An initial attempt I've done (although mostly likely wrong)
Since the PSD is constant, I believe [imath]W(t)[/imath] is simply a delta function [imath]\delta (t)[/imath] by taking the inverse Fourier transform of 1.
[imath] X=\displaystyle\int_0^1 g(t)W(t)dt[/imath] = [imath] \displaystyle\int_0^1 g(t)\delta (t)dt[/imath] = [imath]g(0) = 1[/imath] (hopefully I can use the sifting property of the delta function)
[imath] Y=\displaystyle\int_0^1 h(t)W(t)dt + \displaystyle\int_1^2 h(t)W(t)dt [/imath] = [imath] \displaystyle\int_0^1 h(t)\delta (t)dt + \displaystyle\int_1^2 h(t)\delta (t)dt = 1+1=2[/imath]
So, to calculate the correlation coefficient I would use the formula below, but now I need the variances. Do I use [imath] Var(X) = E(X^2) - (E(X))^2 [/imath]?
An initial attempt I've done (although mostly likely wrong)
Since the PSD is constant, I believe [imath]W(t)[/imath] is simply a delta function [imath]\delta (t)[/imath] by taking the inverse Fourier transform of 1.
[imath] X=\displaystyle\int_0^1 g(t)W(t)dt[/imath] = [imath] \displaystyle\int_0^1 g(t)\delta (t)dt[/imath] = [imath]g(0) = 1[/imath] (hopefully I can use the sifting property of the delta function)
[imath] Y=\displaystyle\int_0^1 h(t)W(t)dt + \displaystyle\int_1^2 h(t)W(t)dt [/imath] = [imath] \displaystyle\int_0^1 h(t)\delta (t)dt + \displaystyle\int_1^2 h(t)\delta (t)dt = 1+1=2[/imath]
So, to calculate the correlation coefficient I would use the formula below, but now I need the variances. Do I use [imath] Var(X) = E(X^2) - (E(X))^2 [/imath]?
