curve defined by the common factor.This case is not typically encountered in applications.However, sometimes the gi(z, w) have a commondata-independent factor (typically of the form zLwN )that can be eliminated from both polynomials.

The number of intersection points of the root locigi(z , w) depends on the polynomial degrees, the arraygeometry, and any symmetries inherent in the poly-nomial coefficients, which in turn depend on thepolynomial selection procedure. Let di denote the de-grees of gi . A bound on the number of intersectionpoints can be based on Bezout’s theorem, which statesthat after accounting for and including roots at infin-ity, the number of common solutions of the homog-enized polynomials, for i = 1 and 2, is d1 × d2. Thepolynomials typically used for direction finding sharedegenerate roots at infinity (i.e., common solutionsinvolving only the highest-order terms of each poly-nomial). By accounting for the multiplicity of theseroots at infinity, we can formulate a bound on thenumber of roots (including multiplicity) in the finitez-w plane. Alternatively, we can apply a result of A.G.

Kouchnirenko [13] to count the number of commonroots of polynomials. For example, an m × n array ge-nerically provides 2(m – 1)(n – 1) roots, with both zand w nonzero.

While this discussion has concentrated on the caseof a unipolar planar array, note that eliminationtheory can be extended to polynomials of more thantwo variables. Hence, in the case of polarization-di-verse planar arrays, where the array-response vectorscan be described as functions of three complex scalars(z, w, and the polarization ratio ρ), the PRIMEmethod may still be applied. In this case, three poly-

FIGURE 11. The polynomial root intersection for multidi-mensional estimation (PRIME) maximum entropy method(MEM), top, versus spectral multiple signal classification(MUSIC) direction estimation with diffuse interference, bot-tom. The PRIME-MEM source-location estimates have fewerpoints captured by the interference when the signal-interfer-ence separation is small, and the source-location estimatesare not as biased when the separation is large.

Interference direction-finding estimatesSource direction-finding estimatesFiltered-source direction-finding estimateInterference elevation

0

1/3

1

–1/3

–2/3

–1

2/3

0

1/3

1

–1/3

–2/3

–1

2/3

Elev

atio

n er

ror (

beam

wid

ths)

Elev

atio

n er

ror (

beam

wid

ths)

1/31/22/31


Spectral MUSIC

PRIME-MEM



nomials must be generated instead of two, but other-wise the technique is similar to that of the unipolarplanar-array case. In fact, the PRIME technique canbe extended to non-planar arrays, where the phasecenters of the subapertures now lie on a three-dimen-sional lattice of points, by adding a third variable cor-responding to the displacement in depth of the sen-sors. The existence of these generalizations makes thePRIME technique powerful.

We have analyzed the asymptotic (in data snap-shots) accuracy of the PRIME versions of algorithmssuch as MUSIC (see the appendix entitled “Asymp-totic Mean-Squared Error for PRIME Direction Esti-mates”), and we can prove that under mild con-straints on array topology, the accuracy of the PRIMEversions of the algorithms can equal that of the spec-tral versions.

Data Examples of Polynomial Root Intersections

Figure 11 shows an example of the performance of aPRIME-based algorithm compared with a two-di-mensional spectral-MUSIC algorithm, in which bothalgorithms attempt to track a source in the presenceof diffuse main-beam interference. In this case, the re-ceived interference-to-signal ratio was roughly 20 dB,and as the range to the desired source decreased, theangular separation between the source and the inter-ference increased. We took the data with the SAS an-tenna by using six subapertures of the vertically polar-ized outer-array configuration. These six subaperturesare shown in the middle configuration in Figure 5.Notice that the number of points where the desiredsource is not resolved is much lower for the PRIMEalgorithm than for the spectral-MUSIC algorithm.Also notice that the spectral-MUSIC algorithm be-gins to exhibit a large amount of bias as the angularextent of the interference increases (as the angularseparation between the desired source and the inter-ference increases), while the PRIME algorithm doesnot suffer from this problem.

