Let {Z} be Gaussian white noise, i.e. {Zt} is a sequence of i.i.d. normal r.v.s each with mean zeroand variance 1. DefineZt,even;- {7/4/2-1-1)/√2, iftis oddXt= =Show that {X} is WN(0, 1) (that is, variables X and Xt+k, k ≥ 1, are uncorrelated with mean zeroand variance 1) but that X₁ and X₁-1 are not i.i.d.

Given that is the Gaussian white noise and random variable is defined as : {¨…

Answered: Let {Z} be Gaussian white noise, i.e.…

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

Let {Z} be Gaussian white noise, i.e. {Zt} is a sequence of i.i.d. normal r.v.s each with mean zero
and variance 1. Define
Zt,
even;
- {7/4/2-1-1)/√2, iftis odd
Xt= =
Show that {X} is WN(0, 1) (that is, variables X and Xt+k, k ≥ 1, are uncorrelated with mean zero
and variance 1) but that X₁ and X₁-1 are not i.i.d.

Expert Solution

Step 1: Given that

Given that $Z subscript t$ is the Gaussian white noise and random variable $X subscript t$ is defined as :

$text X end text subscript t equals space open curly brackets text Z end text subscript t comma space space space space space space space space space space space space space space space space space space space space space space space space space space space space text if t is even; end text space fraction numerator left parenthesis text Z end text squared subscript left parenthesis text t- end text 1 right parenthesis end subscript minus 1 right parenthesis over denominator square root of 2 end fraction comma space text if t is odd end text close curly brackets space space space space space space space space space space space space space space space$

To show that { $text X end text subscript t$ } is a Gaussian white noise process, we need to verify that:

The variable { $text X end text subscript t$ } has mean zero.

The variable { $text X end text subscript t$ } has unit variance.

The variables { $text X end text subscript t$ } and { $text X end text subscript t minus 1 end subscript$ } are not i.i.d.