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Signals and systems : analysis using transform methods and MATLAB Third Edition Michael J. Roberts
Signals and systems : analysis using transform methods and MATLAB Third Edition Michael J. Roberts
Signals and Systems Analysis Using Transform Methods and MATLAB® Michael J. Roberts Professor Emeritus, Department of Electrical and Computer Engineering University of Tennessee Third Edition
SIGNALS AND SYSTEMS: ANALYSIS USING TRANSFORM METHODS AND MATLAB® , THIRD EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2018 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2012, and 2004. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 QVS 22 21 20 19 18 17 ISBN 978-0-07-802812-0 MHID 0-07-802812-4 Chief Product Officer, SVP Products & Markets: G. Scott Virkler Vice President, General Manager, Products & Markets: Marty Lange Vice President, Content Design & Delivery: Betsy Whalen Managing Director: Thomas Timp Brand Manager: Raghothaman Srinivasan/Thomas Scaife, Ph.D. Director, Product Development: Rose Koos Product Developer: Christine Bower Marketing Manager: Shannon O’Donnell Director of Digital Content: Chelsea Haupt, Ph.D. Director, Content Design & Delivery: Linda Avenarius Program Manager: Lora Neyens Content Project Managers: Jeni McAtee; Emily Windelborn; Sandy Schnee Buyer: Jennifer Pickel Content Licensing Specialists: Carrie Burger, photo; Lorraine Buczek, text Cover Image: © Lauree Feldman/Getty Images Compositor: MPS Limited Printer: Quad Versailles All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Roberts, Michael J., Dr. Signals and systems : analysis using transform methods and MATLAB / Michael J. Roberts, professor, Department of Electrical and Computer Engineering, University of Tennessee. Third edition. | New York, NY : McGraw-Hill Education, [2018] | Includes bibliographical references (p. 786–787) and index. LCCN 2016043890 | ISBN 9780078028120 (alk. paper) LCSH: Signal processing. | System analysis. | MATLAB. LCC TK5102.9 .R63 2018 | DDC 621.382/2—dc23 LC record available at https://lccn.loc.gov/2016043890 The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. mheducation.com/highered
To my wife Barbara for giving me the time and space to complete this effort and to the memory of my parents, Bertie Ellen Pinkerton and Jesse Watts Roberts, for their early emphasis on the importance of education.
Preface, xii Chapter 1 Introduction, 1 1.1 Signals and Systems Defined, 1 1.2 Types of Signals, 3 1.3 Examples of Systems, 8 A Mechanical System, 9 A Fluid System, 9 A Discrete-Time System, 11 Feedback Systems, 12 1.4 A Familiar Signal and System Example, 14 1.5 Use of MATLAB® , 18 Chapter 2 Mathematical Description of Continuous-Time Signals, 19 2.1 Introduction and Goals, 19 2.2 Functional Notation, 20 2.3 Continuous-Time Signal Functions, 20 Complex Exponentials and Sinusoids, 21 Functions with Discontinuities, 23 The Signum Function, 24 The Unit-Step Function, 24 The Unit-Ramp Function, 26 The Unit Impulse, 27 The Impulse, the Unit Step, and Generalized Derivatives, 29 The Equivalence Property of the Impulse, 30 The Sampling Property of the Impulse, 31 The Scaling Property of the Impulse, 31 The Unit Periodic Impulse or Impulse Train, 32 A Coordinated Notation for Singularity Functions, 33 The Unit-Rectangle Function, 33 2.4 Combinations of Functions, 34 2.5 Shifting and Scaling, 36 Amplitude Scaling, 36 Time Shifting, 37 Time Scaling, 39 Simultaneous Shifting and Scaling, 43 2.6 Differentiation