Low Rank Tensor STAP filter based on multilinear SVD Maxime Boizard *† , Guillaume Ginolhac * , Frederic Pascal † and Philippe Forster * * SATIE, ENS Cachan 61, av President Wilson F-94230 Cachan, France † SONDRA, Supelec, 3 rue Joliot-Curie F-91192 Gif-sur-Yvette, France Email: maxime.boizard,guillaume.ginolhac,philippe.forster@satie.ens-cachan.fr, frederic.pascal@supelec.fr Abstract—Space Time Adaptive Processing (STAP) is a two- dimensional adaptive filtering technique which uses jointly tem- poral and spatial dimensions to suppress disturbance and to improve target detection. Disturbance contains both the clutter arriving from signal backscattering of the ground and the thermal noise resulting from the sensors noise. In practical cases, the STAP clutter can be considered to have a low rank structure. Using this assumption, a low rank vector STAP filter is derived based on the projector onto the clutter subspace. With new STAP applications like MIMO STAP or polarimetric STAP, the generalization of the classic filters to multidimensional configurations arises. A possible solution consists in keeping the multidimensional structure and in extending the classic filters with multilinear algebra. Using the low-rank structure of the clutter, we propose in this paper a new low-rank tensor STAP filter based on a generalization of the Higher Order Singular Value Decomposition (HOSVD) in order to use at the same time the simple (for example time, spatial, polarimetric, ...) and the combined information (for example spatio-temporal). Results are shown for two cases : classic 2D STAP and 3D polarimetric STAP. In the classic case, vector and tensor filters are equivalent. In the polarimetric case, we show the enhancement of the tensor filter. I. I NTRODUCTION Space Time Adaptive Processing (STAP) is a technique used in airborne phased array radar to detect moving target embedded in an interference background such as jamming or strong clutter [1]. While conventional radars are capable of detecting targets both in the time domain related to target range and in the frequency domain related to target velocity, STAP uses an additional domain (space) related to the target angular localization. The consequence is a two-dimensional adaptive filtering technique which uses jointly temporal and spatial dimensions to suppress disturbances and to improve target detection. The disturbance contains both the clutter arriving from signal backscattering of the ground and the thermal noise resulting from the sensors noise. From the Brennan’s rule formula [2], the STAP clutter can be considered to have a low rank structure. Using this assumption, a low rank vector STAP filter is derived [3], [4] based on the projector onto the orthogonal of the clutter subspace. The STAP data are obtained as a data cube, but they are usually regrouped as vectors to use the space-time infor- mation. With new STAP applications like MIMO (Multiple Input Multiple Output) STAP or polarimetric STAP [5], the generalization of the classic filters to multidimensional con- figurations arises. A possible solution consists in keeping the multidimensional structure and in extending the classic filters with multilinear algebra [6], [7] as it is done in [8]. Using the low-rank structure of the clutter, the HOSVD [9] (High Order Singular Value Decomposition) seems natural to find the tensor projector of the clutter subspace. However, we see in this paper this decomposition is not appropriate because the spatio-temporal dimension is not considered (only the spatial and temporal dimensions separately are seen by HOSVD). We propose a generalization of the HOSVD, which keep the information contained in HOSVD but also add the combined information as spatio-temporal information. To estimate the clutter subspace, we work with a direct data approach instead of the usual covariance approach (both approaches are equiva- lent). Preliminary results are shown for two cases : classic 2D STAP and 3D polarimetric STAP. In the classic case, vector and tensor filters are equivalent. In the polarimetric case, we show the enhancement of the tensor filter. The following convention is adopted: scalars are denoted as italic letters, vectors as lower-case bold-face letters, matrices as bold-face capitals, and tensors are written as bold-face calligraphic letters. We use the superscripts H , for Hermitian transposition and ∗ , for complex conjugation. II. SOME MULTILINEAR ALGEBRA TOOLS This section contains the main multilinear algebra tools used in this paper. Let A, B ∈ C I1×I2×I3 , two 3-dimensional tensors and let a i1i2i3 , b i1i2i3 their elements. We will use the following operators; for more details, especially the case of n-order tensors, we refer the reader to [6], [9]. A. Unfolding Let us start with 2 unfolding operators, which arrange the elements of a tensor in a matrix or a vector: • vector: vec transforms a tensor A into a vector, vec(A) ∈ C I1I2I3 . vec −1 is the inverse operator. • matrix: this operator transforms the tensor A into a matrix. For example, [A] 1 ∈ C I1×I2I3 and [A] 1,2 ∈ C I1I2×I3 . B. Products • scalar product : < A, B >= ∑ i1 ∑ i2 ∑ i3 b ∗ i1i2i3 a i1i2i3 = vec(B) H vec(A)