maximum likelihood methods in radar array signal processing

Maximum Likelihood Methods inRadar Array Signal Processing

A. LEE SWINDLEHURST,MEMBER, IEEE, AND PETRE STOICA,FELLOW, IEEE

We consider robust and computationally efficient maximum like-lihood algorithms for estimating the parameters of a radar targetwhose signal is observed by an array of sensors in interferencewith unknown second-order statistics. Two data models are de-scribed: one that uses the target direction of arrival and signalamplitude as parameters and one that is a simpler, unstructuredmodel that uses a generic target “spatial signature.” An extendedinvariance principle is invoked to show how the less accuratemaximum likelihood estimates obtained from the simple model maybe refined to achieve asymptotically the performance availableusing the structured model. The resulting algorithm requires twoone-dimensional (1-D) searches rather than a two-dimensionalsearch, as with previous approaches for the structured case. If auniform linear array is used, only a single 1-D search is needed. Ageneralized likelihood ratio test for target detection is also derivedunder the unstructured model. The principal advantage of thisapproach is that it is computationally simple and robust to errorsin the model (calibration) of the array response.

Keywords—Antenna arrays, calibration errors, direction of ar-rival, Doppler frequency, extended invariance principle, likelihoodratio detection test, maximum likelihood estimation, radar crosssection, radar signal processing, robust estimation and detection,structured/unstructured modeling.

I. INTRODUCTION

In active radar systems, the primary goal is to detectthe presence and estimate the parameters of targets in thepresence of noise, ground clutter, and electronic counter-measures (e.g., jamming). For airborne radar applications,the parameters of interest include the target signal amplitude(related to its radar cross section), direction of arrival(DOA), range, and Doppler frequency. Even when onlya single target is present, this detection and estimationproblem is quite daunting due to the large volume of theparameter space that must be searched. For this reason,computational ease and efficiency is of the utmost impor-tance.

Manuscript received November 25, 1996; revised May 29, 1997. Thiswork was supported by the Office of Naval Research under ContractN00014-96-1-0934 and by the Swedish Individual Grant program of theSwedish Foundation for Strategic Research.

A. L. Swindlehurst is with the Department of Electrical and ComputerEngineering, Brigham Young University, Provo, UT 84602 USA.

P. Stoica is with the Systems and Control Group, Uppsala University,Uppsala SE-751 03 Sweden.

Publisher Item Identifier S 0018-9219(98)01295-X.

Standard solutions to this problem involve the use ofclassical space-time filters, either data independent (e.g.,as with a delay-and-sum beam former) or “adaptive” (e.g.,using linear constraints, maximum signal-to-interference-plus-noise ratio criteria, etc.) [1]. A discussion of theapplication of such techniques to radar signal processingcan be found in [2]–[5] and references therein. Maximumlikelihood (ML) approaches have not been extensivelyconsidered for this problem, mainly because they are per-ceived to be too computationally complex and becausethere is some reluctance in accepting the required modelingassumptions. Recent work has shown that for a singletarget source in Gaussian interference with unknown spatialcovariance, the ML solution can be obtained via a two-dimensional (2-D) search over target DOA and Doppler,followed by a generalized likelihood ratio test (GLRT) [6],[7].

One of the goals of this paper is to demonstrate that the2-D search required for the ML solution can be replacedwith two simpler one-dimensional (1-D) searches withoutaffecting the asymptotic accuracy of the estimates. If thearray is uniform and linear, then only a single 1-D searchis required. The resulting computational savings have thepotential for making the ML approach more feasible forradar applications. The key idea behind the simplification ofthe algorithm is the use of the so-calledextended invarianceprinciple (EXIP) in estimation first introduced in [8]. Thistechnique involves reparameterizing the ML criterion in away that admits a simple solution and then refining thatsolution by means of a weighted least squares (WLS) fit.In particular, we initially use an unstructured model for thearray response instead of one parameterized by the DOA ofthe target signal. Under this model, the ML solution reducesto a search over only the target Doppler frequency. Refinedestimates of the target parameters, including DOA, are thenobtained via WLS, where the optimal weighting is simplythe Fisher information matrix (FIM) corresponding to theunstructured ML criterion. As will be seen in Section IV,this second step involves a search only for the DOAparameter.

An additional advantage of using an unstructured modelfor the array is that it provides robustness to errors in

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the model for the array response. The array response isusually determined either by empirical measurements, aprocess referred to as arraycalibration, or by makingcertain assumptions about the sensors in the array andtheir geometry (e.g., identical sensors in known locations).Unfortunately, an array cannot be perfectly calibrated, andidealized assumptions made about the array geometry arenever satisfied in practice. Due to changes in antennalocation, temperature, and the surrounding environment, theresponse of the array may be significantly different thanwhen it was last calibrated. Furthermore, the calibrationmeasurements themselves are subject to gain and phaseerrors, and they can only be obtained for discrete angles,thus necessitating interpolation techniques for uncalibrateddirections. For the case of analytically calibrated arraysof nominally identical, identically oriented elements, errorsresult since the elements are not really identical and theirlocations are not precisely known.

Depending on the degree to which the actual arrayresponse differs from its model, detection and estimationperformance may be significantly degraded. For this rea-son, we develop a GLRT for target detection under theunstructured array model, which by its very nature doesnot depend on the availability of array calibration data. Wethen compare the unstructured GLRT with the standard testvia simulation for various levels of array perturbation. Evenin the absence of such perturbations, the unstructured GLRTmay be useful in providing a more rapid “initial scan” of theenvironment prior to application of the structured model.For example, one could apply the unstructured GLRTwith a relatively high false-alarm rate and then use thecomplete model to process data sets that showed an initialdetection. We also show that taking array calibration errorsinto account in the estimation of the target parameters canimprove performance as well. In particular, we demonstratethat information about the second-order statistics of thearray perturbation can be used to “regularize” the EXIPsolution and make it robust to such errors.

Before describing the proposed detection and estimationalgorithms, we introduce the data model assumed in thiswork and the EXIP in Sections II and III, respectively. Thetwo-step estimation procedure employing the EXIP is thendiscussed in Section IV. The GLRT for both the structuredand unstructured models is outlined in Section V and theresults of some simulations are included in Section VI.

II. RADAR DATA MODEL

Suppose an -element antenna array transmits a seriesof pulses using a certain set of transmit weights. Thetransmit weights are usually chosen to steer a beam in somenominal direction, say, . The output of each receivingantenna element is sampled at a number of time instantsfollowing each pulse, the time difference between eachsampling instant and the pulse transmission correspondingto the distance, or range, between the array and a potentialscattering source. The data collected for a given set of trans-mit weights is referred to as acoherent processing interval

(CPI). During a given CPI, a typical radar system collectsthe samples from each range “bin” and interrogates themfor the presence/absence of a target. If present, the DOA,Doppler frequency, and radar cross section of the detectedtarget are estimated. This procedure is continually repeatedfor every range bin from numerous CPI’s in an attempt toprovide constant surveillance over a wide area.

