The notion of normalized coprime factorization (NCF) for sampled-data systems is studied. The problem is approached by the study of linear systems with discrete jumps at periodic time instants. These systems arise in the study of linear systems with sampled-data control and filtering problems. It is shown that the NCF can be obtained through the solution to Riccati equations with jumps. The properties of the Riccati equations and the corresponding properties of the NCF are studied.