TY - JOUR
T1 - Lower bounds for the error decay incurred by coarse quantization schemes
AU - Krahmer, Felix
AU - Ward, Rachel
N1 - Funding Information:
The authors would like to thank Sinan Güntürk for interesting discussions on this topic as well as the Hausdorff Center for Mathematics, Bonn, and the Summer School on “Theoretical Foundations and Numerical Methods for Sparse Recovery” at the RICAM, Linz, where large parts of the project were completed. They gratefully acknowledge the support of a National Science Foundation Postdoctoral Research Fellowship (Ward) and the Charles M. Newman Fellowship at the Courant Institute (Krahmer).
PY - 2012/1
Y1 - 2012/1
N2 - Several analog-to-digital conversion methods for bandlimited signals used in applications, such as ΣΔ quantization schemes, employ coarse quantization coupled with oversampling. The standard mathematical model for the error accrued from such methods 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-αλ) can be achieved by appropriate one-bit Sigma-Delta modulation schemes. However, the best known achievable rate constants α in this setting differ significantly from the general information theoretic lower bound. In this paper, we provide the first lower bound specific to coarse quantization, thus narrowing the gap between existing upper and lower bounds. In particular, our results imply a quantitative correspondence between the maximal signal amplitude and the best possible error decay rate. Our method draws from the theory of large deviations.
AB - Several analog-to-digital conversion methods for bandlimited signals used in applications, such as ΣΔ quantization schemes, employ coarse quantization coupled with oversampling. The standard mathematical model for the error accrued from such methods 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-αλ) can be achieved by appropriate one-bit Sigma-Delta modulation schemes. However, the best known achievable rate constants α in this setting differ significantly from the general information theoretic lower bound. In this paper, we provide the first lower bound specific to coarse quantization, thus narrowing the gap between existing upper and lower bounds. In particular, our results imply a quantitative correspondence between the maximal signal amplitude and the best possible error decay rate. Our method draws from the theory of large deviations.
KW - Approximation by arbitrary nonlinear expressions
KW - Best constants
KW - Circuits
KW - Degree of approximation
KW - Dynamical systems in applications
KW - Harmonic analysis on Euclidean spaces - probabilistic methods
KW - Information and communication
KW - Probability theory - combinatorial probability
KW - Rate of convergence
KW - Widths and entropy
UR - http://www.scopus.com/inward/record.url?scp=81355149771&partnerID=8YFLogxK
U2 - 10.1016/j.acha.2011.06.003
DO - 10.1016/j.acha.2011.06.003
M3 - Letter
AN - SCOPUS:81355149771
SN - 1063-5203
VL - 32
SP - 131
EP - 138
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
IS - 1
ER -