Zero mean white Gaussian noise with a two-sided power spectral density of $4 \mathrm{~W} / \mathrm{kHz}$ is passed through an ideal lowpass filter with a cut-off frequency of $2 \; \mathrm{kHz}$ and a passband gain of $1$, to produce the noise output $n(t)$.
- Obtain the total power in $n(t)$.
- Find the autocorrelation function $\mathrm{E}[n(t)$ $n(t+\tau)$ ] of the noise $n(t)$ as a function of $\tau$.
- Two noise samples are taken at times $t_{1}$ and $t_{2}$. Find the spacing $\left|t_{1}-t_{2}\right|$ so that the product $n\left(t_{1}\right) n\left(t_{2}\right)$ has the most negative expected value and obtain this most negative expected value.