Digital Spectral Analysis

CONTENTS
 
NOTATIONAL CONVENTIONS
GLOSSARY OF KEY SYMBOLS
PREFACE
 
1 INTRODUCTION
1.1 Historical Perspective
1.2 Sunspot Numbers
1.3 A Test Case
1.4 Issues in Spectral Estimation
1.5 How to Use This Text
References
 
2 REVIEW OF LINEAR SYSTEMS AND TRANSFORM THEORY
2.1 Introduction
2.2 Signal Notation
2.3 Continuous Linear Systems
2.4 Discrete Linear Systems
2.5 Continuous-Time Fourier Transform
2.6 Sampling and Windowing Operations
2.7 Relating the Continuous and Discrete Transforms
2.8 The Issue of Scaling for Power Determination
2.9 The Issue of Zero Padding
2.10 The Fast Fourier Transform
2.11 Resolution and the Time-Bandwidth Product
2.12 Extra: Source of Complex-Valued Signals
2.13 Extra: Wavenumber Processing with Linear Spatial Arrays
References
 
3 REVIEW OF MATRIX ALGEBRA
3.1 Introduction
3.2 Matrix Algebra Basics
3.3 Special Vector and Matrix Structures
3.4 Matrix Inverse
3.5 Solution of Linear Equations
3.6 Overdetermined and Underdetermined Linear Equations
3.7 Solution of Overdetermined and Underdetermined Linear Equations
3.8 The Toeplitz Matrix
3.9 The Vandermonde Matrix
References
 
4 REVIEW OF RANDOM PROCESS THEORY
4.1 Introduction
4.2 Probability and Random Variables
4.3 Random Processes
4.4 Substituting Time Averages for Ensemble Averages
4.5 Entropy Concepts
4.6 Limit Spectrum of Test Data
4.7 Extra: Bias and Variance of the Sample Spectrum
References
 
5 CLASSICAL SPECTRAL ESTIMATION
5.1 Introduction
5.2 Summary
5.3 Windows
5.4 Resolution and the Stability-Time-Bandwidth Product
5.5 Autocorrelation and Cross Correlation Estimation
5.6 Correlogram Power Spectral Density (PSD) Estimators
5.7 Periodogram PSD Estimators
5.8 Combined Periodogram/Correlogram Estimators
5.9 Application to Sunspot Numbers
5.10 Conclusion
References
 
6 PARAMETRIC MODELS OF RANDOM PROCESSES
6.1 Introduction
6.2 Summary
6.3 Autoregressive (AR), Moving Average (MA), and Autoregressive-Moving Average (ARMA) Random Process Models
6.4 Relationships Among AR, MA, and ARMA Process Parameters
6.5 Relationship of AR, MA, and ARMA Parameters to ACS
6.6 Spectral Factorization
References
 
7 AUTOREGRESSIVE PROCESS AND SPECTRUM PROPERTIES
7.1 Introduction
7.2 Summary
7.3 Autoregressive Process Properties
7.4 Autoregressive Power Spectral Density Properties
References
 
8 AR SPECTRAL ESTIMATION: BLOCK DATA ALGORITHMS
8.1 Introduction
8.2 Summary
8.3 Correlation Function Estimation Method
8.4 Reection Coecient Estimation Methods
8.5 Least Squares Linear Prediction Estimation Methods
8.6 Estimator Characteristics
8.7 Model Order Selection
8.8 Autoregressive Processes with Observation Noise
8.9 Application to Sunspot Numbers
8.10 Extra: Covariance Linear Prediction Fast Algorithm
8.11 Extra: Modi_ed Covariance Linear Prediction Fast Algorithm
References
 
9 AR SPECTRAL ESTIMATION: SEQUENTIAL DATA ALGORITHMS
9.1 Introduction
9.2 Summary
9.3 Gradient Adaptive Autoregressive Methods
9.4 Recursive Least Squares (RLS) Autoregressive Methods
9.5 Fast Lattice Autoregressive Methods
9.6 Application to Sunspot Numbers
9.7 Extra: Fast RLS Algorithm for Recursive Linear Prediction
References
 
10 ARMA AVERAGE SPECTRAL ESTIMATION
10.1 Introduction
10.2 Summary
10.3 Moving Average Parameter Estimation
10.4 Separate Autoregressive and Moving Average Parameter Estimation
10.5 Simultaneous Autoregressive and Moving Average Parameter Estimation
10.6 Sequential Approach to ARMA Estimation
10.7 A Special ARMA Process for Sinusoids in White Noise
10.8 Application to Sunspot Numbers
References
 
11 PRONY'S METHOD
11.1 Introduction
11.2 Summary
11.3 Simultaneous Exponential Parameter Estimation
11.4 Original Prony Concept
11.5 Least Squares Prony Method
11.6 Modi_ed Least Squares Prony Method
11.7 Prony Spectrum
11.8 Accounting for Known Exponential Components
11.9 Identi_cation of Exponentials in Noise
11.10 Application to Sunspot Numbers
11.11 Extra: Fast Algorithm to Solve Symmetric Covariance Linear Equations
References
 
12 MINIMUM VARIANCE SPECTRAL ESTIMATION
12.1 Introduction
12.2 Summary
12.3 Derivation of the Minimum Variance Spectral Estimator
12.4 Relationship of Minimum Variance and Autoregressive Spectral Estimators
12.5 Implementation of the Minimum Variance Spectral Estimator
12.6 Application to Sunspot Numbers
References
 
13 EIGENANALYSIS-BASED FREQUENCY ESTIMATION
13.1 Introduction
13.2 Summary
13.3 Eigenanalysis of Autocorrelation Matrix for Sinusoids in White Noise
13.4 Eigenanalysis of Data Matrix for Exponentials in Noise
13.5 Signal Subspace Frequency Estimators
13.6 Noise Subspace Frequency Estimators
13.7 Order Selection
References
 
14 SUMMARY OF SPECTRAL ESTIMATORS
Synopsis Table
References
 
15 MULTICHANNEL SPECTRAL ESTIMATION
15.1 Introduction
15.2 Summary
15.3 Multichannel Linear Systems Theory
15.4 Multichannel Random Process Theory
15.5 Multichannel Classical Spectral Estimators
15.6 Multichannel ARMA, AR, and MA Processes
15.7 Multichannel Yule-Walker Equations
15.8 Multichannel Levinson Algorithm
15.9 Multichannel Block Toeplitz Matrix Inverse
15.10 Multichannel Autoregressive Spectral Estimation
15.11 Autoregressive Order Selection
15.12 Experimental Comparison of Multichannel AR PSD Estimators
15.13 Multichannel Minimum Variance Spectral Estimation
15.14 Two Channel Spectral Analysis: Sunspot Numbers and Air Temperature
References
 
16 TWO-DIMENSIONAL SPECTRAL ESTIMATION
16.1 Introduction
16.2 Summary
16.3 Two-Dimensional Linear Systems and Transform Theory
16.4 Two-Dimensional Random Process Theory
16.5 Classical 2-D Spectral Estimation
16.6 Modi_ed Classical 2-D Spectral Estimation
16.7 Two-Dimensional Autoregressive Spectral Estimation
16.8 Two-Dimensional Maximum Entropy Spectral Estimation
16.9 Two-Dimensional Minimum Variance Spectral Estimation Estimator
References

INDEX