The reduced-order model of a time-invariant linear dynamical system, excited by a force of an impulsive type, may be readily obtained using the Ho-Kalman minimal-realization algorithm . The method is based upon a particular factorization of the Hankel matrix in the Markovian representation of the discrete-time process. For stochastic systems, the applicability of the theory has been demonstrated by Akaike  on the assumption that the excitation is a zero-mean white noise of a gaussian type. Some of the most widely known output-only identification methods, such as Eigensystem Realization Algorithm (ERA), Canonical Variate Analysis (CVA), and Balanced Realization (BR)) are based upon the above-mentioned work, with the aid of a robust factorization technique, such as Singular-Value Decomposition (SVD). Notwithstanding the growing popularity of the above methods, some aspects of their applicability are not yet understood. Two points are of particular interest: the first regards the applicability of the theory in highly damped systems; and the second regards its applicability to systems driven by excitations different from the one hypothesized. The aim of the present work is to define a reliable test on the hypotheses. Some numerical and experimental results are presented.