Rectified ALS Algorithm for Multidimensional Harmonic Retrieval

Remy Boyer (1) , Pierre Comon (2)

1 L2S - Laboratoire des signaux et systèmes
2 GIPSA-CICS - GIPSA - Communication Information and Complex Systems

MultiDimensional (MD) Harmonic Retrieval is a challenging multi-parameter estimation problem and is useful for a plethora of operational applications as for instance channel sounding or MIMO radar processing. The MD-harmonic model follows a structured Canonical Polyadic Decomposition (CPD) in the sense that the factors of the CPD are Vandermonde. A standard and popular estimation scheme to derive the CPD is the Alternating Least Squares (ALS) algorithm. Unfortunately, the ALS algorithm does not exploit the a priori known factor structure , which considerably degrades the estimation performance. In this work, a modified ALS-type algorithm is proposed. This new algorithm, called Rectified ALS (RecALS), is able to take into account the Vandermonde structure of the factors. The RecALS algorithm belongs to the Lift-and-Project family and exploits iterated projections on the set of Toeplitz rank-1 matrices. It exhibits a fast convergence and is very accurate in the sense that its Mean Square Error (MSE) is close to the Cramér-Rao Bound for a wide range of Signal to Noise Ratio (SNR).

Mots clés: MultiDimensional Harmonic Retrieval; Toeplitz rank-1 matrix approximation; Index Terms—Structured Canonical Polyadic Decomposition; Vandermonde factors; modified Alternating Least Squares algorithm

Domaines: Sciences de l'ingénieur [physics]; Traitement du signal et de l'image [eess.SP]

Remy Boyer, Pierre Comon. Rectified ALS Algorithm for Multidimensional Harmonic Retrieval. SAM 2016 - 9th IEEE Sensor Array and Multichannel Signal Processing Workshop, Jul 2016, Rio de Janeiro, Brazil. ⟨10.1109/sam.2016.7569678⟩. ⟨hal-01314702⟩