Fundamentals Of Signals And Systems
FUNDAMENTALS OFSIGNALS AND SYSTEMS
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FUNDAMENTALS OFSIGNALS AND SYSTEMSBENOIT BOULETCHARLES RIVER MEDIABoston, Massachusetts
Copyright 2006 Career & Professional Group, a division of Thomson Learning, Inc.Published by Charles River Media, an imprint of Thomson Learning Inc.All rights reserved.No part of this publication may be reproduced in any way, stored in a retrieval system of any type, ortransmitted by any means or media, electronic or mechanical, including, but not limited to, photocopy,recording, or scanning, without prior permission in writing from the publisher.Cover Design: Tyler CreativeCHARLES RIVER MEDIA25 Thomson PlaceBoston, Massachusetts 02210617-757-7900617-757-7951 (FAX)crm.info@thomson.comwww.charlesriver.comThis book is printed on acid-free paper.Benoit Boulet. Fundamentals of Signals and Systems.ISBN: 1-58450-381-5eISBN: 1-58450-660-1All brand names and product names mentioned in this book are trademarks or service marks of theirrespective companies. Any omission or misuse (of any kind) of service marks or trademarks should notbe regarded as intent to infringe on the property of others. The publisher recognizes and respects allmarks used by companies, manufacturers, and developers as a means to distinguish their products.Library of Congress Cataloging-in-Publication DataBoulet, Benoit, 1967Fundamentals of signals and systems / Benoit Boulet.— 1st ed.p. cm.Includes index.ISBN 1-58450-381-5 (hardcover with cd-rom : alk. paper)1. Signal processing. 2. Signal generators. 3. Electric filters. 4. Signal detection. 5. System analysis.I. Title.TK5102.9.B68 2005621.382’2—dc22200501005407 7 6 5 4 3CHARLES RIVER MEDIA titles are available for site license or bulk purchase by institutions, usergroups, corporations, etc. For additional information, please contact the Special Sales Departmentat 800-347-7707.Requests for replacement of a defective CD-ROM must be accompanied by the original disc, yourmailing address, telephone number, date of purchase and purchase price. Please state the nature ofthe problem, and send the information to CHARLES RIVER MEDIA, 25 Thomson Place, Boston,Massachusetts 02210. CRM’s sole obligation to the purchaser is to replace the disc, based on defectivematerials or faulty workmanship, but not on the operation or functionality of the product.
Contents1AcknowledgmentsxiiiPrefacexvElementary Continuous-Time and Discrete-Time Signals and SystemsSystems in EngineeringFunctions of Time as SignalsTransformations of the Time VariablePeriodic SignalsExponential SignalsPeriodic Complex Exponential and Sinusoidal SignalsFinite-Energy and Finite-Power SignalsEven and Odd SignalsDiscrete-Time Impulse and Step SignalsGeneralized FunctionsSystem Models and Basic PropertiesSummaryTo Probe FurtherExercises2Linear Time-Invariant SystemsDiscrete-Time LTI Systems: The Convolution SumContinuous-Time LTI Systems: The Convolution IntegralProperties of Linear Time-Invariant SystemsSummaryTo Probe FurtherExercises3Differential and Difference LTI SystemsCausal LTI Systems Described by Differential EquationsCausal LTI Systems Described by Difference 296v
viContentsImpulse Response of a Differential LTI SystemImpulse Response of a Difference LTI SystemCharacteristic Polynomials and Stability of Differential andDifference SystemsTime Constant and Natural Frequency of a First-Order LTIDifferential SystemEigenfunctions of LTI Difference and Differential SystemsSummaryTo Probe FurtherExercises4Fourier Series Representation of Periodic Continuous-Time SignalsLinear Combinations of Harmonically Related Complex ExponentialsDetermination of the Fourier Series Representation of aContinuous-Time Periodic SignalGraph of the Fourier Series Coefficients: The Line SpectrumProperties of Continuous-Time Fourier SeriesFourier Series of a Periodic Rectangular WaveOptimality and Convergence of the Fourier SeriesExistence of a Fourier Series RepresentationGibbs PhenomenonFourier Series of a Periodic Train of ImpulsesParseval TheoremPower SpectrumTotal Harmonic DistortionSteady-State Response of an LTI System to a Periodic SignalSummaryTo Probe FurtherExercises5The Continuous-Time Fourier TransformFourier Transform as the Limit of a Fourier SeriesProperties of the Fourier TransformExamples of Fourier TransformsThe Inverse Fourier TransformDualityConvergence of the Fourier TransformThe Convolution Property in the Analysis of LTI 192192
