Problem $1(20$pts$)$

Determine the bit rate $\backslash (\backslash$left$($R$\_\{$b$\}\backslash$right$)\backslash ),$ symbol rate $\backslash (\backslash$left$($R$\_\{$S$\}\backslash$right$)\backslash ),$ and transmission bandwidth $\backslash (\backslash$left$($B$\_\{$T$\}\backslash$right$)\backslash )$ of a digital communication system under the following settings.

$($a$)(10$pts$)$

i$.$ The message signal has a bandwidth $\backslash ($ W$=15\backslash$mathrm$\{$kHz$\}\backslash ).$

ii$.$ The sampling frequency is $2\mathrm{.}5$ times the Nyquist rate.

iii. Each sample is quantized using $8$ bits.

iv$.$ The quantized samples are transmitted using $16-$ary PSK modulation.

v$.$ A raised cosine pulse shaping filter with a roll$-$off factor $\backslash ((\backslash$alpha$)\backslash )$ of $0\mathrm{.}25$ is used.

$($b$)(10$pts$)$ The same settings as in $($a$),$ but the system employs Frequency Division Multiplexing $($FDM$)$ to transmit $64$ different message signals simultaneously.