Advanced Digital Signal Processing and Noise Reduction, Second Edition.
`Saeed V. Vaseghi
`Copyright © 2000 John Wiley & Sons Ltd
`ISBNs: 0-471-62692-9 (Hardback): 0-470-84162-1 (Electronic)
`
`Advanced
`Digital Signal
`Processing
`and Noise
`Reduction
`
`Second Edition
`
`IPR PETITION
`US RE48,371
`Sonos Ex. 1029
`
`

`

`Advanced
`Digital Signal
`Processing
`and Noise
`Reduction
`
`Second Edition
`
`SAEED V. VASEGHI
`Professor of Communications and Signal Processing,
`Department of Electronics and Computer Engineering,
`Brunel University, UK
`
`JOHN WILEY & SONS, LTD
`Chichester · New York · Weinheim · Brisbane · Singapore · Toronto
`
`

`

`First Edition published in 1996 jointly by John Wiley & Sons, Ltd. and B. G. Teubner as Advanced
`Signal Processing and Digital Noise Reduction.
`Copyright © 2000 by John Wiley & Sons, Ltd
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`Library of Congress Cataloging-in-Publication Data
`V aseghi, Saeed V.
`Advanced digital signal processing and noise reduction/ Saeed V. Vaseghi.-2nd ed.
`p.cm.
`Includes bibliographical references and index.
`ISBN 0-471-62692-9 (alk.paper)
`1. Signal processing. 2. Electronic noise. 3. Digital Filters (Mathematics) I. Title.
`
`TK5102.9. V37 2000
`621.382 '2----dc21
`
`00-032091
`
`British Library Cataloguing in Publication Data
`
`A catalogue record for this book is available from the British Library
`
`ISBN 0 471 62692 9
`
`Produced from Postscript files supplied by the author.
`Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.
`This book is printed on acid-free paper responsibly manufactured from sustainable forestry, in which at
`least two trees are planted for each one used for paper production.
`
`

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`To my parents
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`With thanks to Peter Rayner, Ben Milner, Charles Ho and Aimin Chen
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`CONTENTS
`
`
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`PREFACE .............................................................................................. xvii
`
`FREQUENTLY USED SYMBOLS AND ABBREVIATIONS.......... xxi
`
`CHAPTER 1 INTRODUCTION...............................................................1
`1.1 Signals and Information...................................................................2
`1.2 Signal Processing Methods ..............................................................3
`1.2.1 Non−parametric Signal Processing .....................................3
`1.2.2 Model-Based Signal Processing ..........................................4
`1.2.3 Bayesian Statistical Signal Processing ................................4
`1.2.4 Neural Networks..................................................................5
`1.3 Applications of Digital Signal Processing .......................................5
`1.3.1 Adaptive Noise Cancellation and Noise Reduction ............5
`1.3.2 Blind Channel Equalisation.................................................8
`1.3.3 Signal Classification and Pattern Recognition ....................9
`1.3.4 Linear Prediction Modelling of Speech.............................11
`1.3.5 Digital Coding of Audio Signals .......................................12
`1.3.6 Detection of Signals in Noise............................................14
`1.3.7 Directional Reception of Waves: Beam-forming ..............16
`1.3.8 Dolby Noise Reduction .....................................................18
`1.3.9 Radar Signal Processing: Doppler Frequency Shift ..........19
`1.4 Sampling and Analog–to–Digital Conversion ...............................21
`1.4.1 Time-Domain Sampling and Reconstruction of Analog
`Signals ..............................................................................22
`1.4.2 Quantisation.......................................................................25
`Bibliography.........................................................................................27
`
`CHAPTER 2 NOISE AND DISTORTION...........................................29
`2.1 Introduction....................................................................................30
`2.2 White Noise ...................................................................................31
`2.3 Coloured Noise ..............................................................................33
`2.4 Impulsive Noise .............................................................................34
`2.5 Transient Noise Pulses...................................................................35
`2.6 Thermal Noise................................................................................36
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`2.7 Shot Noise......................................................................................38
`2.8 Electromagnetic Noise ...................................................................38
`2.9 Channel Distortions .......................................................................39
`2.10 Modelling Noise ..........................................................................40
`2.10.1 Additive White Gaussian Noise Model (AWGN)...........42
`2.10.2 Hidden Markov Model for Noise ....................................42
`Bibliography.........................................................................................43
