Continuous time Gaussian process

[rsingh628]

Hello. This may not be the right place to post, so apologies.
I am stuck on how to begin with the following problem on continuous time Gaussian processes. How do approach the integrals? I do know that W(t) has a constant PSD which would mean the autocorrelation function is a delta function. In the frequency domain, g(t) and h(t) would become sinc functions.

 

[rsingh628]

An initial attempt I've done (although mostly likely wrong)

Since the PSD is constant, I believe $W(t)$ is simply a delta function $\delta (t)$ by taking the inverse Fourier transform of 1.

$X=\displaystyle\int_0^1 g(t)W(t)dt$ = $\displaystyle\int_0^1 g(t)\delta (t)dt$ = $g(0) = 1$ (hopefully I can use the sifting property of the delta function)

$Y=\displaystyle\int_0^1 h(t)W(t)dt + \displaystyle\int_1^2 h(t)W(t)dt$ = $\displaystyle\int_0^1 h(t)\delta (t)dt + \displaystyle\int_1^2 h(t)\delta (t)dt = 1+1=2$

So, to calculate the correlation coefficient I would use the formula below, but now I need the variances. Do I use $Var(X) = E(X^2) - (E(X))^2$?

An initial attempt I've done (although mostly likely wrong)

Since the PSD is constant, I believe $W(t)$ is simply a delta function $\delta (t)$ by taking the inverse Fourier transform of 1.

$X=\displaystyle\int_0^1 g(t)W(t)dt$ = $\displaystyle\int_0^1 g(t)\delta (t)dt$ = $g(0) = 1$ (hopefully I can use the sifting property of the delta function)

$Y=\displaystyle\int_0^1 h(t)W(t)dt + \displaystyle\int_1^2 h(t)W(t)dt$ = $\displaystyle\int_0^1 h(t)\delta (t)dt + \displaystyle\int_1^2 h(t)\delta (t)dt = 1+1=2$

So, to calculate the correlation coefficient I would use the formula below, but now I need the variances. Do I use $Var(X) = E(X^2) - (E(X))^2$?