Waveform Exploitation

Sometimes we can estimate the spatial distribution ofthe interference without the presence of the desiredsource signal. One example of this scenario is a time-gated signal (satisfied by most range-resolving radar

waveforms), in which the interference is a non-gatedbroadband noise source. The received signals in be-tween source pulse returns consist only of interfer-ence and noise signals. As mentioned in Reference 6,waveform knowledge can be a powerful tool in resolv-ing source signals from interference. In the case inwhich the interference has a high interference-to-noise ratio, the maximum-likelihood-optimal esti-mate for the source direction [14] can again be ex-pressed as in Equation 1. The estimate is defined as

ua u R R R a u

a u R a uu

* arg max( ) ˆ ˆ ˆ ( )

( ) ˆ ( ),=

− −

−

Hi x iH

i

1 1

1 (6)

where Ri is the sampled covariance matrix of the in-terference plus noise, and R x is the sampled covari-ance matrix of the desired source plus the interfer-ence. In developing the PRIME technique, we notedthat by driving the denominator of the optimizationexpression to zero, we maximized the overall expres-sion. In this case, unfortunately, we cannot ignore thenumerator of the test statistic, since all of the desiredsource information is contained in the numerator.

An alternate option is to rewrite Equation 6 in theform

arg max( ) ˆ ˆ ˆ ˆ ˆ ( )

( ) ˆ ˆ ( ).

/ / / /

/ /u

a u R R R R R a u

a u R R a u

Hi i x i i

Hi i

− − − −

− −

1 2 1 2 1 2 1 2

1 2 1 2 (7)

The expression can now be viewed as taking theinduced inner product of a normalized whitened ar-ray-response vector ˆ ( ) ( )/R a u a ui w

− ≡1 2 through thewhitened source-plus-interference covariance matrixˆ ˆ ˆ ˆ/ /R R R Ri x i w

− − ≡1 2 1 2 . These terms are referred to aswhitened because the covariance of the interferencecomponent in the source-plus-interference covari-ance matrix is an identity, or spatially white. Thus inexpectation, the test statistic (Equation 7) can bewritten as

arg max( ) ( )

( ) ( )

( ) ( )

( ) ( )

arg max( ) ( )

( ) ( ).

u

u

a u a u

a u a u

a u e e a u

a u a u

a u e e a u

a u a u

wH

w

wH

ws

wH

w wH

w

wH

w

wH

w wH

w

wH

w

+

=

σ 2

(8)



FIGURE 13. GAMMA direction azimuth errors for pointlikemain-beam interference scenario. The desired signal direc-tion can be estimated with strong interference less than one-quarter of a beamwidth away.

Source direction-finding estimatesStandard monopulse estimatesInterference location

0

1/3

1

–1/3

–2/3

–1

2/3

Azi

mut

h er

ror (

beam

wid

ths)

1/61/31/22/31


Here σ s2 is the signal-to-noise ratio of the desired

source, and ew is the whitened array-response vectorof the desired source. An estimate of the vector ew canbe derived from the data by taking the principaleigenvector ew of the matrix Rw . Using this deriva-tion, we can approximate Equation 6 as

arg max( ) ˆ ˆ ( )

( ) ( )

arg max ( ) ˆ .,

u

u

a u e e a u

a u a u

a u e

wH

w wH

w

wH

w

w w= −α

α2 (9)

This last form of the test statistic has eliminated allfunctions of the direction parameters u from the de-nominator by introducing a nuisance scale factor α.This expression defines a test that finds the best whit-ened array-response vector that minimizes the errorvector between the appropriately scaled whitened ar-ray-response vector and the estimated whitenedsource array-response vector.

The solution to the test statistic in Equation 9 is ageneralized monopulse solution. The monopulsetechnique can be thought of as finding the u thatmaximizes Equation 9 when the whitening matrixˆ /R i

−1 2 is the identity matrix, and thus aw(u) = a(u)and, for a single pulse, e xw = . The monopulse tech-

nique projects the error vector αa u ew w( ) – ˆ onto thesubspace spanned by the steering vector of theassumed source direction, and the partial derivative ofthat vector with respect to the desired source param-eters ui . This technique creates an (M + 1)-dimen-sional subspace, where M is the number of source pa-rameters desired. The projected error is then driven tozero by choosing the correct α and u. In the case of nointerference and spatially white noise, the equationsgiving the parameters u decouple into the familiarmonopulse ratio expressions for either linear or pla-nar arrays.

