Abstract-The traditional Bayesian approximation framework for filtering in discrete time systems assumes that the measurement is available at every time instant. But in practice, the measurements could be randomly delayed. In the literature, the problem has been examined and solution is provided by restricting the maximum number of delay to one or two time steps. This paper develops an approach to deal with the filtering problems with an arbitrary number of delays in measurement. Pursuing this objective, traditional Bayesian approximation to nonlinear filtering problem is modified by reformulating the expressions of mean and covariances which appear during the measurement update. We use the cubature quadrature rule to evaluate the multivariate integral expressions for the mean vector and the covariance matrix which appear in the developed filtering algorithm. We compare the new algorithm which accounts for delay with the existing CQKF heuristics on two different examples and demonstrate how accounting for a random delay improves the filtering performance.Index Terms-Nonlinear filtering, Bayesian framework of filtering, Random delay in measurements, Cubature quadrature Kalman filter.

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In this paper, two existing quadrature filters, viz., the Gauss-Hermite filter (GHF) and the sparse-grid Gauss-Hermite filter (SGHF) are extended to solve nonlinear filtering problems with one step randomly delayed measurements. The developed filters are applied to solve a maneuvering target tracking problem with one step randomly delayed measurements.Simulation results demonstrate the enhanced accuracy of the proposed delayed filters compared to the delayed cubature Kalman filter and delayed unscented Kalman filter.