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`CHAPTER 3 PROBABILITY MODELS ..............................................44
`3.1 Random Signals and Stochastic Processes ....................................45
`3.1.1 Stochastic Processes ..........................................................47
`3.1.2 The Space or Ensemble of a Random Process ..................47
`3.2 Probabilistic Models ......................................................................48
`3.2.1 Probability Mass Function (pmf).......................................49
`3.2.2 Probability Density Function (pdf)....................................50
`3.3 Stationary and Non-Stationary Random Processes........................53
`3.3.1 Strict-Sense Stationary Processes......................................55
`3.3.2 Wide-Sense Stationary Processes......................................56
`3.3.3 Non-Stationary Processes..................................................56
`3.4 Expected Values of a Random Process..........................................57
`3.4.1 The Mean Value ................................................................58
`3.4.2 Autocorrelation..................................................................58
`3.4.3 Autocovariance..................................................................59
`3.4.4 Power Spectral Density .....................................................60
`3.4.5 Joint Statistical Averages of Two Random Processes.......62
`3.4.6 Cross-Correlation and Cross-Covariance..........................62
`3.4.7 Cross-Power Spectral Density and Coherence ..................64
`3.4.8 Ergodic Processes and Time-Averaged Statistics .............64
`3.4.9 Mean-Ergodic Processes ...................................................65
`3.4.10 Correlation-Ergodic Processes ........................................66
`3.5 Some Useful Classes of Random Processes ..................................68
`3.5.1 Gaussian (Normal) Process ...............................................68
`3.5.2 Multivariate Gaussian Process ..........................................69
`3.5.3 Mixture Gaussian Process .................................................71
`3.5.4 A Binary-State Gaussian Process ......................................72
`3.5.5 Poisson Process .................................................................73
`3.5.6 Shot Noise .........................................................................75
`3.5.7 Poisson–Gaussian Model for Clutters and Impulsive
`Noise.................................................................................77
`3.5.8 Markov Processes..............................................................77
`3.5.9 Markov Chain Processes ...................................................79
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`ix
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`3.6 Transformation of a Random Process............................................81
`3.6.1 Monotonic Transformation of Random Processes ............81
`3.6.2 Many-to-One Mapping of Random Signals ......................84
`3.7 Summary........................................................................................86
`Bibliography.........................................................................................87
`
`CHAPTER 4 BAYESIAN ESTIMATION.............................................89
`4.1 Bayesian Estimation Theory: Basic Definitions ............................90
`4.1.1 Dynamic and Probability Models in Estimation................91
`4.1.2 Parameter Space and Signal Space....................................92
`4.1.3 Parameter Estimation and Signal Restoration ...................93
`4.1.4 Performance Measures and Desirable Properties of
`Estimators.........................................................................94
`4.1.5 Prior and Posterior Spaces and Distributions ....................96
`4.2 Bayesian Estimation.....................................................................100
`4.2.1 Maximum A Posteriori Estimation .................................101
`4.2.2 Maximum-Likelihood Estimation ...................................102
`4.2.3 Minimum Mean Square Error Estimation .......................105
`4.2.4 Minimum Mean Absolute Value of Error Estimation.....107
`4.2.5 Equivalence of the MAP, ML, MMSE and MAVE for
`Gaussian Processes With Uniform Distributed
`Parameters ......................................................................108
`4.2.6 The Influence of the Prior on Estimation Bias and
`Variance..........................................................................109
`4.2.7 The Relative Importance of the Prior and the
`Observation.....................................................................113
`4.3 The Estimate–Maximise (EM) Method .......................................117
`4.3.1 Convergence of the EM Algorithm .................................118
`4.4 Cramer–Rao Bound on the Minimum Estimator Variance..........120
`4.4.1 Cramer–Rao Bound for Random Parameters ..................122
`4.4.2 Cramer–Rao Bound for a Vector Parameter....................123
`4.5 Design of Mixture Gaussian Models ...........................................124
`4.5.1 The EM Algorithm for Estimation of Mixture Gaussian
