$3.$ Binary data is transmitted over a wireless channel at the rate of $1000$ bits per second. The modulation scheme is BSPK with symbol values of $\backslash (\backslash$pm $1\backslash ).$ Assume the pulse shape $\backslash ($ p$($t$)\backslash )$ is rectangular with value $1$ over the entire symbol interval.

$($a$)$ For the linear modulation scheme described above, sketch the power spectral density of $\backslash (\backslash$tilde$\{$s$\}($t$)\backslash ).$ Be sure to label important frequencies.

$($b$)$ Calculate the average power in $\backslash (\backslash$tilde$\{$s$\}($t$)\backslash ).$

$($c$)$ Suppose the modulation scheme is changed to an $8-$ary system in which the symbol values are $\backslash (\backslash$pm $1,\backslash$pm $3,\backslash$pm $5,\backslash$pm $7\backslash ).$ The system still transmits data at a rate of $1000$ bits per second. The system still uses rectangular pulses with value $1$ that fill the entire symbol interval. By what factor is the bandwidth required for transmission reduced? By what factor is the transmitted power increased?