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 theory is based on very elegant mathematics and leads to many beautiful insights into statistical signal processing. Although prediction is only a part of the more general topics of linear estimation, filtering, and smoothing, this book focuses on linear prediction. This has enabled detailed discussion of 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 spectral processes. This focus and its small size make the book different from many excellent texts which cover the topic, including a few that are actually dedicated to linear prediction. There are several examples and computer-based demonstrations of the theory. Applications are mentioned wherever appropriate, but the focus is not on the detailed development of these applications. The writing style is meant to be suitable for self-study as well as for classroom use at the senior and first-year graduate levels. The text is self-contained for readers with introductory exposure to signal processing, random processes, and the theory of matrices, and a historical perspective and detailed outline are given in the first chapter.
Author(s): P. P. Vaidyanathan
Series: Synthesis Lectures on Signal Processing
Edition: 1
Publisher: Morgan and Claypool Publishers
Year: 2008
Language: English
Pages: 197
The Theory of Linear Prediction......Page 1
Preface......Page 7
Keywords......Page 6
Acknowledgments......Page 9
HISTORY OF LINEAR PREDICTION......Page 15
SCOPE AND OUTLINE......Page 16
Notations......Page 17
PREDICTION ERROR AND PREDICTION POLYNOMIAL......Page 19
THE NORMAL EQUATIONS......Page 20
Expression for the Minimized Mean Square Error......Page 22
The Augmented Normal Equation......Page 23
PROPERTIES OF THE AUTOCORRELATION MATRIX......Page 24
Relation Between Eigenvalues and the Power Spectrum......Page 26
Singularity of the Autocorrelation Matrix......Page 27
Determinant of the Autocorrelation Matrix......Page 29
ESTIMATING THE AUTOCORRELATION......Page 30
CONCLUDING REMARKS......Page 32
DERIVATION OF LEVINSON'S RECURSION......Page 33
The Partial Correlation Coefficient......Page 36
SIMPLE PROPERTIES OF LEVINSON'S RECURSION......Page 37
THE WHITENING EFFECT......Page 40
CONCLUDING REMARKS......Page 43
THE BACKWARD PREDICTOR......Page 45
All-Pass Property......Page 47
Orthogonality of the Optimal Prediction Errors......Page 48
LATTICE STRUCTURES......Page 49
The IIR LPC Lattice......Page 50
Stability of the IIR Filter......Page 51
The Upward and Downward Recursions......Page 52
CONCLUDING REMARKS......Page 53
AUTOREGRESSIVE PROCESSES......Page 55
APPROXIMATION BY AN AR(N) PROCESSES......Page 58
If a Process is AR, Then LPC Will Reveal It......Page 59
Extrapolation of the Autocorrelation......Page 60
AUTOCORRELATION MATCHING PROPERTY......Page 61
POWER SPECTRUM OF THE AR MODEL......Page 64
APPLICATION IN SIGNAL COMPRESSION......Page 72
MA AND ARMA PROCESSES......Page 75
SUMMARY......Page 76
PREDICTION ERROR FOR AN AR PROCESS......Page 79
A MEASURE OF SPECTRAL FLATNESS......Page 82
SPECTRAL FLATNESS OF AN AR PROCESS......Page 86
CASE WHERE SIGNAL IS NOT AR......Page 89
Error Spectrum Gets Flatter as Predictor Order Grows......Page 90
Mean Square Error and Determinant......Page 92
MAXIMUM ENTROPY AND LINEAR PREDICTION......Page 95
Connection to the Notion of Entropy......Page 96
A Direct Maximization Problem......Page 97
CONCLUDING REMARKS......Page 99
INTRODUCTION......Page 101
AUTOCORRELATION OF A LINE SPECTRAL PROCESS......Page 102
Rank Saturation for a Line Spectral Process......Page 104
All Zeros on the Unit Circle......Page 105
Periodicity of a Line Spectral Process......Page 106
FURTHER PROPERTIES OF TIME DOMAIN DESCRIPTIONS......Page 107
PREDICTION POLYNOMIAL OF LINE SPECTRAL PROCESSES......Page 112
SUMMARY OF PROPERTIES......Page 114
IDENTIFYING A LINE SPECTRAL PROCESS IN NOISE......Page 116
Eigenstructure of the Autocorrelation Matrix......Page 117
Computing the Powers at the Line Frequencies......Page 119
LINE SPECTRUM PAIRS......Page 125
CONCLUDING REMARKS......Page 129
FORMULATION OF THE VECTOR LPC PROBLEM......Page 131
BACKWARD PREDICTION......Page 134
LEVINSON'S RECURSION: VECTOR CASE......Page 136
Properties of Matrices Fm f and Fmb......Page 140
Summary of Properties Relating to Levinson's Recursion......Page 142
TRANSFER MATRIX FUNCTIONS IN VECTOR LPC......Page 143
Toward Rearrangement of the Lattice......Page 144
The Symmetrical Lattice......Page 147
THE IIR LATTICE STRUCTURE FOR VECTOR LPC......Page 149
THE NORMALIZED IIR LATTICE......Page 151
THE PARAUNITARY OR MIMO ALL-PASS PROPERTY......Page 152
Unitarity of Building Blocks......Page 154
Propagation of Paraunitary Property......Page 155
Poles of the MIMO IIR Lattice......Page 156
WHITENING EFFECT AND STALLING......Page 158
Review of Matrix Fraction Descripions......Page 160
Relation Between Predictor Polynomials......Page 162
CONCLUDING REMARKS......Page 163
THE ORTHOGONALITY PRINCIPLE......Page 165
CLOSED-FORM SOLUTION......Page 166
CONSEQUENCES OF ORTHOGONALITY......Page 168
SINGULARITY OF THE AUTOCORRELATION MATRIX......Page 170
Appendix B Proof of a Property of Autocorrelations......Page 171
Appendix C Stability of the Inverse Filter......Page 173
Appendix D Recursion Satisfied by AR Autocorrelations......Page 175
Problems......Page 177
References......Page 185
Author Biography......Page 191
Index......Page 193
