I.J. Image, Graphics and Signal Processing, 2013, 11, 35-45 Published Online September 2013 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijigsp.2013.11.04 Copyright © 2013 MECS I.J. Image, Graphics and Signal Processing, 2013, 11, 35-45 A Survey on PARATREE and SUSVD Decomposition Techniques and Their Use in Array Signal Processing Vineet Bhatt Department of Mathematics, HNB Garhwal University, Campus Badshahi Thaul, Tehri Garhwal-249199, Uttarakhand, India vineet.bhatt58@gmail.com Sandeep Kumar* Department of Mathematics, Govt. P.G. College, New Tehri, Tehri Garhwal, Pin: 249 001, Uttarakhand, India drsandeepkumarmath@hotmail.com Abstract —The present manuscript is intended to review few applications of tensor decomposition model in array signal processing. Tensor decomposition models like HOSVD, SVD and PARAFAC are useful in signal processing. In this paper we shall use higher order tensor decomposition in signal processing. Also, a novel orthogonal non-iterative tensor decomposition technique (SUSVD), which is scalable to arbitrary high dimensional tensor, has been applied in MIMO channel estimation. The SUSVD provides a tensor model with hierarchical tree structure between the factors in different dimensions. We shall use a new model known as PARATREE, which is related to PARAFAC tensor models. The PARAFAC and PARATREE both describe a tensor as a sum of rank-1 tensors, but PARATREE has several advantages over PARAFAC, when it is applied as a lower rank approximation technique. PARATREE is orthogonal, fast and reliable to compute, and the order of the decomposition can be adaptively adjusted. The low rank PARATREE approximation has been applied to measure noise suppression for tensor valued MIMO channel sounding measurements. Index Terms —MIMO, SVD, PARATREE, SUSVD, tensor decompositions, signal processing I. INTRODUCTION A higher order tensor is any N-dimensional collection of data. It is generally known as tensor or a multidimensional array. Tensor decompositions and factorizations were initiated by Hitchcock in 1927[1], [2] and later developed by Cattell in 1944 [3] and by Tucker in 1966[4]. Tensor factorizations or decompositions play a fundamental role in enhancing the data and extracting latent components. In the era, tensor is used in a wide variety of applications such as in signal processing [5], data mining [6], neuroscience [7], and many more. In various signal processing applications, instrumental data contains information in more than two dimensions. Recently, researchers have contributed a large amount of research regarding several application areas of well- established matrix operations up to their tensor equivalents. Unfortunately, these extensions from their matrix counterparts are not trivial. For example the SVD has proven to be a powerful tool for analyzing matrix or 2nd-order tensors, its generalization to higher order tensors is not straightforward. There are several approaches for doing this, and none of them is superior in all aspects. Basically there are three fundamental approaches to decompose a higher order tensor, first one is Tucker model (multi-linear SVD or HOSVD) [3], second is CANDECOMP/PARAFAC [8] and the third approach is non- negative tensor factorization. Both the CP and Tucker tensor decomposition can be considered as higher-order generalization of the matrix SVD and PCA respectively. If any tensor decomposes into sum of rank-1 tensor, this type of decomposition is often called “Canonical Decomposition” (CANDECOMP) or “Parallel Factors” model (PARAFAC) [8]. It has been applied in many signal processing applications, such as image recognition, acoustics, wireless channel estimation [9] and array signal processing [10], [11]. Recently, a Tucker-model based HOSVD [12] tensor decomposition subspace technique has also been formulated to improve multidimensional harmonic retrieval problems [13]. In this paper, we are attempting to pursue the contribution of higher order tensor decomposition in signal processing. In signal processing, data obtained from MIMO channel sounding measurements is a good example of tensor-valued data. It is well known that a PARATREE model is an enhanced version of PARAFAC model. Also the PARATREE tensor model is useful in signal processing; it is applied to suppress measurement noise in multidimensional MIMO radio channel measurements. This is Performed by identifying the PARATREE components spanning the noise subspace, and removing their contribution from the channel observation. Therefore in subsection 3 of section II, a novel PARATREE tensor model has been introduced, which is accompanied with SUSVD algorithm. As the rank-1