Consider the system shown in Figure $2($a$),$ which is represented by a block diagram. In this system, the signal $x(t)$ has the frequency spectrum shown in Figure $2($b$),$ and the signal $c(t)$ with fundamental frequency $f_c$ is as depicted in Figure $2($c$).$ The input to the nonlinear circuit is $v_{in}(t)=x(t)+c(t),$ and the output signal $v_{\text{o}\text{u}\text{t}}(t)=a{v}_{in}(t)+b{v}_{in}^2(t)$ of the non$-$linear circuit is passing through a filter with a transfer function $H(f)$ to result in the signal $x_c(t).$

Figure $2($a$).$ The block diagram of AM modulator using non$-$linear circuit.

Figure $2($D$).$ Frequency spectrum or the information signal $x(t)$

$2$

Answer the following questions.

a$)$ Show that the signal $c(t)$ can be rewritten as $c(t)=\frac{4}{}cos({}_{c}t)-\frac{4}{3}cos(3{}_{c}t)+\frac{4}{5}cos(5{}_{c}t)+$cdots by means of using Fourier series expansion. Find the Fourier transform of $c(t).$

b$)$ Investigate whether the square of the periodic signal $c(t),$ represented as $c^2(t),$ retains its periodic nature. Subsequently, simplify $c^2(t)$ and find its Fourier transform.

c$)$ Write the output signal $v_{\text{o}\text{u}\text{t}}(t)$ of the non$-$linear circuit in terms of $x(t),c(t),$ and the critical non$-$negative parameters $a0$ and $b0$ of non$-$linear circuit.

d$)$ Provide a filter $H(f)$ so that its output $x_c(t)$ becomes a conventional AM $($DSB$-$C$)$ signal with carrier frequency $f_c=\frac{1}{T_c}.$

Specify the center frequency, bandwidth, and gain of the provided filter. Then, write the time$-$domain expression for $x_c(t).$

Plot the frequency spectrum $x_c(f)=F\{x_c(t)\},$ and determine the bandwidth of $x_c(t).$

e$)$ Provide a transfer function $H(f)$ so that its output $x_c(t)$ becomes a VSB$-$C modulated signal with the specified bandwidth, and sketch this transfer function.

Then, express $x_c(t)$ in the time$-$domain as a convolution with $h(t)=F^{-1}\{H(f)\},$ and plot its frequency spectrum $x_c(f)=F\{x_c(t)\}.$

f$)$ Design and draw a block diagram for a synchronous demodulator that recovers the information signal $x(t)$ from $x_c(t)$ by utilizing the results from parts $($d$)$ and $($e$)$

$($ I need solutions of each section clearly$)$