$1.$

A message $\backslash (\backslash$mathrm$\{$m$\}(\backslash$mathrm$\{$t$\})=5\backslash$cos $(2000\backslash$pi $\backslash$mathrm$\{$t$\})\backslash )$ is transmitted using the following modulator:

a$)$ Sketch the amplitude spectrum at the output of the modulator. Indicate all relevant numerical values on the spectrum.

The receiver shown below is used to demodulate the signal. Assume that the noise in the channel is white with a power spectral density $\backslash (\backslash$mathrm$\{$S$\}\_\{\backslash$mathrm$\{$n$\}\}(\backslash$mathrm$\{$f$\})=10^\{-8\}\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$Hz$\}\backslash ).$

b$)$ Compute the signal$-$to$-$noise ratio at the output of the BandPass Filter.

c$)$ Compute the signal$-$to$-$noise ratio at the output of the receiver.

d$)$ Compute the figure of merit F $.$

e$)$ Suggest what change must be made to the receiver so that F is improved.

f$)$ Design: In which application would you use this DSB$-$SC modulation scheme instead of another Amplitude Modulation $($AM$)$ scheme. Explain.

Note: B in the figure below is the bandwidth of the message.