REFERENCES

[1] Yao, K.Spherically invariant random processes: Theory andapplications.In V. K. Bhargava, et al., (Eds.), Communications,Information and Network Security, Dordrecht, TheNetherlands: Kluwer Academic Publishers, 2002, ch. 16,315—332.

[2] Gini, F., Greco, M. V., Diani, M., and Verrazzani, L.Performance analysis of two adaptive radar detectorsagainst non-Gaussian real sea clutter data.IEEE Transactions on Aerospace and Electronic Systems,36 (Oct. 2000), 1429—1439.

[3] Farina, A., Gini, F., Greco, M. V., and Verrazzani, L.High resolution sea clutter data: A statistical analysis ofrecorded live data.IEE Proceedings, 144 (June 1997), 121—130.

[4] Conte, E., and Longo, M.Characterisation of radar clutter as a spherically invariantrandom process.IEE Proceedings, Pt. F, 134 (Apr. 1997), 191—197.

[5] Gini, F., and Greco, M.Texture modelling, estimation and validation usingmeasured sea clutter data.Proceedings of the IEE, Radar, Sonar, Navigation, 149(June 2002), 115—124.

[6] Dogand²zic, A., Nehorai, A., and Wang, J.Maximum likelihood estimation of compound-Gaussianclutter and target parameters.In Proceedings of 12th Annual Workshop Adaptive SensorArray Processing (ASAP ’04), Lincoln Laboratory,Lexington, MA, Mar. 2004.

[7] Emmanuelle, J., Ovarlez, J. P., Declercq, D., and Duvaut, P.BORD: Bayesian optimum radar detector.Signal Processing, 83, 6 (2003), 1151—1162.

[8] Lange, K. L., Little, R. J. A., and Taylor, J. M. G.Robust statistical modeling using the t distribution.Journal of the American Statistical Association, 84 (Dec.1989), 881—896.

[9] Abramowitz, M., and Stegun, I. A. (Eds.)Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables.New York: Dover, 1972.

[10] Iskander, D. R., and Zoubir, A. M.Estimation of the parameters of the K-distribution usinghigher order and fractional moments.IEEE Transactions on Aerospace and Electronic Systems,35 (Oct. 1999), 1453—1457.

[11] Kay, S. M.Fundamental of Statistical Signal Processing: EsimationTheory.Prentice Hall Signal Processing Series Alan V.Oppenheim, series editor, Englewood Cliffs, NJ:Prentice-Hall, 1993.

[12] Conte, E., De Maio, A., and Galdi, C.Statistical analysis of real clutter at different rangeresolutions.IEEE Transactions on Aerospace and Electronic Systems,40, 3 (July 2004).

[13] Greco, M., Gini, F., and Diani, M.Robust CFAR detection of random signals incompound-Gaussian clutter plus thermal noise.IEE Proceedings, Pt. F, 148 (Aug. 2001), 227—232.

[14] Griffiths, H. D.Knowledge-based solutions as they apply to the generalradar problem.In Proceedings of the Research Technology Organization,North Atlantic Treaty Organization Lecture Series 233,Knowledge-Based Radar Signal and Data Processing,Rome, Italy, Nov. 6—7, 2000.

[15] Greco, M., Bordoni, F., and Gini, F.X-band sea-clutter nonstationarity: Influence of longwaves.IEEE Journal of Oceanic Engineering, 29 (Apr. 2004),269—283.

[16] Anastassopoulos, V., Lampropoulos, G. A., Drosopoulos, A.,and Rey, N.High resolution radar clutter statistics.IEEE Transactions on Aerospace and Electronic Systems,35 (Jan. 1999), 43—60.

Performance Analysis of Passive Low-Grazing-AngleSource Localization in Maritime EnvironmentsUsing Vector Sensors

We consider the problem of passive estimation of source

direction-of-arrival (DOA) and range using polarization-sensitive

sensor arrays, when the receiver array and signal source are

near the sea surface. The scenario of interest is the case of

low-grazing-angle (LGA) propagation in maritime environments.

We present a general polarimetric signal model that takes into

account the interference of the direct field with the field reflected

from smooth and rough surfaces. Using the Cramer-Rao bound

(CRB) and mean-square angular error (MSAE) bound, we

analyze the performance of different array configurations, which

include an electromagnetic vector sensor (EMVS), a distributed

electromagnetic component array (DEMCA), and a distributed

electric dipole array (DEDA). By computing these bounds, we

show significant advantages in using the proposed diversely

polarized arrays compared with the conventional scalar-sensor

arrays.

I. INTRODUCTION

Tracking targets and radio sources flying near thesea surface is a problem of considerable relevance,mainly because conventional radar systems mayexperience low performance [1]. In this scenario, thesignal arrives at the radar receiver via both a directand an indirect path, the latter produced by reflectionson the sea surface (multipath propagation). The two

Manuscript received May 2, 2005; revised June 5, 2006; releasedfor publication December 18, 2006.

IEEE Log No. T-AES/43/2/903030.

Refereeing of this contribution was handled by E. S. Chornoboy.

This work was supported by the Department of Defense underthe Air Force Office of Scientific Research MURI GrantFA9550-05-1-0443, AFOSR Grant FA9550-05-1-0018, and DARPAfunding under NRL Grant N00173-06-1-G006.

