$(2)[30$ pts$]$ Your unit's engineers have devised a receiver architecture for collecting RF emissions from the Krasnovian communication system in Problem $1.$ Your new boss is $($so far$)$ very impressed with your abilities and has tasked you to validate the proposed receiver design and assess its ability to recover signals from $\backslash ($ x$\_\{$F D M$\}($t$)\backslash ).$

$($a$)[8$ pts$]$ Given the Local Oscillator $($LO$)$ and Band Pass Filter $($BPF$)$ parameters provided below, sketch $($manually$)$ the magnitude spectrum appearing at the output of the filter. Do not attempt to sum signals within the filter passband$-$instead, just overlay them as they would alias into the filter $($relative shape, location, etc.$).$ Be as accurate and to scale as possible. Label all signals, axes, and amplitudes with the proper units.

$($b$/$c$)[8$ pts each$]$ On the next page, sketch the receiver output $($magnitude$)$ spectrum that would result if alternative LO signals $\backslash ($ x$\_\{$L O$+\}($t$)=\backslash$exp $\backslash$left$($j $2\backslash$pi f$\_\{$L O$\}$ t$\backslash$right$)\backslash )$ and $\backslash ($ x$\_\{$L O$-\}($t$)=\backslash$exp $\backslash$left$(-$j $2\backslash$pi f$\_\{$L O$\}$ t$\backslash$right$)\backslash )$ were used. Label all signals, axes, and amplitudes with the proper units. Be as accurate and to scale as possible

$($d$)[6$ pts$]$ Assume that if two or more signals overlap at the filter output, they cannot be recovered due to the aliasing that has occurred. Identify which of the three receiver designs should be used in order to recover the most signals? Justify your answer.

$\backslash (\backslash$left$|$E$\_\{$o$\}($f$)\backslash$right$|\backslash )$ Measured Baseband Message Signal Spectra

$($High Confidence; ALL Magnitudes normalized to $\backslash (\backslash$mathbf$\{1.0\}\backslash )$ volt$)$