$($Sampling$)$ Consider the following system to process the continuous signal in a discrete$-$time system. Let $x_c(t)=\frac{sin(40t)}{t}$ has the Fourier transform as $x_c(j)=1$ for $||40$ and $x_c(j)=0,$ otherwise.

$($a$)$

$($a$)$ Find the condition of the sampling time $T$ so that $x_c(t)$ can be recovered perfectly from the sampling signal $x_p(t)=x_c(t)p(t)$ by drawing the Fourier transform $x_p(j)$ of $x_p(t).$

$($b$)$ Find DTFT $x(e^j)$ of $x[n]=x_c(nT)$ when $T=\frac{1}{100}.$

$($c$)$ Find $y[n]$ and $y_c(t)$ for input $x[n]$ given in $($b$)$ when $H(e^j)=e^{-j}$ for $||<mfrac><mrow><mi></mi></mrow><mrow><mn>2</mn></mrow></mfrac>$ and $H(e^j)=0$ for and $H(e^j)=H(e^{j(+2)}).$

$($d$)$ Find $y[n]$ if $h[n]=(-1{)}^n\frac{sin(\frac{}{2}n)}{n}$ for the input given in $($b$).$

$($Sampling$)$ Consider the following system to process the continuous signal in a discrete$-$time system. Let $x_c(t)=\frac{sin(40t)}{t}$ has the Fourier transform as $x_c(j)=1$ for $||40$ and $x_c(j)=0,$ otherwise.

$($a$)$

$($a$)$ Find the condition of the sampling time $T$ so that $x_c(t)$ can be recovered perfectly from the sampling signal $x_p(t)=x_c(t)p(t)$ by drawing the Fourier transform $x_p(j)$ of $x_p(t).$

$($b$)$ Find DTFT $x(e^j)$ of $x[n]=x_c(nT)$ when $T=\frac{1}{100}.$

$($c$)$ Find $y[n]$ and $y_c(t)$ for input $x[n]$ given in $($b$)$ when $H(e^j)=e^{-j}$ for $||<mfrac><mrow><mi></mi></mrow><mrow><mn>2</mn></mrow></mfrac>$ and $H(e^j)=0$ for and $H(e^j)=H(e^{j(+2)}).$

$($d$)$ Find $y[n]$ if $h[n]=(-1{)}^n\frac{sin(\frac{}{2}n)}{n}$ for the input given in $($b$).$