On the monotonicity and linearity of ideal radix-based A/D converters

Both cyclic and pipelined analog-to-digital converters are getting more and more popular, as they are relatively easy to design and either has a high throughput (pipelined converters) or small area- and power-consumption (cyclic/algorithmic converters). To avoid saturation and to ensure effective digital calibration, in the analog stage(s) of these converters, instead of the ideal two, often a smaller nominal gain (called radix number) is used. In this paper, properties of these radix-based converters are discussed. First, it is shown that these types of converters produce nonmonotonic output. The causes of these phenomena are discussed in detail and a method to avoid nonmonotonicity is suggested. Second, it is shown that even the ideal subradix converters have limited linearity. Lower bound for the differential nonlinearity (DNL) is calculated. The results about monotonicity can be used either to quickly locate or avoid nonmonotonic code transitions in a converter. The derived expressions for the lower bound of the DNL can be used to estimate the minimum required number of cycles (stages) for a converter to push the DNL below the specification.