Consider the second-order real AR process y[t + 2] + ay[t + 1] + a2y[t] =...

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Consider the second-order real AR process y[t + 2] + a₁y[t + 1] + a2y[t] = f[t + 2] where f[t] is a zero-mean white-noise sequence. The difference equation in Tyy [l] = −ā¹Tyy[1 − 1] — ā2™yy[l — 2] —āpryy[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] - r²[1]) 1 (P1, P2): ( = = in terms of p₁, p2 and o a1 = a2 (-a₁ ± √a₁² — 4a2) Tyy [1] = Tyy[2] = 0² (₁ Tyy [0] = 0² a1 1 + a2 = Tyy[0] and a₁ and a2 are known, show that the autocorrelation values a² (c) Using o² = E[y² [t]] =and the results of this problem, show that 1 + a2 -a₂) 1 + a2 a2 [(1 + a2)2 - a²] (1) (2) Tyy[k] + a₁ryy [k - 1] + a2ryy [k − 2] = 0 (3) (4) (5) (6) (7) (8) (d) Using ryy[0] = ² and ryy[1] = −a₁0²/(1 + a2) as initial conditions, find an explicit solution to the Yule-Walker difference equation (9) (10) Consider the second-order real AR process y[t + 2] + a₁y[t + 1] + a2y[t] = f[t + 2] where f[t] is a zero-mean white-noise sequence. The difference equation in Tyy [l] = −ā¹Tyy[1 − 1] — ā2™yy[l — 2] —āpryy[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] - r²[1]) 1 (P1, P2): ( = = in terms of p₁, p2 and o a1 = a2 (-a₁ ± √a₁² — 4a2) Tyy [1] = Tyy[2] = 0² (₁ Tyy [0] = 0² a1 1 + a2 = Tyy[0] and a₁ and a2 are known, show that the autocorrelation values a² (c) Using o² = E[y² [t]] =and the results of this problem, show that 1 + a2 -a₂) 1 + a2 a2 [(1 + a2)2 - a²] (1) (2) Tyy[k] + a₁ryy [k - 1] + a2ryy [k − 2] = 0 (3) (4) (5) (6) (7) (8) (d) Using ryy[0] = ² and ryy[1] = −a₁0²/(1 + a2) as initial conditions, find an explicit solution to the Yule-Walker difference equation (9) (10)

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Part a The YuleWalker equations are a set of linear equations that can be used to estimate the param... View the full answer