PERFORMANCE ANALYSIS OF PASSIVE LOW-GRAZING-ANGLE SOURCE LOCALIZATION USING DIVERSELY POLARIZED ARRAYS Martn Hurtado and Arye Nehorai

ECE Department The University of Illinois at Chicago 851 S. Morgan St. (M/C 154) Chicago, IL 60607, USA. mhurta3@uic.edu, nehorai@ece.uic.edu

ABSTRACT We consider the problem of passive localization of a lowgrazing-angle source employing polarization sensitive sensor arrays. We present a general polarimetric signal model that takes into account the direct eld and the multipath interference produced by reections from smooth and rough surfaces. Applying the Cram r-Rao bound (CRB) and e mean-square angular error (MSAE) bound, we analyze the performance of different array congurations, 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 signicant advantages in using the proposed diversely polarized arrays compared with the conventional scalar-sensor arrays. 1. INTRODUCTION The presence of a signal source and a receiver near a reecting surface is a common scenario in many radar and communication systems; for instance, low-angle radar tracking systems [1] and terrestrial radio links [2]. Under this condition, the received signal can be modeled as the sum of elds arriving through the direct path and the scattered from the surface. A major challenge in multipath propagation at a low-grazing-angle is that the angular separation between direct and reected paths is smaller than the beamwidth of the receiver antenna array; hence, the reected signal cannot be spatially ltered. Most of the previous work in this area has considered scalar-sensor arrays [3]-[6] which cannot resolve very close source arrival paths [7] nor can it provide a good estimation of position parameters: direction-of-arrival (DOA), range, and altitude. We propose to use diversely polarized sensor arrays, which measure more than one electric and/or magnetic component
This work was supported by the grants AFOSR FA9550-05-1-0018 and DARPA/AFOSR FA9550-04-1-0187.

of the eld, to overcome the above difculties and improve the estimation of all the source position parameters. In addition, diversely polarized arrays can be applied to estimate the polarization parameters of the received signal. In this paper, we present a general polarimetric signal model that takes into account the interference of the direct eld with the eld reected by smooth and rough surfaces. Then, using the Cram r-Rao bound (CRB) and the mean-square ane gular error bound (MSAEB ), we analyze the performance of different array congurations, including an electromagnetic vector sensor (EMVS) [8], a distributed electromagnetic component array (DEMCA) [9], [10]; and a distributed electric dipole array (DEDA). We show that the proposed arrays signicantly reduce the parameter error estimation compared with the scalar array. This paper is organized as follows: In Section II, we describe a general measurement model for sensor arrays of any type of polarization. In Section III, we dene the CRB and MSAEB as our measures of system performance. In Section IV, the proposed polarimetric arrays are analyzed via computer simulations. Conclusions are given in Section V. 2. PROBLEM DESCRIPTION AND MODELING In this section, we rst present the measurement model for a smooth surface; then, we extend it for a rough surface. Next, we discuss our statistical assumptions on the signal, multipath interference, and noise. 2.1. Signal Model for Smooth Surfaces Consider a receiver and a signal source at heights hr and hs above a at smooth surface, separated by a distance r on the ground, as shown in Figure 1. We choose a Cartesian coordinate system such that the receiver is located at the origin O. The source elevation angle and the grazing angle of the reected signal are measured from the horizontal plane,

r z
Direct ray Receiver Source

to the direction of propagation. Hence, within a normalization factor, the complex envelope of the direct magnetic eld vector at r is given by
T H d (t) = k E d (t) = [v, h]ps(t)ej2r k/

(5)

hr

Reflected ray

where is the cross-product operator. The components of

hs the direct electric and magnetic eld can be stacked, forming a 6-element complex vector: d (t) = Ed Hd = g(, )V (, )ps(t). (6)

Specular reflection point

where we have dened Figure 1: Low-grazing-angle propagation over a at surface. and the azimuth angle from the x-axis. It can be easily shown that angle and length parameters are related by hs hr tan = , r hs + hr tan = . r (1)
T g(, ) = ej2r k/ ,

(7) . (8)

V (, ) =

h v

v h

These geometrical relationships show that the source location, i.e. range r and altitude hs , can be determined from the bearing angles, assuming the receiver position is known. We assume the propagating eld is a transverse electromagnetic (EM) wave. To describe the direct wave, we dene a right-handed orthonormal triad (k, h, v), where: k = [cos cos , sin cos , sin] h = [ sin , cos , 0]
T T T

Similarly, the components of the EM eld reected from the smooth surface can also be arranged in a vector. However, the reected wave experiences a change of phase and amplitude with respect to the direct signal: r (t) = ej g(, )V (, )0 ps(t ) (9)

v = [ cos sin , sin sin , cos ] .