`Densities .........................................................................125
`4.6 Bayesian Classification ................................................................127
`4.6.1 Binary Classification .......................................................129
`4.6.2 Classification Error..........................................................131
`4.6.3 Bayesian Classification of Discrete-Valued Parameters .132
`4.6.4 Maximum A Posteriori Classification.............................133
`4.6.5 Maximum-Likelihood (ML) Classification.....................133
`4.6.6 Minimum Mean Square Error Classification ..................134
`4.6.7 Bayesian Classification of Finite State Processes ...........134
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`4.6.8 Bayesian Estimation of the Most Likely State
`Sequence.........................................................................136
`4.7 Modelling the Space of a Random Process..................................138
`4.7.1 Vector Quantisation of a Random Process......................138
`4.7.2 Design of a Vector Quantiser: K-Means Clustering........138
`4.8 Summary......................................................................................140
`Bibliography.......................................................................................141
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`CHAPTER 5 HIDDEN MARKOV MODELS.....................................143
`5.1 Statistical Models for Non-Stationary Processes .........................144
`5.2 Hidden Markov Models ...............................................................146
`5.2.1 A Physical Interpretation of Hidden Markov Models .....148
`5.2.2 Hidden Markov Model as a Bayesian Model ..................149
`5.2.3 Parameters of a Hidden Markov Model ..........................150
`5.2.4 State Observation Models ...............................................150
`5.2.5 State Transition Probabilities ..........................................152
`5.2.6 State–Time Trellis Diagram ............................................153
`5.3 Training Hidden Markov Models ................................................154
`5.3.1 Forward–Backward Probability Computation.................155
`5.3.2 Baum–Welch Model Re-Estimation ...............................157
`5.3.3 Training HMMs with Discrete Density Observation
`Models ............................................................................159
`5.3.4 HMMs with Continuous Density Observation Models ...160
`5.3.5 HMMs with Mixture Gaussian pdfs................................161
`5.4 Decoding of Signals Using Hidden Markov Models ...................163
`5.4.1 Viterbi Decoding Algorithm............................................165
`5.5 HMM-Based Estimation of Signals in Noise...............................167
`5.6 Signal and Noise Model Combination and Decomposition.........170
`5.6.1 Hidden Markov Model Combination ..............................170
`5.6.2 Decomposition of State Sequences of Signal and Noise.171
`5.7 HMM-Based Wiener Filters ........................................................172
`5.7.1 Modelling Noise Characteristics .....................................174
`5.8 Summary......................................................................................174
`Bibliography.......................................................................................175
`
`CHAPTER 6 WIENER FILTERS........................................................178
`6.1 Wiener Filters: Least Square Error Estimation ............................179
`6.2 Block-Data Formulation of the Wiener Filter..............................184
`6.2.1 QR Decomposition of the Least Square Error Equation .185
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`6.3 Interpretation of Wiener Filters as Projection in Vector Space ...187
`6.4 Analysis of the Least Mean Square Error Signal .........................189
`6.5 Formulation of Wiener Filters in the Frequency Domain............191
`6.6 Some Applications of Wiener Filters...........................................192
`6.6.1 Wiener Filter for Additive Noise Reduction ...................193
`6.6.2 Wiener Filter and the Separability of Signal and Noise ..195
`6.6.3 The Square-Root Wiener Filter .......................................196
`6.6.4 Wiener Channel Equaliser...............................................197
`6.6.5 Time-Alignment of Signals in Multichannel/Multisensor
`Systems...........................................................................198
`6.6.6 Implementation of Wiener Filters ...................................200
`6.7 The Choice of Wiener Filter Order..............................................201
`6.8 Summary......................................................................................202
`Bibliography.......................................................................................202
`
`CHAPTER 7 ADAPTIVE FILTERS....................................................205
`7.1 State-Space Kalman Filters..........................................................206
`7.2 Sample-Adaptive Filters ..............................................................212
`7.3 Recursive Least Square (RLS) Adaptive Filters ..........................213
`7.4 The Steepest-Descent Method .....................................................219
