This paper presents an alternative, characteristic function based approach for the Bayesian design of estimators for dynamic linear systems and linear detection problems. For a measurement update, the a posteriori characteristic function of the unnormalized conditional probability density function (ucpdf) of the state given the measurement history is obtained as a convolution of the a priori characteristic function of the ucpdf with the characteristic function of the measurement noise. It is shown that this convolution holds for a general measurement structure. Time propagation involves the product of the updated characteristic function of the ucpdf and the characteristic function of the process noise. Some estimation problems are found to be naturally tractable using only characteristic functions, such as the multivariable linear system with additive Cauchy measurement and process noise. It is shown that even the derivation of the Kalman filter algorithm has advantages when formulated using the characteristic function approach. Finally, in some instances the estimation problem can only be formulated in terms of characteristic functions. This is illustrated by a one-update scalar example for symmetric-alpha-stable distributions. (C) 2016 Elsevier Ltd. All rights reserved.