A. Structured Array Model

A target present in a particular range bin during someCPI may be modeled as producing the following base-bandvector signal (after pulse compression and demodulation)[5]–[7]:

(1)

where is the complex amplitude of the signal, is theDoppler shift due to the relative motion between the arrayplatform and the target, and is the response of thearray to a plane wave arriving from azimuthal direction1

The vector is assumed to be scaled such that itsfirst element is unity for all .

Assuming that at most one target is present in any givenrange bin, the sampled vector output of the array at time,or “snapshot,” may be written as

(2)

where is taken to be zero if no target is present andrepresents interference due to clutter, receiver noise, andjamming. Since depends on so many physical variables,deriving a deterministic model for it is problematic andusually impractical. To begin with, we choose to model

as a stationary, temporally white Gaussian randomprocess over the samples of the CPI, with zero-meanand unknown spatial covariance. As explained below,extensions to the case where is not temporally whitewill also be considered. The Gaussian hypothesis is madefor the convenience of the ML approach presented later,but it is not critical in that the ML method derived underthis assumption provides accurate estimates for many otherdata/noise distributions as well.

The above model for is rather simplistic, sinceoften the clutter component cannot be considered to betemporally white. When the interference term is dominatedby clutter, a common approach [5] is to assume that anestimate of both the temporal and spatial statistics ofis available from data in adjacent target-free range binsand CPI’s. However, forming an estimate of the full space-time covariance of typically involves hundreds ofcorrelation measurements, a large amount of secondarydata, and significant computation. In this paper, we proposea computationally efficient alternative. In particular, wesuggest a two-stage operation: 1) estimating a vector au-toregressive (VAR) model for the temporal variation of the

1In principle, �0 could be a vector representing both azimuth andelevation DOA’s. In many applications where this model is used, however,the distance between the (usually airborne) target and array is large enoughthat both may be assumed to be in the same horizontal plane.

422 PROCEEDINGS OF THE IEEE, VOL. 86, NO. 2, FEBRUARY 1998


clutter using secondary data from adjacent range and CPIcells and (2) using the estimated VAR model to prewhitenthe data prior to processing. The computational advantageof this approach stems from the fact that the model forthe temporal statistics of the clutter is embedded in arelatively small number of AR parameters rather than thecomparatively large number of temporal covariance lagsthat may be necessary for the methods of [5]. Additionally,the proposed algorithms estimate the spatial covariancealong with the target parameters, and thus no prior estimateof is required. In the discussion that follows, we willuse the simple assumption of temporal whiteness to derivethe algorithms and other main results of this paper. Thegeneralization of these results using the VAR approach isdescribed in Section IV-C.

With the model of (2) in hand, the problem consideredin this paper may be succinctly stated as follows.

Given a collection of samples as in (2),determine whether or not a target is present, and if present,estimate the parameters

The set represents the domain of the generic parametervector . While the aboveproblem statement implies that detection precedes estima-tion, most solutions to the problem operate in reverse: theparameters are estimated as though a target were presentand then a statistical test is performed to compare thelikelihood of the noise-only and signal-plus-noise models.

B. An Unstructured Array Model

In the discussion above, we have written the arrayresponse explicitly in terms of . Of course, to estimate

, the vector valued function must be known for allof interest. The function can be obtained by means

of an experimental or analytical calibration of the array,but such calibration procedures are never entirely accuratedue to errors in the gain and phase response of the antennaelements, mutual coupling, quantization and interpolationerrors, and drifts in the receiver electronic characteristicsdue to changes in weather, environment, etc. In somecases, these errors can cause the actual and calibratedarray responses to be quite different, which can lead toa significant loss in detection and estimation performance.

Another difficulty with the parameterization of (2) isthe nonlinear dependence of the data on the parameters

and . As a result, most algorithms for estimatingand require some type of nonlinear 2-D search

procedure that may be too time consuming to implement.These problems can be alleviated through the use of a lessstructured but still identifiable model. The price paid is,of course, a loss of performance in situations where themodel of (2) holds, but that loss can be compensated forby the approach to be described in Section IV. In particular,consider the following model for the data, where the signalamplitude and array response have been combined into a

single unstructured vector term:

(3)

The vector , referred to as the “spatial signature” of thetarget, takes the place of and in this model is simplytreated as a set of unknown complex constants. Theparameter vector for this model will be denoted by

and is defined on the domain .In the following sections, we discuss the use of (3) togetherwith (2) in developing estimation and detection algorithmsthat are computationally and statistically efficient and robustto array imperfections.

III. EXTENDED INVARIANCE PRINCIPLE

Suppose that to estimate the radar parameters describedabove, we have a loss function (for example, the MLcriterion) that can be parameterized in terms of eitheror . Denote the resulting criteria as and ,respectively (the subscript explicitly indicates that thecriteria depend on data samples). If an (invertible)mapping existed such that

(4)

the classicalinvariance principleof estimation theory (e.g.,see [9]) could be invoked to prove the equivalence ofminimizing and . Of course, no such function

exists in our problem, since may be factored asonly on a set of measure zero in . One could chooseto constrain the minimization of over this measure-zero set but there would be no advantage, computationalor otherwise, in doing so. Instead, we choose to consideran unconstrained minimization of and apply the so-calledextendedinvariance principle [8], [10] to estimatefrom the estimated .

We present the EXIP by means of the following theorem.Theorem 1: Define

(5)

(6)

and assume that a functionexists satisfying

(7)

Assume that is one to one and well defined over the entiredomain of its input argument. If2

(8)

then

(9)

is asymptotically (for large ) equivalent to the estimate, where

(10)

2The most common way in which (8) is satisfied is if both~�� andf(��)are consistent estimators of~��

0.

SWINDLEHURST AND STOICA: ML METHODS IN SIGNAL PROCESSING 423


and denotes expectation. The term “asymptoticallyequivalent” is to be understood to mean that

where is any consistent vector norm and denotesa random variable that converges to zero in probability ata rate faster than .

Proof: See [8] and [10].The idea behind the EXIP is that if the parameterization

corresponding to is suitably chosen, it may be muchsimpler to solve both (6) and (9) rather than (5) directly.The EXIP approach thus involves the following two-stepprocedure.

1) Minimize the criterion for the unstructuredmodel to obtain the estimate.

2) Use the unstructured estimate in (9) to find anestimate of the structured parameter vector. Thisrefined estimate will asymptotically have the samestatistical properties as the structured estimatefrom(5).

In the next section, we show how the EXIP techniqueapplies for the case of ML loss functions and how it leadsto a computationally efficient (and asymptotically optimal)estimator for .

