Now, assume I have a white noise, [tex]n(t)\tilde \ N(0,1)[/tex], i.e gaussian with zero mean and variance 1, and it goes through a Hilbert filter, i.e we get:(adsbygoogle = window.adsbygoogle || []).push({});

$$ \hat{n}(t) = \int_{-\infty}^{\infty} \frac{1}{t-\tau} n(\tau) d\tau $$

I read that [tex]\hat{n}[/tex] should be also a gaussian, because this is an LTI system.

But when I want to calculate its variance I get zero, so I guess I did something wrong here.

$$ E[\hat{n}^2(t) ] =\int \int_{\mathbb{R}^2} \frac{1}{(t-\tau_1)(t-\tau_2)} E[n(\tau_1)n(\tau_2)]d\tau_1 d\tau_2 $$

Now because the noise has zero mean value and variance 1 we should have: $$E[n(\tau_1)n(\tau_2)]=\delta(\tau_1 - \tau_2)$$

But when I plug I get zero. I read that we should take the principal value of the integral but I don't think it changes this.

Where did I get it wrong?

Thanks in advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# White noise going through a Hilbert filter.

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**