# Question

Consider an AR (2) process which is described by the recursion

Y [n] = a1Y [n– 1] + a2Y [n– 2] + X [n]

Where is an IID random process with zero- mean and variance σ2X.

(a) Show that the autocorrelation function of the AR (2) process satisfies the difference equation,

RYY [k] a1R YY [k – 1] a2R YY = + [k – 2] k = 2, 3, 4 …

(b) Show that the first two terms in the autocorrelation function satisfy

From these two equations, solve for RYY [0] RYY [1] and in terms of a1 a2, and σ2X.

(c) Using the difference equation in part (a) together with the initial conditions in part (b), find a general expression for the autocorrelation function of an AR (2) process.

(d) Use your result in part (c) to find the PSD of an AR (2) process.

Y [n] = a1Y [n– 1] + a2Y [n– 2] + X [n]

Where is an IID random process with zero- mean and variance σ2X.

(a) Show that the autocorrelation function of the AR (2) process satisfies the difference equation,

RYY [k] a1R YY [k – 1] a2R YY = + [k – 2] k = 2, 3, 4 …

(b) Show that the first two terms in the autocorrelation function satisfy

From these two equations, solve for RYY [0] RYY [1] and in terms of a1 a2, and σ2X.

(c) Using the difference equation in part (a) together with the initial conditions in part (b), find a general expression for the autocorrelation function of an AR (2) process.

(d) Use your result in part (c) to find the PSD of an AR (2) process.

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