Find the mean and auto correlation function of Xt Acosomega

Find the mean and auto correlation function of X(t) = Acos(omega t + pi) where omega and pi fixed and A is a random variable with normal density function of zero mean and variance equal to two as X(t) a W.S.S random process A cos(omega t + pi)

Solution

Data.

X(t) = Acos(wt + )

A and are independent and we can spplit\'em.

A = random variable

= [-, ]

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Answer.

Mean and autocorrelation functions provide a partial description of a random process. Only in certain cases (Gaussian), they can provide a fully description.

Mean function:

_X(t) = E {Acos(wt + )} = E{A}E{cos(wt+)} = _AE{cos(wt+)}

solving for = [-, ];

= _A- cos(wt + ) 1/2 d

= -_A/2 sin(wt + )_-|

= -_A/2 [sin(wt + ) - sin(wt - )] -- (Trigonometric identity)

2_A/2 cos(2wt/2) - sin(2/2) = 0

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Autocorrelation function:

R_xx(t₁,t₂) = E{X_t1,X_t2} = E{A²cos(wt₁ +)cos(wt₂ +)} (Trigonometric identity)

= E{A²}E{1/2 [cos(w(t₁-t₂))] + 1/2 [cos[w(t₁+t₂)+2]}

= E{A²} (1/2 E{cos[w(t₁-t₂)]} + 1/2 E{cos[w(t₁+t₂)+2]}

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Analizing both members;

(1/2 E{cos[w(t₁-t₂)]} = nothing random

1/2 E{cos[w(t₁+t₂)+2]} = 0

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then;

= E{A²}1/2 cos[w(t₁-t₂)]

The autocorrelation function is just a function of the time difference = t₁-t₂.

= 1/2 E{A²}cos(w)

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