`7.5 The LMS Filter ............................................................................222
`7.6 Summary......................................................................................224
`Bibliography.......................................................................................225
`
`CHAPTER 8 LINEAR PREDICTION MODELS ..............................227
`8.1 Linear Prediction Coding.............................................................228
`8.1.1 Least Mean Square Error Predictor .................................231
`8.1.2 The Inverse Filter: Spectral Whitening ...........................234
`8.1.3 The Prediction Error Signal.............................................236
`8.2 Forward, Backward and Lattice Predictors..................................236
`8.2.1 Augmented Equations for Forward and Backward
`Predictors........................................................................239
`8.2.2 Levinson–Durbin Recursive Solution .............................239
`8.2.3 Lattice Predictors.............................................................242
`8.2.4 Alternative Formulations of Least Square Error
`Prediction........................................................................244
`8.2.5 Predictor Model Order Selection.....................................245
`8.3 Short-Term and Long-Term Predictors........................................247
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`8.4 MAP Estimation of Predictor Coefficients..................................249
`8.4.1 Probability Density Function of Predictor Output...........249
`8.4.2 Using the Prior pdf of the Predictor Coefficients............251
`8.5 Sub-Band Linear Prediction Model .............................................252
`8.6 Signal Restoration Using Linear Prediction Models...................254
`8.6.1 Frequency-Domain Signal Restoration Using Prediction
`Models ............................................................................257
`8.6.2 Implementation of Sub-Band Linear Prediction Wiener
`Filters..............................................................................259
`8.7 Summary......................................................................................261
`Bibliography.......................................................................................261
`
`CHAPTER 9 POWER SPECTRUM AND CORRELATION ...........263
`9.1 Power Spectrum and Correlation .................................................264
`9.2 Fourier Series: Representation of Periodic Signals .....................265
`9.3 Fourier Transform: Representation of Aperiodic Signals............267
`9.3.1 Discrete Fourier Transform (DFT) ..................................269
`9.3.2 Time/Frequency Resolutions, The Uncertainty Principle
`..................................................................................................269
`9.3.3 Energy-Spectral Density and Power-Spectral Density ....270
`9.4 Non-Parametric Power Spectrum Estimation ..............................272
`9.4.1 The Mean and Variance of Periodograms .......................272
`9.4.2 Averaging Periodograms (Bartlett Method) ....................273
`9.4.3 Welch Method: Averaging Periodograms from
`Overlapped and Windowed Segments............................274
`9.4.4 Blackman–Tukey Method ...............................................276
`9.4.5 Power Spectrum Estimation from Autocorrelation of
`Overlapped Segments.....................................................277
`9.5 Model-Based Power Spectrum Estimation ..................................278
`9.5.1 Maximum–Entropy Spectral Estimation .........................279
`9.5.2 Autoregressive Power Spectrum Estimation ...................282
`9.5.3 Moving-Average Power Spectrum Estimation................283
`9.5.4 Autoregressive Moving-Average Power Spectrum
`Estimation.......................................................................284
`9.6 High-Resolution Spectral Estimation Based on Subspace Eigen-
`Analysis ......................................................................................284
`9.6.1 Pisarenko Harmonic Decomposition...............................285
`9.6.2 Multiple Signal Classification (MUSIC) Spectral
`Estimation.......................................................................288
`9.6.3 Estimation of Signal Parameters via Rotational
`Invariance Techniques (ESPRIT) ...................................292
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`xiii
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`9.7 Summary......................................................................................294
`Bibliography.......................................................................................294
`
`CHAPTER 10 INTERPOLATION.......................................................297
`10.1 Introduction................................................................................298
`10.1.1 Interpolation of a Sampled Signal .................................298
`10.1.2 Digital Interpolation by a Factor of I.............................300
`10.1.3 Interpolation of a Sequence of Lost Samples ................301
`10.1.4 The Factors That Affect Interpolation Accuracy...........303
`10.2 Polynomial Interpolation............................................................304
