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Linear Estimation 1st Edition

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ISBN-13: 978-0130224644
ISBN-10: 0130224642
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Editorial Reviews

From the Inside Flap


The problem of estimating the values of a random (or stochastic) process given observations of a related random process is encountered in many areas of science and engineering, e.g., communications, control, signal processing, geophysics, econometrics, and statistics. Although the topic has a rich history, and its formative stages can be attributed to illustrious investigators such as Laplace, Gauss, Legendre, and others, the current high interest in such problems began with the work of H. Wold, A. N. Kolmogorov, and N. Wiener in the late 1930s and early 1940s. N. Wiener in particular stressed the importance of modeling not just "noise" but also "signals" as random processes. His thought-provoking originally classified 1942 report, released for open publication in 1949 and now available in paperback form under the title Time Series Analysis, is still very worthwhile background reading.

As with all deep subjects, the extensions of these results have been very far-reaching as well. A particularly important development arose from the incorporation into the theory of multichannel state-space models. Though there were various earlier partial intimations and explorations, especially in the work of R. L. Stratonovich- in the former Soviet Union, the chief credit for the explosion of activity in this direction goes to R. E. Kalman, who also made important related contributions to linear systems, optimal control, passive systems, stability theory, and network synthesis.

In fact, least-squares estimation is one of those happy subjects that is interesting not only in the richness and scope of its results, but also because of its mutually beneficial connections with a host of other (often apparently very different) subjects. Thus, beyond those already named, we may mention connections with radiative transfer and scattering theory, linear algebra, matrix and operator theory, orthogonal polynomials, moment problems, inverse scattering problems, interpolation theory, decoding of Reed-Solomon and BCH codes, polynomial factorization and root distribution problems, digital filtering, spectral analysis, signal detection, martingale theory, the so-called Hh theories of estimation and control, least-squares and adaptive filtering problems, and many others. We can surely apply to it the lines written by William Shakespeare about another (beautiful) subject:

"Age does not wither her, nor custom stale,
Her infinite variety."

Though we were originally tempted to cover a wider range, many reasons have led us to focus this volume largely on estimation problems for finite-dimensional linear systems with state-space models, covering most aspects of an area now generally known as Wiener and Kalman filtering theory. Three distinctive features of our treatment are the pervasive use of a geometric point of view, the emphasis on the numerically favored square-root/array forms of many algorithms, and the emphasis on equivalence and duality concepts for the solution of several related problems in adaptive filtering, estimation, and control. These features are generally absent in most prior treatments, ostensibly on the grounds that they are too abstract and complicated. It is our hope that these misconceptions will be dispelled by the presentation herein, and that the fundamental simplicity and power of these ideas will be more widely recognized arid exploited.

The material presented in this book can be broadly categorized into the following topics:

Introduction and Foundations
Chapter 1: Overview
Chapter 2: Deterministic Least-Squares Problems
Chapter 3: Stochastic Least-Squares Problems
Chapter 4: The Innovations Process
Chapter 5: State-Space Models

Estimation of Stationary Processes
Chapter 6: Innovations for Stationary Processes
Chapter 7: Wiener Theory for Scalar Processes
Chapter 8: Recursive Wiener Filters

Estimation of Nonstationary Processes
Chapter 9: The Kalman Filter
Chapter 10: Smoothed Estimators

Fast and Array Algorithms
Chapter 11: Fast Algorithms
Chapter 12: Array Algorithms
Chapter 13: Fast Array Algorithms

Continuous-Time Estimation
Chapter 16: Continuous-Time State-Space Estimation

Advanced Topics
Chapter 14: Asymptotic Behavior
Chapter 15: Duality and Equivalence in Estimation and Control
Chapter 17: A Scattering Theory Approach

Being intended for a graduate-level course, the book assumes familiarity with basic concepts from matrix theory, linear algebra, linear system theory, and random processes. Four appendices at the end of the book provide the reader with background material in all these areas.

