In this study, the authors propose a framework for inversion-based estimation of certain categories of faults in discrete-time linear systems. The fault signal, as an unknown input, is reconstructed from its projections onto two subspaces. The first projection is achieved through an algebraic operation, whereas the second projection is obtained by a dynamic filter whose poles coincide with the transmission zeros of the system. Feedback is then introduced to stabilise this filter as well as to provide an unbiased estimate of the unknown input. Their proposed methodology has two distinctive and practical advantages. First, it represents a unified approach to the problem of inversion of both minimum and non-minimum phase systems as well as systems having transmission zeros on the unit circle. Second, the feedback structure ensures that the proposed scheme is robust to noise. They have shown that the proposed inversion filter is unbiased for certain categories of faults. Finally, they have demonstrated the capabilities and performance of the proposed methodology through several numerical simulation case studies.