Refined error analysis in second-order ΣΔ modulation with constant inputs

Although the technique for sigma-delta (ΣΔ) modulation is well established in practice for performing high-resolution analog-to-digital (A/D) conversion, theoretical analysis of the error between the input signal and the reconstructed signal has remained partial. For modulators of order higher than 1, the only rigorous error analysis currently available that matches practical and numerical simulation results is only applicable to a very special configuration, namely, the standard and ideal k-bit k-loop ΣΔ modulator. Moreover, the error measure involves averaging over time as well as possibly over the input value. At the second order, it is known in practice that the mean-squared error decays with the oversampling ratio λ at the rate O(λ^-5). In this paper. we introduce two new fundamental results in the case of constant input signals. We first establish a framework of analysis that is applicable to all second-order modulators provided that the built-in quantizer has uniformly spaced output levels, and that the noise transfer function has its two zeros at the zero frequency. In particular, this includes the one-bit case, a rigorous and deterministic analysis of which is still not available. This generalization has been possible thanks to the discovery of the mathematical tiling property of the state variables of such modulators. The second aspect of our contribution is to perform an instantaneous error analysis that avoids infinite time averaging. Until now, only a O(λ^-4) type error bound was known to hold in this setting. Under our generalized framework, we provide two types of squared-error estimates; one that is statistically averaged over the input and another that is valid for almost every input (in these sense of Lebesque measure). In both cases, we improve the error bound to O(λ^-4.5), up to a logarithmic factor, for a general class of modulators including some specific ones that are covered in this paper in detail. In the particular case of the standard and ideal two-bit double-loop configuration, our methods provide a (previously unavailable) instantaneous error bound of O(λ^-5), again up to a logarithmic factor.