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Passive Direction Finding Using Airborne Vector Sensors in the Presence of Manifold Perturbations Hasan S. Mir and John D. Sahr
Abstract—This paper studies direction finding of electromagnetic energy sources using passive vector sensor arrays whose manifold is only nominally known. The problem of direction finding is studied in the context of an airborne array, thus facilitating the observation of a ground-based source from multiple look-angles. A calibration algorithm explicitly accounting for the polarization diverse nature of the vector sensor is developed. It is also shown that the direction finding performance provided by a conventional array calibration algorithm is greatly enhanced with usage of the proposed polarization diverse calibration algorithm. Index Terms—Array calibration, direction finding, vector sensor.
I. INTRODUCTION
T
HE physical application studied in this paper is the calibration and usage of airborne vector sensors for performing direction finding of electromagnetic energy sources. Conventional (nonpolarization diverse) sensor arrays consist of identical sensor element types. In contrast, a vector sensor is a polarization diverse array whose “vector” output is a measurement of multiple components of electromagnetic information. A typical full vector sensor consists of two orthogonal triads of dipole and loop antennas with the same phase center, as shown in Fig. 1. The angular resolution of an array is directly related to the size of its aperture. For airborne applications in which a sensor array is mounted on a small aircraft, the physical space available on the airframe is limited and the array aperture is restricted. Usage of a vector sensor for direction finding was first proposed in [1]. The rationale is that because a vector sensor uses multiple components of electromagnetic information, it can offer accurate source location estimates with a smaller aperture. Indeed, if a full vector sensor (which measures the complete electric and magnetic fields) with an arbitrarily small aperture provides sufficient sensitivity, the source location can be estimated by simply calculating the Poynting vector. Thus, vector sensors are useful in situations wherein the available physical space is constrained. A precise characterization of the sensor manifold is required in order to accurately resolve the location of a source. The sensor manifold can be theoretically determined by using knowledge of Manuscript received August 25, 2005; revised February 9, 2006. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Yuri I. Abramovich. This work was sponsored by the Department of Defense under Air Force Contract F19628-00-C-0002. H. S. Mir is with the Massachusetts Institute of Technology (MIT) Lincoln Laboratory, Lexington, MA 02420 USA (e-mail:
[email protected]). J. D. Sahr is with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSP.2006.882056
Fig. 1. A full vector sensor consists of two orthogonal triads. One triad consists of dipoles for measuring the electric field components (E ; E ; E ), while the other triad consists of loops for measuring the magnetic field components (H ; H ; H ).
various sensor array parameters such as the array geometry/configuration. In practice, however, various array anomalies such as sensor pattern variations, sensor coupling, and sensor to receiver electrical cable length differences, cause the actual array manifold to differ, often times significantly, from the theoretical array manifold. This discrepancy necessitates array calibration, for which several techniques have been described in the literature (see [2] and the references therein). Most of the available techniques for array calibration, however, are designed for a conventional sensor array. In this paper, a novel calibration algorithm that explicitly accounts for the polarization diverse nature of the vector sensor is developed. II. ORGANIZATION The remainder of this paper is organized as follows. Section III describes the scenario of interest; Section IV develops the signal model for a polarization diverse array; Section V formulates the proposed polarization diverse calibration algorithm; Section VI presents the simulation results and discussion of applying the calibration algorithm proposed in Section V. III. PROBLEM STATEMENT Consider the scenario of an airborne array on which a signal source can be received from multiple look angles , as
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Fig. 3. Trimmed vector sensors and an eight-channel aircraft configuration. Identical three-channel sensors are mounted on the wings, and a two-channel sensor is mounted on the tip.
Fig. 2. Airborne array scenario. The ground based source is located at a look angle with azimuth and elevation relative to the airborne array. The aircraft can fly around the source and observe it from various look angles.
shown in Fig. 2, where is the azimuth angle and is the elevation angle. In this paper, a high-SNR environment is assumed. The steering vectors can thus be estimated by a single received sample. For moderate-SNR environments, the steering vector can be estimated via signal averaging or as the maximal eigenvector of the received signal’s autocorrelation matrix. The goal is to use the measured steering vectors and corresponding source angle of arrival (AOA) characterized by , combined with knowledge of the array geometry, to perform calibration on the airborne array for the purpose of direction finding. In the following sections, a novel calibration algorithm designed specifically for a polarization diverse array is developed. This calibration algorithm is motivated by the observation that the polarization state is parameterized by a linear two-dimensional (2-D) subspace [3], and based on the approach to calibration used in [2] wherein a transfer-function technique is used to relate the modeled and measured steering vectors. IV. SIGNAL MODEL FORMULATION Initial studies with a vector sensor mounted on an aircraft [4] indicated that some elements of the vector sensor act as “feeds” and are strongly coupled to the airframe, reducing their usefulness for direction finding purposes. The proposed solution to the problem is to use a “trimmed” vector sensor employing only the elements with insignificant airframe interaction. Multiple trimmed vector sensors are sited at strategic locations on the airframe, providing increased aperture for accurate AOA estimates. The concept of distributed and trimmed vector sensors has previously been studied in [5]–[8].
