^{1}

^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

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 (

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

In regards to tensor-based methods for blind spatial signatures estimation, the

In this paper, two tensor-based approaches to the estimation of spatial signatures are presented. By using the signals received on a

The rest of this paper is organized as follows: Section

In the following, we briefly introduce for convenience the basics on operations involving tensors and tensor decompositions, which refer to [

Let

The Tucker decomposition [

In general, the Tucker decomposition is not unique; that is, there are infinite solutions for

The PARAFAC decomposition [

The

The

Throughout this work, special attention is given to dual-symmetric tensors. The PARAFAC decomposition of a given tensor

We consider

In this section, we propose two iterative algorithms to solve the blind spatial signature estimation problem in

With the intention of exploiting the multidimensional structure of the received signal, the noiseless sample covariance matrix (

Considering the case in which the sources are uncorrelated and have unitary variance, we can rewrite (

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

Our goal is to blindly estimate the spatial signature matrices

From the matrix unfoldings of

As discussed in Section

Since (

(

tensor

(

According to (

by fixing

(

(

(

In this approach the factor matrices are treated as independent variables; that is, the dual-symmetry property of the covariance tensor is not exploited. In this case, a final estimate of the spatial signature matrix associated with the

In this section, a link is established between the ALS-ProKRaft algorithm proposed initially in [

The multimode unfolding of the PARAFAC decomposition in (

Equation (

From (

(

of the multimode unfolding matrix

(

(

by applying the multidimensional LS-KRF algorithm on

(

and obtain

(

Note that when compared with conventional ALS-based PARAFAC solutions [

In this section, the blind spatial signature estimation problem is formulated for

In this approach, we consider an

From (

In contrast with (

As mentioned in Section

The spatial signatures of the sources can be estimated from (

For all the previously proposed algorithms, the final estimates for the spatial signature matrices are obtained when the convergence is declared. A usually adopted criterion for convergence is defined as

After the estimation of the spatial signatures matrices

In the following, we discuss the computational complexity of the iterative ALS-Tucker and ALS-ProKRaft algorithms. For the sake of simplicity, the computational complexity of the proposed methods is described in terms of the computational cost of the matrix SVD. For a matrix of size

In this section, we discuss the advantages and disadvantages of the proposed methods to blind spatial signatures estimation in

In contrast, the ALS-Tucker algorithm does not exploit the dual-symmetry property of the data covariance tensor and all factor matrices need to be estimated as independent variables. However, in the ALS-ProKRaft algorithm, only half of the factor matrices are estimated by exploiting the dual-symmetry property of the covariance tensor. Therefore, the ALS-ProKRaft algorithm is more computationally attractive than the ALS-Tucker algorithm. When compared with the state-of-the-art matrix-based algorithms such as MUSIC, ESPRIT, and Propagator Method, the proposed tensor-based algorithms have the advantage of fully exploiting the multidimensional nature of the received signal in less specific scenarios, which leads to more accurate estimates. For instance, the ESPRIT algorithm was formulated for sensor arrays that obey the shift invariance property. On the other hand, the MUSIC algorithm has high computational complexity due to the search of parameters in the spatial spectrum.

In the following, simulation results and performance evaluations of the ALS-Tucker and ALS-ProKRaft algorithms for

In Figures

Total RMSE versus SNR for

Total RMSE versus SNR for

From Figure

Figure

Convergence of the ALS-Tucker and ALS-ProKRaft algorithms.

In the second part of this section, we consider a

Total RMSE versus SNR for

Figure

Total RMSE versus SNR (performance of the ALS-Tucker algorithm for different number of sensors).

In Figure

Total RMSE versus number of samples.

In this paper, two tensor-based approaches based on the Tucker and PARAFAC decompositions have been formulated to solve the blind spatial signatures estimation problem in multidimensional sensor arrays. First, we have proposed a covariance-based generalization of the Tucker decomposition for

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under the PVE Grant no. 88881.030392/2013-01, the productivity Grant no. 303905/2014-0, the postdoctoral scholarship abroad (PDE) no. 207644/2015-2, and the Program Science without Borders, Aerospace Technology supported by CNPq and CAPES for the postdoctoral scolarship abroad (PDE) no. 207644/2015-2.