Signals, systems, transforms, and digital signal processing with MATLAB (1)

Signals, Systems, Transforms, and Digital Signal Processing with MATLAB® Electrical Engineering Signals, Systems, Transforms, and Digital Signal Processing with MATLAB® has as its principal objective simplification without compromise of rigor Graphics, called by the author “the language of scientists and engineers”, physical interpretation of subtle mathematical concepts, and a gradual transition from basic to more advanced topics are meant to be among the important contributions of this book The text establishes a solid background in Fourier, Laplace and z-transforms, before extending them in later chapters After illustrating the analysis of a function through a step-by-step addition of harmonics, the book deals with Fourier and Laplace transforms It then covers discrete time signals and systems, the z-transform, continuousand discrete-time filters, active and passive filters, lattice filters, and continuous- and discrete-time state space models The author goes on to discuss the Fourier transform of sequences, the discrete Fourier transform, and the fast Fourier transform, followed by Fourier-, Laplace, and z-related transforms, including Walsh–Hadamard, generalized Walsh, Hilbert, discrete cosine, Hartley, Hankel, Mellin, fractional Fourier, and wavelet He also surveys the architecture and design of digital signal processors, computer architecture, logic design of sequential circuits, and random signals He concludes with simplifying and demystifying the vital subject of distribution theory Features • Shows how the Fourier transform is a special case of the Laplace transform • Presents a unique matrix-equation-matrix sequence of operations that dispels the mystique of the fast Fourier transform • Examines how parallel processing and wired-in design can lead to optimal processor architecture • Explores the application of digital signal processing technology to real-time processing • Introduces the author’s own generalization of the Dirac-delta impulse and distribution theory • Offers extensive referencing to MATLAB® and Mathematica® for solving the examples Drawing on much of the author’s own research work, this book expands the domains of existence of the most important transforms and thus opens the door to a new world of applications using novel, powerful mathematical tools 90488_Cover.indd Corinthios Signals, Systems, Transforms, and Digital Signal Processing with ® MATLAB Michael Corinthios 90488 4/12/10 10:23 AM Signals, Systems, Transforms, and Digital Signal Processing ® with MATLAB Signals, Systems, Transforms, and Digital Signal Processing ® with MATLAB Michael Corinthios École Polytechnique de Montréal Montréal, Canada 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 or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20150206 International Standard Book Number-13: 978-1-4200-9049-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may 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without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To Maria, Angela, Gis`ele, John v Contents Preface xxv Acknowledgment xxvii Continuous-Time and Discrete-Time Signals and Systems 1.1 Introduction 1.2 Continuous-Time Signals 1.3 Periodic Functions 1.4 Unit Step Function 1.5 Graphical Representation of Functions 1.6 Even and Odd Parts of a Function 1.7 Dirac-Delta Impulse 1.8 Basic Properties of the Dirac-Delta Impulse 1.9 Other Important Properties of the Impulse 1.10 Continuous-Time Systems 1.11 Causality, Stability 1.12 Examples of Electrical Continuous-Time Systems 1.13 Mechanical Systems 1.14 Transfer Function and Frequency Response 1.15 Convolution and Correlation 1.16 A Right-Sided and a Left-Sided Function 1.17 Convolution with an Impulse and Its Derivatives 1.18 Additional Convolution Properties 1.19 Correlation Function 1.20 Properties of the Correlation Function 1.21 Graphical Interpretation 1.22 Correlation of Periodic Functions 1.23 Average, Energy and Power of Continuous-Time Signals 1.24 Discrete-Time Signals 1.25 Periodicity 1.26 Difference Equations 1.27 Even/Odd Decomposition 1.28 Average Value, Energy and Power Sequences 1.29 Causality, Stability 1.30 Problems 1.31 Answers to Selected Problems 2 11 11 12 12 13 14 15 20 21 21 22 22 23 25 25 26 27 28 28 29 30 30 40 Fourier Series Expansion 2.1 Trigonometric Fourier Series 2.2 Exponential Fourier Series 2.3 Exponential versus Trigonometric Series 2.4 Periodicity of Fourier Series 47 47 48 50 51 vii viii Signals, Systems, Transforms and Digital Signal Processing with MATLAB 2.5 2.6 2.7 2.8 2.9 2.10 53 55 55 56 58 58 58 60 60 61 61 64 65 67 70 72 74 74 75 75 77 78 81 83 86 88 89 90 91 92 100 Laplace Transform 3.1 Introduction 3.2 Bilateral Laplace Transform 3.3 Conditions of Existence of Laplace Transform 3.4 Basic Laplace Transforms 3.5 Notes on the ROC of Laplace Transform 3.6 Properties of