Stability Analysis of Bandpass Sigma–Delta Modulators for Single and Dual Tone Sinusoidal Inputs

In this paper, the stability boundaries of higher order 1-bit bandpass (BP) Σ-Δ modulators for single- and dual-sinusoidal inputs are determined using quasi-linear modeling together with the describing-function method. Closed-form mathematical expressions are derived for the stability curves for different quantizer gain values. The stability prediction for the maximum stable input limits for eighth- and tenth-order (i.e., effective fourth- and fifth-order) inverse-Chebyshev- and Butterworth-based BP Σ-Δ modulators for different stopband attenuation and bandwidths are established. The theoretical results are shown to be in good agreement with the simulation results for a variety of input amplitudes, bandwidths, and modulator orders. The proposed mathematical models of the quantizer will immensely help speed up the design of higher order BP Σ-Δ modulators.