Analysis and design of single-bit sigma-delta Modulators using the theory of sliding modes

The existence and stability conditions for a single-bit sigma-delta modulator operated in sliding mode are derived. Based on these conditions, a stable single-bit modulator of arbitrary order is synthesized. Also, an alternative interpretation on the modulator´s noise-shaping mechanism is presented. From the sliding mode aspect, the modulator is regarded as a device that performs frequency-weighted minimization of the modulation error, and the modulator´s loop filter as the weighting function. The noise-shaping equation derived from this aspect helps precisely predict the modulator performance. A fifth-order 2.2-MHz wide-band modulator with an effective resolution of 17 bits is designed as an illustrative example. The simulation result conforms the accuracy of the analysis. In the end, this paper presents a possible extension of single-bit to multibit modulation while still maintaining the sliding-mode operation.