An optimal family of exponentially accurate one‐bit Sigma‐Delta quantization schemes

Abstract

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 . It was recently shown that exponential accuracy of the form O.2 r / can be achieved by appropriate one-bit sigma-delta modulation schemes. By general informationentropy 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 discrete 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.

Root-Exponential Accuracy for Coarse Quantization of Finite Frame Expansions

“…Here C 1 < 1 is a small constant 1 . This work was improved on by Deift et al [5], who showed that the above constant can be pushed to C 1 ≈ 0.102. In order to achieve exponential precision, these works use stable families of r-th-order Σ∆ schemes with approximation errors bounded by C 2 (r)λ −r .…”

“…In this note, we combine the techniques of Blum et al [8] and Güntürk [4]/Deift et al [5] to show that it is possible to achieve root-exponential accuracy in the finite frame setting. In particular, we show that for a family of tight frames of special design that admit themselves as Sobolev duals, and for harmonic frames, root-exponential error rates of O(e −C √ λ ) are achievable.…”

“…A similar issue arises in [9], where the frames are random. On the other hand, in the bandlimited setting, [4] and [5] proposed Σ∆ schemes that do not suffer from an r-dependent constraint on the input sequence. However, the involved constants grow in r.…”

“…By freely optimizing r as a function of λ, [4] and [5] one can balance these effects obtaining exponential precision in λ (measured in the L ∞ norm). In this paper, we use the Σ∆ schemes of [4] and [5] for frame quantization and Sobolev duals as in [8] for linear reconstruction.…”

“…In this paper, we use the Σ∆ schemes of [4] and [5] for frame quantization and Sobolev duals as in [8] for linear reconstruction. Consequently, we can freely optimize r as a function of λ.…”