A new estimation method in modal analysis (Q2734317)

This interesting paper addresses the following problem: estimate \(p\in\mathbb{N}\), \(\vec{b},\vec{z}\in\mathbb{C}^p\) in \(F(t;\vec{b},\vec{z})=\sum_{j=1}^p b_jz_j^t\) from the data \(d_k=F(k;\vec{b},\vec{z})+\varepsilon_k\) \((k=0,1,\ldots,n-1\); \(n\geq 2p)\) where \(\vec{\varepsilon}\sim \mathbb{N}(\vec{0},\sigma^2I)\) and \(t\in\mathbb{R}^+\), \(|\text{arg} z_j|\leq\pi\) \((1\leq j\leq p)\). NEWLINENEWLINENEWLINEIn a number of previous papers the authors proposed an empirical method which subsequently has been given some theoretical underpinning. The procedure is based on the asymptotic properties of the poles of the Padé approximants to the \(\mathbb{Z}\)-transform of the data \(f(z)=\sum_{k=0}^{\infty} d_kz^{-k}\) and strongly uses statistical properties of these approximants. NEWLINENEWLINENEWLINEAn important ingredient is condensation of the random vector of roots of a random polynomial introduced by \textit{J. M. Hammersley} [Proc. 3rd Berkeley Symp. Math. Stat. Prob. 2, 89-111 (1956; Zbl 0074.34302)]. NEWLINENEWLINENEWLINEThe procedure consists of five main steps and the implementation is described in the paper, accompanied by numerical results.