IV. M AXIMUM LIKELIHOOD ESTIMATION

A. Structured Model

The negative log-likelihood function for samples ofdata from the fully parameterized model is easily shown tobe, within a multiplicative and additive constant

(11)

where

(12)

denotes the determinant operation anda conjugatetranspose. The minimization of (11) with respect toand may be performed explicitly. Using standard matrixcalculus results (see, e.g., [11]), the gradient of the criterionwith respect to is easily shown to be

from which it is clear that the ML estimate of is given by3

(13)

where the subscriptdenotes a “structured” estimate. When(13) is substituted into (11), the concentrated criterion

(14)3We assume thatN � m so that the matrixCCC(b; �; !) is invertible

(with probability one) at any point inD� .

results. As shown in [6], [12], and [13], maybe further concentrated with respect toyielding

(15)

(16)

where

(17)

(18)

(19)

and denotes the concentrated criterion. Estimatesof and are obtained by maximizing usingsome type of 2-D search algorithm.

A simpler estimation strategy than the 2-D search of (16)is suggested in [7], where it is proposed thatbe set equalto the pointing angle of the transmit beam andmaximized with respect to only. This is a reasonableidea since it is likely that a measurable return will bepossible only if the target is illuminated by main beamenergy. However, such an approach can in general onlyyield accurate estimates if the transmit beam is sufficientlynarrow.

The statistical properties of the above ML estimator areoutlined in the following theorem.

Theorem 2: The structured ML estimator defined by(14)–(16) is consistent and statistically efficient. Inparticular, the asymptotic variance of the ML (signal)parameter estimates

achieves the Cramer–Rao bound (CRB) [see (20) at thebottom of the next page] where

(21)

(22)

and for brevity we have written and.



The asymptotic variances of the ML DOA and Dopplerestimates are given by

(23)

(24)

Proof: See Appendix A.

B. Unstructured Model

The ML solution for the unstructured model may besimilarly derived. The negative log-likelihood function for

is of the same form as (11) but with replaced by. Minimization with respect to gives

(25)

where the subscript is used to explicitly denote anestimate obtained using the unstructured model. The cri-terion resulting from substitution of (25) into (11) may besimplified as follows:

(26)

(27)

(28)

from which it is clear that the solution for is simply

(29)

The remaining term in (28) is a function of only and canbe written as

(30)

(31)

The unstructured ML estimate of is thus given by

(32)

Since it appears in the ML criteria for both the structuredand unstructured models, we make a note here regardingthe quantity , where

(33)

Neither (16) nor (31) makes sense if . To showthat this cannot occur, define

so that

Then

(34)

(35)

(36)

(37)

since is a projection matrix. Equality in (37) canhold only if belongs to the range of . Due to thepresence of noise and interference in, withprobability one and the ML criterion is well defined.

An interpretation for (32) can be obtained if the problemis viewed as one of estimating the frequency of a singlesinewave observed in distinct but dependent channels.Equation (32) shows that the ML frequency estimate forthis problem is calculated by first spatially prewhiteningthe data and then finding the largest peak in the magnitudeof the Fourier transform averaged over the channels.The derivation above also implies that in principle, arraycalibration data is not needed in detecting the presenceor absence of a target in the radar data set. Given theestimate from (32), one could develop a statistical testfor detection based on the size of

where is a symmetric square root of . If thearray is imprecisely calibrated, such a test may outperform

(20)



the GLRT based on estimates of the target DOA. This issueis further studied in Section V and the simulations.

The statistical properties of the ML estimator for theunstructured model are given in the following theorem.

Theorem 3: The unstructured ML estimator defined by(25), (29), and (32) is consistent and statistically efficientfor the model described by (3). The CRB for the targetparameters

(38)

is given by

(39)

where

(40)

(41)

(42)

(43)

(44)

and are defined in (21) and (22).Proof: See Appendix B.

It is interesting to observe that the asymptotic varianceof is equivalent to that of the Doppler estimate obtainedusing the structured ML approach. From (39), we have that

which is identical to the expression given in (24). This factwill have important ramifications in the development of theEXIP approach below. In particular, becauseand areasymptotically equivalent, we can replaceby in (15)and (16). The resulting estimates forand

(45)

(46)

should have the same asymptotic variance as the struc-tured ML estimates in (15) and (16). This observation isformalized using the EXIP in what follows.

The main result regarding the use of EXIP to findestimates of and from the unstructured estimates

and is summarized by the following theorem.

Theorem 4: Given and obtained from (25),(29), and (32), asymptotically (in ) efficient estimates ofthe radar target parameters in (2) can be found as follows:

(47)

(48)

(49)

(50)

where the subscript denotes an EXIP-based estimate,the subscripts and indicate real and imaginary parts,respectively, and

(51)

(52)

The estimate may be solved for explicitly, leading to acriterion that depends only on

(53)

(54)

Proof: See Appendix C.The simplicity of the final result is quite remarkable,

especially the fact that the EXIP estimate ofis exactlyequivalent to the unstructured estimate. Although itwas shown earlier that both have the same asymptoticvariance (indeed, we showed that and have the sameasymptotic variance, which should also coincide with theasymptotic variance of by the EXIP), the weightingmatrix for EXIP is not block diagonal with respect toand . The off-block diagonal terms just happen to combinein a fortuitous way for (48) to result. It is also interestingto observe that the estimators ofand in (53) and (54)are not just asymptotically equivalent to those predicted by(45) and (46) but are in fact identical. To see this, note that(25)–(28) imply

which demonstrates immediately that, provided (46) and(54) are identical, (45) and (53) are as well. Using thematrix inversion lemma [e.g., see (121) in Appendix B],



we have

Thus, the equation at the bottom of the page proves that(46) and (54) are equivalent as well. Despite the fact thatthese simplified criteria can be obtained without using theEXIP, the result of Theorem 4 is still of interest. Forexample, the weighting of (49) can be regularized to makethe algorithm more robust to array calibration errors. Thisidea is explained in more detail below.

For convenience, we summarize here the two steps ofthe proposed EXIP approach.

a) Estimate the target Doppler frequency from

and then solve for the interference covariance

b) Estimate the target amplitude and DOA using

1) Initial Estimates: The two 1-D searches listed aboveare rendered even more computationally efficient by theavailability of simple initial estimates. For example, asmentioned above, it is likely that the target lies in the mainbeam of the transmit pattern, and hence thatis near the

known nominal transmit direction . Consequently, is agood choice for initializing step b).

Since is just the discrete Fourier transform (DFT)of , the criterion of (32) may be efficiently evaluatedat the frequencies forusing a bank of fast (F)FT’s. For sufficiently large, thevalue of on the FFT grid that maximizes (32) is anexcellent initial estimate for the search in step a). One canthen locally maximize (32) around this initial estimate usingthe computationally efficient zoom or chirp FFT algorithms.In Section V, we will also see that the development of thedetection criterion relies on using (32) evaluated on a gridwith spacing equal to .

2) Solution Via Polynomial Rooting:If the data are col-lected with a uniform linear array (ULA), a further asymp-totic argument can be used to show that the search forinstep b) may be eliminated by reparameterizing the problemin terms of a simple polynomial [14]. In this approach, thepolynomial coefficients are solved for explicitly in closedform, and is found by rooting the resulting first-orderpolynomial equation. The availability of such a procedureis a further advantage of the EXIP approach, since acorresponding simplification is apparently not possible forthe criterion in (16) due to the need for a 2-D search inboth and .