A particular 8-channel trimmed vector sensor configuration studied in this paper is shown in Fig. 3. The two loop antennas measure the and components of the magnetic field, while the vertical dipole measures the component of the electric field. Because of aerodynamic issues and the varying sensor interaction at different locations on the airframe, the trimmed vector sensors on the aircraft are not all identical; the one mounted on the tip of the airframe lacks the vertical dipole. A. Signal Model It is assumed that the vector sensor array is in the far-field of a narrowband signal. Following [9], define the components of the electric and magnetic field received on the array as
(1) and are defined appropriately, and and are the ranges of the polarization angle and phase difference, respectively. It should be pointed out that the additional polarization state parameters and are now required to index the array manifold, along with the previously defined conventional AOA parameters and . However, knowledge of and is generally not of interest, and they are thus considered to be nuisance parameters. Suppose there are sensors of a particular type, and let be the corresponding sensor location matrix. Let be the appropriate row of corresponding to the particular field component that the sensors measure. Define a unit vector in the source direction where
(2)
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The plane wave (far-field) response is defined as
and elevation AOA , where . azimuth AOA The modeled steering vector is constructed based on knowledge of various parameters such as the nominal array geometry and source AOA. Define
(3) where is the signal wavelength. The response/steering vector of the vector sensor array is generated by concatenating the response of identical sensor types .. .
Modeled steering vectors and
(4) Measured array steering vectors
where (5) is the number of distinct field components being meaand sured, or equivalently, the number of different sensor types. The corresponds to the total number size of the vector of sensor elements (channels), and will be denoted by . For the trimmed vector sensor configuration shown in Fig. 3 measuring the three components , , and , the value of equals 3. Letting , , and represent the plane wave response for the sensors on the left wing, right wing, and tip of the aircraft, respectively, the trimmed vector sensor response becomes
for which indeed . For an additive noise scenario, the received signal on the trimmed vector sensor is thus expressed as
where and are vectors whose th elements are and , respectively. In the following development, conventional array calibration is introduced in order to augment its expansion into a calibration algorithm that explicitly accounts for the polarization diverse nature of the vector sensor. A. Conventional Array Calibration Algorithm Conventional array calibration seeks to compute a calibration matrix so that .A calibration matrix is sought such that when it operates on the modeled steering vector, the result is a “good” approximation of the measured steering vector. Unlike previous works, e.g., [10], is not restricted in this paper to be a diagonal matrix. This approach allows not only for more fitting coefficients, but also implicitly accounts for sensor coupling, which is always present for sensor elements with physical separation less than one wavelength [11]. Define (7) One approach to compute the calibration matrix least squares criterion
(6) where is the transmitted signal with power and is the additive noise with variance , which is assumed to be identical for each sensor element. While the hypothesis of equal noise power is certainly a simplifying assumption, it is a good hypothesis for systems in which the receivers have a low noise figure, as the variance of the noise figure amongst a set of such receivers will be small. Furthermore, as stated in Section III, the development in this paper is for an environment in which the SNR is fairly high-thus, it is expected that calibration errors will contribute far more to source AOA estimation errors than the presence of noise. V. ARRAY CALIBRATION Suppose an element sensor array observes a stationary, far-field, narrowband source at known and distinct look angles. Let and represent the modeled and measured array steering vector, respectively, for the source
is to use the (8)
where the subscript denotes the Frobenius norm. The optimal solution for the calibration matrix is computed as
Thus,
(9)
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Because from [12]
the optimal solution for the calibration matrix (9) reduces to
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accounts for, in a global sense, The full matrix angle-dependent sensor gains and various other sensor anomalies such as coupling. The natural extension then of (14) is and to absorb the angle-dependent terms in as well as the remaining manifold perturbaand tions, into corresponding full matrices , so that
(10)
(16)
The calibration matrix reduces the distance between the measured and modeled steering vectors, and is therefore useful in direction finding applications where the modeled steering vector is used to determine the source AOA. It is shown in [2] that this calibration algorithm (which does not assume polarization diversity of the array) yields noticeable performance gains when used with a conventional array. This calibration algorithm can also be applied to a polarization diverse array, with reasonable expectation for some performance gain. However, further performance improvement may be possible if a calibration algorithm explicitly accounting for the polarization diverse nature of the vector sensor is employed. This idea is now further explored.