There is ample material in this book for the instructor to fashion a course to his or her needs and tastes. The authors have used portions of this book as the basis for one-quarter first-year graduate level courses at Stanford University, the University of California at Los Angeles, and the University of California at Santa Barbara; the students were expected to have had some exposure to discrete-time and state-space theory. A typical course would start with Secs.1.1-1.2 as an overview (perhaps omitting the matrix derivations), with the rest of Ch. 1 left for a quick reading (and re-reading from time to time), most of Chs. 2 and 3 (focusing on the geometric approach) on the basic deterministic and stochastic least-squares problems, Ch. 4 on the innovations process, Secs. 6.4-6.5 and 7.3-7.7 on scalar Wiener filtering, Secs. 9.1-9.3, 9.5, and 9.7 on Kalman filtering, Secs. 10.1-10.2 as an introduction to smoothing, Secs. 12.1-12.5 and 13.1-13.4 on array algorithms, and Secs. 16.1-16.4 and 16.6 on continuous-time problems.

More advanced students and researchers would pursue selections of material from Sec. 2.8, Chs. 8, 11, 14, 15, and 17, and Apps. E and R These cover, among other topics, least-squares problems with uncertain data, the problem of canonical spectral factorization, convergence of the Kalman filter, the algebraic Riccati equation, duality, backwards-time and complementary models, scattering, etc. Those wishing to go on to the more recent H¥ theory can find a treatment closely related to the philosophy of the current book (cf. Sec. 1.6) in the research monograph of Hassibi, Sayed, and Kailath (1999).

A feature of the book is a collection of nearly 300 problems, several of which complement the text and present additional results and insights. However, there is little discussion of real applications or of the error and sensitivity analyses required for them. The main issue in applications is constructing an appropriate model, or actually a set of models, which are further analyzed and then refined by using the results and algorithms presented in this book. Developing good models and analyzing them effectively requires not only a good appreciation of the actual application, but also a good understanding of the theory, at both an analytical and intuitive level. It is the latter that we have tried to achieve here; examples of successful applications have to be sought in the literature, and some references are provided to this end.

The development of this textbook has spanned many years. So the material, as well as its presentation, has benefited greatly from the inputs of the many bright students who have worked with us on these topics: J. Omura, P Frost, T Duncan, R. Geesey, D. Duttweiler, H. Aasnaes, M. Gevers, H. Weinert, A. Segall, M. Mort B. Dickinson, G. Sidhu, B. Friedlander, A. Vieira, S. Y Kung, B. Levy, G. Verghese, D. Lee, J. Delosme, B. Porat, H. Lev-Ari, J. Cioffi, A. Bruckstein, T. Citron, Y Bresler, R. Roy, J. Chun, D. Slock, D. Pal, G. Xu, R. Ackner, Y Cho, P Park, T. Boros, A. Erdogan, U. Forsell, B. Halder, H. Hindi, V Nascimento, T. Pare, R. Merched, and our young friend Amir Ghazanfarian (in memoriam) from whom we had so much more to learn.

We are of course also deeply indebted to the many researchers and authors in this beautiful field. Partial acknowledgment is evident through the citations and references; while the list of the latter is quite long, we apologize for omissions and inadequacies arising from the limitations of our knowledge and our energy. Nevertheless, we would be remiss not to explicitly mention the inspiration and pleasure we have gained in studying the papers and books of N. Wiener, R. E. Kalman, and P Whittle.

Major support for the many years of research that led to this book was provided by the Mathematics Divisions of the Air Force Office of Scientific Research and the Army Research Office, by the Joint Services Electronics Program, by the Defense Advanced Research Projects Agency, and by the National Science Foundation. Finally, we would like to thank Bernard Goodwin and Tom Robbins, as well as the staff of Prentice Hall, for their patience and other contributions to this project.