Laplace Transform 3.6.1 Linearity 3.6.2 Differentiation in Time 3.6.3 Multiplication by Powers of Time 3.6.4 Convolution in Time 3.6.5 Integration in Time 3.6.6 Multiplication by an Exponential (Modulation) 3.6.7 Time Scaling 3.6.8 Reflection 3.6.9 Initial Value Theorem 3.6.10 Final Value Theorem 3.6.11 Laplace Transform of Anticausal Functions 3.6.12 Shift in Time 105 105 105 107 110 112 115 116 116 116 117 117 118 118 119 119 119 120 121 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 Dirichlet Conditions and Function Discontinuity Proof of the Exponential Series Expansion Analysis Interval versus Function Period Fourier Series as a Discrete-Frequency Spectrum Meaning of Negative Frequencies Properties of Fourier Series 2.10.1 Linearity 2.10.2 Time Shift 2.10.3 Frequency Shift 2.10.4 Function Conjugate 2.10.5 Reflection 2.10.6 Symmetry 2.10.7 Half-Periodic Symmetry 2.10.8 Double Symmetry 2.10.9 Time Scaling 2.10.10 Differentiation Property Differentiation of Discontinuous Functions 2.11.1 Multiplication in the Time Domain 2.11.2 Convolution in the Time Domain 2.11.3 Integration Fourier Series of an Impulse Train Expansion into Cosine or Sine Fourier Series Deducing a Function Form from Its Expansion Truncated Sinusoid Spectral Leakage The Period of a Composite Sinusoidal Signal Passage through a Linear System Parseval’s Relations Use of Power Series Expansion Inverse Fourier Series Problems Answers to Selected Problems Appendix 1293 royalist, and had a position in the Prefect of Paris Police Fearing for his family’s life of the repercussions of the revolution he moved his family to Arcueil He took personal charge of his son’s education before returning to Paris There, Laplace and Lagrange were among his friends who noticed Augustin-Louis’ mathematical gifts Lagrange’s advice was, however, to give attention to the study of languages before the pursuit of mathematics Cauchy ´ was enrolled between 1802 and 1804 at the Ecole Centrale du Panth´eon where he studied classical languages He then focused progressively on mathematics and in 1805 wrote the entrance examination ´ for the Ecole Polytechnique, was examined by Biot, placed second and attended courses by Lacroix, de Prony and Hachette, with Amp`ere as his analysis tutor ´ Upon graduation from Ecole Polytechnique in 1807 Cauchy was admitted to the highly ´ sought engineering school Ecole des Ponts et Chauss´ees He was appointed to the Ourcq Canal project under the supervision of Pierre Girard In 1810, appointed junior engineer at Cherbourg, he contributed to Napol´eon’s English invasion fleet project by working on the construction of the port of Cherbourg Meanwhile he avidly studied mathematics, inspired by the work of Laplace and Lagrange He explored the properties of polygons and polyhedra and submitted papers in 1811 and 1812 on his discoveries, with the support of Legendre and Malus In Cherbourg Cauchy worked hard on his mathematics research in addition to the long hours spent on his engineering work In 1812 he returned to Paris to live with his parents, suffering from a severe state of depression In Paris he submitted a thesis on symmetric functions which was published in the Journal ´ of the Ecole Polytechnique in 1815 His applications for an academic position were not successful, however, and he had to settle for a return to his Ourcq Canal project Cauchy was a devout Catholic and his attitude to his religion alienated him from his colleagues His aim was for an academic career but he was often turned down A position he applied for at the Bureau des Longitudes went to Legendre Another at the geometry section went to Poinsot Another position went in 1814 to Amp`ere, and a mechanics vacancy which followed Napol´eon’s abdication in April, went to Molard In this last election Cauchy did not receive a single one of the 53 votes cast In the same year, however, he published a memoir on definite integrals that became the foundation of his theory of complex functions With the fall of Napol´eon’s Empire in 1814 the royalist Cauchy finds valuable support from protectors who acceded to power Their influence helped his appointment to assistant ´ professor of analysis, responsible for the second year course, at Ecole Polytechnique The following year life smiled on Cauchy at last He received the Grand Prix of the French Academy of Sciences for his research results on waves His fame was definitely established, however, when he submitted a paper to the Institute solving one of Fermat’s claims on polygonal numbers He was admitted in 1816 to the Academy of Sciences, when political currents led to the dismissal of Carnot and Monge and Cauchy filled one of their positions Cauchy soon married Aloăse de Bure, of a family of reputable Parisian librarians, and they had two daughters In 1817 Cauchy filled a position at the College de France left vacant by Biot’s expedition to Scotland He lectured on methods