(2)

The vector k is pointing from the receiver toward the source; vectors h and v span the plane where the electric and magnetic eld vectors lie. Replacing by in (2), we can also dene a coordinate system for the reected wave. At an observation point r, the complex envelope of the direct electric vector E d (t) C31 is given by [8]
T E d (t) = [h, v]ps(t)ej2r k/

where = 2r/ is the phase shift due to the length difference r between the two paths. For low-grazing angle propagation (r hr , hs ), the path length difference is approximated by r 2hs hr /r. The time delay is given by = r/c, where c is the propagation velocity. The complex reection matrix is 0 = diag(h , v ); expressions of Fresnel reection coefcients can be found in classical textbooks of EM. The scalar g and the matrix V are dened as in (7) and (8), replacing by . Assuming narrowband condition, we can approximate s(t) s(t ). Then, the total eld over the surface is the superposition of the direct and reected eld components: (t) = g(, )V (, ) + ej g(, )V (, )0 ps(t) (10) 2.2. Measurement Models

(3) Consider an array of electric and magnetic sensors with different polarizations. We assume that the array size is much smaller than the source-receiver distance; hence, the incoming wave is a planewave with the same direction of arrival at all the sensors. In the presence of additive noise e(t), the output of an array of m sensors is y(t) = a0 ()s(t) + e(t), t = 1, . . . , N (11)

where the exponential term represents the phase of the planewave at position r, is the transmitted signal wavelength, and s(t) is the complex envelop of the signal. The polarization vector p is dened as p= cos sin sin cos cos j sin (4)

where the angles and are the ellipticity and orientation of the polarization ellipse depicted by the electric eld vector in the plane spanned by h and v. In a plane wave, the electric and magnetic elds are orthogonal to each other and

where y(t) Cm1 is the measurement vector, a0 () Cm1 is the response vector of the sensor array, and =

[, , , , ]T is the vector of the unknown parameters of interest. The response of the lth sensor is given by a0l () = gl (, )l V (, ) + e gl (, )l V (, )0 p (12) for l = 1, . . . , m, where gl , dened in (7), is the phase shift due to the sensor position r l , and l is a 1 6 vector of 1 and 0 entries selecting the component of the EM eld, given by (10), which is being measured by the lth sensor. For example, the selection vector is = [0, 0, 1, 0, 0, 0] for a dipole parallel to the z-axis. Stacking the response of each sensor and arranging the sensor phase shifts in a diagonal matrix G = diag(g1 , . . . , gm ), the array response can be written as a0 () = G(, )V (, ) + ej G(, )V (, )0 p (13) T T where = [1 , . . . , m ]T (see [9], [10] regarding this notation). Note that equation (13) is a general expression for any array, which could be formed by scalar or diversely polarized sensors. 2.3. Rough Surfaces When the surface is smooth, the reected signal is totally coherent with the direct signal; for rough surfaces, the reected signal consists of a coherent component with reduced magnitude and a diffuse component [1]. Then, the specular component for rough surfaces is represented as in (9); however, the reection matrix is replaced by [1] = 0 e8(h sin /)
2

wave. Its covariance P C22 can be decomposed as

(15)

2 2 where either p or u can be zero, I2 is the 2 2 iden tity matrix, is the conjugate transpose, and p is the polarization vector dened as in (4) in terms of the angles (/2, /2] and [/4, /4]. Furthermore, 2 assuming p > 0, this decomposition is unique if and only if | | = /4. The detailed proof is given in [16]. The lemma states that a planewave (t) can be divided 2 into polarized and unpolarized components with powers p 2 and u , respectively. Applying it to the reected wave, we can relate the polarized component to the coherent term, where the polarization vector and power are given by p = 2 2 p and p = s = E|s(t)|2 for any t. The unpolarized component is associated with the diffuse part of the planewave, which we denote u(t) C21 . It follows from (15) that the horizontal and vertical components of the diffuse eld are uncorrelated (the same assertion is deduced from the analysis of real data in [13]). Hence, to consider the reections from a rough surface, the EM eld vector given by (9) is extended as follows:

r (t) = g(, )V (, ) ej ps(t) + u(t) .

