Advanced Signal Processing and Digital Noise Reduction

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1 Advanced Signal Processing and Digital Noise Reduction Saeed V. Vaseghi Queen's University of Belfast UK WILEY HTEUBNER A Partnership between John Wiley & Sons and B. G. Teubner Publishers Chichester New York Brisbane Toronto Singapore Stuttgart Leipzig

2 Preface xvi 1 Introduction Signals and Information Signal Processing Methods Non-parametric Signal Processing Model-based Signal Processing Bayesian Statistical Signal Processing Neural Networks Applications of Digital Signal Processing Adaptive Noise Cancellation and Noise Reduction Blind Channel Equalisation Signal Classification and Pattern Recognition Linear Prediction Modelling of Speech Digital Coding of Audio Signals Detection of Signals in Noise Directional Reception of Waves: Beamforming Sampling and Analog to Digital Conversion Time-Domain Sampling and Reconstruction of Analog Signals Quantisation 20 Bibliography 21 2 Stochastic Processes Random Signals and Stochastic Processes Stochastic Processes The Space or Ensemble of a Random Process Probabilistic Models of a Random Process Stationary and Nonstationary Random Processes Strict Sense Stationary Processes Wide Sense Stationary Processes Nonstationary Processes Expected Values of a Stochastic Process The Mean Value Autocorrelation Autocovariance Power Spectral Density Joint Statistical Averages of Two Random Processes Cross Correlation and Cross Covariance Cross Power Spectral Density and Coherence Ergodic Processes and Time-averaged Statistics 42

3 viii Contents Mean-ergodic Processes Correlation-ergodic Processes Some Useful Classes of Random Processes Gaussian (Normal) Process Multi-variate Gaussian Process Mixture Gaussian Process A Binary-state Gaussian Process Poisson Process Shot Noise Poisson-Gaussian Model for Clutters and Impulsive Noise Markov Processes Transformation of a Random Process Monotonic Transformation of Random Signals Many-to-one Mapping of Random Signals 60 Summary 62 Bibliography 63 3 Bayesian Estimation and Classification Estimation Theory: Basic Definitions Predictive and Statistical Models in Estimation Parameter Space Parameter Estimation and Signal Restoration Performance Measures Prior, and Posterior Spaces and Distributions Bayesian Estimation Maximum a Posterior Estimation Maximum Likelihood Estimation Minimum Mean Squared Error Estimation Minimum Mean Absolute Value of Error Estimation Equivalence of MAP, ML, MMSE and MAVE Influence of the Prior on Estimation Bias and Variance The Relative Importance of the Prior and the Observation Estimate-Maximise (EM) Method Convergence of the EM algorithm Cramer-Rao Bound on the Minimum Estimator Variance Cramer-Rao Bound for Random Parameters Cramer-Rao Bound for a Vector Parameter Bayesian Classification Classification of Discrete-valued Parameters Maximum a Posterior Classification Maximum Likelihood Classification Minimum Mean Squared Error Classification Bayesian Classification of Finite State Processes Bayesian Estimation of the Most Likely State Sequence Modelling the Space of a Random Signal Vector Quantisation of a Random Process Design of a Vector Quantiser: K-Means Algorithm 103

4 ' x Design of a Mixture Gaussian Model The EM Algorithm for Estimation of Mixture Gaussian Densities 105 Summary 108 Bibliography Hidden Markov Models ill 4.1 Statistical Models for Nonstationary Processes Hidden Markov Models A Physical Interpretation of Hidden Markov Models Hidden Markov Model As a Bayesian Method Parameters of a Hidden Markov Model State Observation Models State Transition Probabilities State-Time Trellis Diagram Training Hidden Markov Models Forward-Backward Probability Computation Baum-Welch Model Re-Estimation Training Discrete Observation Density HMMs HMMs with Continuous Observation PDFs HMMs with Mixture Gaussian pdfs Decoding of Signals Using Hidden Markov Models Viterbi Decoding Algorithm HMM-based Estimation of Signals in Noise HMM-based Wiener Filters Modelling Noise Characteristics 136 Summary 137 Bibliography Wiener Filters Wiener Filters: Least Squared Error Estimation Block-data Formulation of the Wiener Filter Vector Space Interpretation of Wiener Filters Analysis of the Least Mean Squared Error Signal Formulation of Wiener Filter in Frequency Domain Some Applications of Wiener Filters Wiener filter for Additive Noise Reduction Wiener Filter and Separability of Signal and Noise Squared Root Wiener Filter Wiener Channel Equaliser Time-alignment of Signals Implementation of Wiener Filters 159 Summary 161 Bibliography Kalman and Adaptive Least Squared Error Filters State-space Kalman Filters Sample Adaptive Filters Recursive Least Squares (RLS) Adaptive Filters 172

