In this talk, a new approach to obtain so-called damping estimates for self-similar solutions to general hyperbolic relaxation systems in one space dimension is presented. Without relying on symmetry assumptions, we extend the damping estimates obtained by Mascia and Zumbrun (2005) for localized perturbations to the nonlocalized case via method of characteristics. These estimates open the door to a general stability theory of shock profiles of hyperbolic relaxation system under nonlocalized perturbations in one space dimension.