Gaussian Channel: Capacity and Explanation
1) Why is there the filter h(t)? What does it represent? Is it a physical filter that we put in the channel (and if yes, why?) or is it used to model the fact that a channel is naturally a low pass filter?


Real channels filter the signal passed through them.

Transmission lines are lossy at high frequencies. Optical fibers have dispersion that smears out high frequency signals. To use a free space channel you have to pass your signal through a transmitter and a receiver that have bandwidth restrictions.

The channel response \$h(t)\$ represents whatever filtering effect the channel has.


  2) Why is the signal X sampled?


To obtain quantifiable information from it.


  3) Should X be band limited or not? And why?


If it wasn't originally, it is after passing it through the ideal band-pass filter \$h(t)\$.