Extrapolation, Interpolation, and Smoothing of Stationary Time Series, with Engineering Applications
2013 Reprint of 1949 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. This is the second book by Norbert Wiener on time series and communication engineering. While the first one, "Cybernetics", treated the subject from a general standpoint and was more philosophical than mathematical, the present volume is more technical than theoretical, and forms a kind of companion piece to the first. It is intended as a tool for engineers working in the field of electrical communication and related subjects. The book consists of an introduction, five chapters, and three appendices. After explaining the general outline of the problem in the introduction, the author gives in Chapter I a review of generalized harmonic analysis which is necessary for the understanding of the following chapters. Chapters II and III are devoted to the problems of prediction and filtering respectively. In Chapter IV there is given a brief account of the theory of multiple prediction, that is, the theory of prediction when we deal with more than one time series at the same time. Finally, in Chapter V there is given a short discussion on the application of similar methods to a problem of approximate differentiation.
Why Read This Book
You should read this book if you want the original, rigorous development of Wiener–Kolmogorov theory that underpins modern DSP: it shows how to optimally extrapolate, interpolate, and smooth stationary stochastic signals and ties the mathematics directly to engineering problems in communications, radar, and audio. You will learn the spectral-factorization and linear-prediction foundations behind many filters and estimators you implement today, gaining a deeper understanding that complements algorithmic or toolbox-driven practice.
Who Will Benefit
Engineers, researchers, and advanced students with a strong mathematical background who need a rigorous foundation in stationary time-series, optimal linear filtering, and spectral methods for communications, radar, or audio/speech applications.
Level: Advanced — Prerequisites: Undergraduate multivariable calculus and linear algebra, basic probability and stochastic processes (random variables, expectation, autocorrelation), and familiarity with Fourier analysis and complex functions.
Key Takeaways
- Derive and apply the Wiener–Kolmogorov optimal linear predictor and smoother for stationary processes
- Perform spectral factorization to produce minimum-phase representations and construct optimal filters
- Relate autocorrelation structure to power spectral densities and use spectral methods for extrapolation and interpolation
- Formulate and solve mean-square estimation problems for noise reduction and signal reconstruction in communication and radar contexts
- Translate theoretical solutions into practical engineering constraints (causality, realizability, and approximation by finite filters)
Topics Covered
- Introduction: statement of the extrapolation/interpolation/smoothing problems
- Chapter I — Generalized Harmonic Analysis and Stationary Processes
- Chapter II — Mean-Square Approximation, Prediction, and Linear Estimation
- Chapter III — Spectral Representations and Power Spectral Density
- Chapter IV — Spectral Factorization and Minimum-Phase Decompositions
- Chapter V — Applications to Communication Engineering (noise reduction, detection, channel considerations)
- Appendix A — Mathematical Preliminaries and Proofs
- Appendix B — Additional Analytical Techniques and Examples
- Appendix C — Tables, Calculations, and Further Notes
Languages, Platforms & Tools
How It Compares
Complementary to modern applied texts like Box & Jenkins' Time Series Analysis (which is more data-driven and statistical) and Haykin's Adaptive Filter Theory (which treats adaptive implementations and modern algorithms); Wiener's book is the original, mathematically rigorous foundation emphasizing stationary-process theory and optimal linear estimation.
