Past exam of the physics A course of the University of Cambridge / 2010 / Paper 2 / Section A / 5 / Solution

A small constant input can repeatedly produce the same analogue-to-digital converter code, so averaging those codes cannot reveal its position within a conversion interval. dither makes adjacent codes toggle; their relative frequencies then encode changes smaller than one conversion step. With adequate oversampling, low-pass filtering and averaging reduce the added, sufficiently uncorrelated noise while retaining the slow signal.

For independent sample errors, the averaged standard deviation scales as $N^{-1/2}$; ideally four times as many samples gain about one effective bit. For example a constant input $0.2\Delta$ rounded to levels separated by $\Delta$ would always return zero. Adding uniform dither on $[-\Delta /2,\Delta /2]$ gives output $\Delta$ with probability $0.2$ and zero otherwise: the expected average is $0.2\Delta$, with error $0.4\Delta /\sqrt{N}$. The benefit concerns the filtered estimate, not a more accurate individual conversion. Excessive dither wastes range, and averaging does not correct calibration bias, correlated drift or arbitrary converter nonlinearity.^[ref]