Dimitri Nion

Tensor Algebra and Multidimensional Harmonic Retrieval in Signal Processing for MIMO Radar

Detection and estimation problems in multiple-input multiple-output (MIMO) radar have recently drawn considerable interest in the signal processing community. Radar has long been a staple of signal processing, and MIMO radar presents challenges and opportunities in adapting classical radar imaging tools and developing new ones. Our aim in this article is to showcase the potential of tensor algebra and multidimensional harmonic retrieval (HR) in signal processing for MIMO radar. Tensor algebra and multidimensional HR are relatively mature topics, albeit still on the fringes of signal processing research. We show they are in fact central for target localization in a variety of pertinent MIMO radar scenarios. Tensor algebra naturally comes into play when the coherent processing interval comprises multiple pulses, or multiple transmit and receive subarrays are used (multistatic configuration). Multidimensional harmonic structure emerges for far-field uniform linear transmit/receive array configurations, also taking into account Doppler shift; and hybrid models arise in-between. This viewpoint opens the door for the application and further development of powerful algorithms and identifiability results for MIMO radar. Compared to the classical radar-imaging-based methods such as Capon or MUSIC, these algebraic techniques yield improved performance, especially for closely spaced targets, at modest complexity.

Decompositions of a Higher-Order Tensor in Block Terms—Part III: Alternating Least Squares Algorithms

In this paper we derive alternating least squares algorithms for the computation of the block term decompositions introduced in Part II. We show that degeneracy can also occur for block term decompositions.

A combination of parallel factor and independent component analysis

Although CPA (canonical/parallel factor analysis) has a unique solution, the actual computation can be made more robust by incorporating extra constraints. In several applications, the factors in one mode are known to be statistically independent. On the other hand, in Independent Component Analysis (ICA), it often makes sense to impose a Khatri-Rao structure on the mixing vectors. In this paper, we propose a new algorithm to impose independence constraints in CPA. Our algorithm implements the algebraic CPA structure and the property of statistical independence simultaneously. Numerical experiments show that our method outperforms in several cases pure CPA, pure ICA, and tensor ICA, a previously proposed method for combining ICA and CPA. We also present a strategy for imposing full or partial symmetry in CPA.

A Tensor Framework for Nonunitary Joint Block Diagonalization

This paper introduces a tensor framework to solve the problem of nonunitary joint block diagonalization (JBD) of a set of real or complex valued matrices. We show that JBD can be seen as a particular case of the block-component-decomposition (BCD) of a third-order tensor. The resulting tensor model fitting problem does not require the block-diagonalizer to be a square matrix: the over- and underdetermined cases can be handled. To compute the tensor decomposition, we build an efficient nonlinear conjugate gradient (NCG) algorithm. In the over- and exactly determined cases, we show that exact JBD can be computed by a closed-form solution based on eigenvalue analysis. In approximate JBD problems, this solution can be used to efficiently initialize any iterative JBD algorithm such as NCG. Finally, we illustrate the performance of our technique in the context of independent subspace analysis (ISA) based on second-order statistics (SOS).

A link between the decomposition of a third-order tensor in rank-(L,L,1) terms and Joint Block Diagonalization

In this paper, we show that the Block Component Decomposition in rank-(L,L,1) terms of a third-order tensor, referred to as BCD-(L,L,1), can be reformulated as a Joint Block Diagonalization (JBD) problem, provided that certain assumptions on the dimensions are satisfied. This JBD-based reformulation leads to a new uniqueness bound for the BCD-(L,L,1). We also propose a closed-form solution to solve exact JBD problems. For approximate JBD problems, this closed-form solution yields a good starting value for iterative optimization algorithms. The performance of our technique is illustrated by its application to blind CDMA signal separation.

A time-frequency technique for blind separation and localization of pure delayed sources

In this paper we address the problem of overdetermined blind separation and localization of several sources, given that an unknown scaled and delayed version of each source contributes to each sensor recording. The separation is performed in the time-frequency domain via an Alternating Least Squares (ALS) algorithm coupled with a Vandermonde structure enforcing strategy across the frequency mode. The latter allows to update the delays and scaling factors of each source with respect to all sensors, up to the ambiguities inherent to the mixing model. After convergence, a reference sensor can be chosen to remove these ambiguities and the Time Difference of Arrival (TDOA) estimates can be exploited to localize the sources individually.