(16)

Then, combining (6) and (16), the total EM eld vector is given by (t) = g(, )V (, ) + ej g(, )V (, ) ps(t) +g(, )V (, )u(t). (17) Despite the fact that the diffuse component carries no useful message, it provides information about the source position through its bearing angles. Hence, this component is considered as signal in the measurement model, instead of as part of the additive noise. Then, the output of an array of m diversely polarized sensors is given by y(t) = A()x(t) + e(t), where x(t) = s(t), uT (t) , A() = [a(), G(, )V (, )] , A() Cm3 is the array response matrix, and a() is dened as in (13), using the reection matrix for a rough surface given in (14). 2.4. Statistical Assumptions We assume that the signals s(t) and u(t), and the noise e(t) are independent identically distributed (i.i.d.) complex circular Gaussian processes with zero mean. In addition, we
T

(14)

where h is the standard deviation of the distribution of the surface heights. The diffuse term accounts for the eld scattered by the irregularities on the surface. We assume that this eld is the result of the contribution of many independent point scatterers. As a consequence of the central limit theorem, the diffuse component can be modeled as a Gaussian random process with zero mean [11], [12]. We also assume that these point scatterers are randomly located. If the position distribution of the scatterers is symmetric around the specular reection point, then the DOA of the diffuse component is concentrated in the direction of the specular component. This assumption is not only intuitive for an homogeneous surface but is also supported by experimental data analyzed in References [12]-[14]. The previous representation of the reected wave does not provide a clear interpretation about its polarization state. To correct this deciency, we apply the following decomposition lemma [15]. Lemma: Let (t) be a 2 1 complex vector whose entries are the horizontal and vertical components of a plane-

t = 1, . . . , N,

(18)

assume that the signals and noise are independent random processes. These processes are completely characterized 2 2 by their covariances, which are respectively s , u I2 , and 2 Im . For simplicity, we assume the same noise variance at each sensor; however, the results can be extended for different covariance structures [8]. Under these assumptions, the output of the array is also an i.i.d. zero-mean complex Gaussian process with covariance matrix Cy () = A()Cx A () + Im ,
2

Ez Ex ,Ey ,Ez Hx , Hy , Hz
x

Ey Ex
x

Hx Hy Hz
(b)

(a)

(19)
Ex Ez Ey

2 2 2 2 2 where Cx = diag(s , u , u ) and = [ T , s , u , 2 ]T is the vector of unknown parameters of the model. The entries of the vector are the parameters of interest, and the powers 2 2 s , u , and 2 are considered nuisance parameters.

Ez Ex Ez
(c) (d)

3. MEASURES OF PERFORMANCE The Cram r-Rao bound (CRB) is a universal lower bound e on the variance of all unbiased estimators of a set of parameters. It is dened as the inverse of the Fisher information matrix (FIM): CRB1 () = FIM() = E 2 ln p(y; ) T (20)

Figure 2: Sketch of the sensor array geometries: (a) EMVS, (b) DEMCA, (c) DEDA, and (d) scalar sensor array (d is the inter-element distance).