where are the unknown calibration parameters. look angles, the calibration parameters will be comFor puted according to the cost function
B. Proposed Polarization Diverse Array Calibration Algorithm
and
(17) To solve for the unknown parameters, define
(18)
Consider two distinct signal polarization vectors (19)
(11) (12) and their respective vector sensor steering vectors, and . According to the observation of Ferrara and Parks [3], the steering vector for an arbitrary polarization at look angle may be expressed as
(13) where . Accordingly, (13) follows from the fact that the polarization state is parameterized by a linear 2-D subspace. and can be thought of as the linear combination Thus, coefficients which are needed to achieve a particular polarization state. From (5), (13) may be expressed as
Let (20) where denotes the Kronecker tensor product, and denotes the element-by-element Hadamard product [13]. Then (17) is equivalent to
(21) By setting to zero and solving for in a manner analogous to that used to solve (8), the optimal solution for the calibration matrices (in the least squares sense) is
(22)
(14) where and are diagonal matrices consisting of terms of the form . Now consider the case where manifold perturbations are present. Conventional array calibration as described in Section V-A approximates the measured steering vector through
(15)
To solve for the coefficients be rewritten as
and
, observe that (16) can
(23) The least squares solution for
and
as such is
(24)
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The optimization of the calibration parameters (17) can now be solved by iterating between (22) and (24). The optimization may , be initialized by selecting where the superscript represents the iteration number. Convergence is achieved for a tolerance level when , where denotes the Frobenius norm [13]. C. Usage in Direction Finding Algorithms It may appear that usage of (16) in a direction finding algorithm would require knowledge of and for every search angle, thereby doubling the dimensionality of the search space. Consider, however, a direction finding estimator of the form
(25) when is the Then measured steering vector is the beamsum estimator [14], and when is the matrix of eigenvectors corresponding to the noise subspace of the received signal is the MuSIC estimator [15]. When (25) is rewritten with the aid of (23) as
(26) it is recognized as a Rayleigh quotient, and the solution for is thus the extremal eigenvector of the matrix
(27) Thus, the existence of an analytical solution for dimensionality of the search space unchanged.
leaves the
VI. SIMULATION RESULTS A. Simulation Setup The trimmed vector sensor configuration in Fig. 3 is used for simulation studies. The aircraft geometry at 90 MHz is shown in Fig. 4. Sensor manifold perturbations are assumed to be caused by near-field scatterers local to the airframe. For every look angle , the measured steering vector is modeled as follows:
.. .
(28)
The model (28) consists of the direct path steering vector , and the multipath component which is modeled by the summation terms. The relative strength of the multipath component is controlled by the parameter , and denotes the number of scatterers. For simulation purposes, and . Observe that the term
Fig. 4. Small aircraft geometry at 90 MHz with three trimmed vector sensor sites (cf. Fig. 3).
