2010/01/27 by Percy Deift, C. Sinan Güntürk, Deift, Percy +4
Computer Science · Engineering · Mathematics · #94A12 #94A14 #94A17 #Analog and Mixed-Signal Circuit Design #Classical Analysis and ODEs (math.CA) #Digital Filter Design and Implementation #FOS: Computer and information sciences #FOS: Mathematics #Image and Signal Denoising Methods #Information Theory (cs.IT) #Numerical Methods and Algorithms #cs.IT #math.CA #math.IT #msc:94A12 #msc:94A14 #msc:94A17
paper · pdf · doi:10.48550/arxiv.1001.5079
35 pages, 3 figures
openalex publication_date 2010/01/27 · arxiv created 2010/01/28 · arxiv updated 2010/02/26 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
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 λ. It was recently shown that exponential accuracy of the form O(2-rλ) 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 ≈ 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.