In this problem we study the effect of Nyquist$-$rate sampling of a cosine signal on its ideal reconstruction. Consider a sinusoidal signal $x_c(t)=cos(2F_{0}t+{}_0).$ It is sampled at a rate of $F_s=100Hz$ and the resulting samples are applied to an ideal DAC to obtain $y_r(t).$

$($a$)$ Determine $y_r(t)$ if $F_0=10,20,$ and $40$ Hz and ${}_0=0$ radians. How does $y_r(t)$ compare with $x_c(t)?$

$($b$)$ Determine $y_r(t)$ if $F_0=50Hz$ and ${}_0=0,\frac{}{3},\frac{}{2},\frac{2}{3},$ and $$ radians. How does $y_r(t)$ compare with $x_c(t)?$

$($c$)$ From your observation of results in $($b$),$ express $y_r(t)$ in terms of ${}_0.$