$7\mathrm{.}25$ A $0-$ to $10-$V$,4-$bit successive approximation A$/$D converter is used to measure an input voltage of $4\mathrm{.}9$ V $.$

a$.$ Determine the binary representation and its analog approximation of the input signal. Explain your answer in terms of the quantization error of the $\frac{A}{D}$ converter.

b$.$ If we wanted to ensure that the analog approximation be within $2\mathrm{.}5$ mV of the actual input voltage, estimate the number of bits required of an $\frac{A}{D}$ converter.