Rectified ALS algorithm for multidimensional harmonic retrieval

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).