4. SIMULATION RESULTS In this section, we analyze the performance of source localization for the following sensor arrays: (a) EMVS [8]; (b) DEMCA [9], [10]; (c) DEDA; and (d) a scalar array. The EMVS consists of 6 co-located sensors, each measuring one component of the EM eld. For a fair comparison, the other three arrays have the same number of sensors, m = 6; however, the sensors are located at different positions, forming a uniform circular array parallel to the horizontal plane, as depicted in Figure 2. The DEMCA has one sensor for each component of the EM eld, as well. The DEDA measures the three components of the electric eld, and the scalar array measures only one component of the EM eld. We carried out computer examples to study the performance of the former arrays. The range between the source and receiver is r = 10,000, the receiver height is hr = 60, and the receiver height hs varies from 0 to 200 ( = 0.3m). Under these conditions, the angular separation between the direct and reected wave is 0.7 . The inter-element distance is d = 0.5. We consider a linear polarized eld ( = 0 ) with the two most signicant orientations: the electric eld is horizontally and vertically polarized, i.e., = 0 and = 90 , respectively. We assume the reections are produced on a seawater surface whose relative complex permittivity is r 80 j240, under calm (h = 0m) and rough (h = 2m) sea state conditions. The 2 signal to noise ratio is SNR = 10 log10 (s / 2 ) = 10dB. For a rough sea state, the power ratio between the signal and 2 2 diffuse multipath component is 10 log10 (s /u ) = 7dB. (Note: this value was selected to approximately match the

If the data have an i.i.d. zero-mean complex Gaussian distribution, the (i, j)th entry of (20) can be written as [17]
1 [FIM()]ij = N tr Cy

Cy 1 Cy C i y j

(21)

where N is the number of snapshots, Cy is the data covariance matrix given by (19), and tr indicates the matrix trace operator. When the desired parameters are a function of the original parameters, i.e. = g(), the CRB is [17] CRB() = g() g() FIM1 () .
T

(22)

If = [r, hs ]T , the function g gives the relation between the length and angle parameters, which can be derived from (1). Estimating the source azimuth and elevation angles is equivalent to estimating the bearing vector k. The angular difference between k and its estimate is called angular error of the direction estimator. A natural measure of the estimator performance is given by the mean-square angular error (MSAE). In [8], it is shown that the MSAE lower bound (MSAEB ) of any unbiased estimator of k is MSAEB (, ) = cos2 CRB() + CRB() (23)

This bound can be considered as an overall measure of error for the source DOA estimation (or reected DOA, replacing by ). Both bounds, the CRB and the MSAEB , are independent of the estimation algorithm, providing a measure of potential performance attainable by the system.

sqrt( MSAE B )

measurements reported in [13].) An array of sensors measuring only one component of the EM eld is not capable of estimating the signal polarization. To compare the scalar array with the other arrays, we assume that the polarization angles and are known. Figure 3 gives the square root of the MSAEB of the source DOA as a function of the source height hs and the signal polarization when the reecting surface is calm seawater. The peaks in the bounds occur when there is strong signal fading produced by the interference of the direct and reected elds. It is shown that the DEMCA performs somewhat better (approximately 1dB) than EMVS because the former exploits the information provided by the differential phase between sensors. The performance difference between these arrays can be increased by enlarging the interelement distance d. In addition, Figure 3 shows that EMVS and DEMCA perform approximately 15dB better than the scalar array. This difference implies that exploiting the polarization aspect of the signal produces an improvement in the estimation of the unknown parameters. The performance of DEDA highly depends on the polarization of the signal as well as on the source azimuth angle , as shown in Figure 4. We can state that EMVS and DEMCA are more robust than DEDA, showing more a stable performance with respect to variations in polarization and azimuth angle of the source. Similar remarks can be obtained computing MSAEB of the reected signal DOA and range CRB. Because of length constraints, those graphs are not shown in the paper Simulations for rough seawater were also performed (see Figure 5 which shows the MSAEB for the source DOA as a function of the source height hs when the source electric eld is horizontally polarized). Results on the bound for EMVS, DEMCA, and DEDA are similar to the smooth surface condition. However, the CRB for the proposed scalar array does not exist, since the Fisher information matrix is singular. The absence of the lower bound means that the depicted scalar array cannot be used in combination with the proposed model for rough surfaces, since the reected wave is represented by two signals with the same bearing and different polarization state. This fact indicates another relevant advantage of diverse polarized arrays. 5. CONCLUSIONS We have addressed the problem of passive source direction of arrival and range estimation using polarimetric sensor arrays located near a reecting surface. We have presented a general measurement model for receive arrays composed of sensors with diverse polarization. In this new polarimetric model, we have proposed to decompose the multipath component into polarized and unpolarized terms, following the decomposition lemma which we have stated in Section 2.3. We have used the Cram r-Rao bound and the mean-square e

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(a)
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(b) Figure 3: Square root of MSAEB for the estimated source direction of arrival versus the source height hs over calm seawater (azimuth angle = 45 ): (a) horizontally and (b) vertically polarized source eld.

