Advanced Signal Processing and Digital Noise Reduction

ISBN-13: 9783322927743 / Niemiecki / Miękka / 2012 / 397 str.

Saeed V. Vaseghi

Spis treści

1 Introduction.- 1.1 Signals and Information.- 1.2 Signal Processing Methods.- 1.2.1 Non-parametric Signal Processing.- 1.2.2 Model-based Signal Processing.- 1.2.3 Bayesian Statistical Signal Processing.- 1.2.4 Neural Networks.- 1.3 Applications of Digital Signal Processing.- 1.3.1 Adaptive Noise Cancellation and Noise Reduction.- 1.3.2 Blind Channel Equalisation.- 1.3.3 Signal Classification and Pattern Recognition.- 1.3.4 Linear Prediction Modelling of Speech.- 1.3.5 Digital Coding of Audio Signals.- 1.3.6 Detection of Signals in Noise.- 1.3.7 Directional Reception of Waves: Beamforming.- 1.4 Sampling and Analog to Digital Conversion.- 1.4.1 Time-Domain Sampling and Reconstruction of Analog Signals.- 1.4.2 Quantisation.- 2 Stochastic Processes.- 2.1 Random Signals and Stochastic Processes.- 2.1.1 Stochastic Processes.- 2.1.2 The Space or Ensemble of a Random Process.- 2.2 Probabilistic Models of a Random Process.- 2.3 Stationary and Nonstationary Random Processes.- 2.3.1 Strict Sense Stationary Processes.- 2.3.2 Wide Sense Stationary Processes.- 2.3.3 Nonstationary Processes.- 2.4 Expected Values of a Stochastic Process.- 2.4.1 The Mean Value.- 2.4.2 Autocorrelation.- 2.4.3 Autocovariance.- 2.4.4 Power Spectral Density.- 2.4.5 Joint Statistical Averages of Two Random Processes.- 2.4.6 Cross Correlation and Cross Covariance.- 2.4.7 Cross Power Spectral Density and Coherence.- 2.4.8 Ergodic Processes and Time-averaged Statistics.- 2.4.9 Mean-ergodic Processes.- 2.4.10 Correlation-ergodic Processes.- 2.5 Some Useful Classes of Random Processes.- 2.5.1 Gaussian (Normal) Process.- 2.5.2 Multi-variate Gaussian Process.- 2.5.3 Mixture Gaussian Process.- 2.5.4 A Binary-state Gaussian Process.- 2.5.5 Poisson Process.- 2.5.6 Shot Noise.- 2.5.7 Poisson-Gaussian Model for Clutters and Impulsive Noise.- 2.5.8 Markov Processes.- 2.6 Transformation of a Random Process.- 2.6.1 Monotonic Transformation of Random Signals.- 2.6.2 Many-to-one Mapping of Random Signals.- Summary.- 3 Bayesian Estimation and Classification.- 3.1 Estimation Theory: Basic Definitions.- 3.1.1 Predictive and Statistical Models in Estimation.- 3.1.2 Parameter Space.- 3.1.3 Parameter Estimation and Signal Restoration.- 3.1.4 Performance Measures.- 3.1.5 Prior, and Posterior Spaces and Distributions.- 3.2 Bayesian Estimation.- 3.2.1 Maximum a Posterior Estimation.- 3.2.2 Maximum Likelihood Estimation.- 3.2.3 Minimum Mean Squared Error Estimation.- 3.2.4 Minimum Mean Absolute Value of Error Estimation.- 3.2.5 Equivalence of MAP, ML, MMSE and MAVE.- 3.2.6 Influence of the Prior on Estimation Bias and Variance.- 3.2.7 The Relative Importance of the Prior and the Observation.- 3.3 Estimate-Maximise (EM) Method.- 3.3.1 Convergence of the EM algorithm.- 3.4 Cramer-Rao Bound on the Minimum Estimator Variance.- 3.4.1 Cramer-Rao Bound for Random Parameters.- 3.4.2 Cramer-Rao Bound for a Vector Parameter.- 3.5 Bayesian Classification.- 3.5.1 Classification of Discrete-valued Parameters.- 3.5.2 Maximum a Posterior Classification.- 3.5.3 Maximum Likelihood Classification.- 3.5.4 Minimum Mean Squared Error Classification.- 3.5.5 Bayesian Classification of Finite State Processes.- 3.5.6 Bayesian Estimation of the Most Likely State Sequence.- 3.6 Modelling the Space of a Random Signal.- 3.6.1 Vector Quantisation of a Random Process.- 3.6.2 Design of a Vector Quantiser: K-Means Algorithm.- 3.6.3 Design of a Mixture Gaussian Model.- 3.6.4 The EM Algorithm for Estimation of Mixture Gaussian Densities.- Summary.- 4 Hidden Markov Models.- 4.1 Statistical Models for Nonstationary Processes.- 4.2 Hidden Markov Models.