A Simultaneous Sparse Approximation Method for Multidimensional Harmonic Retrieval

TL;DR

This paper introduces a sparse-based multidimensional harmonic retrieval method that efficiently estimates signal parameters with high resolution, automatically pairs modes, and is validated through numerical simulations.

Contribution

It proposes a multigrid dictionary refinement technique for large signals, handles multiple modes iteratively, and derives Cramér-Rao bounds for R-D signals.

Findings

Method effectively estimates parameters in noisy environments.

Automatic mode pairing without additional association steps.

Numerical results confirm high resolution and accuracy.

Abstract

In this paper, a sparse-based method for the estimation of the parameters of multidimensional ($R$-D) modal (harmonic or damped) complex signals in noise is presented. The problem is formulated as $R$ simultaneous sparse approximations of multiple 1-D signals. To get a method able to handle large size signals while maintaining a sufficient resolution, a multigrid dictionary refinement technique is associated with the simultaneous sparse approximation problem. The refinement procedure is proved to converge in the single $R$-D mode case. Then, for the general multiple modes $R$-D case, the signal tensor model is decomposed in order to handle each mode separately in an iterative scheme. The proposed method does not require an association step since the estimated modes are automatically "paired". We also derive the Cram\'er-Rao lower bounds of the parameters of modal $R$-D signals. The expressions are given in compact form in the single $R$-D mode case. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed method.