To see how such a reparameterization can be used here,note that for a ULA

where and is the distance in wavelengthsbetween adjacent antenna elements. Because of its structure,

above lies in the nullspace of the followingmatrix:

......

(55)



where is a -dependent variable and its conjugate.The variables and are the coefficients of a first-orderpolynomial whose root is ; the polynomialcoefficients are conjugates of one another since the root lieson the unit circle: , where isan arbitrary scaling. Since does not affect the equation

, the matrix can be thought of as dependingon a single real-valued scalar. For example, scalesuchthat its real part is unity ; then dependsexplicitly only on , which is uniquely related to4 . Toreparameterize (54), define

and note that

The criterion of (54) may thus be rewritten as

(56)

(57)

(58)

The key to the advantage in parameterizing bylies in thefact that, as shown in [14], in (58) may be replacedby

(59)

where is a consistent estimate of without affecting theasymptotic properties of the estimate of (and hence ).With the bracketed part of (59) fixed, the resulting criterionis simply quadratic in , and a closed-form solution canbe obtained. The consistent estimate ofrequired in (59)can be found by solving

(60)

which can also be achieved without a search. Given anestimate of (and hence ), is estimated as

4In certain cases—for example, if� = 1=2 and the target DOA isat end fire (90� relative to broadside)—the real part ofp is zero andcannot be scaled to be unity. For this reason, it is often preferable to usethe constraintjpj2 = 1. In our discussion, however, we choose to setp = 1+ jpi since it simplifies the description of the algorithm.

3) Robustness to Calibration Errors:Although (49) re-duces to (53) and (54), it provides a direct and very naturalway of taking array calibration errors into account and thusmay be preferred in some cases. To see this, suppose that

(61)

where is some unknown perturbation to the array re-sponse. A number of different models could be chosen for(see, for example, [15]–[17]), but for the sake of discussion,assume that is a zero-mean random vector with knowncovariance

Such information may be available from the manufac-turer of the array (e.g., given in terms of gain and phasetolerances) or from the results of several calibration mea-surements.

Under this assumption, the difference betweenand thenominal model is the term , whose covarianceis given by

(62)

Consequently, the performance of the EXIP minimization in(49) could be improved by using a “regularized” weightingmatrix

(63)

where is the standard weighting of Theorem 4 (51)and is an initial estimate of . If the array errors arecircular, then by definition

and . It is easy to show that for this specialcase, simplified criteria like (53) and (54) result, except

with replaced by

The initial estimate of needed for (63) could beobtained in one of the following ways:

• by an initial application of the EXIP algorithm withoutthe regularized weighting;

• using (15) evaluated with the Doppler estimate from(32) and the known transmit direction;

• setting , the first element of , which relieson the assumption previously made that the nominalarray response is scaled such that its first elementis unity.



In our experience with simulated data, the estimate ofobtained from the first method is significantly more

accurate than the other two but requires somewhat morecomputation. If is within a few degrees of , thesecond method will outperform the third, but usually onlyin difficult situations with a weak target where the varianceof the estimate is quite large. Our simulations showthat the improvement gained from the regularization is notvery sensitive to the exact value of; a change in bya factor of three or even more causes little performancedegradation. This also implies that one need not know theprecise value of , and hence that a simple estimate suchas will probably suffice.

It is shown in Section VI that the use of the regularizedweighting offers some improvement in estimation perfor-mance. A similar procedure for making the estimates ofthe structured algorithm (16) robust to array errors appearsto be much more difficult.

C. Extension to Nonwhite Interference

As discussed earlier, when significant clutter is presentin the data, it is unlikely that will be temporally white.In this section, we present a simple technique for applyingthe methods described thus far to the case where istemporally colored. The technique relies on the availabilityof secondary data taken from adjacent range bins andCPI’s that do not contain target energy. Assuming that thestatistics of the interference in the primary and secondarydata are the same, the secondary data is used to derive aVAR filter that whitens . As shown below, the use of thewhitening filter has almost no effect on the implementationof the unstructured ML estimator, but it must be taken intoaccount when using the EXIP approach to find the targetDOA and amplitude.

1) A VAR Prewhitening Filter:To begin, suppose wehave at our disposal a VAR filter of the form

(64)

such that

(65)

is temporally white. Here, denotes the unit delayoperator. The estimation of from the secondarydata will be addressed below. If we preprocess the arraydata with prior to detection and estimation of thetarget parameters, the filtered array output will be of theform

(66)Defining the filtered spatial signature

we have

(67)

and we see that the unstructured ML estimator can bedirectly applied to with no change.

The unstructured algorithm will provide an estimate ofthe Doppler and the filtered spatial signature , whichmust then be fit to the structured model to obtain thetarget DOA and amplitude. A complication arises heredue to the fact that after filtering, the spatial signaturedepends on the target Doppler frequency. This fact inducesa nonlinear dependence of the EXIP criterion on both

and , which makes it unlikely that the parameterscan be estimated without a 2-D search. Hence, in thiscase, the EXIP approach would offer no advantage overthe structured ML solution. However, given the excellentaccuracy of the unstructured Doppler estimate (its variancegoes as ), reasonable estimates forand analogousto (53) and (54) can be obtained using

(68)

(69)

where is the estimate of the spatial covariance ofobtained from (25) using the filtered data and the estimates

and . Once again, only a 1-D search is required toobtain the target parameter estimates.

2) Practical Issues:Below, we outline the computationof the VAR filter using target-free secondary data.Suppose we have samples of data from each ofrange/CPI cells adjacent to the one under study. We willdenote the secondary data by

where and represent the range and CPI of thethdata set. We seek a VAR filter such that

(70)

is temporally white for each. This approach assumes thatthe statistics of the clutter and interference are the same inthe primary and in all secondary data sets, so clearly, onlyrange/CPI cells adjacent to the primary data cell should beused. The order of the filter can be determined usingstandard methods such as those in [18]–[20]. Note that theprocess of filtering the array data as in (66) leads to theloss of data samples at the beginning of the observationwindow, so a penalty term should be applied to keepfairly small.

The standard approach to the problem of estimating theVAR filter coefficients is based on the following least-squares error criterion [20]:

(71)



If we define

(72)

... (73)

(74)

(75)

(76)

(77)

the minimization of (71) can be rewritten in a more compactform

(78)

and, provided that the matrix is full rank, the solutionfor is simply

(79)

A necessary condition for the existence of the matrixinverse in (79) is that have at least as many columnsas rows: , a condition that can easilybe satisfied by choosing a large enough. Of course, anupper bound for exists, effectively determined by howmany adjacent range/CPI cells can be considered to haveidentical clutter and interference statistics.

Some comments regarding the VAR prewhitening ap-proach are given below.

