Sub-step estimation with uniform dither ($\mathrm{Var}(\overline{Q})=f(1-f){\Delta }^2/N$)

Let a constant input lie a fraction $f$ of the way between two consecutive quantizer levels separated by $\Delta$. With independent uniform additive dither spanning one step and nearest-level rounding, the two output codes occur with probabilities $1-f$ and $f$. Their mean equals the input and their sample-average variance is the displayed expression. This simple model shows both the sub-step information and the remaining noise cost.