Analysis of digital gain error compensation in continuous-time cascaded sigma-delta modulators

This paper deals with an in-depth analysis of digital gain error compensation in continuous-time cascaded Sigma-Delta modulators. Illustrated by the example of a 2-1 cascaded modulator, it will be shown that this approach always restores the desired noise-shaping characteristic in cases where, except for the first stage, only first-order modulators are used in the cascade. However, a leakage term still exists which may increase the total in-band noise in case the first integrator of any stage in the cascade exhibits a gain variation. For any other architecture, the digital correction approach fails to truly compensate for gain errors as illustrated by the example of a 2-2 cascaded modulator. Low-order shaped quantization noise may then still determine the total in-band noise.