Variance of white noise and 1/f noise through a linear system

[iVenky]

Hi,

I am trying to solve this math equation on finding the variance of a noise after passing through a system whose impulse response is h(t)
X is the input noise of the system and Y is the output noise after system h(t)
if let's say variance of noise Y is
σ_y²=∫∫Rxx(u,v)h(u)h(v)dudv

where integration limits are from -∞ to +∞. Rxx is the autocorrelation function of noise X. Can you show that if Rxx (τ)=σ_x² δ(τ) (models a white noise), then

σ_y²=σ_x²∫h²(u)du (integration limits are from -∞ to +∞)

and if Rxx (τ)=σ_x² (models a 1/f noise), then

σ_y²=σ_x²(∫h(u)du)² (integration limits are from -∞ to +∞)

I don't understand the math behind statistics that well
Thanks

 

[marcusl]

This sounds like a homework problem. You need to post it in the appropriate Homework forum using the template to show your attempt at a solution.