Digital Signal and Image Processing

Transcription

1 DIGITAL SIGNAL AND IMAGE PROCESSING SERIES Digital Signal and Image Processing using MATLAB 2nd Edition Revised and Updated Volume 2 Advances and Applications The Deterministic Case Gérard Blanchet and Maurice Charbit

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3 Digital Signal and Image Processing using MATLAB

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5 Revised and Updated 2 nd Edition Digital Signal and Image Processing using MATLAB Volume 2 Advances and Applications: The Deterministic Case Gérard Blanchet Maurice Charbit

6 First published 2015 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc St George s Road 111 River Street London SW19 4EU Hoboken, NJ UK USA ISTE Ltd 2015 The rights of Gérard Blanchet and Maurice Charbit to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act Library of Congress Control Number: British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN MATLAB is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book s use or discussion of MATLAB software does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or use of the MATLAB software.

7 Contents Foreword Notations and Abbreviations ix xi Chapter 1 Recap on Digital Signal Processing The sampling theorem Spectral contents Discrete-time Fourier transform (DTFT) Discrete Fourier transform (DFT) Case of random signals Example of the Dual Tone Multi-Frequency (DTMF) Chapter 2 Additional Information About Filtering Filter implementation Examples of filter structures Distributing the calculation load in an FIR filter FIR block filtering FFT filtering Filter banks Decimation and expansion Filter banks Ripple control Principle Programming Chapter 3 Image Processing A little geometry D object Calibration of cameras Pyramidal decompositions Pyramidal decomposition given by Burt and Adelson

8 vi Digital Signal and Image Processing using MATLAB Pyramidal decomposition using a Haar tranformation Stepwise decomposition (lifting scheme) Chapter 4 Numerical Calculus and Simulation Simulation of continuous-time systems Simulation by approximation Exact model simulation Solving of ordinary differential equations (ODEs) Conversion from continuous to discrete time Linear case, continuous-time solution Remarks on the Runge Kutta methods Systems of equations and zero-seeking Zeros of a function using the Newton method Roots of a polynomial with the Newton Raphson method Systems of nonlinear equations Interpolation Thiele s interpolation Another decomposition in continuous fractions Natural cubic splines Solving of linear systems Jacobi method Relaxation method Cholesky factorization Chapter 5 Speech Processing A speech signal model Overview A typology of vocal sounds The AR model of speech production Compressing a speech signal Dynamic Time Warping The DTW algorithm Examples of pathfinding rules Cepstral coefficients Modifying the duration of an audio signal PSOLA Phase vocoder Eliminating the impulse noise The signal model Click detection Restoration

9 Contents vii Chapter 6 Selected Topics Tracking the cardiac rhythm of the fetus Objectives Separating the EKG signals Estimating cardiac rhythms Extracting the contour of a coin Constrained optimization and Lagrange multipliers Equality-constrained optimization Quadratic problem with linear inequality constraints Portfolio optimization Principal Component Analysis (PCA) Determining the principal components Dimension PCA Linear Discriminant Analysis GPS positioning The Viterbi algorithm Convolutional non-recursive encoder Decoding and hard decision Chapter 7 Hints and Solutions 187 H1 Reminders on digital signal-processing H2 Additional information on filtering H3 Image Processing H4 Numerical calculus H5 Speech processing H6 Selected topics Chapter 8 Appendix 243 A1 A few properties of a matrix A2 A few relations for matrices Bibliography 247 Index 251

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11 Foreword This book represents the continuation to Digital Signal and Image Processing: Fundamentals. It is assumed that the reader possesses a good knowledge of the programming language MATLAB and a command of the fundamental elements of digital signal processing: the usual transforms (the Discrete Time Fourier Transform (DTFT), the Discrete Fourier Transform and the z-transform), the properties of deterministic and random signals, and digital filtering. Readers will also need to be familiar with the fundamentals of continuous-spectrum spectral analysis and have a certain amount of mathematical knowledge concerning vector spaces. In order to prevent the reading becoming a penance, we will offer a few reminders of the basics wherever necessary. This book is essentially a collection of examples, exercises and case studies. It also presents applications of digital signal- or image processing, and techniques which were not touched upon in the previous volume. Recap on digital signal processing This section is devoted to the definitions and properties of the fundamental transforms used in digital signal processing: Fourier transform, discrete time Fourier transform and discrete Fourier transform. It concludes with a classic example which enables us to put some known results into practice. Filter implementation This section deals with the structures of filters, the introduction of parallelism into the filtering operations (block filtering and filter banks) and, by way of an example, the Parks McClellan method for FIR filter synthesis (finite impulse response). Image processing The section given over to images offers a few geometrical concepts relating to the representation of 3D objects in a 2D space. Therein, we deal with problems

