International Journal of Antennas and Propagation

Volume 2017 (2017), Article ID 1615962, 11 pages

https://doi.org/10.1155/2017/1615962

## Tensor-Based Methods for Blind Spatial Signature Estimation in Multidimensional Sensor Arrays

^{1}Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, CE, Brazil^{2}Department of Electrical Engineering, University of Brasília, DF, Brasília, Brazil^{3}Institute for Information Technology, Ilmenau University of Technology, Ilmenau, Germany^{4}Fraunhofer Institute for Integrated Circuits IIS, Erlangen, Germany

Correspondence should be addressed to Paulo R. B. Gomes

Received 10 September 2016; Accepted 15 January 2017; Published 15 February 2017

Academic Editor: Elias Aboutanios

Copyright © 2017 Paulo R. B. Gomes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The estimation of spatial signatures and spatial frequencies is crucial for several practical applications such as radar, sonar, and wireless communications. In this paper, we propose two generalized iterative estimation algorithms to the case in which a multidimensional (-D) sensor array is used at the receiver. The first tensor-based algorithm is an -D blind spatial signature estimator that operates in scenarios where the source’s covariance matrix is nondiagonal and unknown. The second tensor-based algorithm is formulated for the case in which the sources are uncorrelated and exploits the dual-symmetry of the covariance tensor. Additionally, a new tensor-based formulation is proposed for an -shaped array configuration. Simulation results show that our proposed schemes outperform the state-of-the-art matrix-based and tensor-based techniques.

#### 1. Introduction

High resolution parameter estimation plays a fundamental role in array signal processing and has practical applications in radar, sonar, mobile communications, and seismology. In light of this, several techniques have been developed to increase the accuracy of the estimated parameters, from which we may cite the classical* Multiple Signal Classification* (MUSIC) [1] and* Estimation of Signal Parameters via Rotational Invariance Technique* (ESPRIT) [2]. However, their performance can be further improved by exploiting the multidimensional structure of the data by means of tensor modeling, which can include several signal dimensions such as space, time, frequency, and polarization. Tensor decompositions have been successfully employed in array signal processing for parameters estimation since they provide better identifiability conditions when compared to conventional matrix-based methods. Another advantage of tensor-based methods is the so-called “tensor gain” which manifests itself with more precise parameter estimates due to the good noise rejection capability of tensor-based signal processing, as shown in [3–6].

In regards to tensor-based methods for blind spatial signatures estimation, the* Parallel Factor *(PARAFAC) analysis decomposition [7] is widely applied due to its well-defined conditions for uniqueness [8]. As seen in [9], an iterative technique for PARAFAC decomposition such as* Trilinear Alternating Least Squares* (TALS) can be applied to estimate the directions of arrival of the sources. Closed-form solutions such as the Standard Tensor ESPRIT (STE) [10] and Closed-Form PARAFAC [11] are also appealing, since these exploit the multidimensional structure in a noniterative fashion. Recently in [12], an iterative algorithm was proposed in a manner similar to Independent Component Analysis (ICA) based on the Orthogonal Procrustes Problem (OPP) and Khatri-Rao factorization [13] for a PARAFAC decomposition with dual-symmetry. This solution exploits the dual-symmetry property of the data tensor and can be applied in covariance-based array signal processing techniques. The method proposed in [14] is based on the Tucker decomposition [15] of a fourth-order covariance tensor and was elaborated for arrays with arbitrary structures, where a priori knowledge about the geometry of the sensor array is not required. However, a limitation of the method in [14] is the necessity of transmitting the same sequence of symbols in different time blocks, which results in a loss of spectral efficiency. The proposed solution is an algorithm for a multidimensional (-D) sensor array in which the different dimensions of the array are exploited, thus dismissing the need to transmit a repeated sequence as in [14], as will be detailed later.

