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.

中文翻译：

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混合稀疏扩展和可分离混合 Prony 方法

信号稀疏性在信号压缩和重建中起着至关重要的作用。Prony-like 方法广泛用于相关应用，例如，有限创新率（FRI）的信号恢复。然而，这些方法通常仅限于单一族的结构化函数展开，即所考虑的结构化函数属于单个算子的特征函数。在本文中，我们研究了混合 Prony 模型 (HPM)，其中函数可以通过两个不同算子的特征函数的组合进行稀疏扩展$\mathcal {L}_1$ 和 $\mathcal {L}_2$ . 虽然类 Prony 方法不能直接处理 HPM，但我们表明，当操作符对时，HPM 可以直接转换为 Peter 和 Plonka 最近考虑的广义 Prony 模型$\lbrace \mathcal {L}_1, \mathcal {L}_2\rbrace$满足一定条件；我们将这些功能称为可分离 HPM (SHPM)。与 Prony 方法类似，SHPM 中的非线性参数估计问题可以转化为求多项式的根问题。给出了示例和数值结果来说明 SHPM。