Given a $5-$bit A$/$D converter, the input signal is $2$ V and the input range is $5$ to $5$ V$.$

$(1)$ What is the resolution of the A$/$D converter $($using the formular that has $2$ n $-1$ as emphasis in the

class?

$(2)$ What is the digital output $($using the algorithm in the lecture$)$ of $2$ V$?$ Please show all steps for full

credit. Hint: shift $-5$ ~ $+5$ V to $0$ ~ $10$ V$,$ then, use the method demonstrated in the lecture.