This paper presents a new approach to reconstruct the plant output of linear time-invariant systems in the case where the available output measurement is quantized. By fitting the quantized measurement data with polynomials in a moving horizon manner, a smooth approximation of plant output is obtained via solving a convex optimization problem. Applying the signal to an observer, the plant output is reconstructed by taking account of the plant dynamics. It is guaranteed that the error between the true and the reconstructed output is bounded. Experimental validation is given by using a DC motor positioning system. It turns out that the proposed approach achieves small reconstruction error and accurate tracking control. In addition, the approach yields a smooth reconstruction signal so that the plant input is well behaved (or smooth) even if PID controller is employed for the plant subject to quantized output measurement.