0018-9251/07/$25.00 c° 2007 IEEE

780 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 2 APRIL 2007

major causes of large errors in the estimation ofradio-source location and direction-of-arrival (DOA)of the target echoes are 1) the angular separationbetween the direct and indirect path is smaller than thebeamwidth of the receiver antenna array, 2) the directand indirect path signals are highly correlated, and3) sea reflections are complex phenomena which arenot easy to model. Hence, these signals are difficultto resolve using conventional scalar arrays [2]. Wepropose using polarized vector sensors to overcomethe above difficulties.In models of the multipath propagation in maritime

environments, the reflected signals can be representedby two components: specular and diffuse. Thisdistinction, somewhat arbitrary, has been used todescribe the reflection phenomena [1, 3—5]. If thesea surface were smooth, specular reflections of thesource signal would impinge on the receiver antenna.This component is amplitude attenuated and phaseshifted with respect to the direct signal. However,because the sea surface is disturbed and irregular, thesource signal is scattered at different angles; hence,the signal arrives at the receiver from directions otherthan and in addition to the specular path. The diffusecomponent represents the signal energy that is notreflected specularly.It is well known that the effects of multipath

propagation are twofold. For one, the interferenceof the direct signals and specular component canproduce fading, which deteriorates the performanceof any DOA estimator [1]. For another, multipathpropagation determines a geometrical relationshipbetween the bearing angles of the signal and thesource location, allowing passive source range andaltitude estimation [6, 7]. Previous work in theseareas has considered scalar-sensor arrays in whichthe reflecting surface is assumed to be smooth (thatis, only the specular component is considered) [6, 7];or the roughness of the surface is modeled byincreasing the sensor noise [8] or representing thediffuse multipath component by a signal componentof random phase [9]. References [10] and [11]studied the application of vector sensors in multipathchannels; however, the signal models disregarded thegeometry of the multipath scenario and properties ofthe reflecting surface.In the work presented here, we consider the

problem of passive localization of a source atlow-grazing angle (LGA) in maritime environments.We propose using polarized vector sensors toimprove the performance of conventional radarsystems and to reduce the estimation error of all thesource parameters, namely DOA, range, altitude,and polarization. Diversely polarized sensor arrays,which measure more than one electric or magneticcomponent of the field, not only are able to resolvedirect and multipath signals from both smooth andrough surfaces, but can also improve the source

position estimate, as we show later. In addition,they can estimate the polarization parameters thatmight be used to discriminate and identify the signalsimpinging on the receiver. These arrays are alsoable to measure depolarization that the signal mayexperience at the reflection.To the best of the authors’ knowledge, this is

the first time that vector sensors have been used forLGA source localization in a maritime environmentand their performance studied. We extend thespecular/diffuse scalar model to be applicable to theframework of diversely polarized arrays and vectorsensors. In this new polarimetric signal model, thefield reflected from a rough surface is decomposedinto polarized and unpolarized components.Then, using the Cramer-Rao bound (CRB) andthe mean-square angular error bound (MSAEB),we analyze the performance of different arrayconfigurations, including an electromagnetic vectorsensor (EMVS) [12], a distributed electromagneticcomponent array (DEMCA) [13, 14], and a distributedelectric dipole array (DEDA). We show that theproposed arrays significantly reduce the parametererror estimation compared with the scalar array.This paper is organized as follows. In Section II,

we characterize the problem and describe a generalmeasurement model for sensor arrays of any type ofpolarization. In Section III, we define the CRB andMSAEB as our measures of system performance.In Section IV, the proposed polarimetric arrays areanalyzed via computer simulations. Section V containsthe conclusions.

II. PROBLEM DESCRIPTION AND MODELING

In this section, we first present the polarimetricmeasurement model for a smooth surface, neglectingthe diffuse scattered field; then we extend this modelto consider the reflections from a rough surface. Next,we discuss our statistical assumptions on the signal,multipath interference, and noise.

A. Signal Model for Smooth Surfaces

In the presence of a sea surface, the signalsreceived by the sensor array are those that comedirectly from the source plus those that are firstreflected by the sea surface. For simplicity, we adopta flat surface representation; however, our modelcan be extended to spherical Earth. Assuming thesurface is smooth, the reflected signal arrives fromthe specular reflection points of the surface, which canbe located by applying Snell’s law of reflection: theangle of reflection is equal to the angle of incidence[15]. In our problem of interest, one specular pointexists, as suggested in Fig. 1. The reflected field is aphase-shifted and amplitude-attenuated replica of the

CORRESPONDENCE 781

Fig. 1. Multipath geometry for flat smooth surface.

direct signal, and the total field is the result of theircoherent sum.Consider a receiver and a signal source at heights

hr and hs above a flat smooth surface, separatedby a distance r on the ground, as shown in Fig. 1.We choose a Cartesian coordinate system such thatthe receiver is located at the origin O. The sourceelevation angle # and the grazing angle à of thereflected signal are measured from the horizontalplane, and the azimuth angle Á from the x-axis. It canbe easily shown that angle and length parameters arerelated by

tan#=hs¡ hrr

, tanà =hs +hrr

: (1)