- 4.2.1 A Physical Interpretation of Hidden Markov Models.- 4.2.2 Hidden Markov Model As a Bayesian Method.- 4.2.3 Parameters of a Hidden Markov Model.- 4.2.4 State Observation Models.- 4.2.5 State Transition Probabilities.- 4.2.6 State-Time Trellis Diagram.- 4.3 Training Hidden Markov Models.- 4.3.1 Forward-Backward Probability Computation.- 4.3.2 Baum-Welch Model Re-Estimation.- 4.3.3 Training Discrete Observation Density HMMs.- 4.3.4 HMMs with Continuous Observation PDFs.- 4.3.5 HMMs with Mixture Gaussian pdfs.- 4.4 Decoding of Signals Using Hidden Markov Models.- 4.4.1 Viterbi Decoding Algorithm.- 4.5 HMM-based Estimation of Signals in Noise.- 4.5.1 HMM-based Wiener Filters.- 4.5.2 Modelling Noise Characteristics.- Summary.- 5 Wiener Filters.- 5.1 Wiener Filters: Least Squared Error Estimation.- 5.2 Block-data Formulation of the Wiener Filter.- 5.3 Vector Space Interpretation of Wiener Filters.- 5.4 Analysis of the Least Mean Squared Error Signal.- 5.5 Formulation of Wiener Filter in Frequency Domain.- 5.6 Some Applications of Wiener Filters.- 5.6.1 Wiener filter for Additive Noise Reduction.- 5.6.2 Wiener Filter and Separability of Signal and Noise.- 5.6.3 Squared Root Wiener Filter.- 5.6.4 Wiener Channel Equaliser.- 5.6.5 Time-alignment of Signals.- 5.6.6 Implementation of Wiener Filters.- Summary.- 6 Kalman and Adaptive Least Squared Error Filters.- 6.1 State-space Kalman Filters.- 6.2 Sample Adaptive Filters.- 6.3 Recursive Least Squares (RLS) Adaptive Filters.- 6.4 The Steepest Descent Method.- 6.5 The LMS Adaptation Method.- Summary.- 7 Linear Prediction Models.- 7.1 Linear Prediction Coding.- 7.1.1 Least Mean Squared Error Predictor.- 7.1.2 The Inverse Filter: Spectral Whitening.- 7.1.3 The Prediction Error Signal.- 7.2 Forward, Backward and Lattice Predictors.- 7.2.1 Augmented Equations for Forward and Backward Predictors.- 7.2.2 Levinson-Durbin Recursive Solution.- 7.2.3 Lattice Predictors.- 7.2.4 Alternative Formulations of Least Squared Error Predictors.- 7.2.5 Model Order Selection.- 7.3 Short-term and Long-term Predictors.- 7.4 MAP Estimation of Predictor Coefficients.- 7.5 Signal Restoration Using Linear Prediction Models.- 7.5.1 Frequency Domain Signal Restoration.- Summary.- 8 Power Spectrum Estimation.- 8.1 Fourier Transform, Power Spectrum and Correlation.- 8.1.1 Fourier Transform.- 8.1.2 Discrete Fourier Transform (DFT).- 8.1.3 Frequency Resolution and Spectral Smoothing.- 8.1.4 Energy Spectral Density and Power Spectral Density.- 8.2 Non-parametric Power Spectrum Estimation.- 8.2.1 The Mean and Variance of Periodograms.- 8.2.2 Averaging Periodograms (Bartlett Method).- 8.2.3 Welch Method ¡Averaging Periodograms from Overlapped and Windowed Segments.- 8.2.4 Blackman-Tukey Method.- 8.2.5 Power Spectrum Estimation from Autocorrelation of Overlapped Segments.- 8.3 Model-based Power Spectrum Estimation.- 8.3.1 Maximum Entropy Spectral Estimation.- 8.3.2 Autoregressive Power Spectrum Estimation.- 8.3.3 Moving Average Power Spectral Estimation.- 8.3.4 Autoregressive Moving Average Power Spectral Estimation.- 8.4 High Resolution Spectral Estimation Based on Subspace Eigen Analysis.- 8.4.1 Pisarenko Harmonic Decomposition.- 8.4.2 Multiple Signal Classification (MUSIC) Spectral Estimation.- 8.4.3 Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT).- Summary.- 9 Spectral Subtraction.- 9.1 Spectral Subtraction.- 9.1.1 Power Spectrum Subtraction.- 9.1.2 Magnitude Spectrum Subtraction.- 9.1.3 Spectral Subtraction Filter: Relation to Wiener Filters.- 9.2 Processing Distortions.- 9.2.1 Effect of Spectral Subtraction on Signal Distribution.- 9.2.2 Reducing the Noise Variance.- 9.2.3 Filtering Out the Processing Distortions.- 9.3 Non-linear Spectral Subtraction.