Signal filtering algorithms in continuous-time systems with uncertain observations

We analyze the least mean-squared error linear filtering problem of a continuous-time wide-sense stationary scalar signal from noisy observations which, in a random way, can consist of signal plus noise or only noise. We assume that the signal is a linear function of the components of the state-vector, and only the system matrix in the state-space model and the crosscovariance function of the state and signal are known. Under the hypothesis that the Bernoulli variables modelling the uncertainty in the observations are independent, with known constant probability of each observation contains signal, we obtain two filtering algorithms to solve this problem: one of them is based on Chandrasekhar-type differential equations and, the other, on Riccati-type ones. The comparison of both algorithms shows that the Chandrasekhar-type one is computationally better than the Riccati-type one. The theoretical results are illustrated by a numerical simulation example.