In this paper, two tensor-based approaches to the estimation of spatial signatures are presented. By using the signals received on a -D sensor array, covariance tensors are calculated and solutions for correlated and uncorrelated sources are presented, respectively. For the former scenario, in which the source’s covariance structure is nondiagonal and unknown, the covariance tensor of the received data is formulated as a Tucker decomposition of order . Such a formulation yields a generalized Tucker model based -D sensor array processing that deals with arbitrary source covariance structures. By assuming uncorrelated sources, we then show that the problem boils down to a PARAFAC decomposition, from which a method that exploits the dual-symmetry property of the covariance tensor is derived by considering the ideas rooted in [12]. For both Tucker and PARAFAC based models, the blind estimation of the spatial signatures is achieved by means of an alternating least squares (ALS) algorithm. The contributions of this paper are twofold: (i) we propose a covariance-based generalization of the Tucker decomposition for the blind spatial signature estimation problem with -D sensor arrays and (ii) we establish a link between dual-symmetry decompositions and techniques based on covariance-based array signal processing for parameter estimation. The performance of the proposed algorithms is evaluated by Monte Carlo simulations, corroborating their gains over competing state-of-the-art matrix-based and tensor-based techniques.

The rest of this paper is organized as follows: Section 2 briefly introduces tensor operations and decompositions. The signal model for an -D sensor array is then presented in Section 3. In Section 4 a novel covariance-based tensor model for the received data is formulated and our blind spatial signature estimation algorithms are formulated. In Section 5 an approach for -shaped sensor arrays is proposed. The computational complexity of the proposed methods is analyzed in Section 6. In Section 7, the advantages and disadvantages of the proposed methods are discussed. Simulation results are provided in Section 8, and the conclusions are drawn in Section 9.

*Notation*. Scalar values are represented by lowercase letters , vectors by bold lowercase letters , matrices by bold uppercase letters , and tensors by calligraphic letters . The symbols , , , and represent the transpose, conjugate transpose, pseudoinverse, and complex conjugate operations, respectively. operator generates a diagonal matrix from a vector . The th row of is represented by , while its th column is represented by . operator converts into a vector , while converts into a matrix. stands for a diagonal matrix constructed from the th row of . stands for the Frobenius norm of a matrix or tensor. “” operator stands for the vector outer product. The Kronecker product is represented by . The Khatri-Rao product between the matrices and , represented by , is defined as

#### 2. Tensor Preliminaries

In the following, we briefly introduce for convenience the basics on operations involving tensors and tensor decompositions, which refer to [16, 17]. Firstly, we present the Tucker decomposition. Then, the PARAFAC decomposition is introduced and issues involving uniqueness are briefly discussed for both cases, which will be useful later. Then, we introduce the dual-symmetry property for these decompositions. The basic material presented in this section is exploited in later sections in the context of our blind spatial signature estimation problem.

##### 2.1. Basic Tensor Operations

Let denote an th order tensor, th entry of which is denoted by . The fibers are the higher-order analogues of matrix rows and columns. The -mode fibers of are vectors of size defined by fixing every index but . The -mode unfolding operation, denoted by , stands for the process of reordering the elements of into a matrix by arranging its -mode fibers to be the columns of the resulting matrix. The -mode product between and a matrix along of the th mode, denoted by , is a tensor of size , obtained by taking the inner product between each -mode fiber and the rows of the matrix ; that is, [16, 17]

##### 2.2. Tucker Decomposition

The Tucker decomposition [15] represents a tensor of order as a multilinear transformation of a core tensor by factor matrices along each mode . In scalar form, the th order Tucker decomposition is given bywhere is th entry of the th mode factor matrix and is th entry of the core tensor . Using -mode product notation, the Tucker decomposition can be written aswhich admits the following factorization in terms of the factor matrices and core tensor:

In general, the Tucker decomposition is not unique; that is, there are infinite solutions for , and that yield the same reconstructed version of the data tensor . However, in special cases where several elements of the core tensor are constrained to be equal to zero, that is, if the core tensor has some sparsity, the number of solutions may be finite, and the associated factor matrices and core tensor become unique up to trivial permutations and scaling ambiguities [18]. The Tucker based methods presented in Sections 4.2 and 5.1 belong to a special category where unique solutions exist.

##### 2.3. PARAFAC Decomposition

The PARAFAC decomposition [7] expresses a tensor as a sum of rank-one tensors; that is, where is the number of factors, also known as the* rank* of the decomposition, and is defined as the minimum number of rank-one tensors that yield exactly.

The th order PARAFAC decomposition (6) can be seen as a special case of the Tucker decomposition (4) with a core tensor and for . The elements of the th order identity tensor are equal to one when all indices are equal and zero elsewhere. Using the -mode product notation, the PARAFAC decomposition can be written aswhile the -mode unfolding of can be expressed as

The th order PARAFAC decomposition is unique up to permutation and scaling ambiguities affecting the columns of factors matrices , if the following sufficient condition is satisfied [19]:where denotes the Kruskal-rank of , defined as the maximum value such that any subset of columns is linearly independent [20].