When the array has subapertures with phase cen-ters that lie on a grid, then just as in the PRIME algo-rithm, the array-response vectors can be characterizedas polynomial functions of complex parameters.Again, for a planar array this characterization requirestwo parameters. If the monopulse projection is doneon the error vector when interference is present in theenvironment, driving the projected error vector tozero norm will generate M + 1 coupled nonlinearpolynomial equations (here, M + 1 equals 3) in thevariables α, z, and w. In this case the projection ma-trix is spanned by the whitened assumed array-re-sponse vector of the source, as well as the whitenedderivative vectors of the assumed source array-re-sponse vector with respect to the desired parametersto be estimated. This projection matrix can be shown

FIGURE 12. Computer simulation of GAMMA-estimateroot-mean-squared error (RMSE) compared to the RMSE ofthe maximum-likelihood-optimal estimation algorithm. TheGAMMA estimate tracks the performance of the optimal es-timator over a wide range of source signal-to-noise ratio(SNR). The Cramér-Rao bound is included for reference.

–22.5

–20.0

–17.5

–15.0

–12.5

–10.0

–7.5

–5.0

5 10 15 20 25 30 35

Ang

le R

MS

E (d

B b

eam

wid

ths)

Source SNR (dB)

Cramér-Rao boundGAMMA algorithmMaximum-likelihood algorithm



to be locally optimal for estimating the desired sourceparameters.

We can solve the equations directly for α becausethey are linear in α, leaving two polynomial equationsin z and w. Just as in the PRIME algorithm, we cansolve these equations by using resultant methods.This new algorithm, GAMMA, has shown statisticalefficiencies virtually equaling that of the true maxi-mum-likelihood estimator (Figure 12), while requir-ing no computationally expensive manifold search.

We note that just as the PRIME algorithm can beextended to polarization-diverse arrays and three di-mensional arrays, so can the GAMMA algorithm. Infact, we generated the polarization-diverse array re-sults in Figure 9 by using the polarization-diverse ver-sion of the GAMMA algorithm.

Data Example of Waveform Exploitation

We applied the GAMMA algorithm to real arraydata. In this test, the desired source was a transmittinghorn placed on the ground, producing a continuous-wave tone. The interference was another ground-based horn, placed some distance from the first,transmitting a noise waveform. We set the power ofthe interference so that the signal-to-interference ra-tio was –20 dB in the narrow frequency cell used toprocess the desired tone. Then we derived the esti-mate of the spatial covariance of the interference byusing frequency cells away from the desired tone cell.Figure 13 shows how the desired signal track developswhile the interference is still well within the mainbeam of the receive antenna.

Conclusion

In this article, we have put forth design concepts forplanar-array direction-finding systems that are in-tended to be robust in the face of main-beam interfer-ence. We presented results of an antenna-subapertureconfiguration study that showed the desirability ofmore complex subaperture configurations than thestandard monopulse configuration, specifically tocounter diffuse main-beam interference. Results fromboth theoretical bounds and tests confirm the advan-tage of using four subapertures in elevation with fiveor more receiver channels. In addition, we showedthat further advantage may be gained by mixing the

polarization of the array subapertures. This gaincomes at a price, however, since processing data froma diversely polarized array when it is faced with thesame interference scenario requires more channels(and therefore more receivers) than unipolar arrays.

The second section of the article presented twonew algorithms for the efficient estimation of source-direction parameters in the presence of main-beam interference. The new algorithms, polynomialroot intersection for multidimensional estimation(PRIME) and generalized adaptive multidimensionalmonopulse algorithm (GAMMA), extend the con-cept of polynomial rooting algorithms to the realm ofplanar arrays. The PRIME family of algorithms al-lows the higher resolution and lower computationalcost of a rooting algorithm when attempting to esti-mate the directions of multiple main-beam sources.The PRIME algorithm’s advantages are demonstratedin the case of a source plus diffuse main-beam inter-ference, in which a PRIME algorithm outperforms aconventional two-dimensional spectral superresolu-tion algorithm. In addition, we showed that theasymptotic mean squared error for techniques such asPRIME–MUSIC can be, under certain benign condi-tions, equal to that of the spectral algorithms theyreplace.

The GAMMA technique is a computationally effi-cient method for calculating a nearly maximum-like-lihood-optimal solution to the direction-estimationproblem in the case of estimatable interference spatialdistribution. We showed that GAMMA is the logicalextension of monopulse-like algorithms to cases inwhich the interference background is no longeruncorrelated sensor to sensor. Finally, we presentedexperimental data that showed the ability to estimatesource-direction parameters in the presence ofpointlike main-beam interference.



R E F E R E N C E S1. L.L. Horowitz, “Airborne Signal Intercept for Wide-Area

Battlefield Surveillance,” Linc. Lab. J., in this issue.2. A.J. Barabell, J. Capon, D.F. DeLong, J.R. Johnson, and K.D.

