Consider the second-order real AR process y[t + 2] + ay[t + 1] + a2y[t] = f[t + 2] where f[t] is a zero-mean white-noise sequence. The difference equation in Tyy [l] = Tyy[1 1] 2yy[l 2] pryy[lp] for 1>0 has a characteristic equation with roots 2 (b) On the other hand, if oy can be expressed as (a) Using the Yule-Walker equations, show that if the autocorrelation values Tyy[lk] = E[y[t - k]y[t 1]] are known, then the model parameters may be determined from Tyy [1] (ryy [0] Tyy [2]) ry [0]-[1]) Tyy [0]ryy [2] ry [1] ry [0] - r[1]) 1 (P1, P2): ( = = in terms of p, p2 and o a1 = a2 (-a a 4a2) Tyy [1] = Tyy[2] = 0 ( Tyy [0] = 0 a1 1 + a2 = Tyy[0] and a and a2 are known, show that the autocorrelation values a (c) Using o = E[y [t]] =and the results of this problem, show that 1 + a2 -a) 1 + a2 a2 [(1 + a2)2 - a] (1) (2) Tyy[k] + aryy [k - 1] + a2ryy [k 2] = 0 (3) (4) (5) (6) (7) (8) (d) Using ryy[0] = and ryy[1] = a0/(1 + a2) as initial conditions, find an explicit solution to the Yule-Walker difference equation (9) (10)