The former geometrical relationships show that thesource location, i.e., range r and altitude hs, canbe determined from the bearing angles, assumingthe receiver position is known. Hence, multipathpropagation provides the geometric informationrequired for passive source localization.We assume the source-receiver and source-reflector

distances are large enough to ensure far-fieldconditions at the receiver and reflecting surface. Then,the propagating field can be considered a transverseelectromagnetic (EM) wave. In order to describethe direct incoming wave, we define a right-handedorthonormal triad (k,h,v), where:

k= [cosÁcos#, sinÁcos#, sin#]T

h= [¡sinÁ, cosÁ, 0]T

v= [¡cosÁsin#, ¡sinÁsin#, cos#]T:(2)

The vector k is pointing from the receiver towardthe source; vectors h and v span the plane where theelectric and magnetic field vectors lie, i.e., the planeorthogonal to k. Since h is parallel to the horizontalplane, it represents the horizontal component ofthe planewave, while v is the vertical component.Replacing # by à in (2), we can also define acoordinate system for the reflected wave.At an observation point r, the complex envelope of

the direct electric vector, denoted by Ed(t) 2C3£1, isgiven by [16]

Ed(t) = [h,v]ps(t)ej2¼rTk=¸ (3)

where the exponential term represents the phase ofthe planewave at position r with respect to the origin

of the reference system, ¸ is the wavelength of thetransmitted signal, and s(t) is the complex envelopeof the transmitted signal. The polarization vector p isdefined as [17]

p=·cos® sin®

¡sin® cos®

¸·cos¯

j sin¯

¸(4)

where the angles ¯ and ® are the ellipticity andorientation of the polarization ellipse depicted by theelectric field vector in the plane spanned by h andv. In a plane wave, the electric and magnetic fieldsare orthogonal to each other and to the direction ofpropagation. Hence, within a normalization factor, thecomplex envelope of the direct magnetic field vectorat r is given by

Hd(t) = k£Ed(t) = [v,¡h]ps(t)ej2¼rTk=¸ (5)

where £ is the cross-product operator. Thecomponents of the direct electric and magnetic fieldcan be stacked, forming a 6-element complex vector:

³d(t) =·EdHd

¸= g(Á,#)V(Á,#)ps(t) (6)

where we have defined

g(Á,#) = ej2¼rTk=¸ (7)

V(Á,#) =·h v

v ¡h

¸: (8)

Similarly, the components of the EM field reflectedfrom the smooth surface can also be arranged in avector. However, the reflected wave experiences achange of phase and amplitude with respect to thedirect signal:

³r(t) = ej±g(Á,Ã)V(Á,Ã)¡0ps(t¡ ¿ ) (9)

where ± = 2¼¢r=¸ is the phase shift due to thelength difference ¢r between the two paths. ForLGA propagation (rÀ hr,hs), the path lengthdifference is approximated by ¢r ¼ 2hshr=r. Thetime delay is given by ¿ =¢r=c, where c is thepropagation velocity. The complex reflectionmatrix ¡0 = diag(°h,°v) represents the amplituderelationship between the incident and reflected electricfields, where the Fresnel reflection coefficients °are functions of the signal polarization, surfacepermittivity ²r, and the grazing angle Ã. Theirexpressions are given by [15]

°h =sináp²r¡ cos2ÃsinÃ+

p²r¡ cos2Ã

°v =²r siná

p²r¡ cos2Ã

²r sinÃ+p²r¡ cos2Ã

:

(10)

For the LGA case, their values are approximately ¡1producing a phase shift close to 180± in the reflected

782 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 2 APRIL 2007

signal. The scalar g and the matrix V are defined as in(7) and (8), replacing # by Ã.Assuming that the inverse of the transmitted

signal bandwidth is much larger than the time delay ¿of the reflected signal with respect to the directsignal (narrowband condition), we can approximates(t)' s(t¡ ¿ ). Then, the total field over the surfaceis the superposition of the direct and reflected fieldcomponents:

³(t) = ³d(t)+ ³r(t)

= [g(Á,#)V(Á,#) + ej±g(Á,Ã)V(Á,Ã)¡0]ps(t):

(11)

B. Measurement Model

Consider an array of electric and magnetic sensorswith different polarizations. We assume the electricsensors are electrically short dipoles aligned withthe axes of the Cartesian coordinate system and themagnetic sensors are small loops orthogonal to theaxis (note that our formulation can be extended toother sensors by considering their directivities in themeasurement model). Therefore, the output of eachsensor is a voltage that is proportional to one of thecomponents of the electric or magnetic field. We alsoassume that the array size is much smaller than thesource-receiver distance; hence, the incoming waveis a planewave with the same DOA at all the sensors.In the presence of additive noise e(t), the output of anarray of m sensors is

y(t) = a0(µ)s(t) + e(t), t= 1, : : : ,N (12)

where y(t) 2 Cm£1 is the measurement vector, a0(µ) 2Cm£1 is the response vector of the sensor array,and µ = [Á,#,Ã,®,¯]T is the vector of the unknownparameters of interest. The response of the lth sensoris given by

a0l(µ) = [gl(Á,#)ÀlV(Á,#) + ej±gl(Á,Ã)ÀlV(Á,Ã)¡0]p

(13)

