Polarization Estimation With a Dipole-Dipole Pair, a ... - PolyU - EIE

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 5, MAY 2012

Polarization Estimation With a Dipole-Dipole Pair, a Dipole-Loop Pair, or a Loop-Loop Pair of Various Orientations Xin Yuan, Student Member, IEEE, Kainam Thomas Wong, Senior Member, IEEE, and Keshav Agrawal

Manuscript received July 21, 2011; revised October 02, 2011; accepted November 11, 2011. Date of publication March 02, 2012; date of current version May 01, 2012. This work was supported by the “Internal Competitive Research Grant” number G-YG38, awarded by the Hong Kong Polytechnic University. X. Yuan and K. T. Wang are with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]). K. Agrawal is with the Department of Electrical Engineering, University of California, Los Angeles, CA 90024 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2012.2189740

among these 15 configurations, even though the effectiveness of polarimetry depends critically on what and how the antennas are employed. This literature gap is filled by the present work. Such dipole/dipole, loop/loop, or dipole/loop antenna-pairs have been much investigated in the open literature: 1) One dipole and one loop, collocated and identically oriented, are labeled a “cocentered orthogonal loop and dipole” (COLD) array. Its physical implementation and electromagnetics are discussed in [20], [24], [43], [49], [52], and [53]. It is used to estimate the arrival-angles and/or the polarization in [14], [39], [48], [57], [63], [64], [68], [76], and [78]: a) specifically in the vertical orientation in [14], [39], [57], [64], [68] (with the corresponding Cramér–Rao bounds available in [63]); b) specifically in the horizontal orientation in [54].2 2) Two identical dipoles: a) When both horizontally but orthogonally oriented in spatial collocation, they have been used as a unit for the estimation of the arrival-angles and/or the polarization in [3]–[5], [7], [11]–[13], [15], [18], [21], [26], [28], [32], [45], [51], [54]–[56], [59], [62], [67], [72], and [74]. b) With one vertical and one horizontal, they are similarly used in [16], [22], [23], [30], [46], [58], and [73]. c) Other orthogonally oriented dipole-pairs are similarly studied in [6], [8], [10], [19], [29], [30], [41], [65], and [79]. Physical implementations of a dipole-pair are presented in [37], [40], [50], [60], [61], [66], and [75]. 3) Two identical loops: a) When horizontally and orthogonally oriented in spatial collocation, they have been used as a unit for the estimation of the arrival-angles and/or the polarization in [34], [38], [42], [44], and [70]. b) Other orthogonally oriented loop-pairs are similarly studied in [41] and [71]. Hardware implementations and electromagnetics are discussed in [1], [27], [31], [36], [69], and [77]. Closed-form estimation formulas are thus available in the open literature for polarization-parameters (with the incident source’s direction-of-arrival is already known) for three out of fifteen possible antenna/orientation configurations. This work,

1A dipole is considered “short,” if it is much shorter than a wavelength. A loop is considered “small,” if its circumference is below a quarter-wavelength.

2An orthogonally (not identically) oriented dipole/loop pair has its electromagnetics investigated in [17], [35], [47].

Abstract—This work aims to estimate the polarization of fully polarized sources, given prior knowledge of the incident sources’ azimuth-elevation directions-of-arrival, using a pair of diversely polarized antennas—two electrically small dipoles, or two small loops, or one each. The pair may be collocated, or spatially separated by a known displacement. Each antenna may orient along any Cartesian coordinate. Altogether, 15 antenna/orientation configurations are thus possible. For each configuration, this paper derives 1) the closed-form polarization-estimation formulas, 2) the associated Cramér–Rao bounds, and 3) the associated computational numerical stability. Index Terms—Antenna arrays, aperture antennas, array signal processing, parameter estimation, polarization estimation.

