# Fourier transform

1. Apr 9, 2012

### peter.a

1. The problem statement, all variables and given/known data

Just something I am working through and am a bit stuck on.

2. Relevant equations

I have taken the fourier transform of an RC circuit which gives me :
Y(ω)=((X(ω))/(1+iωτ))
If i take the voltage across the circuit as white noise then i get:
Y(ω)=σ^²/2π/(1+iωτ))
How can i find the variance function and covariance of this fourier transform
3. The attempt at a solution
I am not sure how to do this

2. Apr 12, 2012

### marcusl

First, welcome to PF. Regarding your questions, why are you trying to find these? Y is a complex function, so what do you even mean by variance? Individual variances of real and imaginary parts? Of its power spectral density |Y|^2?

The variances of Real(Y) and Imag(Y) are the third and second moments of a function known in statistics as a Cauchy distribution, and these moments are undefined. |Y|^2 is the Cauchy distribution itself, and it has undefined first and second moments (mean and variance).

Last edited: Apr 12, 2012
3. Apr 12, 2012

### peter.a

Of the spectral density which is X(ω)/(1+(ωτ)^2)) i had incorectly stated it in the post.