# White noise going through a Hilbert filter.

1. Jun 22, 2013

### MathematicalPhysicist

Now, assume I have a white noise, $$n(t)\tilde \ N(0,1)$$, i.e gaussian with zero mean and variance 1, and it goes through a Hilbert filter, i.e we get:

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

I read that $$\hat{n}$$ 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?