Sigma-Delta modulation is a popular method for analog-to-digital conversion
of bandlimited signals that employs coarse quantization coupled with
oversampling. The standard mathematical model for the error analysis of the
method measures the performance of a given scheme by the rate at which the
associated reconstruction error decays as a function of the oversampling ratio
$\lambda$. It was recently shown that exponential accuracy of the form
$O(2^{-r\lambda})$ can be achieved by appropriate one-bit Sigma-Delta
modulation schemes. By general information-entropy arguments $r$ must be less
than 1. The current best known value for $r$ is approximately 0.088. The
schemes that were designed to achieve this accuracy employ the "greedy"
quantization rule coupled with feedback filters that fall into a class we call
"minimally supported". In this paper, we study the minimization problem that
corresponds to optimizing the error decay rate for this class of feedback
filters. We solve a relaxed version of this problem exactly and provide
explicit asymptotics of the solutions. From these relaxed solutions, we find
asymptotically optimal solutions of the original problem, which improve the
best known exponential error decay rate to $r \approx 0.102$. Our method draws
from the theory of orthogonal polynomials; in particular, it relates the
optimal filters to the zero sets of Chebyshev polynomials of the second kind.

研究了用于 Exponential Accuracy 的一位 Sigma-Delta 调制器设计中涉及到的最小支持反馈滤波器优化问题，采用正交多项式的理论，通过找到弱化的最优解得到了最优解并将最佳的误差下降率从 0.088 提高到了 0.102。