12 x Digital Signal and Image Processing using MATLAB of calibration of cameras. In addition, image compression is also discussed, with the use of examples (pyramidal decompositions, lifting scheme). Digital calculus and simulation This section deals with the algorithms used in most domains in digital processing, and therefore far beyond mere signal processing. It only touches on the domain using a few examples of methods applied to problems of simulation, resolution of differential equations, zero-seeking, interpolation and iterative methods for solving linear systems. Speech processing After a brief introduction to speech production, we will discuss the representation of a speech signal by an autoregressive model, and its application to compression. Next we will give the descriptions of the techniques widely used in this field (Dynamic Time Warping and PSOLA) and, finally, an example of application with decrackling for audio recordings. Selected topics This last chapter presents case studies that go a little further in depth than the examples described in the previous sections. Tracking the cardiac rhythm of the fetus and Extracting the contour of a coin are classic examples of the application of the least squares method. Principal component analysis and linear discriminant analysis are basic methods for the classification of objects (in a very broad sense). The section devoted to optimization under constraints could have been part of the section on numerical methods. The method of Lagrange multipliers is encountered in a multitude of applications. In terms of applications, we present the case of optimization of a stock portfolio. We conclude with the example of the Viterbi algorithm for the hard decoding of convolutional codes. This algorithm is, in fact, a particular case for searching for the shortest possible path in a lattice.

13 Notations and Abbreviations empty set k,n = k n { 1 when t < T/2 rect T (t) = 0 otherwise sinc(x) = sin(πx) { πx 1 when x A 1(x A) = 0 otherwise (a, b] = {x : a < x b} { δ(t) Dirac distribution when t R Kronecker symbol when t Z Re(z) Im(z) x x(t) X(f) i or j = 1 real part of z imaginary part of z integer part of x Fourier transform (x y)(t) continuous time convolution = x(u)y(t u)du R (x y)(t) discrete time convolution (indicator function of A) = u Z x(u)y(t u) = u Z x(t u)y(u) y (n) (t) = dn y(t) dt n, nth order derivative

14 xii Digital Signal and Image Processing using MATLAB x or x I N A A T A H A 1 A # vector x (N N)-dimension identity matrix complex conjugate of A transpose of A transpose-conjugate of A inverse matrix of A pseudo-inverse matrix of A P {X A} E {X} X c = X E {X} var {X} = E { X c 2} E {X Y } probability that X A expectation value of X zero-mean random variable variance of X conditional expectation of X given Y ADC ADPCM AR ARMA BER bps cdf CF CZT DAC DCT d.e./de DFT DTFT DTMF dsp e.s.d./esd FIR FFT FT Analog to Digital Converter Adaptive Differential PCM Autoregressive AR and MA Bit Error Rate Bits per second Cumulative distribution function Clipping Factor Causal z-transform Digital to Analog Converter Discrete Cosine Transform Difference equation Discrete Fourier Transform Discrete Time Fourier Transform Dual Tone Multi-Frequency Digital signal processing/processor Energy spectral density Finite Impulse Response Fast Fourier Transform Continuous Time Fourier Transform

15 Notations and Abbreviations xiii IDFT i.i.d./iid IIR ISI LDA lms MA MAC OTF PAM PCA p.d. ppi p.s.d./psd PSF PSK QAM rls rms r.p./rp SNR r.v./rv STFT TF WSS ZOH ZT Inverse Discrete Fourier Transform Independent and Identically Distributed Infinite Impulse Response InterSymbol Interference Linear discriminant analysis Least mean squares Moving Average Multiplication ACcumulation Optical Transfer Function Pulse Amplitude Modulation Principal Component Analysis Probability Distribution Points per Inch Power Spectral Density Point Spread Function Phase Shift Keying Quadrature Amplitude Modulation Recursive least squares Root mean square Random process Signal to Noise Ratio Random variable Short Term Fourier Transform Transfer Function Wide (Weak) Sense Stationary (Second Order) Process Zero-Order Hold z-transform