This problem is from a book called "System Identification: Theory for the User" by Lennart Ljung.

2E.4. Consider the "state-space description x(t + 1) - fx(t) + w(t) where x, f, h, w, and v are scalars. Iw( and {v() are mutually independent white Gaussian noises with variances Ri and R2, respectively. Show that y(t) can be repre- sented as an ARMA process: y) ay -1)+ay(t - n)- e() +ce(t -1)+ ...+ce(t - n) Determine n, ai, ci, and the variance of e (t) in terms of f, h, Ri, and R2. What is the relationship between e(), w(), and v(t)? 2E.4. Consider the "state-space description x(t + 1) - fx(t) + w(t) where x, f, h, w, and v are scalars. Iw( and {v() are mutually independent white Gaussian noises with variances Ri and R2, respectively. Show that y(t) can be repre- sented as an ARMA process: y) ay -1)+ay(t - n)- e() +ce(t -1)+ ...+ce(t - n) Determine n, ai, ci, and the variance of e (t) in terms of f, h, Ri, and R2. What is the relationship between e(), w(), and v(t)