for l = 1, : : : ,m, where gl, defined in (7), is the phaseshift due to the sensor position rl, and Àl is a 1£ 6vector of 1 and 0 entries selecting the component ofthe EM field, given by (11), which is being measuredby the lth sensor. For example, the selection vector isÀ = [0,0,1,0,0,0] for a dipole parallel to the z-axis.Stacking the response of each sensor and arrangingthe sensor phase shifts in a diagonal matrix G =diag(g1, : : : ,gm), the array response can be written as

a0(µ) = [G(Á,#)¨V(Á,#) + ej±G(Á,Ã)¨V(Á,Ã)¡0]p

(14)

where ¨ = [ÀT1 , : : : ,ÀTm]T (see [13], [14] regarding

this notation). Note that (14) is a general expressionfor any array, which could be formed by scalar or

diversely polarized sensors. The matrices G and ¨take particular values for each array type; for instance,G and ¨ become a 6£ 6 identity matrix when thearray is an EMVS [12].

C. Rough Surfaces

When the surface is smooth, the reflected signalis totally coherent with the direct signal; for roughsurfaces, the reflected signal consists of a coherentcomponent with reduced magnitude and a diffusecomponent. Then, the specular component forrough surfaces is represented as in (9); however, thereflection matrix is replaced by [1]

¡ = ¡0e¡8(¼¾h sinÃ=¸)2 (15)

where ¾h is the standard deviation of the distributionof the surface heights. The exponential term representsthe reduction in the magnitude of the specular fielddue to surface roughness.The diffuse term accounts for the field scattered

by the irregularities on the surface and, hence, notreflected specularly. We assume that this field isthe result of the contribution of many independentpoint scatterers. As a consequence of the central limittheorem, the diffuse component can be modeled as aGaussian random process with zero mean [3, 5]. Wealso assume that these point scatterers are randomlylocated in an area known as glistening surface [18]. Ifthe position distribution of the scatterers is symmetricaround the specular reflection point, then the DOA ofthe diffuse component is concentrated in the directionof the specular component. This assumption is notonly intuitive for a homogeneous surface but is alsosupported by experimental data analyzed in [4], [5],[19].The previous representation of the reflected wave

does not provide a clear interpretation concerning itspolarization state. To correct this deficiency, we applythe following decomposition lemma [20].

LEMMA Let »(t) be a 2£1 complex vector whoseentries are the horizontal and vertical components of aplanewave. Its covariance P» 2 C2£2 can be decomposedas

P» = ¾2pp»p

¤» +¾

2uI2 (16)

where either ¾2p or ¾2u can be zero, I2 is the 2£ 2

identity matrix, “*” is the conjugate transpose, andp» is the polarization vector defined as in (4) in termsof the angles ®» 2 (¡¼=2,¼=2] and ¯» 2 [¡¼=4,¼=4].Furthermore, assuming ¾2p > 0, this decomposition isunique if and only if j¯» j6= ¼=4.The detailed proof is given in [21].

The lemma states that a planewave »(t) can bedivided into polarized and unpolarized componentswith powers ¾2p and ¾

2u , respectively. If ¾

2u = 0, the

CORRESPONDENCE 783

signal is said to be polarized with orientation ®»and ellipticity ¯» . If ¾

2p = 0, it is unpolarized. If both

¾2p6= 0 and ¾2u6= 0, the signal is partially polarized.Applying this lemma to the reflected wave, we

can relate the polarized component to the specularterm, where the polarization vector and power aregiven by p» = ¡p and ¾

2p = ¾

2s = E[js(t)j2] for any

t. The unpolarized component is associated withthe diffuse part of the reflected signal, which wedenote as u(t) 2 C2£1. It follows from (16) that thehorizontal and vertical components of the diffuse fieldare uncorrelated (the same assertion is deduced fromthe analysis of real data in [4]). Hence, to consider thereflections from a rough surface, the EM field vectorgiven by (9) is extended as follows:

³r(t) = g(Á,Ã)V(Á,Ã)[ej±¡ps(t) +u(t)]: (17)

Then, combining (6) and (17), the total EM fieldvector is given by

³(t) = [g(Á,#)V(Á,#) + ej±g(Á,Ã)V(Á,Ã)¡ ]ps(t)

+ g(Á,Ã)V(Á,Ã)u(t): (18)

Despite the fact that the diffuse component carriesno useful message, it provides information about thesource position through its bearing angles. Hence,this component is considered as “signal” in themeasurement model, instead of as part of the additivenoise. Then, the output of an array of m diverselypolarized sensors is given by

y(t) = A(µ)x(t) + e(t), t= 1, : : : ,N (19)

wherex(t) = [s(t), uT(t)]T

A(µ) = [a(µ), B(Á,Ã)]

A(µ) 2Cm£3 is the array response matrix, a(µ)is defined as in (14), using the reflection matrixfor a rough surface given in (15), and B(Á,Ã) =G(Á,Ã)¨V(Á,Ã).

D. Statistical Assumptions

We assume that the signal s(t) and the noisee(t) are independent identically distributed (IID)complex Gaussian processes with zero mean (thisassumption has been universally adopted in thesignal processing community to make these problemstractable [22]). The signal u(t) can also be representedby an IID zero-mean complex Gaussian process [3],[5]. In addition, we assume that these signal andnoise random processes are mutually independent.Then, they are completely characterized by theircovariances, which are respectively ¾2s , ¾

2Im, and ¾2uI2.