ContentsFourier Transforms of Periodic SignalsFilteringSummaryTo Probe FurtherExercises6The Laplace TransformDefinition of the Two-Sided Laplace TransformInverse Laplace TransformConvergence of the Two-Sided Laplace TransformPoles and Zeros of Rational Laplace TransformsProperties of the Two-Sided Laplace TransformAnalysis and Characterization of LTI Systems Using theLaplace TransformDefinition of the Unilateral Laplace TransformProperties of the Unilateral Laplace TransformSummaryTo Probe 5236241243244247248248Application of the Laplace Transform to LTI Differential Systems259The Transfer Function of an LTI Differential SystemBlock Diagram Realizations of LTI Differential SystemsAnalysis of LTI Differential Systems with Initial Conditions Usingthe Unilateral Laplace TransformTransient and Steady-State Responses of LTI Differential SystemsSummaryTo Probe FurtherExercises260264Time and Frequency Analysis of BIBO Stable,Continuous-Time LTI SystemsRelation of Poles and Zeros of the Transfer Function to theFrequency ResponseBode PlotsFrequency Response of First-Order Lag, Lead, and Second-OrderLead-Lag Systems272274276276277285286290296
viiiContents9Frequency Response of Second-Order SystemsStep Response of Stable LTI SystemsIdeal Delay SystemsGroup DelayNon-Minimum Phase and All-Pass SystemsSummaryTo Probe n of Laplace Transform Techniques toElectric Circuit Analysis329Review of Nodal Analysis and Mesh Analysis of CircuitsTransform Circuit Diagrams: Transient and Steady-State AnalysisOperational Amplifier CircuitsSummaryTo Probe FurtherExercises10State Models of Continuous-Time LTI SystemsState Models of Continuous-Time LTI Differential SystemsZero-State Response and Zero-Input Response of aContinuous-Time State-Space SystemLaplace-Transform Solution for Continuous-Time State-Space SystemsState Trajectories and the Phase PlaneBlock Diagram Representation of Continuous-Time State-Space SystemsSummaryTo Probe FurtherExercises11Application of Transform Techniques to LTI FeedbackControl SystemsIntroduction to LTI Feedback Control SystemsClosed-Loop Stability and the Root LocusThe Nyquist Stability CriterionStability Robustness: Gain and Phase MarginsSummaryTo Probe 72373373373381382394404409413413413
Contents12Discrete-Time Fourier Series and Fourier TransformResponse of Discrete-Time LTI Systems to Complex ExponentialsFourier Series Representation of Discrete-Time Periodic SignalsProperties of the Discrete-Time Fourier SeriesDiscrete-Time Fourier TransformProperties of the Discrete-Time Fourier TransformDTFT of Periodic Signals and Step SignalsDualitySummaryTo Probe FurtherExercises1314The elopment of the Two-Sided z-TransformROC of the z-TransformProperties of the Two-Sided z-TransformThe Inverse z-TransformAnalysis and Characterization of DLTI Systems Using the z-TransformThe Unilateral z-TransformSummaryTo Probe FurtherExercises460464465468474483486487487Time and Frequency Analysis of Discrete-Time Signals and Systems497Geometric Evaluation of the DTFT From the Pole-Zero PlotFrequency Analysis of First-Order and Second-Order SystemsIdeal Discrete-Time FiltersInfinite Impulse Response and Finite Impulse Response FiltersSummaryTo Probe FurtherExercises15ixSampling SystemsSampling of Continuous-Time SignalsSignal ReconstructionDiscrete-Time Processing of Continuous-Time SignalsSampling of Discrete-Time Signals498504510519531531532541542546552557
