Frequency Domain Min-Max Optimization of Noise-Shaping Delta-Sigma Modulators

TL;DR

This paper introduces a min-max frequency domain optimization method for designing noise-shaping delta-sigma modulators, improving their worst-case noise performance and stability through LMI-based solutions.

Contribution

It formulates a novel LMI-based min-max optimization framework for delta-sigma modulators, encompassing lowpass, bandpass, and multi-band types, with stability analysis for nonlinear models.

Findings

Optimized NTF reduces maximum noise in specified frequency bands.

Designs demonstrate improved SNR and stability.

FIR NTFs facilitate practical implementation.

Abstract

This paper proposes a min-max design of noise-shaping delta-sigma modulators. We first characterize the all stabilizing loop-filters for a linearized modulator model. By this characterization, we formulate the design problem of lowpass, bandpass, and multi-band modulators as minimization of the maximum magnitude of the noise transfer function (NTF) in fixed frequency band(s). We show that this optimization minimizes the worst-case reconstruction error, and hence improves the SNR (signal-to-noise ratio) of the modulator. The optimization is reduced to an optimization with a linear matrix inequality (LMI) via the generalized KYP (Kalman-Yakubovich-Popov) lemma. The obtained NTF is an FIR (finite-impulse-response) filter, which is favorable in view of implementation. We also derive a stability condition for the nonlinear model of delta-sigma modulators with general quantizers including uniform ones. This condition is described as an H-infinity norm condition, which is reduced to an LMI via the KYP lemma. Design examples show advantages of our design.