On Trajectory-Based Stability Analysis for $1$-bit Sigma-Delta Quantization and its Application to the Second-Order Case

2026-05-16

AI Summary

Existing ℓ¹ stability conditions for high-order one-bit Sigma-Delta quantization systems are overly conservative and fail to accurately characterize the true stability region. This work proposes a dynamical systems approach based on state-trajectory analysis, replacing conventional invariant-set methods, to model the quantization process via nonlinear recurrence relations. By incorporating the structure of sparse feedback filters, the framework captures the system’s evolution more precisely. For the first time, this method yields stability guarantees for second-order and higher systems that strictly improve upon the classical ℓ¹ condition. Specifically, it reduces the required feedback filter length from O(1/(1−‖f‖∞)) to O(1/√(1−‖f‖∞)), substantially expanding the stability boundary and enhancing both efficiency and practical applicability of high-order Sigma-Delta modulators.

Abstract

A state-of-the-art strategy for digitally representing a bandlimited signal $f$ is $ΣΔ$ quantization. $ΣΔ$ quantization schemes choose a bit sequence $(q_n)$ representing the samples $(y_n)$ of $f$ sequentially based on a state sequence $(u_n)$ defined via a recurrence relation of the form \begin{equation*} u_n = (h*u)_n + y_n - q_n, \end{equation*} where $h_j = 0$ for $j\le 0.$ The effectiveness of a quantization scheme crucially depends on the fact that it is stable, i.e. , the state variable remains uniformly bounded in a given class of signals. Thus, a common strategy is to choose $$q_n = \operatorname{sign}((h*u)_n + y_n).$$ It is well known that a sufficient condition for this quantization rule to induce stability is that $$ \|h\|_{\ell^1}+\|f\|_{\infty}\le 2.$$ At the same time, one empirically observes that this condition is conservative and stability holds significantly beyond this bound. In this paper, we address this gap by establishing the first stability guarantees beyond first order that outperform the $\ell^1$ based stability condition. In contrast to many previous approaches, our analysis describes the trajectories of the state variables rather than characterizing the invariant set, an approach that had previously been performed only in some specific example cases. This viewpoint has the main advantage that it makes it possible to treat longer filters, which are difficult to handle through invariant-set analysis because of the resulting high dimensionality. We apply our technique to second-order $ΣΔ$ schemes with sparse feedback filters as proposed by Günturk \cite{gunturk2003one}, showing that the filter length required to guarantee stability significantly improves from the length $O\left(\frac{1}{1-\|f\|_{\infty}}\right)$ needed to apply the $\ell^1$ based criterion to $O\left(\frac{1}{\sqrt{1-\|f\|_{\infty}}}\right)$.

Problem

• Sigma-Delta quantization
• stability analysis
• trajectory-based
• second-order
• feedback filter

Innovation

• trajectory-based analysis
• Sigma-Delta quantization
• stability guarantee
• second-order system
• sparse feedback filter