Unknown differences between system states and observer estimates inherently result in, potentially large, initial errors. This paper deals with the mitigation of this problem in discrete LTI systems by using an approximation of the system state directly calculated from past and present values of its input and output. This information is used in a nonlinear modification of the classical linear state observer structure to reduce the transient error and deviations due to disturbances. We show the stability of this extension in the presence and absence of noise. Additionally, we prove the fulfillment of the separation principle. For the case of noisy measurement, we argue that there is a predictable ideal filter length if the noise variance is known. The ideas are subsequently illustrated in simulation.