On the uncertainty inequality as applied to discrete signals

作     者：Venkatesh, Y.V. Raja, S. Kumar Vidyasagar, G. 

作者机构：Computer Vision and Artificial Intelligence Laboratory Department of Electrical Engineering Indian Institute of Science Bangalore 560012 India Department of Electrical and Computer Engineering Faculty of Engineering National University of Singapore Singapore 117576 Singapore 

出 版 物：《International Journal of Mathematics and Mathematical Sciences》 (Int. J. Math. Math. Sci.)

年 卷 期：2006年第2006卷第1期

页      面：48185-1-48185-页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

摘      要：Given a continuous-time bandlimited signal, the Shannon sampling theorem provides an interpolation scheme for exactly reconstructing it from its discrete samples. We analyze the relationship between concentration (or compactness) in the temporal/spectral domains of the (i) continuous-time and (ii) discrete-time signals. The former is governed by the Heisenberg uncertainty inequality which prescribes a lower bound on the product of effective temporal and spectral spreads of the signal. On the other hand, the discrete-time counterpart seems to exhibit some strange properties, and this provides motivation for the present paper. We consider the following problem: for a bandlimited signal, can the uncertainty inequality be expressed in terms of the samples, using the standard definitions of the temporal and spectral spreads of the signal? In contrast with the results of the literature, we present a new approach to solve this problem. We also present a comparison of the results obtained using the proposed definitions with those available in the literature. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.