For simplicity, we assume the same noise varianceat each sensor; however, the results can be extendedfor different covariance structures [12]. Under these

assumptions, the output of the array is also an IIDzero-mean complex Gaussian process with covariancematrix

Cy(´) = A(µ)CxA¤(µ) +¾2Im (20)

where Cx = diag(¾2s ,¾

2u ,¾

2u) and ´ = [µ

T,¾2s ,¾2u ,¾

2]T isthe vector of unknown parameters of the model. Theentries of the vector µ are the parameters of interest,and the powers ¾2s , ¾

2u , and ¾

2 are considered nuisanceparameters.

III. MEASURE OF PERFORMANCE

In this section, we state the measures ofperformance that we have applied to analyze differentpolarimetric arrays and to contrast them with thescalar-sensor array.The CRB is a universal lower bound on the

variance of all unbiased estimators of a set ofparameters. It is defined as the inverse of the Fisherinformation matrix (FIM), which describes the amountof information that the data provide about unknownparameters:

CRB¡1(´) = FIM(´) =¡E·@2 lnp(y;´)@´@´T

¸: (21)

If the data have an IID zero-mean complex Gaussiandistribution, the (i,j)th entry of (21) can be written as[23]

[FIM(´)]ij =N tr

"C¡1y

@Cy@´i

C¡1y@Cy@´j

#(22)

where N is the number of snapshots, Cy is the datacovariance matrix given by (20), and tr indicates thematrix trace operator. When the desired parameters area function of the original parameters, i.e., º = g(´),the CRB is given by [23]

CRB(º) =@g(´)@´

FIM¡1(´)@g(´)@´

T

: (23)

If º = [r,hs]T, the function g gives the relation

between the length and angle parameters, which canbe derived from (1).Estimating the source azimuth and elevation angles

is equivalent to estimating the bearing vector k. Theangular difference between k and its estimate is calledangular error of the direction estimator. A naturalmeasure of the estimator performance is given by theMSAE. In [12], it has been shown that the MSAElower bound (MSAEB) of any unbiased estimator ofk is

MSAEB(Á,#) =N[cos2#CRB(Á) +CRB(#)]:

(24)

This bound can be considered as an overall measureof error for the source DOA estimation (or reflectedDOA, replacing # by Ã). An important property of

784 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 2 APRIL 2007

TABLE ISummary of Array Properties

Measured DifferentialArray Channels Field Comp. Phase

EMVS 6 6 EM NoDEMCA 6 6 EM YesDED 6 3 E Yes

Scalar array 6 1 Yes

the bound is that it does not depend on the number ofsnapshots, since the CRB is proportional to 1=N.Both bounds, the CRB and the MSAEB, are

independent of the estimation algorithm, providinga measure of potential performance attainable bythe system. Hence, we apply these two bounds toexamine different diversely polarized arrays. Due tothe complexity of the problem, it is not possible toexpress the bounds for the parameters of interest ina compact and closed form. The derivatives of thecovariance matrix Cy required for the calculation ofthe CRB and the MSAEB are given in the appendix.

IV. SIMULATION RESULTS

In this section, we analyze and compare theperformance of source localization for the followingsensor arrays: 1) an EMVS [12], 2) a DEMCA[13, 14], 3) a DEDA, and 4) a scalar array. TheEMVS consists of 6 colocated sensors, eachmeasuring one component of the EM field. For afair comparison, the other three arrays have the samenumber of sensors, m= 6; however, the sensors arelocated at different positions, forming a uniformcircular array parallel to the horizontal plane, asdepicted in Fig. 2. The DEMCA has one sensor foreach component of the EM field, as well. The DEDAmeasures the three components of the electric field,and the scalar array measures only one componentof the EM field. Note that the EMVS provides thecomplete information of the EM field at the receiverlocation; hence, the DOA and the polarization state ofthe incoming signal can be determined. The DEMCA,in addition, measures the differential phases betweenthe sensors, which improves the former estimations.The performance of the DEDA may be deteriorateddepending on the source DOA and polarization, sinceit measures only the electric field. Since the scalararray measures one component of the field, the sourceposition can only be estimated from its data. Becausethe EMVS has all its elements colocated, it is theonly array with no differential phase between itssensors. The properties of each array are summarizedin Table I.We carried out computer examples to study the

performance of the former arrays. The range betweenthe source and receiver is r = 10,000¸, the receiverheight is hr = 60¸, and the source height hs variesfrom 0 to 200¸ (¸= 0:3 m). Under these conditions,

Fig. 2. Sketch of sensor array geometries. (a) EMVS.(b) DEMCA. (c) DEDA. (d) Scalar array. (d is the inter-element

distance.)

the angular separation between the direct and reflectedwave is 0:7±. The inter-element distance is d = 0:5¸.We consider a linear polarized field (¯ = 0±) with thetwo most significant orientations: the electric fieldis horizontally and vertically polarized, i.e., ®= 0±

and ®= 90±, respectively. We assume the reflectionsare produced on a seawater surface whose relativecomplex permittivity is ²r ' 80¡ j240¸, under calm(¾h = 0 m) and rough (¾h = 2 m) sea state conditions.The signal-to-noise ratio is SNR= 10log10(¾

