1 INTRODUCTION

1．1 Characterization of Signals

1．1．1 Deterministic Signals,

1．1．2 Random Signals,Correlation Functions,and Power Spectra,

1．2 Characterization of Linear Time-Invariant Systems

1．2．1 Time-Domain Characterization,

1．2．2 Frequency-Domain Characterization,

1．2．3 Causality and Stability,

1．2．4 Bandpass Systems and Signals,

1．2．5 Inverse Systems,Minimum-Phase Systems,and All-Pass Systems,

1．2．6 Response of Linear Systems to Random Input Signals,

1．3 Sampling of Signals

1．3．1 Time-Domain Sampling of Analog signals,

1．3．2 Sampling the Spectrum of a Discrete-Time Signal,

1．3．3 The Discrete Fourier Transform for Finite-Duration Sequences,

1．3．4 The DFT and IDFT as Matrix Transformations,

1．4 Linear Filtering Methods Based on the DFT

1．4．1 Use of the DFT in Linear Filtering,

1．4．2 Filtering of Long Data Sequences,

1．5 The Cepstrum

1．6 Summary and References

Problems

2 ALGORITHMS FOR CONVOLUTION AND DFT

2．1 Modulo Polynomials

2．2 Circular Convolution as Polynomial Multiplication mod uN-1

2．3 A Continued Fraction of Polynomials

2．4 Chinese Remainder Theorem for Polynomials

2．5 Algorithms for Short Circular Convolutions

2．6 How We Count Multiplications

2．7 Cyclotomic Polynomials

2．8 Elementary Number Theory

2．8．1 Greatest Common Divisors and Euler’s Totient Function,

2．8．2 The Equation ax+by=1,

2．8．3 Modulo Arithmetic,

2．8．4 The Sino Representation of Integers Modulo M,

2．8．5 Exponentials Modulo M,

2．9 Convolution Length and Dimension

2．10 The DFT as a Circular Convolution

2．11 Winograd’s DFT Algorithm

2．12 Number-Theoretic Analogy of DFT

2．13 Number-Theoretic Transform

2．13．1 Mersenne Number Transform,

2．13．2 Fermat Number Transform,

2．13．3 Considerations for Use of NTTs to Perform Circular Convolution,

2．13．4 Use of Surrogate Fields for Complex Arithmetic,

2．14 Split-Radix FFT

2．15 Autogen Technique

2．16 Summary

Problems

3 LINEAR PREDICTION AND OPTIMUM LINEAR FILTERS

3．1 Innovations Representation of a Stationary Random Process

3．1．1 Rational Power Spectra,

3．1．2 Relationships between the Filter Parameters and the Autocorrelation Sequence,

3．2 Forward and Backward Linear Prediction

3．2．1 Forward Linear Prediction,

3．2．2 Backward Linear Prediction,

3．2．3 Optimum Reflection Coefficients for the Lattice Forward and Backward Predictors,