in (28) is identical to (5), except that the polarization state vector is premultiplied by . The 2 2 matrix is a random scattering matrix which models the scatterer-induced transformation of polarization state. is the path length difference which causes a phase offset modeled by the term . For the study presented in this paper, simulations were performed using 200 calibration points randomly sampled over the set of data taken in 90 azimuth and 30 elevation sectors. The presented results are the average of 10 trials. For comparison purposes, direction finding performance is evaluated for the following scenarios. 1) Noise only: It is assumed that noise is the only impairment, and exact knowledge of the sensor manifold is available to perform AOA estimation. 2) No calibration: It is assumed that both noise and sensor manifold perturbations are present. Direction finding is performed using only knowledge of the modeled steering vector. 3) Conventional array calibration: This is the same as Scenario 2, except that direction finding is performed using conventional array calibration (8). 4) Polarization diverse array calibration: This is the same as Scenario 3, except that direction finding is performed using polarization diverse array calibration (21). Scenarios 2 through 4 are also evaluated for the noise-free case and with the usage of synthetic elements or “virtual” sensor elements. To understand synthetic elements, consider an element sensor array and the corresponding element modeled steering vector. Now suppose that an additional sensor elements are included in the modeled steering vector. The modeled
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steering vector now consists of the original element steering vector concatenated with an element vector whose elements are the modeled sensor response at the locations of the synthetic elements. In practice, the placement of synthetic elements is selected in an ad-hoc manner. For arrays in which certain elements are “missing,” a natural choice is to place the synthetic elements at the locations of the real missing elements (see [2] for further discussion on synthetic elements). In this paper, synthetic elements are chosen as the full vector sensor elements (Fig. 1) missing from the trimmed vector sensors. As such, because a full vector sensor consists of six elements, the eightchannel configuration of Fig. 3, which consists of three distinct subarrays, would have a modeled steering vector of length 18 when synthetic elements are used. Three metrics: the AOA estimate rmse, ground range rmse, and calibration residual are used to assess direction finding performance for the above scenarios. The root-mean-square error (rmse) of the AOA estimation error is measured as the rmse of the difference between the elements in the estimated and actual AOAs, so that
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steering vector, and is defined as (33) shown at the bottom of the page. Additionally, the AOA rmse will be compared against the Cramer–Rao lower bound (CRLB), which is derived in the following section. B. Cramer–Rao Lower Bound for a Distributed Vector Sensor Define the real parameter vector signal (6) becomes
. The received
(34) The autocorrelation of the received signal is
(35) Assuming that is a zero-mean Gaussian process uncorrelated with the signal, the Fisher Information Matrix for snapshots is given by [21]
(36)
(29) The CRLB is thus [21] and CRLB
(37) (30) The relevant derivatives are computed as for azimuth and elevation errors, respectively. Directly related to the metric (29) and (30) is that of the ground range rmse at altitude (38)
(31) (39) where (40) (32) The calibration residual is a scaled measure of the distance/ norm (or mean square error) between the measured and modeled
(41) (42)
(33)
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Fig. 5. AOA rmse dependence on SNR. Especially at high SNR, the calibration algorithms offer significantly lower rmses than the No Calibration case. However, because the calibration is not exact, the rmses will eventually plateau. The “Noise Only” case assumes perfect manifold knowledge, and thus continues to approach the CRLB.
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C. Discussion
Fig. 6. Calibration residuals for individual sources. The high calibration residual for the “No Calibration” scenario indicates a poor fit between the actual and assumed theoretical response, and accounts for the large AOA estimation errors in Fig. 7. Usage of synthetic elements yields calibration residuals for the least squares calibration algorithm similar to that of the polarization diverse calibration algorithm. The calibration residual of the polarization diverse calibration algorithm is further reduced by around 20 dB if synthetic elements are used.
Fig. 5 shows the AOA rmse dependence on the array signal-to-noise ratio (SNR), defined as SNR , where for this simulation . • It can be seen that the use of least squares calibration offers significant reductions in AOA rmse compared to the no calibration scenario. • For high SNR, the use of least squares calibration with synthetic elements further reduces the AOA rmse by approximately 8 dB. • The performance of polarization diverse calibration is similar to that of least squares calibration with synthetic elements, and significantly better than the least squares calibration algorithm. • Usage of the polarization diverse calibration algorithm with synthetic elements yields performance quite similar to that of the noise only scenario, indicating a very close match between the measured and calibrated steering vectors. • Note that for the noise only scenario, an exact characterization of the array manifold is available, and the AOA rmse continues to approach the CRLB as the SNR increases. However, for cases in which manifold perturbations are present, the AOA rmse eventually plateaus and remains nonzero, even at infinite SNR. The performance difference between the various calibration algorithms in Fig. 5 can be explained by Fig. 6. • The no calibration scenario has a very high calibration residual, around 0 dB, and hence the AOA rmse levels are very high. • The similarity in calibration residual level between least squares calibration with synthetic elements and polarization diverse calibration accounts for their similar AOA rmse curves.
• Observe that usage of synthetic elements with a calibration algorithm reduces the calibration residual by approximately 20 dB. This translates into a significantly lower AOA rmse, as depicted by Fig. 5. Fig. 7 makes apparent that the use of polarization diverse calibration offers a significant performance advantage over the use of conventional least squares. • Because of the significant difference between the actual and modeled array manifold (as shown in the no calibration scenario in Fig. 6), the AOA errors with no calibration are extremely large compared to the initial estimates. • Using conventional least squares alone significantly reduces the AOA errors, but its performance pales in comparison to polarization diverse calibration, and even least squares calibration with synthetic elements. This is further corroborated in the example direction finding spectra in Fig. 8. The diamond in Fig. 8 represents the true peak location. • That the use of calibration significantly helps in reducing the width of the peak, as compared to the no calibration scenario which has a very spread out peak, is evident in both figures. • In fact, the calibration in many cases performs so well, that the resulting peaks are hardly observable. Table I offers a quantitative algorithm performance comparison. • Perhaps the most informative column is that of the ground range rmse, . From an initial guess error of 502 m, is reduced from over 2 km for the uncalibrated scenario, down to less than 1 m for polarization diverse calibration with synthetic elements.
