Digital Calibration of Amplifier Finite DC Gain and Gain Bandwidth in MASH $\Sigma\Delta$ Modulators

In this brief, a digital background calibration technique is proposed to improve the resolution of discrete-time multistage noise-shaping (MASH) sigma-delta (ΣΔ) modulators. The circuit imperfections of switched-capacitor discrete-time integrators (DTIs) degenerate the modulator output resolution, particularly in MASH structures, due to the quantization noise leakage of the early stages at the modulator output. An accumulative error model is proposed to define the errors induced by the limited dc gain and gain bandwidth (GBW) of DTIs as a function of their outputs. Compensating for these errors is possible through two steps, which are the DTI output digital approximation and adaptive filter correction. To identify the adaptive filter parameters, 1-b pseudorandom noise test signals are injected to the inputs of the DTIs. Then, the effect of these test signals is investigated by using a correlation-based least-mean-square algorithm. Simulation results confirm the effectiveness of the proposed method to eliminate the errors associated with the finite dc gain and GBW. Consequently, relaxed amplifiers with lower power consumption can be used.