Problem $1$:

A random message signal $M(t)$ is used to modulate a sinusoidal carrier, resulting in a DSB$-$SC signal $U(t)$ with a power spectral density given below $($only the positive frequency components are plotted, and the negative frequency components and positive frequency components are symmetric with respect to zero frequency$).U(t)$ is transmitted through a channel with a power attenuation of $10$ dB $.$

$($a$)$ Compute the power of the received signal $R(t)$ in the absence of noise.

$($b$)$ The channel in part $($a$)$ is affected by additive white Gaussian noise with a power spectral density of $\frac{N_0}{2}=0\mathrm{.}001\frac{W}{H}z.$ A synchronous demodulator is used at the receiver. Compute the signal$-$to$-$noise ratio at the output of the demodulator, in dB $.$

$($c$)$ If USB$-$AM modulation is used instead with the same additive white Gaussian noise with $\frac{N_0}{2}=0\mathrm{.}001\frac{W}{H}z,$ find the power of the received signal $R_{\text{U}\text{S}\text{B}}(t),$ and the $SNR$ at the output of the demodulator, in dB $.$