2s =¾

2) =10 dB. For a rough sea state, the power ratio betweenthe signal and diffuse multipath component is10log10(¾

2s =¾

2u) = 7 dB. (Note: this value was selected

to approximately match the measurements reportedin [4].)An array of sensors measuring only one

component of the EM field is not capable ofestimating the signal polarization. To compare thescalar array with the other arrays, we assume thatthe polarization aspects ® and ¯ are known, and theparameters of interest are the bearing angles Á, #,and Ã.Fig. 3 gives the square root of the MSAEB of

the source DOA as a function of the source heighths and the signal polarization when the reflectingsurface is calm seawater. The peaks in the boundsoccur when there is strong signal fading produced bythe interference of the direct and reflected fields. TheDEMCA performs somewhat better (approximately1 dB) than EMVS because the former exploitsthe information provided by the differential phasebetween sensors. The performance difference betweenthese arrays can be increased by enlarging the

CORRESPONDENCE 785

Fig. 3. Square root of MSAEB for source DOA versus thesource height hs over calm seawater (azimuth angle Á= 45±).

(a) Horizontal. (b) Vertical polarized field.

inter-element distance d. In addition, Fig. 3 showsthat EMVS and DEMCA perform approximately15 dB better than the scalar array. This differenceimplies that exploiting the polarization aspect of thesignal produces an improvement in the estimationof the unknown parameters. The performance ofDEDA depends highly on the polarization of thesignal as well as on the source azimuth angle Á,as shown in Fig. 4. We can state that EMVS andDEMCA are more robust than DEDA, showinga stable performance with respect to variations inpolarization and azimuth angle of the source.The bounds of the MSAEB of the reflected signal

DOA are similar to the results depicted in Figs. 3 and4; thus, they are not explicitly shown here.Fig. 5 plots the square root of the range CRB

(normalized with respect to the number of samplesN) as a function of the source height hs for a calmseawater surface. The CRB range follows the samepattern of the MSAEB of the source DOA; however,it is several orders of magnitude larger. This result isa consequence of the relationship between the range

Fig. 4. Square root of MSAEB for the source DOA versussource azimuth angle Á over calm seawater (source height

hs = 50¸) for horizontal polarized field.

Fig. 5. Square root of source range CRB, normalized withrespect to number of samples, versus source height hs over calmseawater (azimuth angle Á= 45±) for horizontal polarized field.

r and the angles # and Ã: small errors in the bearingangles produce large errors in the range, see (1).Simulations for rough seawater were also

performed (see Fig. 6, which shows the MSAEBfor the source DOA as a function of the sourceheight hs when the electric field is horizontally andvertically polarized). Results on the bound for EMVS,DEMCA, and DEDA are similar to the smooth surfacecondition. However, the CRB for the proposed scalararray does not exist, since the FIM is singular. Theabsence of the lower bound means that the depictedscalar array cannot be used in combination withthe proposed model for rough surfaces, since thereflected wave is represented by two signals with thesame bearing and different polarization state. Thisfact indicates another relevant advantage of diversepolarized arrays.By numerical examples, we also have found that

when the field is linearly polarized (¯ = 0±) withthe electric field horizontal or vertical polarized(®= 0,90±), the off-diagonal entries of the FIM

786 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 2 APRIL 2007

Fig. 6. Square root of MSAEB for source DOA versus sourceheight hs over rough sea water (azimuth angle Á= 45

±).(a) Horizontal polarized field. (b) Vertical polarized field.

between bearing and polarization angles are zero.Thus, the FIM is block diagonal and the CRB of thebearing angles can be calculated independently ofthe knowledge of the polarization. It can be said thatthe bearing angles Á, #, and à are decoupled from ®and ¯ [23]. However, this result cannot be extended toother states of polarization.

V. CONCLUSION

We have addressed the problem of passiveDOA and range estimation of a source at LGA inmaritime environments. To improve the performanceof conventional scalar systems, we proposed theapplication of different vector-sensor configurations.In order to represent the multipath propagation overthe sea surface, we extended the specular/diffusescalar model for its application in the vector-sensorframework. In this new polarimetric model, weproposed decomposing the reflected field intopolarized and unpolarized components, following thedecomposition lemma that we stated in Section IIC.

We used the CRB and the MSAEB as performancemeasures for studying polarimetric arrays. In addition,we analyzed and compared different arrays bycomputing the former bounds for the source range anddirection-of-arrival estimation. We showed that it ispossible to significantly reduce the error estimationof the unknown parameters when the full EMinformation is exploited using EMVS or DEMCA.Furthermore, we showed that the performances ofEMVS and DEMCA are more stable than that ofDEDA, since they are independent of the polarizationstate and azimuth angle of the source.

