We introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the AAA algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method (MPM) applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices that are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the MPM is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the MPM, but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the MPM for noisy data and for signal approximation by short exponential sums.

中文翻译：

---

从 ESPRIT 到 ESPIRA：通过迭代有理逼近估计信号参数

我们介绍了一种基于迭代有理近似 (ESPIRA) 的稀疏指数和信号参数估计新方法。我们的算法使用 AAA 算法对给定等距信号值的离散傅里叶变换进行有理逼近。我们表明 ESPIRA 可以解释为应用于 Loewner 矩阵的矩阵铅笔法 (MPM)。这些 Loewner 矩阵与通常用于信号恢复的 Hankel 矩阵密切相关。由于通过索引集的自适应选择构建 Loewner 矩阵，MPM 是稳定的。ESPIRA 对精确数据的恢复结果与 ESPRIT 和 MPM 相似，但计算量更少。此外，对于噪声数据和短指数和的信号近似，ESPRIT 和 MPM 的性能明显优于 ESPRIT 和 MPM。