- 9.4 Implementation of Spectral Subtraction.- 9.4.1 Application to Speech Restoration and Recognition.- Summary.- 10 Interpolation.- 10.1 Introduction.- 10.1.1 Interpolation of a Sampled Signal.- 10.1.2 Digital Interpolation by a Factor of I.- 10.1.3 Interpolation of a Sequence of Lost Samples.- 10.1.4 Factors that Affect Interpolation.- 10.2 Polynomial Interpolation.- 10.2.1 Lagrange Polynomial Interpolation.- 10.2.2 Newton Interpolation Polynomial.- 10.2.3 Hermite Interpolation Polynomials.- 10.2.4 Cubic Spline Interpolation.- 10.3 Statistical Interpolation.- 10.3.1 Maximum a Posterior Interpolation.- 10.3.2 Least Squared Error Autoregressive Interpolation.- 10.3.3 Interpolation Based on a Short-term Prediction Model.- 10.3.4 Interpolation Based on Long-term and Short-term Correlations.- 10.3.5 LSAR Interpolation Error.- 10.3.6 Interpolation in Frequency-Time Domain.- 10.3.7 Interpolation using Adaptive Code Books.- 10.3.8 Interpolation Through Signal Substitution.- Summary.- 11 Impulsive Noise.- 11.1 Impulsive Noise.- 11.1.1 Autocorrelation and Power Spectrum of Impulsive Noise.- 11.2 Stochastic Models for Impulsive Noise.- 11.2.1 Bernoulli-Gaussian Model of Impulsive Noise.- 11.2.2 Poisson-Gaussian Model of Impulsive Noise.- 11.2.3 A Binary State Model of Impulsive Noise.- 11.2.4 Signal to Impulsive Noise Ratio.- 11.3 Median Filters.- 11.4 Impulsive Noise Removal Using Linear Prediction Models.- 11.4.1 Impulsive Noise Detection.- 11.4.2 Analysis of Improvement in Noise Detectability.- 11.4.3 Two-sided Predictor.- 11.4.4 Interpolation of Discarded Samples.- 11.5 Robust Parameter Estimation.- 11.6 Restoration of Archived Gramophone Records.- Summary.- 12 Transient Noise.- 12.1 Transient Noise Waveforms.- 12.2 Transient Noise Pulse Models.- 12.2.1 Noise Pulse Templates.- 12.2.2 Autoregressive Model of Transient Noise.- 12.2.3 Hidden Markov Model of a Noise Pulse Process.- 12.3 Detection of Noise Pulses.- 12.3.1 Matched Filter.- 12.3.2 Noise Detection Based on Inverse Filtering.- 12.3.3 Noise Detection Based on HMM.- 12.4 Removal of Noise Pulse Distortions.- 12.4.1 Adaptive Subtraction of Noise pulses.- 12.4.2 AR-based Restoration of Signals Distorted by Noise Pulses.- Summary.- 13 Echo Cancellation.- 13.1 Telephone Line Echoes.- 13.1.1 Telephone Line Echo Suppression.- 13.2 Adaptive Echo Cancellation.- 13.2.1 Convergence of Line Echo Canceller.- 13.2.2 Echo Cancellation for Digital Data Transmission over Subscriber’s Loop.- 13.3 Acoustic Feedback Coupling.- 13.4 Sub-band Acoustic Echo Cancellation.- Summary.- 14 Blind Deconvolution and Channel Equalisation.- 14.1 Introduction.- 14.1.1 The Ideal Inverse Channel Filter.- 14.1.2 Equalisation Error, Convolutional Noise.- 14.1.3 Blind Equalisation.- 14.1.4 Minimum and Maximum Phase Channels.- 14.1.5 Wiener Equaliser.- 14.2 Blind Equalisation Using Channel Input Power Spectrum.- 14.2.1 Homomorphic Equalisation.- 14.2.2 Homomorphic Equalisation using a Bank of High Pass Filters.- 14.3 Equalisation Based on Linear Prediction Models.- 14.3.1 Blind Equalisation Through Model Factorisation.- 14.4 Bayesian Blind Deconvolution and Equalisation.- 14.4.1 Conditional Mean Channel Estimation.- 14.4.2 Maximum Likelihood Channel Estimation.- 14.4.3 Maximum a Posterior Channel Estimation.- 14.4.4 Channel Equalisation Based on Hidden Markov Models.- 14.4.5 MAP Channel Estimate Based on HMMs.- 14.4.6 Implementations of HMM-Based Deconvolution.- 14.5 Blind Equalisation for Digital Communication Channels.- 14.6 Equalisation Based on Higher-Order Statistics.- 14.6.1 Higher-Order Moments.- 14.6.2 Higher Order Spectra of Linear Time-Invariant Systems.- 14.6.3 Blind Equalisation Based on Higher Order Cepstrum.- Summary.- Frequently used Symbols and Abbreviations.