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

A state-of-the-art strategy for digitally representing a bandlimited signal $f$ is $\Sigma\Delta$ quantization. $\Sigma\Delta$ 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 $\Sigma\Delta$ schemes with sparse feedback filters as proposed by G\"unturk \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 $O\left(\frac{1}{\sqrt{1-\|f\|_{\infty}}}\right)$.