$$x[n]=0.6x[n-1]+w[n]$$ $$y[n]=x[n]+v[n]$$ $$\sigma_w^2=0.25, $$$$\sigma_v^2=0.5$$
Find the expression for the Kalman filter equation at convergence and the corresponding mean square error.
I know Kalman filter equation is $$\hat{x}_k=A\hat{x}_{k-1}+Bu_k+K_k(y_k-C(A\hat{x}_{k-1}+Bu_k))$$
But here $u_k=0$ then it reduces to $$\hat{x}_k=A\hat{x}_{k-1}+K_k(y_k-C(A\hat{x}_{k-1}))$$
But how to calculate Kalman gain from given variances? Please help me in solving this.