Senne, “Performance Comparison of Superresolution ArrayProcessing Algorithms,” Project Report TST-72, Lincoln Labo-ratory (9 May 1984, rev. 15 June 1998).

3. H.L. Van Trees, Detection, Estimation, and Modulation Theory,Pt. I (Wiley, New York, 1968).

4. P. Beckman and A. Spizzichino, The Scattering of Electromag-netic Waves from Rough Surfaces (Pergamon Press, New York,1963).

5. M. Wax and I. Ziskind, “On Unique Localization of MultipleSources by Passive Sensor Arrays,” IEEE Trans. Acoust. SpeechSignal Process. 37 (7), 1989, pp. 996–1000.

6. K.W. Forsythe, “Utilizing Waveform Features for AdaptiveBeamforming and Direction Finding with Narrowband Sig-nals,” Linc. Lab. J., in this issue.

7. J.E. Evans, J.R. Johnson, and D.F. Sun, “Applications of Ad-vanced Signal Processing Techniques to Angle of Arrival Esti-mation in ATC Navigation and Surveillance Systems,” Techni-cal Report 582, Lincoln Laboratory (23 June 1982), DTIC#ADA-118306.

8. R. Schmidt, “Multiple Emitter Location and Signal ParameterEstimation,” Proc. RADC Spectrum Estimation Workshop,Rome, N.Y., 3–5 Oct. 1979, pp. 243–258.

9. A.J. Weiss and B. Friedlander, “Direction Finding for Di-versely Polarized Signals Using Polynomial Rooting,” IEEETrans. Signal Process. 41 (5), 1993, pp. 1893–1905.

10. T.-Y. Li, T. Sauer, and J.A. York, “The Cheater’s Homotopy: AnEfficient Procedure for Solving Systems of Polynomial Equa-tions,” SIAM J. Numer. Anal. 26 (5), 1989, pp. 1241–1251.

11. B.L. van der Waerden, Modern Algebra, Vols. 1 and 2 (Ungar,New York, 1953).

12. Discriminants, Resultants, and Multidimensional Determinants,I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky,(Birkhauser, Boston, 1994).

13. A.G. Kouchnirenko, “Polyèdres de Newton et Nombres deMilnor,” Invent. Math. 32, 1976, pp. 1–31.

14. E.J. Kelley and K.W. Forsythe, “Adaptive Detection and Pa-rameter Estimation for Multidimensional Signal Models,”Technical Report 848, Lincoln Laboratory (19 Apr. 1989),DTIC #ADA-208971.

15. G.F. Hatke, “Conditions for Unambiguous Source LocationUsing Polarization Diverse Arrays,” 27th Asilomar Conf. on Sig-nals, Systems & Computers 2, Pacific Grove, Calif., 1–3 Nov.1993, pp. 1365–1369.

16. C.P. Mathews and M.D. Zoltowski, “Eigenstructure Tech-niques for 2-D Angle Estimation with Uniform Linear Ar-rays,” IEEE Trans. Signal Process. 42 (9), 1994, pp. 2395–2407.



array of N elements.There exists a beamformer that transforms the array-response vector a(u) for any value of u into abeamspace array-response vector ar(u), where allcomponents of ar(u) are strictly real and independentof u. Consider the spatial covariance matrix Rxx of thebeamformed outputs from the array due to K-planewave signals plus spatially white noise. This matrixwill have the form

R A U R A U Ixx r ss rH

n= +( ) ( ) ,σ 2 (1)

where σn2 is the noise power, Rss is the K × K

source covariance matrix, and

A U a u a ur r r K( ) ( ), , ( )≡ [ ]1 K

is the collection of K array-response vectors for the Ksources. The matrix Ar(U) is real, since ar(ui ) is realfor all i. Now consider the case in which the sourcecovariance matrix is real (an example of this is if thesources are independent). In this case, the matrix Rxxis real and symmetric.

Because the sample covariance matrix is a sufficientstatistic for estimating the direction parameters U forGaussian signals [6], it is sufficient to study the infor-mation in the spatial covariance matrix in order todetermine whether K Gaussian sources can beuniquely localized. One method of determining anecessary condition on the number of signals K is tosimply count the number of real parameters thatmust be estimated to determine U, and compare thatto the number of real scalars available in estimatingthe covariance matrix Rxx . Reference 15 shows that ifthere is any chance of ambiguity in estimating any ofthe parameters (source directions or source correla-tions), then there must be ambiguity in estimatingthe source directions. In the case of a real Rss , thenumber of parameters available in Rxx is (N

2 + N)/2.The number of real parameters involved in the model

of the spatial covariance matrix as stated in Equation1 is 2K + (K 2 + K)/2 + 1, with 2K parameters corre-sponding to the directions U, (K 2 + K )/2 correspond-ing to the elements of the unknown (but real sym-metric) source covariance matrix, and one parameterfor the noise power.

