I'm trying to solve a random process problem:(adsbygoogle = window.adsbygoogle || []).push({});

"Let [itex]x(n) = v(n) + 3v(n-1)[/itex] where [itex]v(n)[/itex] is a sequence of independent random variables with mean µ and variance s². Whats its mean and autocorrelation function? Is this process stationary? Justifiy."

To discover the mean, I applied the expected value at x(n):

[itex] E[x(n)] = E[ v(n) ] + 3 E[v(n-1)] = \mu + 3\mu = 4\mu[/itex]

To obtain the autocorrelation function:

[itex]r(n_1,n_2) = E[x(n_1)x(n_2)] = E[v(n_1)v(n_2) +3v(n_1)v(n_2 -1) + 3v(n_1-1)v(n_2) + 9v(n_1 -1)v(n_2-1)] [/itex]

Since v(n) is an independent sequence..

[itex]r(n_1,n_2) = E[v(n_1)] E[v(n_2)] + 3E[v(n_1)] E[v(n_2-1)] + 3 E[v(n_1-1)] E[v(n_2)] + 9 E[v(n_1 -1 ) ] E[v(n_2 -1)] [/itex]

But [itex]E[v(n)] = \mu[/itex], then

[itex]r(n_1,n_2) = \mu^2 + 3\mu^2 + 3 \mu^2 + 9 \mu^2 = 16\mu^2 [/itex]

I'm afraid that I made mistake evaluating the autocorrelation function..So, is there any error? If yes, could someone correct me? Thanks!

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# Mean and autocorrrelation function

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