Recursive state estimation problems for explicit and implicit time-invariant linear systems, both for systems with and without unknown inputs, can be formulated as a single problem usually referred to as descriptor Kalman filtering. Solutions to this problem have been proposed in the literature; however, these solutions either neglect possible contributions of future dynamics to the current estimate or make unnecessary assumptions on the structure of the system, In this paper, the authors propose a solution to this problem which leads to a constructive method lifting these unnecessary assumptions. This method uses a generalization of the shuffle algorithm.