I. INTRODUCTION

P

OLARIMETRY is the measurement and the interpretation of the polarization of transverse waves [2]. The polarization state of an incident wave could reveal attributes intrinsic to the emitter or the reflector, e.g., a star’s photosphere asymmetry due to “limb polarization” [33]. Commonly used for polarization-estimation are electrically short dipoles and small loops.1 Such a dipole (loop), when oriented along a Cartesian coordinate, would measure the electric-field (magnetic-field) component of the incident transverse wavefield along that axis. Polarization is bivariate; hence, a minimum of two diversely polarized antennas are needed for polarimetry of a fully polarized wavefield. If limiting the choice of antennas to linearly polarized antennas (i.e., dipole(s) or/and loop(s)) aligned along some Cartesian axes, there exist

possible antenna/

orientation configurations, because two components are here chosen, out of a total of six electromagnetic components. The open literature presently offers no comprehensive comparison

0018-926X/$31.00 © 2012 IEEE

YUAN et al.: POLARIZATION ESTIMATION WITH A DIPOLE-DIPOLE PAIR, A DIPOLE-LOOP PAIR, OR A LOOP-LOOP PAIR OF VARIOUS ORIENTATIONS

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in Section III, will provide closed-form estimation formulas for the other twelve. Section IV will derive and will compare the corresponding Cramér–Rao lower bounds in polarization-estimation for these 15 antenna/orientation configurations. Lastly, Section V will contrast these 15 configurations’ intrinsic numerical stability, with respect to the matrix-inversion implicitly involved in the polarization estimation, given prior knowledge of the direction-of-arrival.

would exist between these two antennas. Each of these 15 configurations would have a 2 1 array-manifold:

II. ANTENNA-PAIR’S ARRAY MANIFOLD

The question is: which of these fifteen antenna-pair options would be “best,” to estimate and unambiguously. This work will investigate this issue, with regard to the numerical stability, the estimation accuracy, and the estimation validity-region of each antenna/orientation configuration. Note that a number of relationships exist among the array manifolds of these 15 configurations:

A fully polarized transverse electromagnetic wave is characterized by its six-component electromagnetic-field vector [(2. 5)–(2.8) and (2.11)–(2.12) in [9] with the noise there set to zero]:

(2)

(1)

where signifies the emitter’s elevation-angle denotes measured from the positive -axis, the corresponding azimuth-angle measured from the posrefers to the auxiliary polaritive -axis, symbolizes the polarization ization angle, phase difference, , and the superscript denotes transdepends on only the incident source’s position. Note that direction-of-arrival (DOA) and enjoys a block-radialoid structure, whereas depends on only the sources’ polarization state. If the receive-antennas are either dipoles and/or loops, and if these receive-antennas orient along some Cartesian axes, then the array manifold would equal the two corresponding components of the six-element vector in (1). Thus, different antenna/orientation configurations are possible for such an antenna-pair. For this selection of 2 out of 6 elements, it may be represented by a 2 6 selection-matrix , which has a “1” on each row, but zeroes elsewhere. All subsequent analysis will allow these two receive-antennas to be spatially separated or collocated—the latter obviously represents a degenerate case of the former. More mathematically, let one antenna be located at the Cartesian origin, , with the other antenna at a known location without loss of generality. Then, a spatial phase-factor of

III. POLARIZATION-ESTIMATION FORMULAS FOR ALL 15 ANTENNA/ORIENTATION CONFIGURATIONS Eigen-based parameter-estimation algorithms typically involve an intermediate step, that estimates each incident source’s steering vector, correct to within an unknown complex-value scalar . That is, available (for each incident source)3 is the estimate , from which and are to be estimated. (This approximation becomes equality in noiseless or asymptotic cases.) Hence, there exist two equations and two unknowns. Algebraic manipulation of the two equations yields the estimation formulas of and . For example, consider the pair (i.e., configuration 1A of Table I). The two equations are the two rows of

(3) 3This does NOT presume only a single source impinging upon the receiver. There could be multiple sources. Moreover, these several sources could possibly be cross-correlated, broadband, and/or time-varying.

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TABLE I POLARIZATION-ESTIMATION FORMULAS OF THE 15 ANTENNA/ORIENTATION CONFIGURATIONS

The two unknowns are and , with , and (thus ) already known.4 Table I lists the estimation formulas of and , for each of the 15 antenna/orientation configurations.5 There, . These estimation-formulas are applicable for the entire vaand . These lidity-region of estimation-formulas are new to the open literature, to the best knowledge of the present authors, except those few antenna/orientation configurations with references cited in Table I. It is noteworthy that the vertically oriented “OLD array” (i.e., configuration) has estimation-formulas indepenthe dent of the source’s direction-of-arrival; hence, and there require no prior information of and . This independence applies to no other antenna/orientation configuration. This vertically oriented “OLD array” is unique in this regard, because: 1) Its array manifold is independent of the azimuth-angle . 2) Though its array manifold depends on the elevation-angle through (and only through) , this factor is common to both entries in the array manifold. Hence, the ratio between these two entries would be independent of . AND CRB , FOR ALL IV. CRAMÉR–RAO BOUNDS, CRB 15 ANTENNA/ORIENTATION CONFIGURATIONS OF SECTION III

To avoid unnecessary distraction from the present objective to compare among the fifteen antenna/orientation configura-

1 = 1 = 1 = 0 here, (3) gives the “COLD” array oriented along

4If the x-axis.