`10.2.1 Lagrange Polynomial Interpolation ...............................305
`10.2.2 Newton Polynomial Interpolation .................................307
`10.2.3 Hermite Polynomial Interpolation .................................309
`10.2.4 Cubic Spline Interpolation.............................................310
`10.3 Model-Based Interpolation ........................................................313
`10.3.1 Maximum A Posteriori Interpolation ............................315
`10.3.2 Least Square Error Autoregressive Interpolation ..........316
`10.3.3 Interpolation Based on a Short-Term Prediction Model
`..................................................................................................317
`10.3.4 Interpolation Based on Long-Term and Short-term
`Correlations..................................................................320
`10.3.5 LSAR Interpolation Error..............................................323
`10.3.6 Interpolation in Frequency–Time Domain ....................326
`10.3.7 Interpolation Using Adaptive Code Books....................328
`10.3.8 Interpolation Through Signal Substitution ....................329
`10.4 Summary....................................................................................330
`Bibliography.......................................................................................331
`
`CHAPTER 11 SPECTRAL SUBTRACTION.....................................333
`11.1 Spectral Subtraction...................................................................334
`11.1.1 Power Spectrum Subtraction .........................................337
`11.1.2 Magnitude Spectrum Subtraction..................................338
`11.1.3 Spectral Subtraction Filter: Relation to Wiener Filters .339
`11.2 Processing Distortions ...............................................................340
`11.2.1 Effect of Spectral Subtraction on Signal Distribution...342
`11.2.2 Reducing the Noise Variance ........................................343
`11.2.3 Filtering Out the Processing Distortions .......................344
`11.3 Non-Linear Spectral Subtraction ...............................................345
`11.4 Implementation of Spectral Subtraction ....................................348
`11.4.1 Application to Speech Restoration and Recognition.....351
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`11.5 Summary....................................................................................352
`Bibliography.......................................................................................352
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`CHAPTER 12 IMPULSIVE NOISE ....................................................355
`12.1 Impulsive Noise .........................................................................356
`12.1.1 Autocorrelation and Power Spectrum of Impulsive
`Noise ............................................................................359
`12.2 Statistical Models for Impulsive Noise......................................360
`12.2.1 Bernoulli–Gaussian Model of Impulsive Noise ............360
`12.2.2 Poisson–Gaussian Model of Impulsive Noise...............362
`12.2.3 A Binary-State Model of Impulsive Noise ....................362
`12.2.4 Signal to Impulsive Noise Ratio....................................364
`12.3 Median Filters ............................................................................365
`12.4 Impulsive Noise Removal Using Linear Prediction Models .....366
`12.4.1 Impulsive Noise Detection ............................................367
`12.4.2 Analysis of Improvement in Noise Detectability ..........369
`12.4.3 Two-Sided Predictor for Impulsive Noise Detection ....372
`12.4.4 Interpolation of Discarded Samples ..............................372
`12.5 Robust Parameter Estimation.....................................................373
`12.6 Restoration of Archived Gramophone Records.........................375
`12.7 Summary....................................................................................376
`Bibliography.......................................................................................377
`
`CHAPTER 13 TRANSIENT NOISE PULSES....................................378
`13.1 Transient Noise Waveforms ......................................................379
`13.2 Transient Noise Pulse Models ..................................................381
`13.2.1 Noise Pulse Templates .................................................382
`13.2.2 Autoregressive Model of Transient Noise Pulses ........383
`13.2.3 Hidden Markov Model of a Noise Pulse Process.........384
`13.3 Detection of Noise Pulses ..........................................................385
`13.3.1 Matched Filter for Noise Pulse Detection ....................386
`13.3.2 Noise Detection Based on Inverse Filtering .................388
`13.3.3 Noise Detection Based on HMM .................................388
`13.4 Removal of Noise Pulse Distortions..........................................389
`13.4.1 Adaptive Subtraction of Noise Pulses ...........................389
`13.4.2 AR-based Restoration of Signals Distorted by Noise
`Pulses ...........................................................................392
`13.5 Summary....................................................................................395
