In many practical applications the state variables are defined on a compact set of the state space. For estimating such variables constrained particle filters have been successfully applied to nonlinear systems. For the saturated system the measurement information can be used during the sampling procedure to obtain particles that approximate the true state of the system. This can be achieved by using a detection function, which detects the saturation as it occurs. In this paper we propose the Saturated Particle Filter algorithm which incorporates the measurements into the importance sampling procedure through the detection function. The new filter is applied to the Lindley-type stochastic process, where the stochastic process depends on an exogenous parameter. This parameter changes during the simulation. Furthermore, the system is corrupted with high measurement noise. The simulations show that our new filter achieves better performance than the standard Constrained SIR filter, while it preserves low computational complexity.