$($b$)$ Below figure $($a$)$ shows a system for filtering continuous$-$time signal using a discrete$-$time filter. The input signal $x_a(t)$ is first converted to discrete$-$time signal $x[n]$ using an ideal ADC with sampling rate $F_S=\frac{1}{T}*H(e^j)$ is a lowpass filter with cutoff frequency of $\frac{}{4}.$ The filter $y[n]$ is then reconstructed to analogue signal $y_r(t)$ using an ideal DAC. Assume the spectrum $x_a(j)$ of $x_a(t)$ is given in $($b$).$

$($a$)$

$($b$)$

i$)$ Find the maximum value of sampling period $T$ to avoid aliasing in the ADC; $(3\text{'})$

ii$)$ If $\frac{1}{T}=20kHz,$ sketch the spectrum of $x[n]$ and $y[n]$;$(6^{\text{'}})$

iii$)$ Using sampling frequency in ii$),$ determine the spectrum of reconstructed signal $y_r(t).(3\text{'})$