of integration where he presented with rigorous analysis the conditions of convergence of infinite series and his definition of an integral His cours d’Analyse on the development of basic theorems of calculus as precisely as possible, was ´ designed for the students of Ecole Polytechnique In 1826 he presented Sur un nouveau genre de calcul analogue au calcul infinitesimal on the foundations of his famous theorem of residues, and in 1829 in Lecons sur le Calcul Diff´erentiel he defined for the first time a complex function of a complex variable The analysis course given by Cauchy is decried by his students as well as his colleagues It is this same course, however, which was published in 1821 and 1823, to become the reference 1294 Signals, Systems, Transforms and Digital Signal Processing with MATLAB on analysis of the 19th century, replacing intuition by analytical rigor Pioneering precise definitions are given to notions such as limits, continuity and series convergence Some false conclusions such as his proof that the limit of a series of continuous functions is continuous attest to his pioneering effort in exploring an unknown new domain The French Revolution of 1830, also known as the July Revolution, saw the overthrow of King Charles X, the French Bourbon monarch, and the ascension to the throne of his cousin Louis Philippe, the Duc d’Orl´eans, who himself, after 18 precarious years on the throne, would in turn be overthrown Such political turmoil in Paris and the years of strife and hard work made Cauchy, a staunch loyalist to the House of Bourbon, decide to absent himself for a while from France He left Paris in September 1830 and spent a short time in Switzerland, leaving behind in Paris his wife and two daughters There he was an enthusiastic helper in setting up the Acad´emie Helv´etique but the project collapsed due to political conflicts The July Revolution in France required that Cauchy swear an oath of allegiance to the new regime and, when he failed to return to Paris to so, he lost all his positions In Turin during 1831 he accepted an offer from the King of Piedmont for a chair of theoretical physics He taught in Turin and Menbrea who attended his courses, commented, “very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them His presentations were obscure clouds, illuminated from time to time by flashes of pure genius of the 30 who enrolled with me, I was the only one to see through it.” In 1833 Cauchy moved from Turin to Prague in order to tutor the grandson of Charles X However, the prince showed very little interest in mathematics Cauchy became annoyed and screamed and yelled The queen sometimes said to him, “too loud, not so loud.” Cauchy moved back to Paris in 1838 and regained his position at the Academy However, he did not regain his teaching positions, having refused to take the oath Later in 1839 a position at the Bureau Des Longitudes became vacant Although Cauchy was elected, he was not allowed to attend any meetings or receive a salary, again because of his refusal to take the oath The mathematics chair at Coll`ege de France became vacant in 1843 Cauchy should have easily won on account of his exceptional scientific profile, but apparently due to his religious and political views he was not chosen Henceforth, Cauchy’s mathematical contributions declined Cauchy made landmark contributions to mathematical physics, mathematical astronomy, and differential equations His four volume text Exercises d’Analyse et de Physique Mathemtique was published between 1840 and 1847 He stubbornly stuck to his religious and political views to the dismay of colleagues Cauchy never took an administrative post and was disrespectful and condescending toward some young scientists such as Abel and Galois, disregarding and even losing theses of great scientific value His colleagues begrudged him the political influence that led to his admission to the Academy and resented his intransigence and religious bigotry He was disliked, regarded as arrogant, recognizing no one else’s contribution Abel wrote of him after his visit to the Institute in 1826, “Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.” He was accused by his colleagues of liberally copying without citation the results of others They referred to Cauchy as cochon In the last few years of his life he had a dispute with Duhamel regarding a result on inelastic shocks Cauchy claimed to be the first to give the results in 1832, but Poncelet referred to his own work on the subject in 1826 Even though Cauchy was proved wrong he would never admit it Cauchy died on May 27, 1857 His last words were “Men pass away, but their deeds abide.” Many terms in mathematics bear his name, the Cauchy integral theorem, in the Appendix 1295 theory of complex functions, the Cauchy–Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy–Riemann equations, and the Cauchy sequences A book of his collective