We have a linear system with observation as follows:

$x(t+1)=Ax(t)+Bu(t)+w(t)$

$y(t)=Cx(t)+z(t)$

for given information $y(0\sim t)$, we can construct the dynamics of $\hat{x}(t)$ using the Kálmán filter equations.

What if we are given information $y(0\sim t-N)$ where $N$ is a constant?