$(20)$ Figure $1$ shows an overall system for filtering a continuous$-$time signal using a discrete$-$time filter. The frequency response of reconstruction filter $H_r(j)$ is shown in Figure $2.$ The continuous$-$time Fourier transform $($CTFT$)$ of $x_c(t),$ i$.$e$.x_c(j)$ is shown in Figure $3.$

Figure $1$

Figure $2$

Figure $3$

$($a$)(5)$ Sketch the CTFT of $x_s(t),$ i$.$e$.x_s(j)$ and discrete$-$time Fourier transform $($DTFT$)$ of $x[n],$ i$.$e$.x(e^j)$ when the sampling frequency, $\frac{1}{T}=20kHz.$

$($b$)(10)$ Consider a discrete$-$time lowpass filter in Figure $4.$ Note that for a certain range of values of $T,$ the overall system, with input $x_c(t)$ and output $y_r(t),$ is equivalent to a continuous$-$time lowpass filter with frequency response $H_{\text{e}\text{f}\text{f}}(j)$ sketched in Figure $5.$ Determine the range of $T$ for which aforementioned statement becomes true when $x_c(j)$ is band$-$limited to $||2{10}^4$ as shown in Figure $3.$

$($c$)(5)$ For the range of values determined in $($b$),$ sketch ${}_c$ as a function of $\frac{1}{T}.$