5/25/2010 7 Gaussian Filtering G i k l ffi i t l d f th 2D G iGaussian kernel coefficients are samp led from the 2D Gaussian function Where σis the standard deviation of the distribution
Filtrage Linéaire: filtre binomia l • Approximation du filtre de Gauss: moyenne pondérée des voisins – filtre 3x3 – filtre 5x5 • avantage – rapide – plus robuste que moyenneur • inconvénient – un plus complexe à calculer 10 Filtrage Linéaire: filtre gaussien • Largeur de la fenêtre
TRAITEMENT D’IMAGES FILTRAGE SPATIAL Max Mignotte Département d’Informatique et de Recherche Opérationnelle FILTRE GAUSSIEN (PASSE-BAS) Gaussienne(x,y
1 2 Etude sous MATLAB 1 Cr´eer sous MATLAB un script vide nomm´e TP3Partie1 m 2 Ouvrir l’image cameraman tif, la stocker dans une matrice X1et la renormaliser 3 Appliquer sur l’image X1un bruit blanc Gaussien de variance σ2 = 0 01 et stocker le r´esultat dans une matrice X2
Traitement d'images Convolution – filtrage passe-bas, passe-haut Séance 7 Moy5x5 Filtre gaussien σ=2 21 Filtres de lissage non-linéaires : filtre médian
Les objectifs du cours sur le traitement des images (partie 1) sont de: 1 Présenter quelques notions sur les différents types de bruit pouvant influener le ontenu d’une image 2 Présenter des approches de filtrage linéaire 1 filtre moyenneur uniforme 2 filtre gaussien (et oneption d’un filtre gaussien) 3
C and MATLAB functions for implementing various DSP operations The practical slant of the book makes the concepts more concrete Use The book may be used at the junior or senior level It is based on a junior-level DSP course that I have taught at Rutgers since 1988 The assumed background is only a first course on linear systems
3 The process of image convolution A convolution is done by multiplying a pixel’s and its neighboring pixels color value by a matrix Kernel: A kernel is a (usually) small
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INTRODUCTION TO
Signal
Processing
INTRODUCTION TO
Signal
Processing
Sophocles J. Orfanidis
Rutgers University
To my lifelong friend George Lazos
Copyright © 2010 by Sophocles J. Orfanidis
This book was previously published by Pearson Education, Inc. Copyright © 1996-2009 by Prentice Hall, Inc. Previous ISBN 0-13-209172-0. All rights reserved. No parts of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopy- ing, recording or otherwise, without the prior written permission of the author.
MATLAB
?R is a registered trademark of The MathWorks, Inc.
Web page:www.ece.rutgers.edu/~orfanidi/i2sp
Contents
Preface xiii
1 Sampling and Reconstruction 1
1.1 Introduction, 1
1.2 Review of Analog Signals, 1
1.3 Sampling Theorem, 4
1.3.1 Sampling Theorem, 6
1.3.2 Antialiasing Prefilters, 7
1.3.3 Hardware Limits, 8
1.4 Sampling of Sinusoids, 9
1.4.1 Analog Reconstruction and Aliasing, 10
1.4.2 Rotational Motion, 27
1.4.3 DSP Frequency Units, 29
1.5 Spectra of Sampled Signals
,29
1.5.1 Discrete-Time Fourier Transform, 31
1.5.2 Spectrum Replication, 33
1.5.3 Practical Antialiasing Prefilters, 38
1.6 Analog Reconstructors
,42
1.6.1 Ideal Reconstructors, 43
1.6.2 Staircase Reconstructors, 45
1.6.3 Anti-Image Postfilters, 46
1.7 Basic Components of DSP Systems, 53
1.8 Problems, 55
2 Quantization 61
2.1 Quantization Process, 61
2.2 Oversampling and Noise Shaping
,65
2.3 D/A Converters, 71
2.4 A/D Converters, 75
2.5 Analog and Digital Dither
,83
2.6 Problems, 90
3 Discrete-Time Systems 95
3.1 Input/Output Rules, 96
3.2 Linearity and Time Invariance, 100
3.3 Impulse Response, 103
vii viiiCONTENTS
3.4 FIR and IIR Filters, 105
3.5 Causality and Stability, 112
3.6 Problems, 117
4 FIR Filtering and Convolution 121
4.1 Block Processing Methods, 122
4.1.1 Convolution, 122
4.1.2 Direct Form, 123
4.1.3 Convolution Table, 126
4.1.4 LTI Form, 127
4.1.5 Matrix Form, 129
4.1.6 Flip-and-Slide Form, 131
4.1.7 Transient and Steady-State Behavior, 132
4.1.8 Convolution of Infinite Sequences, 134
4.1.9 Programming Considerations, 139
4.1.10 Overlap-Add Block Convolution Method, 143
4.2 Sample Processing Methods, 146
4.2.1 Pure Delays, 146
4.2.2 FIR Filtering in Direct Form, 152
4.2.3 Programming Considerations, 160
4.2.4 Hardware Realizations and Circular Buffers, 162
4.3 Problems, 178
5 z-Transforms 183
5.1 Basic Properties, 183
5.2 Region of Convergence, 186
5.3 Causality and Stability, 193
5.4 Frequency Spectrum, 196
5.5 Inversez-Transforms, 202
5.6 Problems, 210