3．2．4 Relationship of an AR Process to Linear Prediction,

3．3 Solution of the Normal Equations

3．3．1 Levinson-Durbin Algorithm,

3．3．2 The Schur Algorithm,

3．4 Properties of the Linear Prediction-Error Filters

3．5 AR Lattice and ARMA Lattice-Ladder Filters

3．5．1 AR Lattice Structure,

3．5．2 ARMA Processes and Lattice-Ladder Filters,

3．6 Wiener Filters for Filtering and Prediction

3．6．1 FIR Wiener Filter,

3．6．2 Orthogonality Principle in Linear Mean-Square Estimation,

3．6．3 IIR Wiener Filter,

3．6．4 Noncausal Wiener Filter,

3．7 Summary and References

Problems

4 LEAST-SQUARES METHODS FOR SYSTEM MODELING AND FILTER DESIGN

4．1 System Modeling and Identification

4．1．1 System Identification Based on FIR(MA)System Model,

4．1．2 System Identification Bascd on All-Pole(AR)System Model,

4．1．3 System Identification Based on Pole-Zero(ARMA)System Model,

4．2 Least-Squares Filter Design for Prediction and Deconvolution

4．2．1 Least-Squares Linear Prediction Filter,

4．2．2 FIR Least-Squares Inverse Filters,

4．2．3 Predictive Deconvolution,

4．3 Solution of Least-Squares Estimation Problems

4．3．1 Definition and Basic Concepts,

4．3．2 Matrix Formulation of Least-Squares Estimation,

4．3．3 Cholesky Decomposition,

4．3．4 LDU Decomposition,

4．3．5 QR Decomposition,

4．3．6 Gram-Schmidt Orthogonalization,

4．3．7 Givens Rotation,

4．3．8 Householder Reflection,

4．3．9 Singular-Value Decomposition,

4．4 Summary and References

Problems

5 ADAPTIVE FILTERS

5．1 Applications of Adaptive Filters

5．1．1 System Identification or System Modeling,

5．1．2 Adaptive Channel Equalization,

5．1．3 Echo Cancellation in Data Transmission over Telephone Channels,

5．1．4 Suppression of Narrowband Interference in a Wideband Signal,

5．1．5 Adaptive Line Enhancer,

5．1．6 Adaptive Noise Cancelling,

5．1．7 Linear Predictive Coding of Speech Signals,

5．1．8 Adaptive Arrays,

5．2 Adaptive Direct-Form FIR Filters

5．2．1 Minimum Mean-Square-Error Criterion,

5．2．2 The LMS Algorithm,

5．2．3 Properties of the LMS Algorithm,

5．2．4 Recursive Least-Squares Algorithms for Direct-Form FIR Filters,

5．2．5 Properties of the Direct-Form RLS Algorithms,

5．3 Adaptive Lattice-Ladder Filters

5．3．1 Recursive Least-Squares Lattice-Ladder Algorithms,

5．3．2 Gradient Lattice-Ladder Algorithm,

5．3．3 Properties of Lattice-Ladder Algorithms,

5．4 Summary and References

Problems

6 RECURSIVE LEAST-SQUARES ALGORITHMS FOR ARRAY SIGNAL PROCESSING

6．1 QR Decomposition for Least-Squares Estimation

6．2 Gram-Schmidt Orthogonalization for Least-Squares Estimation

6．2．1 Least-Squares Estimation Using the MGS Algorithm,

6．2．2 Physical Meaning of the Quantities in the MGS Algorithm,

6．2．3 Time-Recursive Form of the Modified Gram-Schmidt Algorithm,

6．2．4 Variations of the RMGS Algorithm,

6．2．5 Implementation of the RMGS Algorithm Using VLSI Arrays,and Its Relationship to the Least-Squares Lattice Algorithm,

6．3 Givens Algorithm for Time-Recursive Least-Squares Estimation

6．3．1 Time-Recursive Givens Algorithm,

6．3．2 Givens Algorthm without Square Roots,

6．3．3 The CORDIC Approach to Givens Transformations,

6．4 Recursive Least-Squares Estimation Based on the Householder Transformation

6．4．1 Block Time-Recursive Least-Squares Estimation Using the Householder Transformation,

6．5 Order-Recursive Least-Squares Estimation Algorithms

6．5．1 Fundamental Relations of ORLS Estimation,

6．5．2 Canonical Structures for ORLS Estimation Algorithms,

6．5．3 Variations in the Basic Processing Cells of ORLS Algorithms,

6．5．4 Systematic Investigation and Derivation of ORLS Algorithms,

6．6 Summary and References

Problems

7 QRD-BASED FAST ADAPTIVE FILTER ALGORITHMS

7．1 Background

7．1．1 Signal Flow Graphs,

7．1．2 QRD-based RLS,Revisited,

7．1．3 Residual Extraction,

7．2 QRD Lattice

7．3 Multichannel Lattice

7．4 Fast QR Algorithm

7．5 Multichannel Fast QR Algorithm

7．6 Summary and References

Problems

8 POWER SPECTRUM ESTIMATION

8．1 Estimation of Spectra from Finite-Duration Observations of Signals

8．1．1 Computation of the Energy Density Spectrum,

8．1．2 Estimation of the Autocorrelation and Power Spectrum of Random Signals：The Periodogram,

8．1．3 Use of the DFT in Power Spectrum Estimation,

8．2 Nonparametric Methods for Power Spectrum Estimation

8．2．1 Bartlett Method:Averaging Periodograms,

8．2．2 Welch Method:Averaging Modified Periodograms,

8．2．3 Blackman and Tukey Method:Smoothing the Periodogram,

8．2．4 Performance Characteristics of Nonparametric Power Spectrum Estimators,

8．2．5 Computational Requirements of Nonparametric Power Spectrum Estimates,

8．3 Parametric Methods for Power Spectrum Estimation

8．3．1 Relationships Between the Autocorrelation and the Model Parameters,

8．3．2 Yule-Walker Method for the AR Model Parameters,

8．3．3 Burg Method for the AR Model Parameters,

8．3．4 Unconstrained Least-Squares Method for the AR Model Parameters,

8．3．5 Sequential Estimation Methods for the AR Model Parameters,

8．3．6 Selection of AR Model Order,

8．3．7 MA Model for Power Spectrum Estimation,

8．3．8 ARMA Model for Power Spectrum Estimation,

8．3．9 Experimental Results,

8．4 Minimum-Variance Spectral Estimation

8．5 Eigenanalysis Algorithms for Spectrum Estimation

8．5．1 Pisarenko Harmonic Decompsition Method,

8．5．2 Eigendecomposition of the Autocorrelation Matrix for Sinusoids in White Noise,

8．5．3 MUSIC Algorithm,

8．5．4 ESPRIT Algorithm,

8．5．5 Order Selection Criteria,

8．5．6 Experimental Results,

8．6 Summary and References

Problems

9 SIGNAL ANALYSIS WITH HIGHER-ORDER SPECTRA

9．1 Use of Higher-Order Spectra in Signal Processing

9．2．1 Moments and Cumulants of Random Signals,

9．2 Definition and Properties of Higher-Order Spectra

9．2．2 Higher-Order Spectra (Cumulant Spectra),

9．2．3 Linear Non-Gaussian Processes,

9．2．4 Nonlinear Processes,

9．3 Conventional Estimators for Higher-Order Spectra

9．3．1 Indirect Method,

9．3．2 Direct Method,

9．3．3 Statistical Properties of Conventional Estimators,

9．3．4 Test for Aliasing with the Bispectrum,

9．4 Parametric Methods for Higher-Order Spectrum Estimation

9．4．1 MA Methods,

9．4．2 Noncausal AR Methods,

9．4．3 ARMA Methods,

9．4．4 AR Methods for the Detection of Quadratic Phase Coupling,

9．5 Cepstra of Higher-Order Spectra

9．5．1 Preliminaries,

9．5．2 Complex and Differentical Cepstra,

9．5．3 Bicepstrum,

9．5．4 Cepstrum of the Power Spectrum,

9．5．5 Cepstrum of the Bicoherence,

9．5．6 Summary of Cepstra and Key Observation,

9．6 Phase and Magnitude Retrieval from the Bispectrum

9．7 Summary and Refefences

Problems

REFERENCES

INDEX