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`Bibliography.......................................................................................395
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`CHAPTER 14 ECHO CANCELLATION ...........................................396
`14.1 Introduction: Acoustic and Hybrid Echoes ................................397
`14.2 Telephone Line Hybrid Echo .....................................................398
`14.3 Hybrid Echo Suppression ..........................................................400
`14.4 Adaptive Echo Cancellation ......................................................401
`14.4.1 Echo Canceller Adaptation Methods.............................403
`14.4.2 Convergence of Line Echo Canceller............................404
`14.4.3 Echo Cancellation for Digital Data Transmission.........405
`14.5 Acoustic Echo ............................................................................406
`14.6 Sub-Band Acoustic Echo Cancellation......................................411
`14.7 Summary....................................................................................413
`Bibliography.......................................................................................413
`
`CHAPTER 15 CHANNEL EQUALIZATION AND BLIND
`DECONVOLUTION....................................................416
`15.1 Introduction................................................................................417
`15.1.1 The Ideal Inverse Channel Filter ...................................418
`15.1.2 Equalization Error, Convolutional Noise ......................419
`15.1.3 Blind Equalization.........................................................420
`15.1.4 Minimum- and Maximum-Phase Channels...................423
`15.1.5 Wiener Equalizer...........................................................425
`15.2 Blind Equalization Using Channel Input Power Spectrum........427
`15.2.1 Homomorphic Equalization ..........................................428
`15.2.2 Homomorphic Equalization Using a Bank of High-
`Pass Filters ...................................................................430
`15.3 Equalization Based on Linear Prediction Models......................431
`15.3.1 Blind Equalization Through Model Factorisation.........433
`15.4 Bayesian Blind Deconvolution and Equalization ......................435
`15.4.1 Conditional Mean Channel Estimation .........................436
`15.4.2 Maximum-Likelihood Channel Estimation...................436
`15.4.3 Maximum A Posteriori Channel Estimation .................437
`15.4.4 Channel Equalization Based on Hidden Markov
`Models..........................................................................438
`15.4.5 MAP Channel Estimate Based on HMMs.....................441
`15.4.6 Implementations of HMM-Based Deconvolution .........442
`15.5 Blind Equalization for Digital Communication Channels.........446
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`15.5.1 LMS Blind Equalization................................................448
`15.5.2 Equalization of a Binary Digital Channel......................451
`15.6 Equalization Based on Higher-Order Statistics .........................453
`15.6.1 Higher-Order Moments, Cumulants and Spectra ..........454
`15.6.2 Higher-Order Spectra of Linear Time-Invariant
`Systems ........................................................................457
`15.6.3 Blind Equalization Based on Higher-Order Cepstra .....458
`15.7 Summary....................................................................................464
`Bibliography.......................................................................................465
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`INDEX .....................................................................................................467
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`

`Advanced Digital Signal Processing and Noise Reduction, Second Edition.
`Saeed V. Vaseghi
`Copyright © 2000 John Wiley & Sons Ltd
`ISBNs: 0-471-62692-9 (Hardback): 0-470-84162-1 (Electronic)
`
`Noise-free signal space
`fh
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`fh
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`Noisy signal space
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`After subtraction of
`the noise mean
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`fh
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`fl
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`11
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`SPECTRAL SUBTRACTION
`
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`11.1 Spectral Subtraction
`11.2 Processing Distortions
`11.3 Non-Linear Spectral Subtraction
`11.4 Implementation of Spectral Subtraction
`11.5 Summary
`
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`
`
`S
`
`pectral subtraction is a method for restoration of the power spectrum
`or the magnitude spectrum of a signal observed in additive noise,
`through subtraction of an estimate of the average noise spectrum from
`the noisy signal spectrum. The noise spectrum is usually estimated, and
`updated, from the periods when the signal is absent and only the noise is
`present. The assumption is that the noise is a stationary or a slowly varying
`process, and that the noise spectrum does not change