In this problem we are interested in comparing the performance of DSBSC modulation scheme with the performance of an FM modulation scheme. In both cases, the message signal is $m(t)=A_{m}cos(2D_{m}t)$ and the modulated signal $x(t)\{$whether FM or DSBSC $\}$ is corrupted by additive white noise with power spectral density $\frac{N_o}{2}.$ The received signal is $r(t)=x(t)+n_w(t).$ The following parameters are given:

$A_m=1$ volts, $f_m=10Hz,No={10}^{-5}$watt$\frac{s}{H}z$

a$)$ System #$1$: DSBSC

The transmitted signal is $x_{DSB}(t)=A_{d}m(t)cos(2P{f}_{c}t)$

$f_c=50MHz$

$A_d=10$ volts

After passing through the front end BPF $($assume ideal. What is the BW of this filter?$)$ the received signal is demodulated using a coherent demodulator as discussed in class.

Sketch the receiver structure and find the SNR at the output of the demodulator.

b$)$ System #$2$: FM

$x_{FM}(t)=A_{f}cos(2f_{c}t+2k_f{\displaystyle \underset{0}{\overset{t}{}}}m()d)$

The transmitted signal is

$f_c=50MHz$

$k_f=30H\frac{z}{?}$ volts

$A_f=10$ volts

After passing through the front end BPF $($assume ideal. What is the BW of this filter?$)$ the received signal is demodulated using the standard FM demodulator discussed in class. Sketch the receiver structure and find the SNR at the output of the demodulator.

c$)$ For this part of the question, choose the value of $A_f$ so that $x_{FM}(t)$ has the same power as $x_{DSB}(t).$ What would the minimum required bandwidth of $x_{FM}(t)$ so that the SNR at the output of the FM demodulator be $6$ times the SNR at the output of the coherent DSBSC demodulator.