The Analytical Signal and Related Problem
In many communication systems it is necessary to use the concepts of instantaneous amplitude and phase of signals. These concepts are not naturally defined, and the classical Fourier theory shows a contradiction between the ideas of frequency and that of
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TIME AND FREQUENCY REPRESENTATION OF SIGNALS AND SYSTEMS
EDITED BY G. LONGO
UNIVERSITA DI TRIESTE B. PICINBONO ... LABORATOIRE DES SIGNAUX ET SYSTEMES
SPRINGER-VERLAG WIEN GMBH
Le spese di stampa di questo volume sono in parte coperte da contributi del Consiglio Nazionale delle Ricerche.
This volume contains 23 illustrations.
This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 1989 by Springer-Verlag Wien Originally published by Springer Verlag Wien-New York in 1989
pp. 11-68 and 125-170: copyright not transferred.
In order to make this volume available as economically and as rapidly as possible the authors' typescripts have been reproduced in their original forms. This method unfortunately has its typographical limitations but it is hoped that they in no way distract the reader.
ISBN 978-3-211-82143-5 DOI 10.1007/978-3-7091-2620-2
ISBN 978-3-7091-2620-2 (eBook)
PREFACE
This volume contains the text of four lectures given at the CISM from 21 to 25 September, 1987, under the general title "Time and frequency representation of signals and systems". We present our thanks to the administration of the CISM which made this meeting possible. It was an extremely pleasant week with many opportunities for meetings and discussions. We hope that this book will reflect the atmosphere of the lectures as well as their content. The concept of time and frequency representation of signals is as old as the first notation of written music. If we consider the musical line of a single voice instrument such as a flute or oboe, we see a succession of notes which must be played in time. At each time instant the exact note being played can be detected. As each note is associated with a particular fundamental frequency, or eventually with some harmonic frequencies, we can say that the score is a time and frequency representation of a "signal", which is the whole piece of music. Those interested in both music and physics can attempt to present a mathematical model of the phenomenon. For this purpose we have to find the appropriate theory, which is obviously the Fourier representation of signals and systems. Using this theory, we call s(t) the signal representing the whole musical piece and, as the frequency aspect, at the basis of the theory of harmony, is fundamental, we calculate the Fourier transform S(v) of this signal. But as this Fourier transform is completely independent of time, there is no way of finding which note is to be played at a particular time instant, although the ear makes this recognition very easily. In other words, there is a mathematical contradiction in talking about an instantaneous frequency, even though musical notation seems to use this concept. The physicist cannot consider this situation as satisfactory, and as a result he must introduce some tools allowing the simultan
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