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The Theory of Linear Prediction Copyright© 2008byMorgan&Claypool Allrightsreserved.Nopartofthispublicationmay bereproduced, storedinaretrievalsystem, ortransmittedin anyformorbyanymeans---electronic,mechanical,photocopy,recording,oranyotherexceptforbriefquotations inprintedreviews,withoutthepriorpermissionofthepublisher. TheTheoryofLinearPrediction P.P.Vaidyanathan www.morganclaypool.com ISBN:1598295756 paperback ISBN:9781598295757 paperback ISBN:1598295764 ebook ISBN:9781598295764 ebook DOI:10.2200/S00086ED1V01Y200712SPR03 APublicationintheMorgan&ClaypoolPublishersseries SYNTHESISLECTURESONSIGNALPROCESSING#3 Lecture#3 SeriesEditor:JoséMoura,CarnegieMellonUniversity SeriesISSN ISSN1932-1236 print ISSN1932-1694 electronic The Theory of Linear Prediction P.P.Vaidyanathan CaliforniaInstituteofTechnology SYNTHESISLECTURESONSIGNALPROCESSING#3 M &C & Morgan Claypool Publishers ToUsha,Vikram,andSagarandmyparents. vi ABSTRACT Linear prediction theory has had a profound impact in the field of digital signal processing. Although the theory dates back to the early 1940s, its influence can still be seen in applications today.The theoryis basedon very elegantmathematicsand leadstomany beautifulinsightsinto statisticalsignalprocessing.Although predictionisonlyapartofthemoregeneraltopicsoflinear estimation, filtering, and smoothing, this book focuses on linear prediction. This has enabled detailed discussionof a number of issues that are normally not found in texts. For example, the theory of vector linear prediction is explained in considerable detail and so is the theory of line spectralprocesses.Thisfocusanditssmallsizemakethebookdifferentfrommanyexcellenttexts which cover the topic, including a few that are actually dedicatedto linear prediction. There are several examples and computer-baseddemonstrationsof the theory. Applications are mentioned whereverappropriate,but thefocusisnoton thedetaileddevelopmentoftheseapplications.The writing style is meant to be suitable for self-studyas well as for classroom use at the senior and first-yeargraduatelevels.Thetextisself-containedforreaderswithintroductoryexposuretosignal processing,randomprocesses,andthetheoryofmatrices,andahistoricalperspectiveanddetailed outlinearegiveninthefirstchapter. KEYWORDS Linearpredictiontheory,vectorlinearprediction,linearestimation,filtering,smoothing, linespectralprocesses,Levinson’srecursion,latticestructures,autoregressivemodels vii Preface Linear prediction theory has had a profound impact in the field of digital signal processing. Although the theory dates back to the early 1940s, its influence can still be seen in applications today.The theoryis basedon veryelegant mathematicsand leadsto many beautifulinsightsinto statisticalsignalprocessing.Although predictionisonlyapartofthemoregeneraltopicsoflinear estimation, filtering, and smoothing, I have focused on linear prediction in this book. This has enabledmetodiscussindetailanumberofissuesthatarenormallynotfoundintexts.Forexample, the theory of vector linear prediction is explained in considerable detail and so is the theory of linespectralprocesses.Thisfocusanditssmall sizemakethe bookdifferentfrommany excellent textsthatcoverthetopic,includingafewthatareactuallydedicatedtolinearprediction.Thereare several examples and computer-baseddemonstrationsof the theory. Applications are mentioned whereverappropriate,butthefocusisnotonthedetaileddevelopmentoftheseapplications. The writing style is meant to be suitable for self-studyas well as for classroom use at the seniorandfirst-yeargraduatelevels.Indeed,thematerialhereemergedfromclassroomlecturesthat I hadgiven overthe yearsatthe CaliforniaInstituteofTechnology.So,the text isself-contained for readers with introductory exposure to signal processing, random processes,and the theory of matrices.AhistoricalperspectiveandadetailedoutlinearegiveninChapter1. ix Acknowledgments The pleasant academic environment provided by the California Institute of Technology and the generous support from the National Science Foundation and the Office of Naval Research have beencrucialindevelopingsomeoftheadvancedmaterialscoveredinthisbook. During my ‘‘young’’ days, I was deeply influenced by an authoritative tutorial on linear filtering by Prof. Tom Kailath (1974) and a wonderful tutorial on linear prediction by John Makhoul (1975). These two articles, among other excellent references,have taught me a lot and so has the book by Anderson and Moore (1979). My ‘‘love for linear prediction’’ was probably kindledby thesethree references.The small contributionI have made here would not have been possiblewereitnotforthesereferencesandotherexcellentonesmentionedintheintroductionin Chapter1. It is impossibleto reduceto wordsmy gratitudeto Usha, who has shown infinite patience during my busy days with research and book projects. She has endured many evenings and weekendsofmy ‘‘disappearance’’to work.Her sinceresupport and the enthusiasmandlove from myuncomplainingsonsVikramandSagararemuchappreciated! P.P.Vaidyanathan CaliforniaInstituteofTechnology