5To facilitate the subsequent exposition, the orthogonal-loop-and-dipole (OLD) configurations of 1A, 1B and 1C in Table I will be identified as the “OLD array.”

tions, a very simple statistical data model will be used here for the Cramér–Rao bound derivation: the received signal is a pure tone at unity-power, a known frequency of , and a known initial phase of . At the th time-instant of , the antenna-pair collects the 2 1 data-vector, (4) refers to the time-sampling period, denotes a where 2 1 vector of spatio-temporally uncorrelated zero-mean complex-value Gaussian additive noise, with an unknown determin, where repreistic covariance-matrix of senting the known noise-variance at each antenna. With number of time-samples, the collected data-set equals

(5)

symbolizes the where noise vector with a Kronecker product, represents a , and spatio-temporal covariance matrix of denotes an identity matrix. Therefore, , i.e., a complex-value Gaussian vector with mean and covariance . The to-be-estimated and are modeled as deterministic. Collect all deterministic unknown entities into the 2 1 vector


of (FIM)

. The resulting 2

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2 Fisher information matrix

(6) has its

th entry equal to (see [25, (8.34)]): (17)

(7) where

denotes the real-value part of the entity inside represents the trace operator, and (8) (9) (10) (11)

where

. The elements of the FIM equal

(12)

(13)

(14)

(15) where , and , with symbolizing the th entry of the matrix in . The polarization-estimation Cramér–Rao bounds equal

(16)

Tables II lists the closed-form formulas for CRB and CRB , explicitly in terms of the data-model parameters. These Cramér–Rao bounds would be unchanged by any prior known spatial separation between the two receive-antennas, and these Cramér–Rao bounds are independent also of the value of the frequency, if prior known. This is because the values of the spatial separation and of the frequency would in , which is canceled in the course of affect only calculating the FIM. Fig. 1(a)–(c) plots for the three “OLD arrays” of configurations and CRB 1A-1C of Table I. Fig. 2(a) and (b) plots CRB of the horizontally oriented dipole-pair (i.e., configuration 2A). Fig. 2(c) and (d) does the same for the horizontally loop-pair (i.e., configuration 2B). Each remaining configuration has and dependent on three or more independent variables, thus not fully representable by any three-dimensional graph. Below are some qualitative observations on these derived Cramér–Rao bounds. 1) The horizontally oriented “OLD arrays” ( of configuration 1A, and of configuration 1B) offer finite Cramér-Rao bounds for sources impinging not horizontally. Please see Figs. 1(a) and (b). In contrast, the vertically oriented “OLD array” ( of configuration 1C) offers finite Cramér-Rao bounds for sources impinging horizontally. Please see Fig. 1(c). 2) The horizontally oriented “OLD arrays” ( of configuration 1A, and of configuration 1B) has a independent of the polarization parameters of and . The vertically oriented “OLD array” ( of configuration 1C) has both and independent of the azimuth-angle . 3) The Cramér-Rao bounds of the “OLD array” are related to the Cramér–Rao bounds of the other horizontally oriented “OLD array” of , by substituting by . This is because . Similar relationships exist between configurations 3A and 4A, between configurations 3B and 4B, between configurations 6A and 7A, and between configurations 6B and 7B. 4) For configurations 3A-7B, both and depend on the direction-of-arrival (i.e., and ) and the polarization (i.e., and ). Other dependencies are listed in Table III. 5) as for all 15 configurations.

