457 research outputs found
An Optimal Family of Exponentially Accurate One-Bit Sigma-Delta Quantization Schemes
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
can be achieved by appropriate one-bit Sigma-Delta
modulation schemes. By general information-entropy arguments must be less
than 1. The current best known value for 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 . 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.Comment: 35 pages, 3 figure
Output Filter Aware Optimization of the Noise Shaping Properties of {\Delta}{\Sigma} Modulators via Semi-Definite Programming
The Noise Transfer Function (NTF) of {\Delta}{\Sigma} modulators is typically
designed after the features of the input signal. We suggest that in many
applications, and notably those involving D/D and D/A conversion or actuation,
the NTF should instead be shaped after the properties of the
output/reconstruction filter. To this aim, we propose a framework for optimal
design based on the Kalman-Yakubovich-Popov (KYP) lemma and semi-definite
programming. Some examples illustrate how in practical cases the proposed
strategy can outperform more standard approaches.Comment: 14 pages, 18 figures, journal. Code accompanying the paper is
available at http://pydsm.googlecode.co
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