and Integration, 47 2.7 Even and Odd Signals, 49 Combinations of Even and Odd Signals, 51 Derivatives and Integrals of Even and Odd Signals, 53 2.8 Periodic Signals, 53 2.9 Signal Energy and Power, 56 Signal Energy, 56 Signal Power, 58 2.10 Summary of Important Points, 60 Exercises, 61 Exercises with Answers, 61 Signal Functions, 61 Shifting and Scaling, 62 Derivatives and Integrals of Functions, 66 Generalized Derivative, 67 Even and Odd Functions, 67 Periodic Signals, 69 Signal Energy and Power of Signals, 70 Exercises without Answers, 71 Signal Functions, 71 Scaling and Shifting, 71 Generalized Derivative, 76 Derivatives and Integrals of Functions, 76 Even and Odd Functions, 76 Periodic Functions, 77 Signal Energy and Power of Signals, 77 Chapter 3 Discrete-Time Signal Description, 79 3.1 Introduction and Goals, 79 3.2 Sampling and Discrete Time, 80 3.3 Sinusoids and Exponentials, 82 Sinusoids, 82 Exponentials, 85 3.4 Singularity Functions, 86 The Unit-Impulse Function, 86 The Unit-Sequence Function, 87 CONTENTS iv
Contents v The Signum Function, 87 The Unit-Ramp Function, 88 The Unit Periodic Impulse Function or Impulse Train, 88 3.5 Shifting and Scaling, 89 Amplitude Scaling, 89 Time Shifting, 89 Time Scaling, 89 Time Compression, 90 Time Expansion, 90 3.6 Differencing and Accumulation, 94 3.7 Even and Odd Signals, 98 Combinations of Even and Odd Signals, 100 Symmetrical Finite Summation of Even and Odd Signals, 100 3.8 Periodic Signals, 101 3.9 Signal Energy and Power, 102 Signal Energy, 102 Signal Power, 103 3.10 Summary of Important Points, 105 Exercises, 105 Exercises with Answers, 105 Functions, 105 Scaling and Shifting Functions, 107 Differencing and Accumulation, 109 Even and Odd Functions, 110 Periodic Functions, 111 Signal Energy and Power, 112 Exercises without Answers, 113 Signal Functions, 113 Shifting and Scaling Functions, 113 Differencing and Accumulation, 114 Even and Odd Functions, 114 Periodic Signals, 115 Signal Energy and Power, 116 Chapter 4 Description of Systems, 118 4.1 Introduction and Goals, 118 4.2 Continuous-Time Systems, 119 System Modeling, 119 Differential Equations, 120 Block Diagrams, 124 System Properties, 127 Introductory Example, 127 Homogeneity, 131 Time Invariance, 132 Additivity, 133 Linearity and Superposition, 134 LTI Systems, 134 Stability, 138 Causality, 139 Memory, 139 Static Nonlinearity, 140 Invertibility, 142 Dynamics of Second-Order Systems, 143 Complex Sinusoid Excitation, 145 4.3 Discrete-Time Systems, 145 System Modeling, 145 Block Diagrams, 145 Difference Equations, 146 System Properties, 152 4.4 Summary of Important Points, 155 Exercises, 156 Exercises with Answers, 156 System Models, 156 Block Diagrams, 157 System Properties, 158 Exercises without Answers, 160 System Models, 160 System Properties, 162 Chapter 5 Time-Domain System Analysis, 164 5.1 Introduction and Goals, 164 5.2 Continuous Time, 164 Impulse Response, 164 Continuous-Time Convolution, 169 Derivation, 169 Graphical and Analytical Examples of Convolution, 173 Convolution Properties, 178 System Connections, 181 Step Response and Impulse Response, 181 Stability and Impulse Response, 181 Complex Exponential Excitation and the Transfer Function, 182 Frequency Response, 184 5.3 Discrete Time, 186 Impulse Response, 186 Discrete-Time Convolution, 189