6 Transfer Functions 214
6.1 Equivalent Descriptions of Digital Filters, 214
6.2 Transfer Functions, 215
6.3 Sinusoidal Response, 229
6.3.1 Steady-State Response, 229
6.3.2 Transient Response, 232
6.4 Pole/Zero Designs, 242
6.4.1 First-Order Filters, 242
6.4.2 Parametric Resonators and Equalizers, 244
6.4.3 Notch and Comb Filters, 249
6.5 Deconvolution, Inverse Filters, and Stability, 254
6.6 Problems, 259
CONTENTSix
7 Digital Filter Realizations 265
7.1 Direct Form, 265
7.2 Canonical Form, 271
7.3 Cascade Form, 277
7.4 Cascade to Canonical, 284
7.5 Hardware Realizations and Circular Buffers, 293
7.6 Quantization Effects in Digital Filters, 305
7.7 Problems, 306
8 Signal Processing Applications 316
8.1 Digital Waveform Generators, 316
8.1.1 Sinusoidal Generators, 316
8.1.2 Periodic Waveform Generators, 321
8.1.3 Wavetable Generators, 330
8.2 Digital Audio Effects, 349
8.2.1 Delays, Echoes, and Comb Filters, 350
8.2.2 Flanging, Chorusing, and Phasing, 355
8.2.3 Digital Reverberation, 362
8.2.4 Multitap Delays, 374
8.2.5 Compressors, Limiters, Expanders, and Gates, 378
8.3 Noise Reduction and Signal Enhancement, 382
8.3.1 Noise Reduction Filters, 382
8.3.2 Notch and Comb Filters, 398
8.3.3 Line and Frame Combs for Digital TV, 409
8.3.4 Signal Averaging, 421
8.3.5 Savitzky-Golay Smoothing Filters
, 427
8.4 Problems, 453
9 DFT/FFT Algorithms 464
9.1 Frequency Resolution and Windowing, 464
9.2 DTFT Computation, 475
9.2.1 DTFT at a Single Frequency, 475
9.2.2 DTFT over Frequency Range, 478
9.2.3 DFT, 479
9.2.4 Zero Padding, 481
9.3 Physical versus Computational Resolution, 482
9.4 Matrix Form of DFT, 486
9.5 Modulo-NReduction, 489
9.6 Inverse DFT, 496
9.7 Sampling of Periodic Signals and the DFT, 499
9.8 FFT, 504
9.9 Fast Convolution, 515
9.9.1 Circular Convolution, 515
9.9.2 Overlap-Add and Overlap-Save Methods, 520
9.10 Problems, 523
xCONTENTS
10 FIR Digital Filter Design 532
10.1 Window Method, 532
10.1.1 Ideal Filters, 532
10.1.2 Rectangular Window, 535
10.1.3 Hamming Window, 540
10.2 Kaiser Window, 541
10.2.1 Kaiser Window for Filter Design, 541
10.2.2 Kaiser Window for Spectral Analysis, 555
10.3 Frequency Sampling Method, 558
10.4 Other FIR Design Methods, 558
10.5 Problems, 559
11 IIR Digital Filter Design 563
11.1 Bilinear Transformation, 563
11.2 First-Order Lowpass and Highpass Filters, 566
11.3 Second-Order Peaking and Notching Filters, 573
11.4 Parametric Equalizer Filters, 581
11.5 Comb Filters, 590
11.6 Higher-Order Filters, 592
11.6.1 Analog Lowpass Butterworth Filters, 594
11.6.2 Digital Lowpass Filters, 599
11.6.3 Digital Highpass Filters, 603
11.6.4 Digital Bandpass Filters, 606
11.6.5 Digital Bandstop Filters, 611
11.6.6 Chebyshev Filter Design
, 615
11.7 Problems, 628
12 Interpolation, Decimation, and Oversampling 632
12.1 Interpolation and Oversampling, 632
12.2 Interpolation Filter Design
, 638
12.2.1 Direct Form, 638
12.2.2 Polyphase Form, 640
12.2.3 Frequency Domain Characteristics, 645
12.2.4 Kaiser Window Designs, 647
12.2.5 Multistage Designs, 649
12.3 Linear and Hold Interpolators
, 657
12.4 Design Examples
, 661
12.4.1 4-fold Interpolators, 661
12.4.2 Multistage 4-fold Interpolators, 667
12.4.3 DAC Equalization, 671
12.4.4 Postfilter Design and Equalization, 674
12.4.5 Multistage Equalization, 678
12.5 Decimation and Oversampling
, 686
12.6 Sampling Rate Converters
, 691
12.7 Noise Shaping Quantizers
, 698
12.8 Problems, 705
CONTENTSxi
13 Appendices 713
A Random Signals
, 713 A.1 Autocorrelation Functions and Power Spectra, 713
A.2 Filtering of Random Signals, 717
B Random Number Generators, 719
B.1 Uniform and Gaussian Generators, 719
B.2 Low-Frequency Noise Generators
, 724
B.3 1/fNoise Generators
, 729
B.4 Problems, 733
C Complex Arithmetic in C, 736
D MATLAB Functions, 739
References 758
Index 775
Preface
This book provides an applications-oriented introduction to digital signal processing written primarily for electrical engineering undergraduates. Practicing engineers and graduate students may also find it useful as a first text on the subject. Digital signal processing is everywhere. Today"s college students hear "DSP" all the time in their everyday life-from their CD players, to their electronic music synthesizers, to the sound cards in their PCs. They hear all about "DSP chips", "oversampling digitalquotesdbs_dbs8.pdfusesText_14
Gaussian Filtering - University of Auckland