Hybrid Sparse Expansion and Separable Hybrid Prony Method

Abstract

Signal sparsity plays an essential role in signal compression and reconstruction. Prony-like methods are widely used in the relevant applications e.g., the recovery of signals with finite rate of innovation (FRI). However, these methods are usually limited to the structured function expansions of a single family, i.e., the considered structured functions belonging to the eigenfunctions of a single operator. In this paper, we investigate hybrid Prony models (HPMs), wherein functions can be expanded sparsely by the combination of eigenfunctions of two different operators $\mathcal {L}_1$ and $\mathcal {L}_2$ . Although the Prony-like methods cannot directly deal with HPMs, we show that HPMs can be directly converted to the generalized Prony model recently considered by Peter and Plonka when the operator pair $\lbrace \mathcal {L}_1, \mathcal {L}_2\rbrace$ satisfies certain conditions; we refer to these functions as separable HPMs (SHPMs). Similar to the Prony method, the problem of nonlinear parameter estimation in SHPMs can be converted to the problem of finding the roots of polynomials. Examples and numerical results are given to illustrate SHPMs.