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The Theory of Linear Prediction

File : pdf, 2.7 MB, 198 pages

TOC

1. Introduction
1.1 History of Linear Prediction
1.2 Scope and Outline
1.2.1 Notations

2. The Optimal Linear Prediction Problem
2.1 Introduction
2.2 Prediction Error and Prediction Polynomial
2.3 The Normal Equations
2.3.1 Expression for the Minimized Mean Square Error
2.3.2 The Augmented Normal Equation
2.4 Properties of the Autocorrelation Matrix
2.4.1 Relation Between Eigenvalues and the Power Spectrum
2.4.2 Singularity of the Autocorrelation Matrix
2.4.3 Determinant of the Autocorrelation Matrix
2.5 Estimating the Autocorrelation
2.6 Concluding Remarks

3. Levinson’s Recursion
3.1 Introduction
3.2 Derivation of Levinson’s Recursion
3.2.1 Summary of Levinson’s Recursion
3.2.2 The Partial Correlation Coefficient
3.3 Simple Properties of Levinson’s Recursion
3.4 The Whitening Effect
3.5 Concluding Remarks

4. Lattice Structures for Linear Prediction
4.1 Introduction
4.2 The Backward Predictor
4.2.1 All-Pass Property
4.2.2 Orthogonality of the Optimal Prediction Errors
4.3 Lattice Structures
4.3.1 The IIR LPC Lattice
4.3.2 Stability of the IIR Filter
4.3.3 The Upward and Downward Recursions
4.4 Concluding Remarks

5. Autoregressive Modeling
5.1 Introduction
5.2 Autoregressive Processes
5.3 Approximation by an AR(N) Processes
5.3.1 If a Process is AR, Then LPCWill Reveal It
5.3.2 Extrapolation of the Autocorrelation
5.4 Autocorrelation Matching Property
5.5 Power Spectrum of the AR Model
5.6 Application in Signal Compression
5.7 MA and ARMA Processes

6. Prediction Error Bound and Spectral Flatness
6.1 Introduction
6.2 Prediction Error for an AR Process
6.3 A Measure of Spectral Flatness
6.4 Spectral Flatness of an AR Process
6.5 Case Where Signal Is Not AR
6.5.1 Error Spectrum Gets Flatter as Predict or Order Grows
6.5.2 Mean Square Error and Determinant
6.6 Maximum Entropy and Linear Prediction
6.6.1 Connection to the Notion of Entropy
6.6.2 A Direct Maximization Problem
6.6.3 Entropy of the Prediction Error
6.7 Concluding Remarks

7. Line Spectral Processes
7.1 Introduction
7.2 Autocorrelation of a Line Spectral Process
7.2.1 The Characteristic Polynomial
7.2.2 Rank Saturation for a Line Spectral Process
7.3 Time Domain Descriptions

7.3.1 Extrapolation of Autocorrelation
7.3.2 All Zeros on theUnit Circle
7.3.3 Periodicity of a Line Spectral Process
7.3.4 Determining the Parameters of a Line Spectral Process
7.4 Further Properties of Time Domain Descriptions
7.5 Prediction Polynomial of Line Spectral Processes
7.6 Summary of Properties
7.7 Identifying a Line Spectral Process in Noise
7.7.1 Eigenstructure of the Autocorrelation Matrix
7.7.2 Computing the Powers at the Line Frequencies
7.8 Line Spectrum Pairs
7.9 Concluding Remarks

8. Linear Prediction Theory for Vector Processes
8.1 Introduction
8.2 Formulation of the Vector LPC Problem
8.3 Normal Equations: Vector Case
8.4 Backward Prediction
8.5 Levinson’s Recursion: Vector Case
8.6 Properties Derived from Levinson’s Recursion
8.6.1 Properties of Matrices F f m and Fb

8.6.2 Monotone Properties of the Error Covariance Matrices
8.6.3 Summary of Properties Relating to Levinson’s Recursion
8.7 Transfer Matrix Functions in Vector LPC
8.8 The FIR Lattice Structure for Vector LPC
8.8.1 Toward Rearrangement of the Lattice
8.8.2 The Symmetrical Lattice
8.9 The IIR Lattice Structure for Vector LPC
8.10 The Normalized IIR Lattice
8.11 The Paraunitary orMIMO All-Pass Property
8.11.1 Unitarity of Building Blocks
8.11.2 Propagation of Paraunitary Property
8.11.3 Poles of the MIMO IIR Lattice
8.12 Whitening Effect and Stalling
8.13 Properties of Transfer Matrices in LPC Theory
8.13.1 Review of Matrix Fraction Descripions
8.13.2 Relation Between Predictor Polynomials
8.14 Concluding Remarks

A. Linear Estimation of Random Variables
A.1 The Orthogonality Principle
A.2 Closed-Form Solution
A.3 Consequences of Orthogonality
A.4 Singularity of the Autocorrelation Matrix
B. Proof of a Property of Autocorrelations
C. Stability of the Inverse Filter
D. Recursion Satisfied by AR Autocorrelations

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