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To tailor time series models to a particular physical problem and to follow the working of various techniques for processing and analyzing data, one must understand the basic theory of spectral (frequency domain) analysis of time series. This classic book provides an introduction to the techniques and theories of spectral analysis of time series. In a discursive style, and with minimal dependence on mathematics, the book presents the geometric structure of spectral analysis. This approach makes possible useful, intuitive interpretations of important time series parameters and provides a unified framework for an otherwise scattered collection of seemingly isolated results.The books strength lies in its applicability to the needs of readers from many disciplines with varying backgrounds in mathematics. It provides a solid foundation in spectral analysis for fields that include statistics, signal process engineering, economics, geophysics, physics, and geology. Appendices provide details and proofs for those who are advanced in math. Theories are followed by examples and applications over a wide range of topics such as meteorology, seismology, and telecommunications.Topics covered include Hilbert spaces; univariate models for spectral analysis; multivariate spectral models; sampling, aliasing, and discrete-time models; real-time filtering; digital filters; linear filters; distribution theory; sampling properties ofspectral estimates; and linear prediction. Key Features* Hilbert spaces* univariate models for spectral analysis* multivariate spectral models* sampling, aliasing, and discrete-time models* real-time filtering* digital filters* linear filters* distribution theory* sampling properties of spectral estimates* linear prediction
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THESPECTRALANALYSIS OFTIMESERIES
This is Volume 22 in P R O B A B I L I T Y A N D M A T H E M A T I C A L STATISTICS A Series of M o n o g r a p h s and Textbooks Editors: Z. W. Birnbaum and E. Lukacs
A complete list of titles in this series appears at the end of this volume.
THESPECTRALANALYSIS OFTIMESERIES LambertH. Koopmans Department of Mathematics and Statistics University of New Mexico Albuquerque, New Mexico 87131
ACADEMICPRESS San Diego Boston New York London Sydney Tokyo Toronto
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Copyright 9 1995, 1974 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Academic Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495
United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW1 7DX
International Standard Book Number: 0-12-419251-3 Library of Congress Catalog Card Number: 73-7441
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Contents
xi
PREFACE
xiii
ACKNOWLEDGMENTS
xv
PREFACETO THE SECOND EDITION
Chapter 1 Preliminaries 1.1 Introduction 1.2 Time Series and Spectra 1.3 Summary of Vector Space Geometry 1.4 Some Probability Notations and Properties
Chapter 2 Models for Spectral Analysis-The
Univariate Case
2.1 Introduction 2.2 The Wiener Theory of Spectral Analysis 2.3 Stationary and Weakly Stationary Stochastic Processes 2.4 The Spectral Representation for Weakly Stationary
Stochastic Processes-A
1 1 13 26
Special Case
29 30 37 39
2.5 The General Spectral Represent