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Improved Array Interpolation for Reduced Bias in DOA Estimation for GNSS

Array signal processing has been the focus of particular interest in recent decades. Recently antenna arrays were incorporate to GNSS receivers seeking to improve overall precision and robustness against interference as well as spoofing. In multi-antenna GNSS receivers, direction of arrival (DOA) estimation of the impinging signals onto the antenna array is an important processing step in order to introduce knowledge of the DOAs of the satellites into beamforming algorithms and to achieve interference and spoofing detection. 

One form of array construction is of particular interest, the uniform array. In this type of array, the antennas with equivalent radiation characteristics are equidistant from its neighbours. Array manifold vectors of such arrays present a so-called Vandermonde structure. A Vandermonde or consequently a centro-hermitian structure allows the application of a wide range of array signal processing techniques for DOA estimation such as the Forward Backward Averaging (FBA) [1], Spatial Smoothing (SPS) [2] as well as Root-MUSIC [3] or polynomial rooting techniques. FBA and SPS are of particular interest for GNSS systems since they aim to “decorrelate” strongly correlated or even coherent signals, such as multipath or spoofing signal components, which are responsible for introducing significant biases in the final position estimation.

Unfortunately, due to mutual coupling of the single elements and imprecise antenna placement within the array, an array manifold resembling a uniform array with a Vandermonde structure is not always achievable. Thus, in real implementations the response of an imperfect array needs to be mapped into the response of a uniform array in order to apply the previously mentioned array signal techniques. This is known in the literature as array interpolation [4], [5], [6], [7], [8], [9], [10] and [11]. Array interpolation is usually done by calculating a transform matrix that shapes the response of the array accordingly. However, unless the system operates with a very large number of antennas, there usually are not enough degrees of freedom to properly shape the response over the entire array manifold. These limitations are usually addressed by calculating different transform matrices for each spatial response region or sector of the manifold. Even when the manifold is divided into narrow regions/sectors the transformation is not perfect and thus introduces a bias in the final DOA, time-delay and carrier phase estimations. Different methods for calculating these transformation matrices exist in the literature, the most popular method consists of a simple least squares fit of the real manifold into the manifold of the uniform array. This method, however, only takes into account the structure of the array itself and does not take into account the signal received over the manifold or the structure of the noise over the manifold.

In this work, we present an alternative way of calculating the transform matrices by taking into account the estimates of information on the signal received and the spatial structure of the noise during the measurements. The extra information introduced into the calculation of the results significantly decreased bias when performing DOA estimation over the transformed dataset even when containing highly correlated signals like multipath or spoofing. After applying the transform FBA and SPS can “decorrelate” multipath and spoofing signals and thus multipath mitigation and spoofing detection can be enhanced significantly. Our approach makes it possible to exploit the Vandermode or centro-hermitian structure of the interpolated array by low-cost high-resolution DOA estimation methods like Unitary ESPRIT [12].

References:
[1] S. Pillai and B. H. Kwon, “Forward/backward Spatial Smoothing Techniques for Coherent Signal Identification ,” IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 37, pp. 8–9, January 1989.
[2] J. E. Evans, J. R. Johnson, and D. F. Sun, “Application of advanced signal processing techniques to angle of arrival estimation in ATC navigation and surveillance system,” Massachusetts Institute of Technology, Tech. Rep., 1982.
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[8] Lau, G. Cook, and Y. Leung, “An improved array interpolation approach to DOA estimation in correlated signal environments,” in Acoustics, Speech, and Signal Processing, 2004. Proceedings. (ICASSP ’04). IEEE International Conference on, 2004.
[9] B. Lau, M. Viberg, and Y. Leung, “Data-adaptive array interpolation for DOA estimation in correlated signal environments,” in Acoustics, Speech, and Signal Processing, 2005. Proceedings. (ICASSP ’05). IEEE International Conference on, 2005.
[10] M. Buhren, M. Pesavento, and J. F. Bohme, “A new approach to array interpolation by generation of artificial shift invariances: interpolated ESPRIT,” in Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP ’03). 2003 IEEE International Conference on, 2003.
[11] A.Weiss and M. Gavish, “The interpolated ESPRIT algorithm for direction finding,” in Electrical and Electronics Engineers in Israel, 1991. Proceedings, 17th Convention of, 1991.
[12] M. Haardt and J. A. Nossek, “Unitary ESPRIT: how to obtain increased estimation accuracy with a reduced computational burden,” IEEE Transactions on Signal Processing, vol. 43, no. 5, May 1995.A