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Figure 4: Square root of MSAEB for the estimated source direction of arrival versus the source azimuth angle over calm seawater (source height hs = 50).

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sqrt( MSAE B )

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[7] E. R. Ferrara and T. M. Parks, Direction nding with an array of antennas having diverse polarizations, IEEE Trans. Antennas Propagat., vol. 31, pp. 231-236, Mar. 1983. [8] A. Nehorai and E. Paldi, Vector-sensor array processing for electromagnetic source localization, IEEE Trans. Signal Process., vol. 42, pp. 376-398, Feb. 1994. [9] C. M. S. See and A. Nehorai, Source localization with distributed electromagnetic component sensor array processing, Int. Symp. on Signal Processing and Its Applications, 1-4 July 2003, vol. 1, pp. 177-180. [10] C. M. S. See and A. Nehorai, Source localization with partially calibrated distributed electromagnetic component sensor array, Workshop on Statistical Signal Processing, 28 Sept.-1 Oct. 2003, pp. 458-461. [11] C. Beard, I. Katz, and L. Spetner, Phenomenological vector model of microwave reection from the ocean, IEEE Trans. Antennas Propagat., vol. 4, pp. 25-30, Apr. 1956. [12] A. Straiton and C. Tolbert, Measurement and analysis of instantaneous radio height-gain curves at 8.6 millimeters over rough surfaces, IEEE Trans. Antennas Propagat., vol. 4, pp. 346-351, July 1956. [13] C. Beard, Coherent and incoherent scattering of microwaves from the ocean, IEEE Trans. Antennas Propagat., vol. 9, pp. 470-483, Sep. 1961. [14] T. Lo and J. Litva, Characteristics of diffuse multipath at low grazing angles in naval environments, Int. Geoscience and Remote Sensing Symp., IGARSS 91, June 1991, vol. 3, pp. 1259-1263. [15] B. Hochwald and A. Nehorai, Polarimetric modeling and parameter estimation with applications to remote sensing, IEEE Trans. Signal Process., vol. 43, pp. 1923-1935, Aug. 1995. [16] B. Hochwald, Aspects of vector-sensor proccessing, Ph. D. dissertation, Yale University, New Haven, CT, Dec. 1995. [17] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Englewood Cliffs, NJ: Prentice Hall, 1993.

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Figure 5: Square root of MSAEB for the estimated source direction of arrival versus the source height hs over rough sea water (azimuth angle = 45 ) for horizontally polarized source eld. angular error bound as performance measures for studying polarimetric arrays. In addition, we have analyzed and compared different arrays by computing the former bounds for the source range and direction of arrival estimation. We have shown that it is possible to signicantly reduce the error estimation of the unknown parameters when the full EM information is exploited using EMVS or DEMCA. Furthermore, we have shown that the performances of EMVS and DEMCA are more stable than that of DEDA, since they are independent of the polarization state and azimuth angle of the source. 6. REFERENCES [1] D. K. Barton, Low angle radar tracking, Proc. IEEE, vol. 62, pp. 687-703, June 1974. [2] A. R. Webster and T. S. Merritt, Multipath angles-ofarrival on a terrestrial microwave link, IEEE Trans. Commun., vol. 38, pp. 25-30, Jan. 1990. [3] T. Griesser and C. A. Balanis, Oceanic low-angle monopulse radar tracking errors, IEEE J. Ocean. Eng., vol. 12, Jan. 1987. [4] T. Lo and J. Litva, Use of a highly deterministic multipath signal model in low-angle tracking, IEE Proc. F., vol. 138, pp. 163-171, Apr. 1991. [5] M. Hamilton and P. M. Schultheiss, Passive ranging in multipath dominant environments, Part I: known multipath parameters, IEEE Trans. Signal Process., vol. 40, pp. 1-12, Jan. 1992. [6] J. K. Jao, A matched array beamforming technique for low angle radar tracking in multipath, IEEE National Radar Conference, Mar. 1994, pp. 171-176.