In this paper, a novel approach for parameter identification of linear time invariant (LTI) systems using matrix pencils and ESPRIT-type methods is presented. The relations between Hankel matrices formed from the truncated impulse response and the companion matrix of the poles of the system are fully investigated. Specifically, it is shown that, up to a constant Hankel matrix, every Hankel matrix of finite rank is a power of a companion matrix. Thus, the identification of the system poles reduces directly to solving a generalized eigenvalue problem constructed from two shifted Hankel matrices of the impulse response. Next, we derive an ESPRIT method for the system identification problem. The most significant poles are the solution of an eigenvalue problem of a matrix formed from the singular value decomposition of augmented Hankel matrices of the truncated impulse response of the system. This approach can also be applied for system order reduction. Finally, a generalization for system identification of multi-output single-input linear systems is provided.