a$.$A sensor in a control system has a voltage range $3\mathrm{.}5$ V to $+3\mathrm{.}5$ V$.$ When the signal is converted from a continuous$-$time signal to a discrete$-$time signal, a $3-$bit quantisation system is used. The binary code $000$ is assigned to $3\mathrm{.}5$ V and the binary code $111$ is assigned to $+3\mathrm{.}5$ V$.$

i$.$Where is the midpoint or $0$ V level located in the binary codes and what is the quantisation interval of the system?

ii$.$What is the binary code output from the system when the input is $2\mathrm{.}1$ V and what quantisation error has been introduced in the conversion process?

iii.The input range of the system is modified so it is now in the range $0$ V to $+3\mathrm{.}5$ V$.$ The binary code $000$ is assigned to $0$ V and the binary code $111$ is assigned to $+3\mathrm{.}5$ V$.$ State what the quantisation error is for an input signal of $2\mathrm{.}1$ V with this reduced input signal range. Compare you answer to that of part $($ii$)$ and comment on any similarities or differences between the errors.

i$.$It is now specified that the original system in $($a$)$ with a voltage range of $3\mathrm{.}5$ V to $+3\mathrm{.}5$ V$,$ must not have a quantisation error greater than $10$ mV$.$

Calculate the minimum number of bits that should be used to represent the quantised signal?

ii$.$Data from the sensor is to be relayed in $$real$$ time $($that is$,$ as a live transmission$)$ over a communication system that operates at up to $3600$ bit s$1.$ The original signal contains useful information in frequencies up to $400$ Hz$.$ Discuss whether the sampling rate ensures the transmitted discrete$-$time signal contains the full range of frequencies in the continuous$-$time signal.