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TABLE II CRAMÉR–RAO BOUNDS FOR POLARIZATION-ESTIMATES FOR THE 15 ANTENNA/ORIENTATION CONFIGURATIONS

V. CONDITION NUMBERS ASSOCIATED WITH THE ESTIMATION-FORMULAS OF SECTION III The estimate

denotes the 2 1 data collected at the time-instant where . The above requires an inversion of the 2 2 matrix .6 The numerical robustness in computing thus depends of , where on the condition number denotes the largest-magnitude eigenvalue of the matrix

could be construed as (18)

6If g were estimated without the above matrix inversion, this condition-number analysis would be irrelevant.


e ;h e ;h e ;h

M = M= M=

M=

M=

M=

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Fig. 1. (a) (2 ) ( )CRB( ) = (sin ) (2 ) ( )CRB( ) plotted versus and , for configuration 1A (i.e., the horizontally oriented “OLD” array, f g). (b) (2 ) ( )CRB( ) = (sin ) (2 ) ( )CRB( ) plotted versus and , for configuration 1B (i.e., the horizontally oriented “OLD” array, f g). (c) (2 ) ( )CRB( ) = (sin ) (2 ) ( )CRB( ) plotted versus and , for configuration 1C (i.e., the vertically oriented “OLD” array, f g).

, and symbolizes the smallest-magnitude eigenvalue. If approaches singularity, would approach infinity. The numerically most stable matrix-inversion corresponds to . Table IV lists the condition number for each of the 15 different composition/orientation configurations, derived by depends only on is the present authors. Because independent of and , but depends only on and . is attained by only the three “OLD array” The ideal , and . compositions/orientations of For the other 12 configurations (i.e., configurations 2A-2B to 7A-7B), Fig. 3(a)–(f) plots their condition numbers versus and versus . Each figure shows the values of and where . These are summarized in the two right-most columns of Table IV.

Below are some qualitative observations: 6) The condition numbers are all independent of the spatial separation between the two component-antennas, and independent of the frequency. This is because the matrix always has a unity condition number, regardless of the aforemenis unaffected by these entities. tioned entities. Hence, 7) For and only for the three “OLD array” configurations (i.e., a dipole and a loop in an identical orientation), . The reason is as follows: From (1) (19) with referring to the th entry of the vector inside gives the square brackets. The radialoid structure of

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M =

e ;e M= h ;h

e ;e

M )=( )CRB() plotted versus and , for configuration h ;h g). (d) (2M )=( )CRB() plotted versus and , for

Fig. 2. (a) (2 ) ( )CRB( ) plotted versus and , for configuration 2A (i.e., f g). (b) (2 2A (i.e., f g). (c) (2 ) ( )CRB( ) plotted versus and , for configuration 2B (i.e., f g). configuration 2B (i.e., f

, where symbolizes a 2 2 identity matrix. Hence, the condition number equals 1.7 8) For all the dipole-pair configurations and all the loop-pair configurations (i.e., configurations 2A-2B, 3A-3B, 4A-4B), each “A” configuration (such as 2A) can lead to the corresponding “B” configuration (such as 2B), by for the switching with . More precisely, with “A” configuration and with for the “B” config, where

uration, then

denotes

and are similar a 90 -rotation matrix. That is, matrices. Hence, configuration 2A must have the same 7Below is a more intuitive explanation: For a dipole and a loop, they would most “completely” measure the incident electromagnetic field, if the dipole and the loop are identically oriented. As an example, consider a vertically oriented dipole/loop pair, i.e., f g. The vertically oriented loop measures the -axis magnetic field, which contains partial information of the -axis and -axis components of the electrical field, complementary to the vertical dipole’s -axis electric field measurement. This complementarity (between the dipole’s measurement and the loop’s measurement) would disappear for any of the 12 non-“OLD-array” configurations.

z y

e ;h

z

h

x

condition number as configuration 2B. Similar reasoning applies between 3A and 3B, and between 4A and 4B. 9) For non-identically oriented dipole/loop antenna-pairs (i.e., configurations 5A-5B to 7A-7B), each “A” configuration (such as 5A) can lead to the corresponding “B” configuration (such as 5B), also by switching with . Consider configurations 5A and 5B, , where

denotes a reflection matrix. Hence,

and have the same condition number. Consequentially, configuration 5A must have the same condition number as configuration 5B, Similar reasoning applies between 6A and 6B, and between 7A and 7B. 10) The condition number of configurations 3A-3B, with a shift in , leads to the condition number of configurations 4A-4B. This is because the -axis in 3A-3B corresponds to the -axis in 4A-4B. Similarly, this holds between configurations 6A-6B on one hand and configurations 7A-7B on the other hand.