xContentsSummaryTo Probe FurtherExercises16Introduction to Communication SystemsComplex Exponential and Sinusoidal Amplitude ModulationDemodulation of Sinusoidal AMSingle-Sideband Amplitude ModulationModulation of a Pulse-Train CarrierPulse-Amplitude ModulationTime-Division MultiplexingFrequency-Division MultiplexingAngle ModulationSummaryTo Probe FurtherExercises17System Discretization and Discrete-Time LTI State-Space ModelsControllable Canonical FormObservable Canonical FormZero-State and Zero-Input Response of a Discrete-TimeState-Space Systemz-Transform Solution of Discrete-Time State-Space SystemsDiscretization of Continuous-Time SystemsSummaryTo Probe 99604605605617618621622625628636637637Appendix A: Using MATLAB645Appendix B: Mathematical Notation and Useful Formulas647Appendix C: About the CD-ROM649Appendix D: Tables of Transforms651Index665
ContentsxiList of LecturesLecture 1:Lecture 2:Lecture 3:Lecture 4:Lecture 5:Lecture 6:Lecture 7:Lecture 8:Lecture 9:Lecture 10:Lecture 11:Lecture 12:Lecture 13:Lecture 14:Lecture 15:Lecture 16:Lecture 17:Lecture 18:Lecture 19:Lecture 20:Lecture 21:Lecture 22:Lecture 23:Lecture 24:Lecture 25:Lecture 26:Lecture 27:Lecture 28:Lecture 29:Lecture 30:Lecture 31:Lecture 32:Lecture 33:Lecture 34:Lecture 35:Lecture 36:Lecture 37:Lecture 38:Lecture 39:Lecture 40:Lecture 41:Lecture 42:Lecture 43:Lecture 44:Signal ModelsSome Useful SignalsGeneralized Functions and Input-Output System ModelsBasic System PropertiesLTI systems: Convolution SumConvolution Sum and Convolution IntegralConvolution IntegralProperties of LTI SystemsDefinition of Differential and Difference SystemsImpulse Response of a Differential SystemImpulse Response of a Difference System; Characteristic Polynomialand StabilityDefinition and Properties of the Fourier SeriesConvergence of the Fourier SeriesParseval Theorem, Power Spectrum, Response of LTI System to Periodic InputDefinition and Properties of the Continuous-Time Fourier TransformExamples of Fourier Transforms, Inverse Fourier TransformConvergence of the Fourier Transform, Convolution Property andLTI SystemsLTI Systems, Fourier Transform of Periodic SignalsFilteringDefinition of the Laplace TransformProperties of the Laplace Transform, Transfer Function of an LTI SystemDefinition and Properties of the Unilateral Laplace TransformLTI Differential Systems and Rational Transfer FunctionsAnalysis of LTI Differential Systems with Block DiagramsResponse of LTI Differential Systems with Initial ConditionsImpulse Response of a Differential SystemThe Bode PlotFrequency Responses of Lead, Lag, and Lead-Lag SystemsFrequency Response of Second-Order SystemsThe Step ResponseReview of Nodal Analysis and Mesh Analysis of CircuitsTransform Circuit Diagrams, Op-Amp CircuitsState Models of Continuous-Time LTI SystemsZero-State Response and Zero-Input ResponseLaplace Transform Solution of State-Space SystemsIntroduction to LTI Feedback Control SystemsSensitivity Function and TransmissionClosed-Loop Stability AnalysisStability Analysis Using the Root LocusThey Nyquist Stability CriterionGain and Phase MarginsDefinition of the Discrete-Time Fourier SeriesProperties of the Discrete-Time Fourier SeriesDefinition of the Discrete-Time Fourier 1367381387394400404409425430435
xiiContentsLecture 45:Lecture 46:Lecture 47:Lecture 48:Lecture 49:Lecture 50:Lecture 51:Lecture 52:Lecture 53:Lecture 54:Lecture 55:Lecture 56:Lecture 57:Lecture 58:Lecture 59:Lecture 60:Lecture 61:Lecture 62:Lecture 63:Lecture 64:Lecture 65:Lecture 66:Lecture 67:Lecture 68:Lecture 69:Lecture 70:Properties of the Discrete-Time Fourier TransformDTFT of Periodic and Step Signals, DualityDefinition and Convergence of the z-TransformProperties of the z-TransformThe Inverse z-TransformTransfer Function Characterization of DLTI SystemsLTI Difference Systems and Rational Transfer FunctionsThe Unilateral z-TransformRelationship Between the DTFT and the z-TransformFrequency Analysis of First-Order and Second-Order SystemsIdeal Discrete-Time FiltersIIR and FIR FiltersFIR Filter Design by WindowingSamplingSignal Reconstruction and AliasingDiscrete-Time Processing of Continuous-Time SignalsEquivalence to Continuous-Time Filtering; Sampling ofDiscrete-Time SignalsDecimation, Upsampling and InterpolationAmplitude Modulation and Synchronous DemodulationAsynchronous DemodulationSingle Sideband Amplitude ModulationPulse-Train and Pulse Amplitude ModulationFrequency-Division and Time-Division Multiplexing; Angle ModulationState Models of LTI Difference SystemsZero-State and Zero-Input Responses of Discrete-Time State ModelsDiscretization of Continuous-Time LTI 46552556558577583586591595617622628