5 6.4 The Steepest Descent Method The LMS Adaptation Method 181 Summary 182 Bibliography Linear Prediction Models Linear Prediction Coding Least Mean Squared Error Predictor The Inverse Filter: Spectral Whitening The Prediction Error Signal Forward, Backward and Lattice Predictors Augmented Equations for Forward and Backward Predictors Levinson-Durbin Recursive Solution Lattice Predictors Alternative Formulations of Least Squared Error Predictors Model Order Selection Short-term and Long-term Predictors MAP Estimation of Predictor Coefficients Signal Restoration Using Linear Prediction Models Frequency Domain Signal Restoration 209 Summary 212 Bibliography Power Spectrum Estimation Fourier Transform, Power Spectrum and Correlation Fourier Transform Discrete Fourier Transform (DFT) Frequency Resolution and Spectral Smoothing Energy Spectral Density and Power Spectral Density Non-parametric Power Spectrum Estimation The Mean and Variance of Periodograms Averaging Periodograms (Bartlett Method) Welch Method :Averaging Periodograms from Overlapped and Windowed Segments Blackman-Tukey Method Power Spectrum Estimation from Autocorrelation of Overlapped Segments Model-based Power Spectrum Estimation Maximum Entropy Spectral Estimation Autoregressive Power Spectrum Estimation Moving Average Power Spectral Estimation Autoregressive Moving Average Power Spectral Estimation High Resolution Spectral Estimation Based on Subspace Eigen Analysis Pisarenko Harmonic Decomposition Multiple Signal Classification (MUSIC) Spectral Estimation Estimation of Signal Parameters via Rotational Invariance

6 xi Techniques (ESPRIT) 238 Summary 240 Bibliography Spectral Subtraction Spectral Subtraction Power Spectrum Subtraction Magnitude Spectrum Subtraction Spectral Subtraction Filter: Relation to Wiener Filters Processing Distortions Effect of Spectral Subtraction on Signal Distribution Reducing the Noise Variance Filtering Out the Processing Distortions Non-linear Spectral Subtraction Implementation of Spectral Subtraction Application to Speech Restoration and Recognition 257 Summary 259 Bibliography Interpolation Introduction Interpolation of a Sampled Signal Digital Interpolation by a Factor of / Interpolation of a Sequence of Lost Samples Factors that Affect Interpolation Polynomial Interpolation Lagrange Polynomial Interpolation Newton Interpolation Polynomial Hermite Interpolation Polynomials Cubic Spline Interpolation Statistical Interpolation Maximum a Posterior Interpolation Least Squared Error Autoregressive Interpolation Interpolation Based on a Short-term Prediction Model Interpolation Based on Long-term and Short-term Correlations LSAR Interpolation Error Interpolation in Frequency-Time Domain Interpolation using Adaptive Code Books Interpolation Through Signal Substitution 289 Summary 291 Bibliography Impulsive Noise Impulsive Noise Autocorrelation and Power Spectrum of Impulsive Noise Stochastic Models for Impulsive Noise Bernoulli-Gaussian Model of Impulsive Noise Poisson-Gaussian Model of Impulsive Noise 299

7 X 'i Contents A Binary State Model of Impulsive Noise Signal to Impulsive Noise Ratio Median Filters Impulsive Noise Removal Using Linear Prediction Models Impulsive Noise Detection Analysis of Improvement in Noise Detectability Two-sided Predictor Interpolation of Discarded Samples Robust Parameter Estimation Restoration of Archived Gramophone Records 311 Summary 312 Bibliography Transient Noise Transient Noise Waveforms Transient Noise Pulse Models Noise Pulse Templates Autoregressive Model of Transient Noise Hidden Markov Model of a Noise Pulse Process Detection of Noise Pulses Matched Filter Noise Detection Based on Inverse Filtering Noise Detection Based on HMM Removal of Noise Pulse Distortions Adaptive Subtraction of Noise pulses AR-based Restoration of Signals Distorted by Noise Pulses 324 Summary 327 Bibliography Echo Cancellation Telephone Line Echoes Telephone Line Echo Suppression Adaptive Echo Cancellation Convergence of Line Echo Canceller Echo Cancellation for Digital Data Transmission over Subscriber's Loop Acoustic Feedback Coupling Sub-band Acoustic Echo Cancellation 339 Summary 341 Bibliography Blind Deconvolution and Channel Equalisation Introduction The Ideal Inverse Channel Filter Equalisation Error, Convolutional Noise Blind Equalisation Minimum and Maximum Phase Channels 349

8 xiii Wiener Equaliser Blind Equalisation Using Channel Input Power Spectrum Homomorphic Equalisation Homomorphic Equalisation using a Bank of High Pass Filters Equalisation Based on Linear Prediction Models Blind Equalisation Through Model Factorisation Bayesian Blind Deconvolution and Equalisation Conditional Mean Channel Estimation Maximum Likelihood Channel Estimation Maximum a Posterior Channel Estimation Channel Equalisation Based on Hidden Markov Models MAP Channel Estimate Based on HMMs Implementations of HMM-Based Deconvolution Blind Equalisation for Digital Communication Channels Equalisation Based on Higher-Order Statistics Higher-Order Moments Higher Order Spectra of Linear Time-Invariant Systems Blind Equalisation Based on Higher Order Cepstrum 379 Summary 385 Bibliography 385 Frequently used Symbols and Abbreviations 388 Index 391

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