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TABLE III CHARACTERISTICS OF CRB( ) AND CRB( )

TABLE IV CONDITION-NUMBERS FOR TABLE I’S POLARIZATION-ESTIMATION FORMULAS

11) For each antenna/orientation configuration, the condition number is symmetric with respect to along along the -axis. the -axis, and with respect to

VI. CONCLUSION Investigated herein are all 15 possible antenna/orientation configurations that measure any 2 of the 6 components of the electromagnetic-field vector. Obtained for each antenna/orientation configuration are the closed-form polarization-estimation formulas, the corresponding validity region for unambiguous estimation, the corresponding Cramér–Rao bounds, and the

corresponding condition number as a function of the azimuth-elevation direction-of-arrival. The same derived results would apply, whether each pair is collocated or spatially separated. Among these 15 configurations, especially advantageous is the vertically oriented “OLD array” (i.e., the configuration), which requires no prior information of the source’s direction-of-arrival for polarization, but offers an ideal condition number of unity, and has a finite Cramér-Rao bounds except for near-vertically incident sources. For the horizontally configuration or the oriented “OLD arrays” (i.e., the configuration), they also offer the ideal condition number of unity, and have finite Cramér–Rao bounds except for near-horizontally incident sources.

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DS2

DS2

Fig. 3. (a) Condition number of for configuration 2A (i.e., fe ; e g) and the configuration 2B (i.e., fh ; h g). (b) The condition number of for for configuration 4A (i.e., fe ; e g) and the configconfiguration 3A (i.e., fe ; e g) and the configuration 3B (i.e., fh ; h g). (c) The condition number of uration 4B (i.e., fh ; h g). (d) The condition number of for configuration 5A (i.e., fe ; h g) and the configuration 5B (i.e., fh ; e g). (e) The condition number of for configuration 6A (i.e., fe ; h g) and the configuration 6B (i.e., fe ; h g). (f) The condition number of for configuration 7A (i.e., fe ; h g) and the configuration 7B (i.e., fe ; h g).

DS2

DS2

DS2

DS2


ACKNOWLEDGMENT The authors would like to thank Prof. L. Yeh for valuable discussions.

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Xin Yuan (SM’09) received the B.Eng. degree in electronic information engineering and the M.Eng. in information and communication engineering from Xidian University, Xi’an, China, in 2007 and 2009, respectively, and the Ph.D. degree in electronic and information engineering from the Hong Kong Polytechnic University, Hong Kong, in 2012. His research interest lies in diversely polarized antenna-array signal processing.

Kainam Thomas Wong (SM’01) received the B.S.E. degree in chemical engineering from the University of California, Los Angeles, in 1985, the B.S.E.E. degree from the University of Colorado, Boulder, in 1987, the M.S.E.E. degree from Michigan State University, East Lansing, in 1990, and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1996. He was a Manufacturing Engineer at the General Motors Technical Center, Warren, MI, from 1990 to 1991 and a Senior Professional Staff Member in the Johns Hopkins University Applied Physics Laboratory, Laurel, MD, from 1996 to 1998. From 1998 to 2006, he had been a faculty member at Nanyang Technological University, Singapore, the Chinese University of Hong Kong, and the University of Waterloo, Waterloo, ON, Canada. Since 2006, he has been with the Hong Kong Polytechnic University as an Associate Professor. His research interest includes sensor-array signal processing and signal processing for communications. Dr. Wong was conferred the Premier’s Research Excellence Award by the Canadian province of Ontario in 2003. He was an Associate Editor for the IEEE SIGNAL PROCESSING LETTERS from 2006 to 2010, and Circuits, Systems, and Signal Processing from 2007 to 2009. He has been an associate editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY since 2007, and the IEEE TRANSACTIONS ON SIGNAL PROCESSING since 2008.

Keshav Agrawal received the B.Tech. degree in electrical engineering from the Indian Institute of Technology, Kanpur, India, in 2011. He is currently pursuing the M.S.E.E. degree in the Department of Electrical Engineering, University of California, Los Angeles. He was a Research Assistant at Hong Kong Polytechnic University in the Summer of 2011. His research interest includes diversely polarized antennaarray signal processing.