A signal $x(t)=30sinc(60t)$ is applied to a nonlinear modulator which is depicted in the following figure. Assume the carrier frequency, $f_c,$ is $1$ kHz and the nonlinear device has the response governed by: $v_{\text{o}\text{u}\text{t}}=a_1{v}_{in}+a_2(v_{in}{)}^3.$

$($a$)$ Calculate and sketch the spectrum of $x(t).$

$($b$)$ Determine the output signal $v_{\text{o}\text{u}\text{t}}.$

$($c$)$ Give the conditions on the required center frequency $f_0$ and the bandwidth of the bandpass filter in relation to $f_c$ and the bandwidth of the signal $x(t)$ such that this system will produce a standard DSB signal.

$($d$)$ Determine the output DSB signal $s(t).$

$($e$)$ Determine values of $a_1$ and $a_2$ such that the maximum value of the amplitude of the output signal $s(t)$ is $10.$

Fig. P$.1$