In order for the parameters to be estimateduniquely, their number must be less than or equal tothe number of scalars uniquely determined in esti-mating Rxx from the data, which we have determinedto be (N 2 + N )/2. Thus a necessary condition is

N NK

K K2 2

22

21

+ ≥ + + + . (2)

Now consider K = N – 1. Then Equation 2 requiresthat 2 ≥ 2N, which will not hold true for the mini-mum number of antennas needed to estimate thedirection of one signal. Now consider K = N – 2. Inthis case, Equation 2 holds for all values of N. Hencea necessary condition for unique localization is thatthe number of sources must be less than or equal toN – 2.

A P P E N D I X A :M A X I M U M N U M B E R O F S O U R C E SU N I Q U E L Y L O C A L I Z A B L E W I T H A

C E N T R O S Y M M E T R I C A R R A Y



a(α, β) denote the array response of a planar arrayas a function of the angles v = (α, β). For applications,we can think of α = 2πuxdx/λ and b = 2πuydy /λ,where (ux , uy , uz) are the direction cosines for a signalarriving with wavelength λ at a planar x-y array whoseelements lie on a rectangular grid with x-spacing dxand y-spacing dy . We can write the spectral-MUSICfunction as

a v a v

a v P a v

H

HN

( ) ( )

( ) ˆ ( ),

where PN denotes the noise-space projector based onthe sample covariance matrix. The v-coordinates oflocal maxima provide angle estimates. We let R =[ESEN]Λ[ESEN]H express the eigenanalyzed true co-variance matrix with ordered eigenvalues in the di-agonal matrix Λ and associated normalized eigenvec-tors in the columns of the signal-subspace matrix ESand the noise-subspace matrix EN .

We approximate the noise-subspace projectorasymptotically in the number of samples L as

ˆ ,P P E B E B EN N S NH

NH

SH≈ + +

where B is an M × (M – S ) array with independent,mean-zero, complex circular Gaussian entries ofcovariance

E B Ljkj

j

2 12

= –

( – )

λ λ

λ λ

for 1 ≤ j ≤ S and 1 ≤ k ≤ M – S, and where

λ λ λ λ λ1 1≥ ≥ > = = ≡+L LS S M

are the ordered eigenvalues of the spatial covarianceand S is the signal-subspace dimension. Letv v v∆ ≡ – 0 express the estimation error, where v0 isthe true direction of arrival. We then define thematrix

D ≡

L S–

( ) ( )–, ,

–.1

2 2diag 1

1 S

λ λ

λ λ

λ λ

λ λL

Let aα and aβ be α and β partial derivatives of a(α, β ).Also let ℜ ⋅( ) denote the real part of its argument. Weuse the asymptotic approximation of the noise-spaceprojector given above and the notation just intro-duced to write the asymptotic variance of the estima-tion error as [16]

E

P P

P P

T HS S

H

HN

HN

HN

HN

v v a E DE a

a a a a

a a a a

∆ ∆[ ] ≈ ×

ℜ

1

21

( )

.

–

α α α β

β α β β

Let z equal eiα and w equal to eiβ be the unit circle el-ements corresponding to the angles v equal to(α, β ). Henceforth, z and w are allowed to take valuesoff the unit circle, in which case their phases corre-spond to angle estimates. The true angles of a signalare denoted v0 equal to (α0, β 0), and (z0, w0) =( , ) .e ei iα β0 0

PRIME utilizes arrays whose elements lie on a pla-nar grid. The elements can be indexed by integerpairs. Let the (j, k)th element have response γjkz jwk,where

–( – ),

( – ).

