Sigma-delta (/spl Sigma//spl Delta/) quantization and finite frames

The K-level Sigma-Delta (/spl Sigma//spl Delta/) scheme with step size /spl delta/ is introduced as a technique for quantizing finite frame expansions for /spl Ropf//sup d/. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that /spl Sigma//spl Delta...

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Bibliographic Details
Published in:IEEE transactions on information theory 2006-05, Vol.52 (5), p.1990-2005
Main Authors: Benedetto, J.J., Powell, A.M., Yilmaz, O.
Format: Article
Language:English
Subjects:
Applied sciences
Approximation
Counting
Delta-sigma modulation
Dictionaries
Errors
Estimating techniques
Exact sciences and technology
Finite frames
Frames
Hilbert space
Information systems
Information, signal and communications theory
Mathematical analysis
Mathematics
Modulation
Modulation, demodulation
Optimization
Phase change materials
Pulse modulation
Quantization
Sampling, quantization
Sigma-Delta quantization
Signal and communications theory
Signal processing
Signal processing algorithms
Statistical analysis
Systems, networks and services of telecommunications
Telecommunications
Telecommunications and information theory
Transmission and modulation (techniques and equipments)
Upper bound
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Summary:The K-level Sigma-Delta (/spl Sigma//spl Delta/) scheme with step size /spl delta/ is introduced as a technique for quantizing finite frame expansions for /spl Ropf//sup d/. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that /spl Sigma//spl Delta/ quantization of a unit-norm finite frame expansion in /spl Ropf//sup d/ achieves approximation error where N is the frame size, and the frame variation /spl sigma/(F,p) is a quantity which reflects the dependence of the /spl Sigma//spl Delta/ scheme on the frame. Here /spl par//spl middot//spl par/ is the d-dimensional Euclidean 2-norm. Lower bounds and refined upper bounds are derived for certain specific cases. As a direct consequence of these error bounds one is able to bound the mean squared error (MSE) by an order of 1/N/sup 2/. When dealing with sufficiently redundant frame expansions, this represents a significant improvement over classical pulse-code modulation (PCM) quantization, which only has MSE of order 1/N under certain nonrigorous statistical assumptions. /spl Sigma//spl Delta/ also achieves the optimal MSE order for PCM with consistent reconstruction.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2006.872849