Advances in Gabor Analysis
The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract har monic analysis to basic applications. T
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John J. Benedetto University of Maryland
Editorial Advisory Board Akram Aldroubi NIH, Biomedical Engineering/ Instrumentation Ingrid Daubechies Princeton University Christopher Heil Georgia Institute of Technology James McClellan Georgia Institute of Technology Michael Unser NIH, Biomedical Engineering/ Instrumentation M. Victor Wickerhauser Washington University
Douglas Cochran Arizona State University Hans G. Feichtinger University of Vienna Murat Kunt Swiss Federal Institute ofTechnology, Lausanne Wim Sweldens Lucent Technologies Bell Laboratories Martin Vetterii Swiss Federal Institute of Technology, Lausanne
Applied and Numerical Harmonic Analysis Published titles Cooper: Introduction to Partial Differential Equations with MA nAB (ISBN 0-8176-3967-5)
j, M,
C.E. D'Attellis and EM Femandez-Berdaguer: Wavelet Theory and Harmonic Analysis in Applied Sciences (ISBN 0-8176-3953-5) H,G, Feichtinger and T Strohmer: Gabor Analysis and Algorithms (ISBN 0-8176-3959-4) TM. Peters, j,HT, Bates, G.B, Pike, p, Munger, and j,C, Williams: Fourier Transforms and Biomedical Engineering (ISBN 0-8176-3941-1) AI. Saichev and WA Woycz:yns~: Distributions in the Physical and Engineering Sciences (ISBN 0-8176-3924-1)
R. Tolimierei and M, An: Time-Frequency Representations (ISBN 0-8176-3918-7) G,T Herman: Geometry of Digital Spaces (ISBN 0-8176-3897-0)
A Prochazka, j, Uhlir, P,j,w. Rayner, and N,G, Kingsbury: Signal Analysis and Prediction (ISBN 0-8176-4042-8) j, Ramanathan: Methods of Applied Fourier Analysis (ISBN 0-8176-3963-2)
A Teolis: Computational Signal Processing with Wavelets (ISBN 0-8176-3909-8) w.o, Bray and tv. Stanojevic: Analysis of Divergence (ISBN 0-8176-4058-4) GT, Herman and A Kuba: Discrete Tomography (ISBN 0-8176-4101-7) j,j, Benedetto and P,j,S,G, Ferreira: Modem Sampling Theory (ISBN 0-8176-4023-1)
A Abbate, C,M, DeCusatis, and PK Das: Wavelets and Subbands (ISBN 0-8176-4136-> 0 is small, then most of the energy is concentrated on T, and T may indeed be considered the essential support of f. For f = 0 we obtain the exact support of f. As a first idea to modify the uncertainty principle of Benedicks we may replace the exact support in Theorem 2.4.1 by the essential support. In this way we obtain a quantitative version of the uncertainty principle. The following theorem of Donoho and Stark [9] gives a precise expression to Metatheorem B which says that a signal must occupy a region of area at least one in the time-frequency plane.
Theorem 2.5.1. (Donoho-Stark) Suppose f E L2(~d) \ {O} is cT-concentrated on T ~ ~d and j is co-concentrated on 0 ~ ~d. Then (2.5.1) The hypothesis of this theorem can be written in the form
r Joer IRf(x,wW dxdw < 4f~llfll~. JTc Metatheorem C suggests that an inequality similar to (2.5.1) holds for the ambiguity function and Wigner distribution respectively. Lemma 2.5.1. (Weak uncertainty principle for the STFT) Let f and g E L2(~d). If
//)A(f,g)(x,w)1 2 dxdw = / for some U ~ ~2d and c
ilV
gf (x,w)1 2 dxdw 2 (1- c)
2 0, then lUI 2 1 - c.
Ilfll~llgll~ (2.5.2)
Proof: Note