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Fig. 7. Source AOA estimation errors. The polarization diverse calibration algorithm performs extremely well. When used with synthetic elements, the resulting AOA estimation errors are concentrated around the origin.
TABLE I PERFORMANCE OF CALIBRATION ALGORITHMS FOR A POLARIZATION-DIVERSE ARRAY. d VALUES CORRESPOND TO h = 3000 m
the source of a performance bottleneck when used in an actual system.
VII. CONCLUSION Vector sensors are useful in applications where limited physical space necessitates sensor arrays with reduced aperture. Direction finding performance in the presence of sensor manifold perturbations was assessed for a configuration of multiple trimmed vector sensors applicable to a small aircraft. A novel polarization diverse calibration algorithm was developed and shown to yield a very significant performance improvement over calibration algorithms for conventional arrays. It was also shown that additional performance improvement can be achieved by incorporating synthetic elements. Fig. 8. Example direction finding spectra. The diamond represents the true source location. The peak width when using a calibration algorithm is significantly narrower and closer to the true source location than the No Calibration case.
• It should be emphasized at this point that the results presented in this paper use simulated, rather than experimental data, and may thus be somewhat optimistic. • What the simulation results do show nonetheless is that a significant performance gain over conventional array calibration algorithms is possible with usage of the proposed polarization diverse calibration algorithm. • Furthermore, the promising results of the polarization diverse calibration algorithm indicates that it should not be
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[7] K. T. Wong, L. Li, and M. D. Zoltowski, “Root-MUSIC-based direction-finding and polarization-estimation using diversely-polarized possibly-collocated antennas,” IEEE Antennas Wireless Propagat. Lett., vol. 3, pp. 129–132, 2004. [8] K. T. Wong and A. K.-Y. Lai, “Inexpensive upgrade of base-station dumb-antennas by two magnetic loops for ‘blind’ adaptive downlink beamforming,” IEEE Antennas Propagat. Mag., vol. 47, pp. 189–193, February 2005. [9] K. T. Wong and M. D. Zoltowski, “Closed-form direction finding and polarization estimation with arbitrarily spaced electromagnetic vectorsensors at unknown locations,” IEEE Antennas Propagat., vol. 48, pp. 671–681, May 2000. [10] G. H. Whipple, C. M. Keller, and K. W. Forsythe, “Performance characterization of external array self calibration algorithms using experimental data,” in Proc. 34th Asilomar Conf. Signal, Syst. Comput., Oct. 2000, pp. 623–628. [11] W. Stutzman and G. Thiele, Antenna Theory and Design. New York: Wiley, 1997. [12] L. Trefethen and D. Bau, Numerical Linear Algebra. Singapore: SIAM, 1997. [13] T. Moon and W. Stirling, Mathematical Methods and Algorithms for Signal Processing. Upper Saddle River, NJ: Prentice–Hall. [14] H. Mir et al., “Self-calibration of an airborne array,” in Proc. 38th Asilomar Conf. Signal, Syst. Comput., Nov. 2004, pp. 2340–2344. [15] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Antennas Propagat., vol. 34, pp. 276–280, Mar. 1986. [16] J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J., vol. 7, pp. 308–313, 1965.
[17] R. Fletcher, Practical Methods of Optimization. New York: Wiley, 1980. [18] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes in C. Cambridge, U.K.: Cambridge University Press, 1988. [19] S. Haykin, Adaptive Filter Theory. Upper Saddle River, NJ: Prentice Hall, 1996. [20] D. Manolakis, V. Ingle, and S. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering and Array Processing. New York: McGraw-Hill, 2000. [21] H. Van Trees, Optimum Array Processing (Detection, Estimation, and Modulation Theory, Part IV). New York: Wiley-Interscience, 2002. Hasan S. Mir was born on June 27, 1982. He received the B.S. (cum laude), M.S., and Ph.D. degrees in electrical engineering from the University of Washington, Seattle, in 2000, 2001, and 2005 respectively. Currently, he is a member of the Air Defense Techniques group at MIT Lincoln Laboratory. His research interests are in array signal processing. Dr. Mir was the recipient of the MIT Lincoln Laboratory Ph.D. fellowship.
John D. Sahr, photograph and biography not available at the time of publication.