• As mentioned, the standard “fully adaptive” ap-proaches for dealing with spatially and temporallycorrelated clutter involve estimating and invertingthe covariance associated with all theinterference terms in an observation window [5]. Inessence, these techniques attempt to whiten the datain both space and time, and they are computationallymore involved than the above VAR approach.

• The spatial covariance of the whitened interferencedata could also be estimated using the secondary dataand then used to whiten the interference spatiallyas well. However, this whitening will of course beimperfect owing to finite-sample effects and the factthat the statistics of the interference in the primaryand secondary data may (at least slightly) differ fromone another. This observation, along with the factthat the estimation methods introduced in this papercan handle spatially correlated interference in the pri-mary data without any problem, suggests that spatialprewhitening is not necessary or even desirable.

• In principle, it is possible to compute the covariance ofthe error in and incurred by using an estimated

to prewhiten the data. This error covariance couldbe used to regularize the EXIP least-squares fit in thesame way that information about the covariance of thecalibration errors was used in Section IV-C3.

V. DETECTION

In this section, we discuss the binary hypothesis testingproblem that arises when determining whether or not atarget is present in a given data set. In particular, it isnecessary to decide which of the following two hypothesesis valid for :

The most common approach to solving this type of probleminvolves first estimating the signal parameters as if a targetwere present and then comparing the likelihood of(withthe true parameters replaced by estimates) to that of.This is referred to as a GLRT.

A. GLRT for Structured Model

If and represent the probability densities of thedata corresponding to the two hypotheses, the GLRT forthe structured ML model can be formulated as

(80)

where are the structured ML estimates andis some threshold. The threshold is chosen to achieve

some desired performance, most often specified in terms ofa fixed probability of false alarm (PFA). Since andare Gaussian, the logarithm of both sides of (80) is usuallytaken, in which case the GLRT takes the form

(81)

where we have replaced with its ML estimate undereach of the hypotheses. The parameter estimates in (81)can be replaced by the EXIP estimates and , whichhave the same asymptotic variance but are computationallysimpler to obtain.

If we use PFA as the criterion for choosing , we mustbe able to find the probability that exceeds the thresholdwhen it should not, i.e., when holds. This in turnrequires that the distribution ofunder be determined, avery difficult problem due to the complicated dependence ofthe test statistic on the data. However, it can be shown thatfor general models (not necessarily of the form consideredhere) and under quite general conditions, theasymptoticdistribution of the GLRT quantity is chi-squared () withdegrees of freedom equal to the difference between thenumber of free parameters under and [20].

In the structured ML problem, the number of extra freeparameters is four, corresponding to two parameters for(a real and imaginary part) and one each forand . Acomplication arises here due to the fact that under, theparameters and are undefined, which renders the generalresult of [20] inapplicable. The approach taken in [13] toovercome this problem is to compute the distribution ofconditionedon and , in which case the number of extra



parameters is two. Thus, conditioned onand , is arandom variable. Since the test statistic is computed

by maximizing over and , order statistics must be usedto find the unconditional density for and the desired PFAthreshold. This too is quite a difficult problem in the generalcase. A reasonable heuristical approximation was proposedin [13], where was treated as a known quantity equal to

and order statistics applied only to samples oftakenin . It was shown in [13] that under these assumptions, agiven PFA could be obtained by choosing

(82)

In the discussion below, we apply the GLRT to the un-structured model and derive a threshold test for a given PFAsimilar to the one above. The derivation is considerablyeasier due to the simplicity of the unstructured model,however, and the resulting test is much more robust incases where and calibration errors are present.Additionally, the unstructured test obtained below can bereadily extended to the case of temporally correlated noise,unlike the structured GLRT, which in that case requires a2-D search (see Section IV-C2).

B. GLRT for Unstructured Model

For the unstructured model, the two competing hypothe-ses are

The resulting GLRT is similar in form to (81), withreplaced by

(83)

From (31) and (33), we have that

and thus the problem is to determine such that

(84)

occurs under with a given PFA . Since the logarithmis monotonic, the inequality of (84) is equivalent to thecondition

(85)

which for large can further be approximated as

(86)

since approaches zero as increases.Under , we have that

w.p. (87)

(88)

With

(89)

(87) implies that the asymptotic distributions ofand coincide and hence that (84) is asymptoticallyequivalent to

(90)

Since is an -element zero-mean circular Gaussianrandom vector with covariance , is the sumof the squares of zero-mean real-valued independentGaussian random variables with variance . Thus

(91)

for every , and hence is (asymptotically) the maximumvalue of a collection of random variables. Notethat, conditioned on , the number of extra free parame-ters under for the unstructured model is (a realand imaginary part for each element of, which meansthat (91) is in agreement with the general result of [20]mentioned earlier.

Define to be the th element of the vectorand observe that under

which is simply the sum of the periodograms corre-sponding to each element of . Since the elements ofare independent circular white Gaussian processes,and can be shown to be (asymptotically) indepen-dent random variables whenever or[21]. If we restrict the evaluation of our detection variableto the DFT frequenciesthen and are independent for and

is asymptotically distributed as the maximum valueof a collection of independent random variables.In what follows, we let denote the maximizer ofthe criterion evaluated on the aforementioned DFTfrequency grid.

To find the threshold needed to achieve a given ,define

(92)



where denotes the probability of a generic event. Formaximized over the DFT frequencies, we have

Thus, given , we can calculate

and determine using (92) and a table of probabil-ities.

The GLRT detection procedure derived in this section issummarized in the following.

Given a collection of data , suppose it isdesired to decide between the two hypotheses

with probability of false alarm , where is temporallywhite with unknown spatial covariance. A GLRT thatasymptotically achieves this goal is outlined below.

1) Compute the threshold by solving

where is defined by (92).

2) Compute the sample covarianceand calculate theDFT at the frequencies for

.

3) Find the DFT frequency that maximizes the vector

periodogram ; call this frequency .

4) Decide that is true if

otherwise assume that holds.

The case of temporally correlated clutter and interferencecan be handled using the “whitening approach” outlined inSection IV-C. The performance degradation induced by theimperfections in the estimated whitening filter is expectedto be minor.

1) Finite Sample Behavior of the GLRT:As emphasizedalready, the detection tests developed in Sections V-Aand V-B are valid asymptotically in . However, theapplication of the tests in situations where is relativelysmall bears somewhat further discussion. We raise twoissues here. First, the FFT grid used to evaluate thedetection statistics is quite coarse when is small.Consequently, if a target is present, its Doppler frequency isless likely to be near the center of any FFT bin, and hence

the peak in the detection criterion due to the target maybe artificially reduced. This in turn decreases the ability ofthe tests to detect a weak target. One way to overcome thisproblem is to obtain a finer frequency grid by zero-paddingthe data before taking the FFT. The asymptotic distributionof the resulting test statistic is more difficult to obtain (sincethe samples of the periodogram are no longer independent)and the asymptotic probability of false alarm is no longerguaranteed, but this may be a reasonable price to pay,especially considering that the asymptotic arguments areless accurate for small anyway.