Nj k

N1

2

1

2≤ ≤

The γjk are not assumed to have any pattern varia-tions. The array-response vector has apparent size N2,for example, when N is even. Some of the γjk, how-ever, are allowed to vanish. This concession is takeninto account by dropping these elements from the ar-ray-response vector so that its actual length is denotedM, as above. The array response can then be written

A P P E N D I X B :A S Y M P T O T I C M E A N – S Q U A R E D E R R O R F O R

P R I M E D I R E C T I O N E S T I M A T E S



as a vector a(z, w) with monomial entries satisfying

a a( , ) ( , )e e e ei iNz iNwα β α β α β≡

for appropriately chosen Nz and Nw. To rephrase two-dimensional spectral-MUSIC accuracy results interms of the monomial vector a(z, w) and its deriva-tives, we define

G

a E DE a

a E DE a

G a a a a G

≡

=

≡

≡

ee

z w z w

C P

i

i

HS S

H

HS S

H

N z wH

N z wH

α

β

δ α β α β

0

0

00

0 0 0 0

0 0 0 0

( ) ( )

( ) ( )

[ , ] [ , ] .

, ,

, ,

Then the asymptotic accuracy of two-dimensionalMUSIC can be written as

E CTNv v∆ ∆[ ] ≈ ℜδ

21( ) .–

This expression can be simplified slightly when thearray is centrosymmetric. Centrosymmetry holdswhen γ γjk j k= – – . For example, if the γjk are all ei-ther one or zero, then the elements are paired about acommon phase center, with each element of a pair atopposite ends of a diameter through the phase center.Of course there can be an element at the phase centerthat can be considered paired with itself. For cen-trosymmetric arrays and signals that are not fully cor-related, CN is real.

One version of PRIME is based on a polynomialbuilt from the denominator of the MUSIC function.We let

F z w z w z w P z wJ K TN( , ) ( , ) ˆ ( , )– –≡ a a1 1 ,

where J and K are chosen to make F (z, w) a polyno-mial. With infinite samples (i.e., P PN N= ), F(z, w)has double roots in z at z = z0 for fixed w = w0. Simi-larly, F (z, w) has double roots in w for fixed z = z0.Thus the partials Fz and Fw have a common root at(z0, w0). The method of intersecting these partials iscalled the partial-derivative method. Angle estimates(α, β) are obtained from the angular components of(z, w). The estimates have the same asymptotic accu-

racy as those of two-dimensional spectral MUSIC asdescribed above.

Another implementation of PRIME is based onchoosing noise-subspace probes (a special case of sub-space division PRIME-MUSIC). This method pro-vides polynomials of smaller degree than those ob-tained from partial derivatives. For two M-vectors, qi,we define two polynomials

g z w P z w ii iH

N( , ) ˆ ( , ) , .≡ q a for = 1, 2

The common roots provide estimates of (z0, w0).Let ρδ ≡ − −( , )z z w w0 0 represent the estimation er-ror. Let PE denote the projector onto the two-dimensional subspace of the (true) noise subspacespanned by PNqi. Also, let

C PEH

z wH

E z w≡G a a a a G[ , ] [ , ] .

Then, asymptotically, the estimation error is complexcircular Gaussian with variance

E HE

HC[ .] –v v G Gδ δ δ≈ 1

Note that CE ≤ CN (hence C CN E– –1 1≤ ) because PE ≤

PN as positive semi-definite Hermitian matrices.Equality is achieved if the span of q and the span ofaz(z0, w0), aw(z0, w0) project onto the same two-di-mensional subspace of the noise space.

The variance of the angle estimates is expressed bythe 2 × 2 real matrix δ / ( )–2 1ℜ CE . Now,

δ δ δ2 2 2

1 1 1( ) ( ) ,– – –ℜ ≤ ℜ ≤ ℜC C CN N E

where the left-hand and right-hand expressions repre-sent the asymptotic error variances of two-dimen-sional spectral MUSIC and PRIME based onnoise-space probes, respectively. It follows that theasymptotic accuracy of two-dimensional MUSIC isgenerally better. However, for the proper choice ofnoise-space probes and centrosymmetric planar ar-rays, the accuracies are identical. In particular, whenthe array is centrosymmetric and the noise subspace istwo-dimensional, the accuracies are identical.



. is a staff member in the Ad-vanced Techniques group. Hisresearch interests center onadaptive-array processing, withan emphasis on efficient super-resolution algorithms formultidimensional arrays,including polarization-diversearrays. He also studies perfor-mance bounds for adaptivearrays and the application ofspace-time adaptive processingto multipath interferencemitigation.

Gary joined Lincoln Labora-tory in 1990, after graduatingfrom Princeton Universitywith a Ph.D. in electricalengineering. He also holds aB.S. degree from the Univer-sity of Pennsylvania and anM.A. degree from PrincetonUniversity, both in electricalengineering. He is an associateeditor of IEEE Transactions onSignal Processing.

superresolution source location with planar arrays

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