The Generalized Operator Based Prony Method

Abstract

AbstractThe generalized Prony method is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator A. However, this procedure requires the evaluation of higher powers of the linear operator A that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method that essentially simplify the acquisition of the needed samples and, at the same time, can improve the numerical stability of the method. The first extension regards the change of operators from A to $$\varphi (A)$$ φ ( A ) , where $$\varphi $$ φ is a suitable operator-valued mapping, such that A and $$\varphi (A)$$ φ ( A ) possess the same set of eigenfunctions. The goal is now to choose $$\varphi $$ φ such that the powers of $$\varphi (A)$$ φ ( A ) are much simpler to evaluate than the powers of A. The second extension concerns the choice of the sampling functionals. We show how new sets of different sampling functionals $$F_{k}$$ F k can be applied with the goal being to reduce the needed number of powers of the operator A (resp. $$\varphi (A)$$ φ ( A ) ) in the sampling scheme and to simplify the acquisition process for the recovery method.