Analogue to digital conversion comprises both sampling and quantisation.

Samples of a continuous time signal $$are equal to its values at times $,$ where $$is the sampling period and $$is an integer, $.$

$1.$ This question considers two different quantisers. Each has a gain of one. The output of the mid$-$step quantiser is equal to the nearest integer to the value of the input. The output of the mid$-$rise quantiser is equal to $0\mathrm{.}5$ for input values in the range $0$ to $1,1\mathrm{.}5$ for input values in the range $1$ to $2,$ and so on$.$