Acknowledgmentswish to acknowledge the contribution of Dr. Maier L. Blostein, emeritus professor in the Department of Electrical and Computer Engineering at McGillUniversity. Our discussions over the past few years have led us to the currentcourse syllabi for Signals & Systems I and II, essentially forming the table of contents of this textbook.I would like to thank the many students whom, over the years, have reportedmistakes and suggested useful revisions to my Signals & Systems I and II coursenotes.The interesting and useful applets on the companion CD-ROM were programmed by the following students: Rafic El-Fakir (Bode plot applet) and Gul PilJoo (Fourier series and convolution applets). I thank them for their excellent workand for letting me use their programs.Ixiii
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Prefacehe study of signals and systems is considered to be a classic subject in thecurriculum of most engineering schools throughout the world. The theory ofsignals and systems is a coherent and elegant collection of mathematical results that date back to the work of Fourier and Laplace and many other famousmathematicians and engineers. Signals and systems theory has proven to be anextremely valuable tool for the past 70 years in many fields of science and engineering, including power systems, automatic control, communications, circuit design, filtering, and signal processing. Fantastic advances in these fields havebrought revolutionary changes into our lives.At the heart of signals and systems theory is mankind’s historical curiosity andneed to analyze the behavior of physical systems with simple mathematical models describing the cause-and-effect relationship between quantities. For example,Isaac Newton discovered the second law of rigid-body dynamics over 300 yearsago and described it mathematically as a relationship between the resulting forceapplied on a body (the input) and its acceleration (the output), from which onecan also obtain the body’s velocity and position with respect to time. The development of differential calculus by Leibniz and Newton provided a powerful tool formodeling physical systems in the form of differential equations implicitly relatingthe input variable to the output variable.A fundamental issue in science and engineering is to predict what the behavior, or output response, of a system will be for a given input signal. Whereas science may seek to describe natural phenomena modeled as input-output systems,engineering seeks to design systems by modifying and analyzing such models.This issue is recurrent in the design of electrical or mechanical systems, where asystem’s output signal must typically respond in an appropriate way to selectedinput signals. In this case, a mathematical input-output model of the system wouldbe analyzed to predict the behavior of the output of the system. For example, in theTxv
xviPrefacedesign of a simple resistor-capacitor electrical circuit to be used as a filter, the engineer would first specify the desired attenuation of a sinusoidal input voltage of agiven frequency at the output of the filter. Then, the design would proceed by selecting the appropriate resistance R and capacitance C in the differential equationmodel of the filter in order to achieve the attenuation specification. The filter canthen be built using actual electrical components.A signal is defined as a function of time representing the evolution of a variable. Certain types of input and output signals have special properties with respectto linear time-invariant systems. Such signals include sinusoidal and exponentialfunctions of time. These signals can be linearly combined to form virtually anyother