IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007 5337 Higher Order Direction Finding From Arrays With Diversely Polarized Antennas: The PD-2q-MUSIC Algorithms Pascal Chevalier, Anne Ferréol, Laurent Albera, and Gwénaël Birot Abstract—Fourth-order (FO) and, a short while ago, th-order, , high-resolution methods exploiting the information con- tained in the FO and the th-order, , statistics of the data, respectively, are now available for direction finding of non-Gaussian signals. Among these methods, the -MUSIC methods, , are the most popular. These methods are asymp- totically robust to a Gaussian background noise whose spatial coherence is unknown and offer increasing resolution and ro- bustness to modeling errors jointly with an increasing processing capacity as increases. However, these methods have been mainly developed for arrays with identical sensors only and cannot put up with arrays of diversely polarized sensors in the presence of diversely polarized sources. In this context, the purpose of this paper is to introduce, for arbitrary values of , , three extensions of the -MUSIC method, able to put up with arrays having diversely polarized sensors for diversely polarized sources. This gives rise to the so-called polarization diversity -MUSIC (PD- -MUSIC) algorithms. For a given value of , these algo- rithms are shown to increase the resolution, the robustness to modeling errors, and the processing capacity of the -MUSIC method in the presence of diversely polarized sources. Besides, some PD- -MUSIC algorithms are shown to offer increasing performances with when resolution in both direction of arrival and polarization is required. Index Terms—Direction finding (DF), direction of arrival (DOA), higher order, identifiability, polarization diversity, under- determined mixtures, virtual array (VA), 2q-MUSIC. I. INTRODUCTION D URING the last two decades, fourth-order (FO) direction finding (DF) methods [1], [8], [28], [31], exploiting the information contained in the FO statistics of the observations, have been developed for non-Gaussian signals. Among these methods, the FO extension of the well-known MUSIC method [30], called 4-MUSIC [28], is the most popular. These methods are asymptotically robust to a Gaussian noise whose spatial co- herence is unknown and generate a virtual increase of both the Manuscript received November 21, 2006; revised March 26, 2007. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Dr. Andreas Jakobsson. P. Chevalier and A. Ferréol are with the Thales-Communications, 92704 Colombes Cedex, France (e-mail: pascal.chevalier@fr.thalesgroup.com; anne.ferreol@fr.thalesgroup.com). L. Albera and G. Birot are with the INSERM U642 LTSI, Université Rennes 1, 35042 Rennes Cedex, France (e-mail: laurent.albera@univ-rennes1.fr; gwenael.birot@univ-rennes1.fr). 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/TSP.2007.899367 number of sensors and the effective aperture of the considered array [4], [10]. This introduces the FO virtual array (VA) con- cept presented in [10] and [4]. A consequence of this property is that, despite of their higher variance [2], FO DF methods allow for the processing of more sources than sensors and an increase of both the resolution and, at least for several poorly angularly separated sources, the robustness to modeling errors of second-order (SO) methods [6]. To still increase the resolu- tion power of DF methods, their robustness to modeling errors and the number of sources to be processed from a given array of sensors, while keeping their robustness to a Gaussian back- ground noise whose spatial coherence is unknown, the MUSIC method has been extended recently [6] to an arbitrary even order , . This gives rise to the so-called -MUSIC method, which exploits the information contained in the th-order sta- tistics of the observations. This method is shown in [6] to have resolution, robustness to modeling errors (for several poorly an- gularly separated sources), and processing capacity increasing with . These results are directly related to the higher order extension, presented in [3], of the FO VA concept. This con- cept allows to explain why, despite of their higher variance, -MUSIC methods with may offer better performances than 2-MUSIC or 4-MUSIC methods when some resolution is required. This is, in particular, the case in the presence of several sources, when the latter are poorly angularly separated or in the presence of modeling errors inherent in operational contexts. However, both 4-MUSIC [28] and -MUSIC, [6] algorithms have been mainly developed for arrays with iden- tical sensors, and cannot put up, in the presence of arbitrary polarized sources, with arrays of diversely polarized sensors. The exploitation of arrays with diversely polarized sensors is very advantageous since for such arrays, multiple signals may be resolved on the basis of polarization as well as direction of arrival (DOA). This added information improves both DOA accuracy and resolution in general [14], [36], [38] and also increases robustness to modeling errors [15]. However, most of methods which are currently available for DF from arrays with diversely polarized sensors exploit only the information contained in the SO statistics of the observations. Among these SO methods, we find extensions to array with diversely polarized, and possibly collocated, sensors of SO methods such as MUSIC [11], [41], pencil-MUSIC [20], root-MUSIC [13], [37], [42], ESPRIT [21]–[25], [39], [40], [44], subspace fitting [32], MODE [26], and maximum likelihood [29], [43] methods, respectively. Note that a comparative performance analysis of MUSIC and pencil-MUSIC methods for such arrays 1053-587X/$25.00 © 2007 IEEE