Question $1$: Nyquist Bandwidth and Project $2(20$ points$)$

A communication system has a bandwidth of $250$ MHz and adopt raised$-$cosine pulse shaping with the roll$-$off factor, $=0\mathrm{.}35$ and BPSK modulation.

What is the data rate it can achieve?

What is the highest achievable data rate of the system.

Illustrate and annotate the waveforms and spectra in the above two cases with corresponding roll off factors.

A raised$-$cosine pulse reaches its peak of $1$ at $t=0.$ Tabulate the time, $t,$ when a raised$-$cosine pulse reaches the secondary largest peak and its values through your codes in Project $2?$

$\backslash$table$[[,$The value of the secondary largest peak$],[1\mathrm{.}0,],[0\mathrm{.}75,],[0\mathrm{.}5,],[0\mathrm{.}25,],[0,]]$

Sketch the waveform of raised cosine when the bit stream is $-1,1,-1,1,1,-1,1,-1,-1.$

Label the eye diagrams generated from the homework with five cases of $A(0,0\mathrm{.}3),B(0\mathrm{.}5,0\mathrm{.}1),C(1,0\mathrm{.}1),D(0\mathrm{.}25,0\mathrm{.}3),$ and $E(0\mathrm{.}75,0\mathrm{.}3)$ where the first element is alpha and second noise level.