works entitled Oeuvres completes d’Augustin Cauchy (1882-1870) was published in 27 volumes References: J J O’Connor and E F Robertson, January 1997, MacTutor History of Mathematics http://www-history.mcs.st-andrews.ac.uk/Biographies/Cauchy.html http://www.bibmath.net/bios/index.php3?action=affiche&quoi=cauchy A.23 Niels Henrik Abel (1802–1829) FIGURE A.26 Niels Niels Henrik Abel.jpg) Henrik Abel (www.commons.wikimedia.org/wiki/Image: Born: August, 1802 in Frindoe, Norway; died: April 6, 1829 in Froland, Norway Niels Henrik Abel was a Norwegian mathematician who in a short life span of 26 years became a pioneer who advanced modern mathematics of his time by leaps and bounds He was born, one of seven children, in the small village of Frindoe, near Stavanger, Norway, where his father was a poor Protestant minister in the diocese of Christiansand Abel’s life was spent in poverty In 1815 he studied at the Cathedral School in Christiania In 1817 the mathematics teacher Bernt Holmboe, newly arrived at the school, was impressed by the young Abel’s mathematical talent Soon, Abel began to study university level mathematics texts and, within a year, Abel was reading the works of Euler, Newton, Lagrange, Laplace and Gauss In 1820 Abel’s father died, and he had to support his mother and family Thanks to Holmboe’s help Abel received a scholarship to remain at his school and was able to enter the university of Christiania (Oslo) in 1821 Holmboe, moreover, collected contributions from his colleagues enabling Abel to pursue his studies At the university 1296 Signals, Systems, Transforms and Digital Signal Processing with MATLAB of Christiania, Christopher Hansteen, professor of astronomy, provided Abel with both financial and moral support Abel obtained a preliminary degree from the university in 1822 and continued his research independently, with further subsidies obtained by Holmboe While in his final year at school, Abel had begun working on the solution of quintic equations (of the fifth order) by radicals He published papers in 1823 in the new periodical Magazin for Naturvidenskaberne, edited by Hanseen, on functional equations and integrals In his paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation Abel was given a small grant allowing him to visit Degen and other mathematicians in Copenhagen There he met Christine Kemp who soon became his fianc´ee Upon his return to the university of Christiania, his friends urged the Norwegian government to grant him a fellowship to study in Germany and France While awaiting the royal decree in 1824 he published, as a pamphlet at his own expense, his proof that no algebraic solution exists for the general quintic equation He sent the pamphlet to Gauss, whom he intended to visit in Gă ottingen while on his travels Gauss disregarded it, failing to recognize its true value In August 1825 Abel received the fellowship from the Norwegian government allowing him to travel abroad, and he set out with four friends visiting at first mathematicians in Norway and Denmark In Copenhagen he was given a letter of introduction to August Leopold Crelle He spent the winter of 1825 to 1826 with his Norwegian friends in Berlin, where Crelle, a civil engineer and avid mathematician, became his close friend and mentor With Abel’s enthusiastic support, Crelle published the Journal fă ur die Reine und Angewandte Mathematik (“Journal for pure and applied mathematics”) Its first volume appeared in 1826 and contained an article by Abel elaborating on his results of the quintic equation titled Recherches sur les fonctions elliptiques, together with six other papers by Abel on equation theory, functional equations, integration in finite forms and theoretical mechanics Abel had planned to visit Gauss in Găottingen on the way to Paris However, learning that Gauss was displeased to receive his work on the the general quintic equation, he decided to cancel his trip to Gă ottingen He arrived in Paris in 1826 where he sought the most renowned mathematicians and continued work on transcendental and elliptic functions He developed what was to be known as Abel’s theorem on the integrals of algebraic functions This theorem was the foundation of what came to be known as the theory of Abelian integrals and Abelian functions His presentations did not evoke in Paris the enthusiasm he had hoped for, for he was a new unknown entity on the scene He showed his treatise on a class of transcendental functions to Cauchy who brushed the young man aside with disdain Abel submitted his memoir to the Academy of Sciences on the sum of integrals of a given algebraic function, which is a generalization of Euler’s relation on elliptic integrals, hoping to make known his recent discoveries He waited in vain for a response Meanwhile he was diagnosed as having tuberculosis Heavily in debt, Abel returned to Norway To subsist he tutored schoolchildren, received a small