Let us consider a pair signal-observation ((xn,yn),n 0) where the unobserved signal (xn) is a Markov chain and the observed component is such that, given the whole sequence (xn), the random variables (yn) are independent and the conditional distribution of yn only depends on the corresponding state variable xn. The main problems raised by these observations are the prediction and filtering of (xn). We introduce sufficient conditions allowing to obtain computable filters using mixtures of distributions. The filter system may be finite or infinite dimensional. The method is applied to the case where the signal xn = Xn is a discrete sampling of a one dimensional diffusion process: Concrete models are proved to fit in our conditions. Moreover, for these models, exact likelihood inference based on the observation (y0,...,yn) is feasable.