Error estimates for the ESPRIT algorithm

Abstract

Let zj := efj (j = 1,…, M) with fj ∈[−φ, 0]+i[−π, π) and small φ ≥ 0 be distinct nodes. With complex coefficients cj ≠ 0, we consider an exponential sum h(x) := c1 ef1 x +… +cM efM x (x ≥ 0). Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Determine all parameters of h, if N noisy sampled values hk := h(k) + ek (k = 0,…,N − 1) with N≫2M are given, where ek are small error terms. This parameter identification problem is a nonlinear inverse problem which can be efficiently solved by the ESPRIT algorithm. In this paper, we present mainly corresponding error estimates for the nodes zj (j = 1,…, M). We show that under appropriate conditions, the results of the ESPRIT algorithm are relatively insensitive to small perturbations on the sampled data.