A necessary and sufficient condition for mismatch shaping in multi-bit DACs

Mismatch-shaping digital-to-analog converters (DACs) are enabling components in most of today´s high performance delta-sigma data converters. These multi-bit DACs ensure that the noise created by static component mismatches in the DAC is spectrally shaped so that most of its power resides outside of the signal band. Although most mismatch-shaping DACs adhere to a common architecture consisting of a digital encoder and a bank of 1-bit DACs, the theoretical analysis of the DAC noise has been tailored to the specific implementations. This paper presents a necessary and sufficient condition for a general multi-bit DAC to produce DAC noise that has desired properties in its PSD. As an example application, this paper uses the condition to analyze and compare several mismatch-shaping DACs.