grant from his university and obtained a substitute teaching position In spite of his illness and poverty he produced several papers on the theory of equations, later to be referred to as the theory of Abelian equations with Abelian groups In a short time he developed the theory of elliptic functions independently of the work of Karl Gustav Jacobi Legendre saw the new ideas in the papers that Abel and Jacobi were writing and said “Through these works you two will be placed in the class of the foremost analysts of our times.” By now his fame had spread over all mathematics centers A group from the French Academy submitted a request to grant him a suitable position to King Bernadotte of Appendix 1297 Norway-Sweden Meanwhile Crelle worked on securing for him a position of professor in Berlin In the fall of 1828 Abel’s health took a turn for the worst, and a trip on a sled to visit his fianc´ee at Froland near Christmas time aggravated his condition Crelle persisted more intensely in his efforts to obtain an appointment for Abel in Berlin He succeeded and wrote to Abel on April 8, 1829 that his dream had come true It was too late; Abel had died on April 6, at the age of 26 References: http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Abel.html http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Abel.html A.24 Johann Peter Gustav Lejeune Dirichlet (1805–1859) FIGURE A.27 Johann Peter Gustav Lejeune Dirichlet Born: February 13, 1805 in Dă uren, French Empire (now Germany), died: May 5, 1859 in Gă ottingen, Hanover (now Germany) Lejeune Dirichlet was born in Dă uren (now in Germany) His grandfather lived in the town of Richelette in Belgium, whence the name “Lejeune Dirichlet” (“le jeune de Richelette” (the young one from Richelette) His father was the postmaster of Dă uren, situated about halfway between Aachen and Cologne After two years at the Gymnasium he attended the Jesuit College in Cologne and there he had the good fortune to be taught by Georg Ohm His first paper to the Paris Academy in July 1825 established instantly his reputation since it tackled the famous Fermat’s last theorem The theorem claimed that for n > there are no nonzero integers x, y, z such that xn + y n = z n The cases n = and n = had been proven by Euler and Fermat Dirichlet’s paper presented a partial proof for the case n = 5, which was completed by Legendre who was one of the referees Dirichlet also completed his own proof almost at the same time, and presented at a later date a full proof 1298 Signals, Systems, Transforms and Digital Signal Processing with MATLAB for the case n = 14 Dirichlet studied mathematics at Găottingen university where he was a student of Karl Gauss and Karl Jacobi He also studied briefly in Paris where he benefited considerably from his contacts with Fourier, Biot, Laplace, Lacroix, Legendre, and Poisson Joseph Fourier greatly motivated his interest in expansions using trigonometric series In 1826 he returned to Germany and taught at Breslau and later at the Military Academy in Berlin He then moved to the university of Berlin to stay for 27 years before returning to Găottingen university to fill the chair left vacant by Gauss death Dirichlet’s work on number theory was an extension of Gauss developments, and Dirichlets book, the Vorlesungen u ăber Zahlentheorie (1863; Lectures on Number Theory), is comparable in its depth and extent to Gauss’ Disquisitiones Dirichlet made many significant discoveries in number theory and his solution of a problem related to primes was a pioneering effort in applying analytical techniques to solve problems in number theory In 1829 Dirichlet was able to define the sufficient conditions for the existence of Fourier series to converge Fourier also initiated the interest of Dirichlet in mathematical physics, motivating his interest in multiple integrals and the boundary-value problem This came to be known as the Dirichlet problem, concerning the solution of partial differential equations, such as encountered in the study of heat flow, electrostatics, and in many other areas of physics The growth of a more rigorous understanding of analysis owes to Dirichlet what is essentially the modern definition of the concept of a function In 1831, Dirichlet married Rebecca Henriette Mendelssohn Bartholdy, who came from a distinguished family, being a granddaughter of the philosopher Moses Mendelssohn and a sister of the composer Felix Mendelssohn Jacobi, who taught at Kă onigsberg, was one of Dirichlet’s lifelong friends, and they had great respect for each other’s contributions, in particular in number theory Ferdinand Eisenstein, Leopold Kronecker, and Rudolf Lipschitz were Dirichlet’s students After his death on May 5, 1859 in Găottingen, Dirichlets lectures and landmark results in number theory were collected, edited and published by his friend and fellow mathematician Richard Dedekind under the title Vorlesungen