 Sub-step estimation with uniform dither

Let a constant input be $x=(k+f)\Delta$ with $0\le f\le 1$, between consecutive quantization levels $k\Delta$ and $(k+1)\Delta$. Add independent centered uniform dither $U\sim \mathrm{Uniform}[-\Delta /2,\Delta /2]$ before nearest-level rounding, and assume that neither code clips.

The upper code is selected exactly when $U\ge (1/2-f)\Delta$. This subinterval occupies fraction $f$ of the uniform distribution's support, so $Q=k\Delta +\Delta B$ with $B$ a Bernoulli random variable of parameter $f$. Thus $\mathbb{E}[Q]=x$ and $\mathrm{Var}(Q)=f(1-f){\Delta }^2$. Averaging $N$ independent conversions givesFor example, an input one quarter of a step above the lower code produces the upper code one quarter of the time. Its averaged code can estimate a sub-step voltage, with a finite remaining variance. Centering, independence and absence of clipping are assumptions of this calculation; a shifted dither distribution can bias the result.