Let X be a random process. X(t) = A*Cos(wt+θ) ; where A and w are constants. The only random thing is θ. Lets say θ has a probability density function,
f(θ)= 1/2pi for 0<θ<2pi and zero elsewhere.
My confusion starts here.
Auto correlation function R(t) is defined as E[X(t) * X(t + τ)]. fine! Afterall closer the values X(t) and X(t+τ) are, higher will the "correlation" value. It makes sense till now.
Then, the text says that
R(t) = ∫X(t)*X(t+τ)*f(θ)dθ [integration limits from 0 to 2pi]. I dont understand why we have f(θ) here. θ was present within X(t) i.e inside Cos term, I dont see why its effect is in multplication like this.
Ofcourse since f(θ) is 1/2pi, its like finding the mean of ∫X(t)*X(t+τ)dθ afterall that is what expectancy is.
But still I feel hole.What if pdf of θ was piece wise continuous like
f(θ) = 1/2pi ; 0<θ<pi = 1/6pi ; pi<θ<4pi = 0 ; elsewhere
Kindly help me understand why we have f(θ) term multiplied there. Thanks!
Let's change the notation a little bit. Let $g(t,\theta) = A\cos(\omega t + \theta) = X(t)$. My guess is that the text was supposed to be,
$$R(t) = E[X(t)X(t+\tau)] = E[g(t,\theta)g(t+\tau,\theta)] = \int g(t,\theta)g(t+\tau,\theta)f(\theta)\,d\theta.$$
I think they simply substituted $X(t)$ for $g(t,\theta)$ and $X(t+\tau)$ for $g(t+\tau,\theta)$ in the expression above, hiding the dependence of $X(t)$ and $X(t+\tau)$ on $\theta$.