$$

The message $m(t)=2cos(400t)+4sin(500t+\frac{}{3})$ modulates the carrier $C(t)=$Acos$(8000t),$ using DSB$-$AM$.$ Find the time domain and frequency domain representation of the modulated signal and plot the spectrum $($Fourier transform$)$ of the modulated signal.

In a DSB$-$AM system, the carrier $c(t)=$Acos$(2f_{c}t),$ and the message signal is given by $m(t)=sinc(t)+si{n}^2(t).$ Find the freguency domain representation and the bandwidth of the modulated signal.

A message signal $m(t)=cos(2000t)+2cos(4000t)$ modulates the carrier $,c(t)=100cos(2fct),$ where $f_c=1MHz$ to produce the DSB signal $u(t)=m(t)C(t).$

$(1)$ Determine the expression for the upper$-$sideband $($USB$)$ signal.

$(2)$ Sketch the spectrum of the USB signal.

Consider a bandpass signel $S(t)=50cos(22000t)+100cos(20000t)+150cos(18000t),$ with carrier freguency $f_c=10000Hz.$ Let $S_e(t)$ be the complex envelope of $S(t).$ Please find $S_l(t)$ and $.$

Consider a bandpass signal $x(t)$ with spectrum

$+(3+j4)(f-310000)+(5+j6)(f-320000)+(1-j2)(f+300000)$

$+(3-j4)(f+310000)+(5-j6)(f+320000)$

$(1)$ Is $x(t)$ a periodic signal? Explain your answer.

$(2)$ Find the complex envelope of $x(t).$