APPENDIX

Given that the sensor array output y(t) is azero-mean Gaussian random process, the (i,j)th entryof the FIM for N snapshots is

[FIM(´)]ij =N tr

"C¡1y

@Cy@´i

C¡1y@Cy@´j

#:

where ´ = [Á,#,Ã,®,¯,¾2,¾2s ,¾2u]T is the vector of

real parameters. The relevant derivatives of the datacovariance are listed next. When ´ represents thesignal and noise power:

@Cy@¾2

= Im

@Cy@¾2s

= aa¤

@Cy@¾2u

=G(Á,Ã)¨·I3¡uuT (u£)T(u£) I3¡uuT

¸¨TGT(Á,Ã)

where (u£) is a matrix operator performing thecross product with vector u, defined in [12]. When´ represents any of the parameters of interest (i.e., Á,#, Ã, ®, ¯):

@Cy@´

=@A

@´CxA

¤+ACx@A¤

@´:

The derivative of the array response with respect tothe polarization angles ® and ¯ is

@A

@´=·[G(Á,#)¨V(Á,#) + ej±G(Á,Ã)¨V(Á,Ã)¡ ]

@p@´,0¸

where 0 is a matrix whose entries are zeros, and thederivatives of the polarization vector are

@p@®

=·¡sin®cos¯+ j cos®sin¯¡cos®cos¯¡ j sin®sin¯

¸@p@¯

=·¡cos®sin¯+ j sin®cos¯sin®sin¯+ j cos®cos¯

¸:

The derivative with respect to the bearing angles Á, #,and à is

@A

@´=·@a@´,@B

@´

¸:

CORRESPONDENCE 787

The derivatives of the vector a and the matrix B are

@a@Á

=

·@G(Á,#)@Á

¨V(Á,#) +G(Á,#)¨@V(Á,#)@Á

+ej±@G(Á,Ã)@Á

¨V(Á,Ã)¡ + ej±G(Á,Ã)¨@V(Á,Ã)@Á

¡

¸p

@a@#

=

·@G(Á,#)@#

¨V(Á,#) +G(Á,#)¨@V(Á,#)@#

+ej±G(Á,Ã)¨V(Á,Ã)@¡

@#

¸p

@a@Ã

=

·@ej±

@ÃG(Á,Ã)¨V(Á,Ã)¡ + ej±

@G(Á,Ã)@Ã

¨V(Á,Ã)¡

+ej±G(Á,Ã)¨@V(Á,Ã)@Ã

¡ + ej±G(Á,Ã)¨V(Á,Ã)@¡

@Ã

¸p

@B

@Á=@G(Á,Ã)@Á

¨V(Á,Ã) +G(Á,Ã)¨@V(Á,Ã)@Á

@B

@#= 0

@B

@Ã=@G(Á,Ã)@Ã

¨V(Á,Ã) +G(Á,Ã)¨@V(Á,Ã)@Ã

:

The derivatives of matrix G are

@G(Á,#)@Á

= (j2¼ cos#=¸)diag(rT1h, : : : ,rTmh)¯G(Á,#)

@G(Á,#)@#

= (j2¼=¸)diag(rT1v, : : : ,rTmv)¯G(Á,#)

where ¯ represents the elementwise product. Thederivatives of matrix V are (see [12])

@V(Á,#)@Á

=·vsin#¡ucos# ¡hsin#

¡hsin# ¡vsin#+ucos#

¸@V(Á,#)@#

=·0 ¡u¡u 0

¸:

The expressions of @G(Á,Ã)=@Ã and @V(Á,Ã)=@Ãcan be found replacing # by Ã. The derivatives of theexponential term are

@ej±

@#=

j2¼hr¸cos2#

ej±,@ej±

@Ã=¡j2¼hr¸cos2Ã

ej±

where ± = 2¼hr(tan#¡ tanÃ)=¸. The derivative of thereflection matrix ¡ is

@¡

@Ã=@¡0@Ã

e¡8(¼¾h sinÃ=¸)2

¡¡0e¡8(¼¾h sinÃ=¸)2(4¼¾h=¸)

2 sinà cosÃ

where

@¡0@Ã

= 2

2664%cosá sinÃ%sinÃ+ %2

0

0²(²¡ 1)cosá sinÃ%(²sinÃ+ %)2

3775where %=

p²¡ cos2Ã.

MARTIN HURTADOARYE NEHORAIDept. of Electrical and Systems EngineeringWashington University in St. LouisBryan Hall, Room 201Campus Box 1127One Brookings DriveSt. Louis, MO 63130E-mail: (nehorai@ese.wust.edu)

REFERENCES

[1] Barton, D. K.Low angle radar tracking.Proceedings of the IEEE, 62 (June 1974), 687—703.

[2] Ferrara, E. R., and Parks, T. M.Direction finding with an array of antennas havingdiverse polarizations.IEEE Transactions on Antennas Propagation, 31 (Mar.1983), 231—236.

[3] Beard, C., Katz, I., and Spetner, L.Phenomenological vector model of microwave reflectionfrom the ocean.IEEE Transactions on Antennas Propagation, 4 (Apr.1956), 25—30.

[4] Beard, C.Coherent and incoherent scattering of microwaves fromthe ocean.IEEE Transactions on Antennas Propagation, 9 (Sept.1961), 470—483.

[5] Straiton, A., and Tolbert, C.Measurement and analysis of instantaneous radioheight-gain curves at 8.6 millimeters over rough surfaces.IEEE Transactions on Antennas Propagation, 4 (July1956), 346—351.

[6] Hamilton, M., and Schultheiss, P. M.Passive ranging in multipath dominant environments,Part I: Known multipath parameters.IEEE Transactions on Signal Processing, 40 (Jan. 1992),1—12.

