Variance of filtered white noise

Question

I was asked a question, as posted here, and the answer given is (A) i.e. $\frac{3}{2} A^2 N_0$.

My solution steps was:

1. Finding the mean of the output process: Since input is gaussian the output will be also Gaussian, as per property of Gaussian variable.

2. Finding the mean square value (MSV) of the process i.e. $\mathbb{E}(Y^2)$.

3. Variance = MSV$-\mathbb{E}(Y)^2$

But I am confused do mean of output process will be zero? Since the input is zero-mean white Gaussian noise if yes how?

Is there any other way to solve the question?

Answer 1

The output process is clearly zero-mean because the LTI system cannot add a mean to the zero-mean input process. The variance of the filtered process is given by

$$E\{Y^2\}=\frac{N_0}{2}\int_{-\infty}^{\infty}|h(t)|^2dt\tag{1}$$

which results in $3A^2N_0/2$.