The second small- issue we discuss has to do with theasymptotic approximation

(93)

used in going from (85) to (86). In principle, such a stepmight seem unnecessary. Indeed, as explained above, it ispossible to choose a threshold

(94)

such that

with a certain desired probability under . The thresh-old could then be found exactly by inverting (94)

(95)

We use the notation to differentiate this threshold fromthe one obtained using the approximation of (93), whichresults in . For a fixed , increases veryslowly with [22], and it can indeed be shown that

Despite the fact that the use of (95) would reduce ourreliance on asymptotic arguments, there are some subtlereasons to prefer the test developed using the approximationof (93). First, when is relatively small, the solutionfor in (95) may not be defined for common valuesof . In particular, the desired threshold isnot guaranteed to be less than . In such cases, theequivalence between the tests in (84) and (85) breaksdown, and the derivation of a threshold for (84) wouldrequire the finite sample (nonasymptotic) distribution of

to be found. Using instead of (95) avoids thisdifficulty and also provides a well-defined test in all cases,although one could question its performance for small.As demonstrated in Section VI, however, simulations showthat the simpler threshold performs quite well forsmall , essentially as well as the GLRT developed forthe structured array model. But more than that, and quitesurprisingly, it turns out that yields much better resultsthan the more “exact” threshold derived using (95)—at leastwhen is large enough that is well defined. The reasonbehind this interesting result can be explained as follows.

• Since for all , it can easily beshown that .



• Although is asymptotically , it wasshown in (37) that , implying that infinite samples, the random variables tendto be smaller than random variables (whichhave a nonzero probability of exceeding ).

Consequently, the lower threshold provided by tendsto compensate for the fact that , and hence ,are smaller than predicted by their asymptotic distributions.

VI. SIMULATION RESULTS

We present below the results of several simulationscomparing the detection and estimation performance of thestructured and unstructured ML estimators. The followingparameters were common to all of the simulations.

• The array was composed of nominally identi-cal, uniformly spaced, unit-gain antennas separated byone-half wavelength.

• If present, the calibration errors were generated asGaussian random variables with covariance ,which amounts to assuming that the calibration errorsare independent and identically distributed from an-tenna to antenna. The termcalibration error is usedbelow to refer to the quantity , which is thestandard deviation of the total perturbation (both realand imaginary parts) for a given sensor.

• The target was located at (broadside), withamplitude and Doppler shift .

• The covariance of the interference was set atfor various , where is a randomly chosen

positive definite matrix scaled and regularized so thatits determinant is unity. The same was used inall of the simulations; its maximum and minimumeigenvalues were 2.36 and 0.095, corresponding to acondition number of about 25. The signal-to-noise ratio(SNR) was defined to be 20 dB, as the targetamplitude and nominal antenna gains were both unity.

• In some of the simulations, a point source “jammer”with a DOA of 5 was introduced into the data. Thejammer signal was assumed to be temporally white(circular) Gaussian noise.

• The PFA was set to for the detection tests.

In what follows, the term “unstructured GLRT” (or simplyUGLRT) will be used to refer to the GLRT obtained for theunstructured array model. UML will refer to the process ofestimating the target parameters via the unstructured MLalgorithm and the EXIP. A corresponding meaning is to beattached to the acronyms SGLRT and SML.

A. Verification of Asymptotics

The purpose of the first example we consider is to“validate” the asymptotic arguments presented earlier. Inparticular, we provide insight into the size of required toachieve the asymptotic regime. The results presented hereare indicative of numerous simulation runs. Fig. 1 showsthe PFA for the structured and unstructured GLRT’s versus

Fig. 1. PFA for structured and unstructured GLRT’s versus num-ber of snapshots.

the number of snapshots taken from the array with no signalpresent. The results are based on 1000 trials conducted foreach value of . The number of trials was not enoughto verify if the desired was reached, butour main purpose was to compare the threshold behaviorof the structured and unstructured GLRT’s. While bothtests approach a low value of with increasing , the“threshold ” for the structured GLRT appears to be afew (two to three) snapshots smaller than that for theunstructured GLRT.

The empirical behavior of the SML and UML parameterestimates is compared with the CRB in the next exam-ple. Figs. 2–4 show the root mean square (rms) error ofthe DOA, Doppler, and amplitude estimates, respectively,versus for a case with 0-dB SNR and an ideal array (nocalibration errors). Each data point on the plots is calculatedfrom the results of 1000 independent experiments. It isclear from the figures that there is essentially no differencebetween the quality of the SML and UML estimates for allvalues of . Thus, for these cases at least, the asymptoticequivalence of the two methods is apparent with verylittle data. Somewhat more data are required for theirperformance to reach the CRB, but there is little excesserror for . Similar results have also been obtained atlower SNR’s, though plots for these cases are not includedhere since they are almost identical to Figs. 2–4.

B. Robust Detection

Fig. 5 compares the detection probabilities of the SGLRTand UGLRT at various SNR (500 trials at each SNR) for acase involving an array with a calibration error of 0.1. Sincethe unperturbed array is assumed to have unit gain sensors,this corresponds to standard deviations in the array’s gainand phase of 20 dB and 0.1 rad, respectively. A differentperturbation was applied to the array at each trial, so thecurves in Fig. 5 represent the detection probability averagedover uniform linear arrays with this level of calibrationerror. The three curves for the SGLRT show its performancefor “pointing” errors of 0 , 1 , and 2 . Recall that the



Fig. 2. DOA estimation performance and CRB versus number ofsnapshots, 0-dB SNR.

Fig. 3. Doppler estimation performance and CRB versus numberof snapshots, 0-dB SNR.

SGLRT is implemented by evaluating the test statisticat , the pointing angle of the transmit beam. Apointing error occurs if the actual target DOA is slightlydifferent from . Fig. 5 demonstrates that the SGLRT isvery sensitive to the assumption that . No suchdifficulty is observed with the UGLRT, of course, since itdoes not use DOA information. Even with no pointing error,we see that the performance of the SGLRT deterioratesat high SNR, since the presence of the calibration errorsreduces the relative height of the peak value of the teststatistic evaluated over . At higher SNR, there is lessof a chance that the noise will artificially push the peakover the threshold. This deterioration is more evident withpointing errors, since the peak has been reduced evenfurther.

The above experiment was repeated with a strong jammeradded to the data, and the resulting probabilities of detectionare plotted in Fig. 6. The jammer was located at a DOA of5 , and the jammer-to-signal power ratio (JSR) was heldfixed at 20 dB. Since the jammer is assumed to broadcast

Fig. 4. Amplitude estimation performance and CRB versus num-ber of snapshots, 0-dB SNR.

Fig. 5. Probability of detection versus SNR for various pointingerrors.

temporally white noise, its net statistical effect is to makea large rank one contribution to . The presence of thejammer causes a slight reduction in probability of detectionat low SNR but appears to lead to an “improvement” inperformance for the SGLRT at higher SNR. This improve-ment is only artificial, however, since it is the randomcontribution of the jammer that is pushing the peak of thedetection criterion above the threshold. We will see belowthat the estimation performance of SML in these high SNRregions is very poor.