[7] Jao, J. K.A matched array beamforming technique for low angleradar tracking in multipath.In IEEE National Radar Conference, Mar. 1994, 171—176.

[8] Lo, T., and Litva, J.Use of a highly deterministic multipath signal model inlow-angle tracking.IEE Proceedings, Pt. F, 138 (Apr. 1991), 163—171.

[9] Griesser, T., and Balanis, C. A.Oceanic low-angle monopulse radar tracking errors.IEEE Journal of Oceanic Engineering, 12 (Jan. 1987).

[10] Chua, P. H., See, C. M. S., and Nehorai, A.Vector-sensor array processing for estimating angles andtimes of arrival of multipath communication signals.IEEE International Conference on Acoustics, Speech,Signal Processing (ICASSP), May 1998, 3325—3328.

[11] Rahamim, D., Tabrikian, J., and Shavit, R.Source localization using vector sensor array in amultipath environment.IEEE Transactions on Signal Processing, 52 (Nov. 2004),3096—3103.

[12] Nehorai, A., and Paldi, E.Vector-sensor array processing for electromagnetic sourcelocalization.IEEE Transactions on Signal Processing, 42 (Feb. 1994),376—398.

788 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 43, NO. 2 APRIL 2007

[13] See, C. M. S., and Nehorai, A.Source localization with distributed electromagneticcomponent sensor array processing.International Symposium on Signal Processing and ItsApplications, vol. 1, July 1—4, 2003, 177—180.

[14] See, C. M. S., and Nehorai, A.Source localization with partially calibrated distributedelectromagnetic component sensor array.In Workshop on Statistical Signal Processing, Sept.28—Oct. 1, 2003, 458—461.

[15] Harrington, R. F.Time Harmonic Electromagnetic Fields.New York: McGraw Hill, 1961.

[16] Compton, R.The tripole antenna: An adaptive array with fullpolarization flexibility.IEEE Transactions on Antennas Propagation, 29 (Nov.1981), 944—952.

[17] Giuli, D.Polarization diversity in radar.Proceedings of the IEEE, 74 (Feb. 1986), 245—269.

[18] Beckmann, P., and Spizzichino, A.The Scattering of Electromagnetic Waves from RoughSurfaces.Norwood, MA: Artech House, 1963.

[19] Lo, T., and Litva, J.Characteristics of diffuse multipath at low grazing anglesin naval environments.In International Geoscience and Remote SensingSymposium, (IGARSS ’91), vol. 3, June 1991,1259—1263.

[20] Hochwald, B., and Nehorai, A.Polarimetric modeling and parameter estimation withapplications to remote sensing.IEEE Transactions on Signal Processing, 43 (Aug. 1995),1923—1935.

[21] Hochwald, B.Aspects of vector-sensor proccessing.Ph.D. dissertation, Yale University, New Haven, CT, Dec.1995.

[22] Krim, H., and Viberg, M.Two decades of array signal processing research.IEEE Signal Processing Magazine, 13 (July 1996), 67—94.

[23] Kay, S. M.Fundamentals of Statistical Signal Processing: EstimationTheory.Englewood Cliffs, NJ: Prentice-Hall, 1993.

Comparison of Detectability of Radar CompressionWaveforms in Classic Passive Receivers

Several types of passive receivers can be used to detect

radar emissions over considerable distances. Previous work has

been performed which compares the detectability of bi-phase

waveforms to a classic rectangular pulse. It is also of interest

to compare the detectability of equal-bandwidth/equal-energy

FMCW, quadriphase, and polyphase waveforms. The quadriphase

waveform is the quadriphase version of the bi-phase Welti code.

The polyphase waveforms are the P1, P2, P3, and P4 codes.

Relative detection ranges are predicted using a simulation of

the square-law, wideband crystal video, and channelized passive

receivers possessing typical bandwidth, noise-floor, and loss

parameters. A comparison is given showing the relative LPI/LPD

attributes of each waveform and the relative detection attributes

of each receiver.

I. INTRODUCTION

Many have published open literature workon the problem of intercepting radar waveforms[1—7]. Schrick and Wiley [3] suggest a rapidlyswept superheterodyne receiver used under certainconditions. Modern intercept predictions usuallyrely on knowing the exact conditions, channelcharacteristics, signal and noise statistics, and otherparameters to make informed estimates as to theminimum detectable signal levels. However, therestill remains a role for computer simulations whichallow the user the ability to rapidly change noisestatistics, receiver bandwidths, radar waveforms, andother parameters yielding various estimates on passivereceiver behavior.In a previous paper, Gross and Chen [8] compared

the detectability of the rectangular pulse and bi-phaseWelti codes. It was shown that the Welti codedwaveform was much less detectable in the widebandand channelized receivers than the rectangular pulse.Since the pulse and Welti code were designed to havethe same total energy, both were equally detectable inthe square-law receiver.It is also of interest to compare the detectability

of the rectangular pulse with several compressionwaveforms such as the classic FMCW, Weltiquadriphase, and polyphase waveforms. The collection

Manuscript received January 4, 2005; revised September 2, 2005and August 4, 2006; released for publication November 20, 2006.

IEEE Log No. T-AES/43/2/903031.

Refereeing of this contribution was handled by V. C. Chen.

0018-9251/07/$25.00 c° 2007 IEEE

CORRESPONDENCE 789