u ăber Zahlentheorie (Lectures on Number Theory) References: http://en.wikipedia.org/wiki/Johann Peter Gustav Lejeune Dirichlet http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Dirichlet.html http://www.bookrags.com/biography/johann-peter-gustav-lejeune-dirichlet-wom/ A.25 Pafnuty Lvovich Chebyshev (1821–1894) Born: May 16, 1821 in Okatovo, Russia; died: December 8, 1894 in St Petersburg, Russia Pafnuty Lvovich Chebyshev was born in Okatovo, a village west of Moscow He was one of nine children of Lev Pavlovich Chebyshev and Agrafena Ivanovna Chebysheva His father was a wealthy landowner who in his earlier military career had fought as an officer against Napol´eon’s invading armies Pafnuty Lvovich had a physical handicap with one limb weaker than the other, causing him to limp and he had to walk with a stick He was thus unable to pursue an officer’s career and early on replaced child’s play and sports with a passion Appendix 1299 FIGURE A.28 Pafnuty Lvovich Chebyshev for constructing mechanisms He was tutored at home by his mother, his cousin Avdotia Kvintillianovna Soukhareva and a music teacher whom he later acknowledged for teaching him the importance of analysis, precision and harmony From his mother he learned the basic skills of reading and writing, while his cousin, acting as a governess, taught him French and arithmetic Later in life his fluency in French helped him in his publications and in his visits to France French at the time was a natural language for formulating mathematics and communicating with European mathematicians In 1832, the family moved to Moscow mainly to attend to the education of their sons He was tutored at home; his mathematics and physics lessons were given by P N Pogorelski, one of the most renowned teachers in Moscow In 1837, Chebyshev began his studies of mathematics at Moscow university In courses on mechanics his professor was Nikolai Dmetrievich Brashman, who taught a wide range of subjects covering applied mechanics, mechanical engineering, hydraulics and probability theory Later Chebyshev would cite the great influence Brashman had on developing his areas of research In 1841 Chebyshev was awarded the silver medal for his work “Calculation of the roots of equations.” In this contribution Chebyshev proposed an approximation for the solution of algebraic equations of the nth degree based on Newton’s algorithm Chebyshev proceeded to a master’s degree program under Brashman’s supervision In 1843 he published a paper on multiple integrals in French in Liouville’s journal In 1846 he received his master’s degree upon defending his thesis “An Attempt to an Elementary Analysis of Probabilistic Theory.” In 1847 Chebyshev became assistant professor of mathematics at St Petersburg university In 1849, in his doctorate dissertation, he defended his results on the theory of congruences and became professor at St Petersburg university in 1860 The Paris academy elected him corresponding member in the same year, and full foreign member in 1874 In 1872, after 25 years of teaching at St Petersburg university, he became professor emeritus In 1882 he left the university to devote his life to research In 1893, he was elected honorable member of the St Petersburg Mathematical Society He died November 26, 1894, in St Petersburg References: http://www-history.mcs.st-andrews.ac.uk/Biographies/Chebyshev.html http://en.wikipedia.org/wiki/Pafnuty Chebyshev http://www.britannica.com/eb/article-9022729/Pafnuty-Lvovich-Chebyshev 1300 Signals, Systems, Transforms and Digital Signal Processing with MATLAB http://www.bibmath.net/bios/index.php3 A.26 Paul A.M Dirac The Nobel Prize in Physics 1933 The following is a reproduction, with gratitude and acknowledgment for the permission granted by the Nobel Foundation, of the Foundation’s official biography on the life of Paul A.M Dirac, as it appears on the site: http://nobelprize.org/nobel prizes/physics/laureates/1933/dirac-bio.html FIGURE A.29 Paul A.M Dirac (www.commons.wikimedia.org/wiki/Paul A.M Dirac) Paul Adrien Maurice Dirac was born on 8th August, 1902, at Bristol, England, his father being Swiss and his mother English He was educated at the Merchant Venturer’s Secondary School, Bristol, then went on to Bristol university Here, he studied electrical engineering, obtaining the B.Sc (Engineering) degree in 1921 He then studied mathematics for two years at Bristol university, later going on to St.John’s College, Cambridge, as a research student in mathematics He received his Ph.D degree in 1926 The following year he became a Fellow of St.John’s College and, in 1932, Lucasian Professor of Mathematics at Cambridge Dirac’s work has been concerned with the mathematical and theoretical aspects of quantum mechanics He began work on the new quantum mechanics as soon as it was introduced by Heisenberg in 1928 — independently producing a mathematical equivalent which consisted essentially