*Article* **Prony Method for Two-Generator Sparse Expansion Problem**

**Abdulmtalb Hussen 1,\* and Wenjie He <sup>2</sup>**


**Abstract:** In data analysis and signal processing, the recovery of structured functions from the given sampling values is a fundamental problem. Many methods generalized from the Prony method have been developed to solve this problem; however, the current research mainly deals with the functions represented in sparse expansions using a *single generating function*. In this paper, we generalize the Prony method to solve the sparse expansion problem for *two generating functions*, so that more types of functions can be recovered by Prony-type methods. The two-generator sparse expansion problem has some special properties. For example, the two sets of frequencies need to be separated from the zeros of the Prony polynomial. We propose a *two-stage least-square detection method* to solve this problem effectively.

**Keywords:** Prony method; exponential sums; eigenfunctions; eigenvalues; sparse expansion; generating function; Hankel matrix; short time Fourier transform; least-square method