C. Robust Estimation

We study here the estimation performance of SML andUML for the same scenario described in the previoussection for the detection tests. The only difference in thisexample is that the calibration error for the array wasincreased to 0.2, corresponding to errors of14 dB in gainand 0.2 rad in phase. The target parameters were estimatedby maximizing the SML and UML criteria on a grid withspacing in and 0.05 in over the interval [ 7 ,



Fig. 6. Probability of detection versus SNR for various pointingerrors, strong jammer included.

Fig. 7. DOA estimation performance versus SNR, no jammerpresent.

7 ]. The frequency resolution was increased by choosingand zero-padding the data to the required length.

In each case, was chosen large enough so that the errordue to the grid quantization would be several times smallerthan the error predicted by the CRB. This was also thecriterion used in choosing the grid spacing for the DOAparameter.

The regularized weighting of (63) was used for UML,but the value of used for the weighting was varied tostudy the sensitivity of the method to knowledge of thecalibration error statistics. The initial estimate ofrequiredfor the regularization was obtained by first applying UMLwith the standard weighting. Figs. 7–9 show the rms errorversus SNR with no jammer present for the target DOA,Doppler, and amplitude estimates, respectively. Figs. 10–12show the results when a jammer at 5is present with JSR

dB. The dash-dot line indicates the performance ofUML with no regularization, while the solid line gives theresults for the correct weighting. The symbols “x” and “o”on the plots correspond to the results for regularized UML

Fig. 8. Doppler estimation performance versus SNR, no jammerpresent.

Fig. 9. Amplitude estimation performance versus SNR, no jam-mer present.

implemented with overestimated by a factor of two andfour, respectively. It is seen that the performance improve-ment offered by the regularization is very insensitive toexact knowledge of this parameter.

With or without regularization, the UML method isseveral times more accurate than SML at high SNR. Thepresence of the calibration error reduces the peak of the2-D SML criterion at the true and , and produces alocal maximum that is far removed from them. The UMLtechnique does not suffer from this problem since at highSNR, it is able accurately to estimate without beingaffected by the array perturbation. With an accurate estimateof , the error in the DOA parameter is correspondinglyreduced. The difference between the methods is smaller atlow SNR but not insignificant. If the calibration error isreduced to 0.1, as in the detection examples above, all ofthe algorithms perform essentially identical at low SNR butSML still degrades rapidly as the SNR is increased above0 dB. Last, note that the CRB is included in each plot toindicate the best performance that would be possible if nocalibration errors were present.



Fig. 10. DOA estimation performance versus SNR, strong jam-mer present.

Fig. 11. Doppler estimation performance versus SNR, strongjammer present.

VII. CONCLUSIONS

We have presented several techniques for detection andestimation in radar problems where a target is observed byan antenna array in noise with unknown spatial covariance.Two data models were introduced: one that uses a struc-tured model with the target signal amplitude and DOA asparameters and one that is a simpler, unstructured model forthe array response. ML solutions were derived for each caseand the corresponding CRB’s established. The EXIP wasused to show how the simpler unstructured solution couldbe processed to estimate the target parameters with the sameasymptotic accuracy as the more complicated structuredalgorithm.

Several advantages of the EXIP approach were observed.First, statistically efficient estimates of the target parameterscan be obtained using two 1-D searches rather than one2-D search. Since the estimates are often determined byevaluating the given criterion on a grid, this can providesubstantial computational savings. Furthermore, if a uni-form linear array is used, one of the 1-D searches can

Fig. 12. Amplitude estimation performance versus SNR, strongjammer present.

be eliminated for further reduction of the computationalload. Second, the two-step approach using the unstructuredmodel provides a simple and effective way of taking arraycalibration errors into account in the estimation of the targetDOA. The resulting DOA estimates can be significantlymore accurate and much less sensitive to such errors. Last,the use of an unstructured model for the array responseallows for the development of a likelihood ratio detectiontest that does not depend on the accuracy or the availabilityof array calibration data, nor on the availability of priorinformation about the target DOA. Consequently, the testis robust in situations where such information is impreciselyknown. Our simulations indicate that when the array isperfectly calibrated, little degradation results from usingthe simpler two-step algorithm over the direct structuredapproach. When calibration or pointing errors are present,the simpler algorithm offers much better performance.

In summary, the use of an unstructured array modeltogether with the EXIP provides a very simple procedurefor applying ML methods to radar signal processing prob-lems. The computational simplicity and robustness of theproposed estimation and detection algorithms make themworthy of serious consideration for real radar applications.

APPENDIX APROPERTIES OF THESTRUCTURED ML ESTIMATOR

A. Consistency

We first establish the consistency of the structured MLalgorithm. To begin with, we have

(96)

(97)



If we let denote the Kronecker delta, the limit of thefirst term in (97) is simply since

ifotherwise.

By the law of large numbers [23], the second term con-verges uniformly to zero with probability one since iszero mean. Consequently

ifotherwise

(98)

Letting , the ML criterion in (16) is thusasymptotically given by

with probability one. At , the Cauchy–Schwartzinequality may be used to obtain the following upper boundfor the limiting ML criterion:

where Equality is achievediff which in turn is possible iff sincewe implicitly assume an “unambiguous” array.

Although the limiting ML criterion has a global maxi-mum at the true parameter values, establishing consistencyis complicated by the fact that the criterion is discontinuousin . This problem can be solved using arguments similarto those in Lemma 1 and Theorem 1 of [24]. It was shownin [24] that consistency is achieved provided thatcan be written as the sum of two components

that satisfy the following conditions:

1) uniformly in ;

2) for any , there exist and such that

for all and .

Using (16) and the definition for , we may writeas (99) and (100), shown at the bottom of the

page, where . If we associatewith (99) and with (100), it can be

shown that conditions 1) and 2) listed above are satisfied(a proof of this result will not be given here, however).Using reasoning similar to that for the proof of Theorem 1in [24], it follows that

(101)

(102)

The consistency of the estimates forand is easilyestablished by substituting and into (15) and (12)and (13).

B. Cramer–Rao Bound

Since the structured ML estimates are consistent, theywill be statistically efficient, and their asymptotic variancewill be given by the CRB. Under the structured model, theobservations are independent circularGaussian random variables with meanand covariance , where we have dropped the subscriptfor brevity. In this case, element of the FIM for theobservations can be shown to be equal to [21], [25], [26]

(103)

Since and depend on different elements of, itis clear that will be block diagonal with respect tothe signal and noiseparameters. In particular, the first term of (103) will givea nonzero result only for the noise block, while the secondterm is only nonzero for the signal block. Since we areconcerned with the CRB only for the signal parameters, weneed only consider the second term

(104)

(99)

(100)



The partial derivatives needed to compute (104) are givenbelow

(105)

(106)

(107)

(108)

where

The expression for the CRB, shown in (109) at the bottomof the page, is obtained by substituting (105)–(108) into(104) where

(110)

(111)

and .To find the variance of the ML Doppler and DOA

estimates, we use the following formula for the inverse ofa partitioned symmetric matrix:

(112)

where is the lower 2 2 blockof . For convenience, define

and let the subscriptsand denotereal and imaginary parts, respectively. Then

The estimate variances are thus given by

(113)

(114)

APPENDIX BPROPERTIES OF THEUNSTRUCTUREDML ESTIMATOR

A. Consistency

The consistency of the unstructured ML algorithm isestablished in a manner similar to the structured case.Following (96)–(98), we have

ifotherwise

and hence

ifotherwise.