of a noncommutative algebra for calculating atomic properties — and wrote a series of papers on the subject, published mainly in the Proceedings of the Royal Society, leading up to his relativistic theory of the electron (1928) and the theory of holes (1930) This latter theory required the existence of a positive particle having the same mass and charge as the known (negative) electron This, the positron was discovered experimentally at a later date (1932) by C D Anderson, while its existence was likewise proved by Blackett and Occhialini (1933) in the phenomena of “pair production” and “annihilation.” Appendix 1301 The importance of Dirac’s work lies essentially in his famous wave equation, which introduced special relativity into Schrăodingers equation Taking into account the fact that, mathematically speaking, relativity theory and quantum theory are not only distinct from each other, but also oppose each other, Dirac’s work could be considered a fruitful reconciliation between the two theories Dirac’s publications include the books Quantum Theory of the Electron (1928) and The Principles of Quantum Mechanics (1930; 3rd ed 1947) He was elected a Fellow of the Royal Society in 1930, being awarded the Society’s Royal Medal and the Copley Medal He was elected a member of the Pontifical Academy of Sciences in 1961 Dirac has travelled extensively and studied at various foreign universities, including Copenhagen, Gă ottingen, Leyden, Wisconsin, Michigan, and Princeton (in 1934, as Visiting Professor) In 1929, after having spent five months in America, he went round the world, visiting Japan together with Heisenberg, and then returned across Siberia In 1937 he married Margit Wigner, of Budapest Paul A.M Dirac died on October 20, 1984 From Nobel Lectures, Physics 1922-1941, Elsevier Publishing Company, Amsterdam, 1965 This biography was first published in the book series Les Prix Nobel It was later edited and republished in Nobel Lectures 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Transforms, and Digital Signal Processing with MATLAB® has as its principal objective simplification without compromise of rigor Graphics, called by the author “the language of scientists and engineers”, physical interpretation of subtle mathematical concepts, and a gradual transition from basic to more advanced topics are meant to be among the important contributions of this book The text establishes a solid background in Fourier, Laplace and z-transforms, before extending them in later chapters After illustrating the analysis of a function through a step-by-step addition of harmonics, the book deals with Fourier and Laplace transforms It then covers discrete time signals and systems, the z-transform, continuousand discrete-time filters, active and passive filters, lattice filters, and continuous- and discrete-time state space models The author goes on to discuss the Fourier transform of sequences, the discrete Fourier transform, and the fast Fourier transform, followed by Fourier-, Laplace, and z-related transforms, including Walsh–Hadamard, generalized Walsh, Hilbert, discrete cosine, Hartley, Hankel, Mellin, fractional Fourier, and wavelet He also surveys the architecture and design of digital signal processors, computer architecture, logic design of sequential circuits, and random signals He concludes with simplifying and demystifying the vital subject of distribution theory Features • Shows how the Fourier transform is a special case of the Laplace transform • Presents a unique matrix-equation-matrix sequence of operations that dispels the mystique of the fast Fourier transform • Examines how parallel processing and wired-in design can lead to optimal processor architecture • Explores the application of digital signal processing technology to real-time processing • Introduces the author’s own generalization of the Dirac-delta impulse and distribution theory • Offers extensive referencing to MATLAB® and Mathematica® for solving the examples Drawing on much of the author’s own research work, this book expands the domains of existence of the most important transforms and thus opens the door to a new world of applications using novel, powerful mathematical tools 90488_Cover.indd Corinthios Signals, Systems, Transforms, and Digital Signal Processing with ® MATLAB Michael Corinthios 90488 4/12/10 10:23 AM ... Signals, Systems, Transforms, and Digital Signal Processing ® with MATLAB Signals, Systems, Transforms, and Digital Signal Processing ® with MATLAB Michael Corinthios... impulse δ (t) 8 Signals, Systems, Transforms and Digital Signal Processing with MATLAB FIGURE 1.12 Approximation of the unit step function and its derivative A simple sequence and the limiting... electrical system and electric circuits in particular Similarly, homologies exist between 14 Signals, Systems, Transforms and Digital Signal Processing with MATLAB hydraulic, heat transfer and other