signal, which is the basis of the Fourier series representation of periodic signals and the Fourier transform representation of aperiodic signals.The Fourier representation opens up a whole new interpretation of signals interms of their frequency contents called the frequency spectrum. Furthermore, in thefrequency domain, a linear time-invariant system acts as a filter on the frequencyspectrum of the input signal, attenuating it at some frequencies while amplifying itat other frequencies. This effect is called the frequency response of the system.These frequency domain concepts are fundamental in electrical engineering, as theyunderpin the fields of communication systems, analog and digital filter design, feedback control, power engineering, etc. Well-trained electrical and computer engineers think of signals as being in the frequency domain probably just as much asthey think of them as functions of time.The Fourier transform can be further generalized to the Laplace transform incontinuous-time and the z-transform in discrete-time. The idea here is to definesuch transforms even for signals that tend to infinity with time. We chose to adoptthe notation X( jω ), instead of X(ω ) or X( f ), for the Fourier transfor
THE CD-ROM THAT ACCOMPANIES THE BOOK MAY BE USED ON A SINGLE PC ONLY. THE LICENSE DOES NOT PERMIT THE USE ON A NETWORK (OF ANY . This book is printed on acid-free paper. Benoit Boulet. Fundamentals of Signals and Systems. ISBN: 1-58450-381-5 . Causal LTI Systems Described by Difference Equations 96 Contents v.
Signals And Systems by Alan V. Oppenheim and Alan S. Willsky with S. Hamid Nawab. John L. Weatherwax January 19, 2006 wax@alum.mit.edu 1. Chapter 1: Signals and Systems Problem Solutions Problem 1.3 (computing P and E for some sample signals)File Size: 203KBPage Count: 39Explore further(PDF) Oppenheim Signals and Systems 2nd Edition Solutions .www.academia.eduOppenheim signals and systems solution manualuploads.strikinglycdn.comAlan V. Oppenheim, Alan S. Willsky, with S. Hamid Signals .www.academia.eduSolved Problems signals and systemshome.npru.ac.thRecommended to you based on what's popular Feedback
Signals and Systems In this chapter we introduce the basic concepts of discrete-time signals and systems. 8.1 Introduction Signals specified over a continuous range of t are continuous-time signals, denoted by the symbols J(t), y(t), etc. Systems whose inputs and outputs are continuous-time signals are continuous-time systems.
a more general class of continuous time signals. Fourier Transform is at the heart of modern communication systems We have defined the spectrum for a limited class of signals such as sinusoids and periodic signals. These kind of spectrum are part of Fourier series representation of signals ( ) ( )
Signals & Systems CREC Dept. of ECE Page 10 LECTURE NOTES ON SIGNALS AND SYSTEMS (15A04303) II B.TECH – I SEMESTER ECE PREPARED BY Mr. K. JAYASREE ASSOCIATE PROFESSOR DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING . Lathi, “Linear Systems and Signals”, Second Ed
2. Simulate the generation of signals and operations on them. 3. Illustrate Gibbs phenomenon 4. Analyze the signals using Fourier, Laplace and Z transforms. COURSE OUTCOMES 1. Understand the applications of MATLAB and to generate matrices of various dimension 2. Generate the various signals and sequences and perform operations on signals. 3.
tation in Signals and Systems, Oppenheim and Willsky with Nawab, 2nd Edition, Prentice Hall, 1997. 2.1 SIGNALS, SYSTEMS, MODELS, PROPERTIES Throughout this text we will be considering various classes of signals and systems, developing models for them and studying their properties.
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Chapter 2 - Signals, Signs and Pavement Markings PA Driver’s Manual - 7-SIGNALS Traffic signals are installed at intersections to control the movement of vehicles and pedestrians. Traffic signals are arranged in either vertical lines or horizontal lines. When they are arranged vertically, red is always on top and green on the bottom.
