Let a certain channel has a uniform double$-$sided noise power spectral density $S_n(f)=0\mathrm{.}5{10}^{-3}\frac{W}{H}z.$ An amplitude$-$modulated signal $x_c(t)=800[1+0\mathrm{.}5m_n(t)]cos(2f_{c}t)$ is transmitted through this channel. Suppose $m(t)=5cos(10{10}^{3}t).$ The carrier frequency is $100$ MHz $.$ The input signal to the receiver first passes through an appropriate ideal bandpass filter and then is fed into an envelope detector for demodulation. Assume the envelope detector operates above the threshold.

$($a$)(5$ pts$)$ What is the transmission efficiency $?$

$($b$)(2$ pts$)$ What is the input $SNRSN{R}_i$ of the demodulator?

$($c$)(2$ pts$)$ What is the output $SNRSN{R}_o$ of the demodulator?

$($d$)(2$ pts$)$ What is the channel SNR $SN{R}_c?$

$($e$)(4$ pts$)$ What is the Detection gain and the Figure of Merit of the system? What is the relationship between them and the transmission efficiency $?$

$($f$)(5$ pts$)$ Sketch the figure of synchronous detection and envelope detection via $SN\frac{R_o}{S}N{R}_i.$ Compare the two detection methods and analyze their pros and cons, and point out how do we choose them.

$($g$)(2$ pts$)$ Compare the noise performance of DSB$-$SC$,$ SSB$-$SC and AM $($SSB$-$LC$).$ Give your reasons.