T Kailath
Stanford, CA

A. H. Sayed
Westwood, CA

B. Hassibi
Murray Hill, NJ

From the Back Cover

This original work offers the most comprehensive and up-to-date treatment of the important subject of optimal linear estimation, which is encountered in many areas of engineering such as communications, control, and signal processing, and also in several other fields, e.g., econometrics and statistics. The book not only highlights the most significant contributions to this field during the 20th century, including the works of Wiener and Kalman, but it does so in an original and novel manner that paves the way for further developments. This book contains a large collection of problems that complement it and are an important part of piece, in addition to numerous sections that offer interesting historical accounts and insights. The book also includes several results that appear in print for the first time.


  • Takes a geometric point of view.
  • Emphasis on the numerically favored array forms of many algorithms.
  • Emphasis on equivalence and duality concepts for the solution of several related problems in adaptive filtering, estimation, and control.
    • These features are generally absent in most prior treatments, ostensibly on the grounds that they are too abstract and complicated. It is the authors' hope that these misconceptions will be dispelled by the presentation herein, and that the fundamental simplicity and power of these ideas will be more widely recognized and exploited. Among other things, these features already yielded new insights and new results for linear and nonlinear problems in areas such as adaptive filtering, quadratic control, and estimation, including the recent Hà theories.

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Product Details

  • Paperback: 854 pages
  • Publisher: Prentice Hall; 1 edition (April 10, 2000)
  • Language: English
  • ISBN-10: 0130224642
  • ISBN-13: 978-0130224644
  • Product Dimensions: 6.9 x 1.7 x 9.1 inches
  • Shipping Weight: 2.8 pounds (View shipping rates and policies)
  • Average Customer Review: 4.3 out of 5 stars  See all reviews (6 customer reviews)
  • Amazon Best Sellers Rank: #896,171 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

16 of 17 people found the following review helpful By Jeffrey Andrews on February 5, 2001
Format: Paperback
Kailath, Sayed, and Hassibi do an excellent job of explaining what is a fairly complicated subject. This book is best-suited for scholars who desire a deep understanding of estimation theory. Engineers who want to quickly understand how to implement a Kalman Filter might be better off buying Adaptive Filter Theory by Simon Haykin.
The first chapter provides a good overview of the book, although it makes the most sense once the subject matter of the rest of the book has been digested a bit. A consistent framework emphasizing innovations (or the new information which appears at any iteration) is used throughout the book, and both continuous and discrete-time techniques for stochastic estimation are given nearly equal treatment, although the real-world engineer is likely to be interested in the latter.
Professor Kailath's articulate nature and knack for the clever anecdote or one-liner shines throughout the book, making it, while very mathematical in nature, quite readable for the motivated student.
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13 of 14 people found the following review helpful By Nicolas Chapados on September 17, 2001
Format: Paperback
This is one of the best engineering textbooks I have read, period. Although the subject matter is not for the faint-hearted, the authors' attention to pedagogical details shine throughout (repetition is the key to learning). The Kalman filter is introduced naturally as a consequence of a general framework for obtaining the best linear estimator of a random variable given others (earlier observations), and the geometric intuition is stressed repeatedly.
No important issue is omitted, including a very complete treatment of numerical issues and fast algorithms. My only gripe is with the assumption that all model parameters are KNOWN; in other words, the important aspect system identification (parameter estimation, learning, or whatever you call it in your field) is left to other textbooks.
Moreover, and no minor accomplishment, is the amazingly small number of typographical errors (at least up to where I have read so far; a bit over half the book), which is remarkable given the dense mathematical contents.
All in all, I would give it 6 stars if possible. Everything is there: it transmits a deep intuition for the matter, a places it in its historical context through interesting and amusing notes; it leaves the reader fulfilled but not overwhelmed.
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4 of 4 people found the following review helpful By Shadow on September 25, 2005
Format: Paperback
This is an excellent text that covers estimation theory from a modern point of view. It will be especially interesting to anyone with a graduate degree in physics because Kailath, et al derive the theory of linear estimation from a point of view very similar to that of modern quantum mechanics - they even use similar bra/ket notation!

Basic and advanced statistical mathematics is somewhat an implied prerequisite for understanding this text. From what I have seen, I honestly find nothing negative to critique - its probably one of the best technical textbooks I have in my large library.
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