Splitting the criterion into two terms as

(115)

(116)

we again have two functions that satisfy the consis-tency conditions of [24] given in Appendix A. Thus,

and furthermore

The consistency of follows from substitutinginto (25).

(109)



B. Cramer–Rao Bound

The data in the unstructured modelare independent circular Gaussian random variables withmean and covariance . As for thestructured case studied in Appendix A, the unstructuredCRB is block diagonal with respect to the signal

and noise ( ) parameters. TheFIM is computed exactly as in (104), with replaced by

, and the required partial derivatives are given by

(117)

(118)

(119)

Substituting (117)–(119) into (104) yields the expressionfor the FIM shown in (120) at the bottom of the page.

An explicit expression for the CRB can be obtained byinverting (120). To this end, we make use of (112) and thematrix inversion lemma

(121)

by associating with the upper 2 2 block ofand with the scalar in the lower right corner. We beginby noting that

where we have used the notation. Furthermore

(122)

(123)

(124)

where . Thus

(125)

(126)

(127)

which is the lower right element of (39). Together, (124)and (127) also establish the validity of the lower left andupper right blocks of (39). The remaining block also easilyfollows

APPENDIX CPROOF OF THEOREM 4

The theorem is proved by applying the general EXIPresult in Theorem 1 to the criteria of the structuredand unstructured ML radar target parameter esti-mators. In particular, a function will be defined thatsatisfies (7) and relates the SML and UML parameters, andit will be shown that the WLS minimization of (9) reducesto (47)–(50), (53), and (54). Thus, Theorem 1 states thatthe estimates of and , obtained respectively by(47), (48), (53), and (54), are asymptotically as accurate asthose obtained by the structured ML algorithm and henceare statistically efficient.

The function relating the elements of and isdefined implicitly by

(128)

(129)

(130)

where

These equations, together with the consistency of bothalgorithms established in Appendixes A and B, satisfyconditions (7) and (8) of Theorem 1.

Since ML criteria are used in this problem, the optimalweighting in (10) is seen to be equal to the FIM, evaluatedin this case for the unstructured model. It is shown inAppendix B that the FIM for the unstructured model isblock diagonal with respect to the signal and noiseparameters, so the criterion of (9) will be composed of twoterms, one involving only and the other only the elementsof . These two terms may thus be minimized separately.Since the parameterization for is the same under boththe structured and unstructured models, the term involving

can be made to be zero by simply setting .This establishes (47).

(120)



The remaining term involving is found by substituting(129) and (130) into (9)

(131)

where the weighting is given by the FIM in (120)evaluated at the UML parameter estimates and as before,the subscripts and denote real and imaginary parts,respectively.

Define

so that

where is the upper left block of the FIMevaluated at . If we minimize (131) with respect to ,a simple calculation yields

(132)

which when substituted into (131) gives the followingcriterion involving only and :

(133)

(134)

Define

and set the derivative with respect toequal to zero to obtain

Solving for yields

and thus

(135)

which is (53).Since , we see from (132) that

, which verifies (48). The fact that alsoproves that (131) is equal to (49). Equation (50) followsfrom (49) if we combine real and imaginary parts, and(54) is obtained by substituting (53) into (50). Thus, allof the statements in Theorem 4 are verified and the proofis complete.

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A. Lee Swindlehurst (Member, IEEE) receivedthe B.S.(summa cum laude) and M.S. degreesin electrical engineering from Brigham YoungUniversity, Provo, UT, in 1985 and 1986, re-spectively, and the Ph.D. degree in electricalengineering from Stanford University, Stanford,CA, in 1991.

From 1983 to 1984, he was with Eyring Re-search Institute, Provo, as a Scientific Program-mer writing software for a Minuteman missilesimulation system. During 1984–1986, he was a

Research Assistant in the Department of Electrical Engineering, BrighamYoung University. During 1985–1988, he was with the InformationSystems Laboratory at Stanford University. From 1986 to 1990, he alsowas with ESL, Inc., Sunnyvale, CA, where he was involved in the designof algorithms and architectures for a variety of radar and sonar signal-processing systems. He joined the Faculty of the Department of Electricaland Computer Engineering, Brigham Young University, in 1990, wherehe currently is an Associate Professor. During 1996–1997, he held a jointappointment as a Visiting Scholar at both Uppsala University, Uppsala,Sweden, and the Royal Institute of Technology, Stockholm, Sweden. Hisresearch interests include sensor array signal processing, detection, andestimation theory, system identification, and control.

Dr. Swindlehurst was an Associate Editor for IEEE TRANSACTIONS ON

SIGNAL PROCESSING. He received an Office of Naval Research GraduateFellowship for 1985–1988.

Petre Stoica(Fellow, IEEE) received the M.Sc.and D.Sc. degrees in automatic control fromthe Bucharest Polytechnic Institute, Romania, in1972 and 1979, respectively.

Since 1972, he has been with the Departmentof Automatic Control and Computers of theBucharest Polytechnic Institute, where he is aProfessor of system identification and signalprocessing. He spent 1992, 1993, and the firsthalf of 1994 with the Systems and ControlGroup of Uppsala University, Sweden, as a

Guest Professor. In the second half of 1994, he held a Chalmers 150th An-niversary Visiting Professorship with the Applied Electronics Department,Chalmers University of Technology, Gothenburg, Sweden. Currently, he isaffiliated with the Systems and Control Group of Uppsala University as aDocent. He has published more than 310 papers in journals and conferencerecords on these topics. He also has published six book chapters and sevenbooks. Most recently, he is the coauthor (with R. Moses) ofIntroductionto Spectral Analysis(Englewood Cliffs, NJ: Prentice-Hall, 1997). He ison the editorial boards of four journals in the field. His scientific interestsinclude system identification, time-series analysis and prediction, statisticalsignal and array processing, spectral analysis, and radar signal processing.

Dr. Stoica is a corresponding member of the Romanian Academy and afellow of the Royal Statistical Society. In 1993, he received an honorarydoctorate degree from Uppsala University. He was the corecipient (withA. Nehorai) of the 1989 IEEE ASSP Senior Award for contributions toarray processing. He received the 1996 Technical Achievement Awardfrom the IEEE Signal Processing Society for fundamental contributionsto statistical signal processing as well as several other awards and prizes.He is listed inWho’s Who in the World.



maximum likelihood methods in radar array signal processing

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