Table of Contents
ISRN Applied Mathematics
Volume 2012, Article ID 630702, 7 pages
http://dx.doi.org/10.5402/2012/630702
Research Article

Sign Data Derivative Recovery

1Louisiana Accelerator Center, The University of Louisiana at Lafayette, Lafayette, LA 70504-4210, USA
2Ion Beam Modification and analysis Laboratory, Department of Physics, University of North Texas, Denton, TX 76203, USA

Received 2 November 2011; Accepted 29 November 2011

Academic Editors: J. Shen and F. Zirilli

Copyright © 2012 L. M. Houston et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given only the signs of signal plus noise added repetitively or sign data, signal amplitudes can be recovered with minimal variance. However, discrete derivatives of the signal are recovered from sign data with a variance which approaches infinity with decreasing step size and increasing order. For industries such as the seismic industry, which exploits amplitude recovery from sign data, these results place constraints on processing, which includes differentiation of the data. While methods for smoothing noisy data for finite difference calculations are known, sign data requires noisy data. In this paper, we derive the expectation values of continuous and discrete sign data derivatives and we explicitly characterize the variance of discrete sign data derivatives.