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Show that for an spectral density is AR(1) process, X¢ pX;-1 + e, with |ø| < 1 the

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

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I am stuck on an AR (1) process spectral density question for a time series ( please see the image for the question and my answer). Would you please explain the next steps for solving this question?

Show that for an AR(1) process, X, = pX;-1 + et, with |p| < 1 the
spectral density is
g2
f(ω)
1+ ф2 — 2 сos(2по)
Xt = pxt-1 + e,
with ø| < 1
.2
|k|
Vk =
k = 0,±1,±2
X.
(1-ф2)
-ikw
f(ω)
1+
2п
Deika
k=1
i=1

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