This paper concerns the state estimation problem for linear discrete-time non-Gaussian systems. It is known that filters based on quadratic functions of the measurements processes (Quadratic Filter) improve the estimation accuracy of the optimal linear filter. In order to enlarge the class of systems, which can be processed by a Quadratic Filter, we rewrite the system model by introducing an output injection term. The resulting filter, named the Feedback Quadratic Filter, can be applied also to non asymptotically stable systems. We prove that the performance of the Feedback Quadratic Filter depends on the gain parameter of the output term, which can be chosen so that the estimation error is always less than or equal to the Quadratic Filter. (C) 2017 Elsevier Ltd. All rights reserved,