Throughout this work, special attention is given to dual-symmetric tensors. The PARAFAC decomposition of a given tensor of order is said to have dual-symmetry if defined as follows:Note that this definition also applies to Tucker decomposition by simply replacing the identity tensor by an arbitrary core tensor of order .

#### 3. Signal Model

We consider snapshots originating from the superposition of far-field narrowband signal sources sampled by a -dimensional sensor array of size , where is the size of the th array dimension, . The matrix collects the samples received by the sensor array, which can be factored as [10] where (i) is the spatial signature matrix of the -D array for and ;(ii) is the spatial signature matrix of the th dimension;(iii) is the array response in the th dimension to the th planar wavefront () which is function of the spatial frequency ;(iv) is the matrix containing the signal transmitted by the sources;(v) is the additive white Gaussian noise (assumed uncorrelated to the source signals).From (11), the sample covariance matrix of the signals received at the sensor array is given by where is the sample covariance matrix of the source signals and is the noise variance.

#### 4. Tensor-Based Methods for Blind Spatial Signature Estimation

In this section, we propose two iterative algorithms to solve the blind spatial signature estimation problem in -D sensor arrays. Initially, a novel multidimensional structure is formulated from the covariance matrix of the received data. Then, an alternating least squares- (ALS-) based algorithm for a Tucker decomposition of order is proposed. Finally, we derive a link between the method in [12] and a covariance-based blind spatial signature estimation problem.

##### 4.1. Novel Covariance Tensor

With the intention of exploiting the multidimensional structure of the received signal, the noiseless sample covariance matrix (12), given by , is interpreted as a multimode unfolding of the noiseless covariance tensor of order , defined aswhere is the source covariance tensor, which has dimensions, each of size . Note that this tensor is dual-symmetric; that is, the factor matrix related to ()th dimension is equal to , and . The th frontal slice of is a diagonal matrix whose main diagonal is given by the th column of the covariance matrix . For instance, considering for the sake of notation, the following expression satisfies the relationship previously cited:where the matrix denotes the th frontal slice of the covariance tensor obtained by fixing its last two modes. The tensor follows a dual-symmetric Tucker decomposition of order with factor matrices and , and core tensor .

Considering the case in which the sources are uncorrelated and have unitary variance, we can rewrite (13) aswhere is the identity tensor of order in which each dimension has size . In this case, the covariance tensor follows a dual-symmetric PARAFAC decomposition of order .

In general, the Tucker decomposition does not impose restrictions on the core tensor structure, which makes this model more flexible. In the context of this paper, this characteristic reflects an arbitrary and unknown structure for the source’s covariance which can also be estimated from (13). In contrast, the PARAFAC decomposition (15) denotes a particular case of the Tucker decomposition when the sources’ signals are uncorrelated and the source covariance matrix is perfectly known (i.e., diagonal). In practice, this may not hold.

##### 4.2. ALS-Tucker Algorithm

Our goal is to blindly estimate the spatial signature matrices and which refer to the different dimensions of the sensor array from the covariance tensor . For the sake of simplicity, from this point on, we consider . In matrix-based notation, the Tucker decomposition (13) allows the following factorization in terms of its factor matrices and core tensor in accordance with (5):wherewhile and , , denote the -mode unfolding of the covariance tensor and the core tensor , respectively.

From the matrix unfoldings of , an ALS based algorithm is formulated to estimate the desired factor matrices. An estimate of the spatial signature matrix , associated with the th dimension of the covariance tensor, is obtained by solving the following least squares (LS) problem: whose analytic solution is given by

As discussed in Section 4.1, the Tucker decomposition does not impose restrictions on the structure of the core tensor and its estimation becomes necessary. Let be an unknown matrix of arbitrary structure. The following LS problem is formulated from the vectorization of the sample covariance matrix : from which an estimate of can be obtained as where .

Since (19) and (21) are nonlinear functions of the parameters to be estimated, the blind spatial signature estimation problem can be solved using a classical ALS iterative solution [21, 22]. The basic idea of the algorithm is to estimate one factor matrix at each step while the others remain fixed at the values obtained in previous steps. This procedure is repeated until convergence. The proposed generalized ALS-Tucker algorithm for -D sensor arrays is summarized in Algorithm 1.