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Let {$x_t$} be the process given by $x_t$=$A\cos(wt)+B\sin(wt)$ where $A$ and $B$ are uncorrelated random variables with mean $0$ and variance $1$ and $w$ is a fixed frequency in the interval $[0, \pi]$. Find its autocovariance function at lag $k$, where $k=0,1,2,...$

I did:

$$\gamma{(k)}=E(x_tx_{t-k})=E[(A\cos(wt)+B\sin(wt))(A\cos(w(t-k))+B\sin(w(t-k)))]$$

Is there a way to make this easier to compute?

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  • $\begingroup$ Simply apply the formula for the covariance of linear combinations: that is, covariance is a bilinear function of the coefficients. $\endgroup$
    – whuber
    Feb 2, 2019 at 18:46
  • $\begingroup$ I found the duplicate with this search, which may have other helpful links. $\endgroup$
    – whuber
    Feb 2, 2019 at 20:23

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