Contents vi Derivation, 189 Graphical and Analytical Examples of Convolution, 192 Convolution Properties, 196 Numerical Convolution, 196 Discrete-Time Numerical Convolution, 196 Continuous-Time Numerical Convolution, 198 Stability and Impulse Response, 200 System Connections, 200 Unit-Sequence Response and Impulse Response, 201 Complex Exponential Excitation and the Transfer Function, 203 Frequency Response, 204 5.4 Summary of Important Points, 207 Exercises, 207 Exercises with Answers, 207 Continuous Time, 207 Impulse Response, 207 Convolution, 209 Stability, 213 Frequency Response, 214 Discrete Time, 214 Impulse Response, 214 Convolution, 215 Stability, 219 Exercises without Answers, 221 Continuous Time, 221 Impulse Response, 221 Convolution, 222 Stability, 224 Discrete Time, 225 Impulse Response, 225 Convolution, 225 Stability, 228 Chapter 6 Continuous-Time Fourier Methods, 229 6.1 Introduction and Goals, 229 6.2 The Continuous-Time Fourier Series, 230 Conceptual Basis, 230 Orthogonality and the Harmonic Function, 234 The Compact Trigonometric Fourier Series, 237 Convergence, 239 Continuous Signals, 239 Discontinuous Signals, 240 Minimum Error of Fourier-Series Partial Sums, 242 The Fourier Series of Even and Odd Periodic Functions, 243 Fourier-Series Tables and Properties, 244 Numerical Computation of the Fourier Series, 248 6.3 The Continuous-Time Fourier Transform, 255 Extending the Fourier Series to Aperiodic Signals, 255 The Generalized Fourier Transform, 260 Fourier Transform Properties, 265 Numerical Computation of the Fourier Transform, 273 6.4 Summary of Important Points, 281 Exercises, 281 Exercises with Answers, 281 Fourier Series, 281 Orthogonality, 282 Forward and Inverse Fourier Transforms, 286 Relation of CTFS to CTFT, 293 Numerical CTFT, 294 System Response, 294 Exercises without Answers, 294 Fourier Series, 294 Forward and Inverse Fourier Transforms, 300 System Response, 305 Relation of CTFS to CTFT, 306 Chapter 7 Discrete-Time Fourier Methods, 307 7.1 Introduction and Goals, 307 7.2 The Discrete-Time Fourier Series and the Discrete Fourier Transform, 307 Linearity and Complex-Exponential Excitation, 307 Orthogonality and the Harmonic Function, 311 Discrete Fourier Transform Properties, 315 The Fast Fourier Transform, 321 7.3 The Discrete-Time Fourier Transform, 323 Extending the Discrete Fourier Transform to Aperiodic Signals, 323 Derivation and Definition, 324 The Generalized DTFT, 326 Convergence of the Discrete-Time Fourier Transform, 327 DTFT Properties, 327
Contents vii Numerical Computation of the Discrete-Time Fourier Transform, 334 7.4 Fourier Method Comparisons, 340 7.5 Summary of Important Points, 341 Exercises, 342 Exercises with Answers, 342 Orthogonality, 342 Discrete Fourier Transform, 342 Discrete-Time Fourier Transform Definition, 344 Forward and Inverse Discrete-Time Fourier Transforms, 345 Exercises without Answers, 348 Discrete Fourier Transform, 348 Forward and Inverse Discrete-Time Fourier Transforms, 352 Chapter 8 The Laplace Transform, 354 8.1 Introduction and Goals, 354 8.2 Development of the Laplace Transform, 355 Generalizing the Fourier Transform, 355 Complex Exponential Excitation and Response, 357 8.3 The Transfer Function, 358 8.4 Cascade-Connected Systems, 358 8.5 Direct Form II Realization, 359 8.6 The Inverse Laplace Transform, 360 8.7 Existence of the Laplace Transform, 360 Time-Limited Signals, 361 Right- and Left-Sided Signals, 361 8.8 Laplace-Transform Pairs, 362 8.9 Partial-Fraction Expansion, 367 8.10 Laplace-Transform Properties, 377 8.11 The Unilateral Laplace Transform, 379 Definition, 379 Properties Unique to the Unilateral Laplace Transform, 381 Solution of Differential Equations with Initial Conditions, 383 8.12 Pole-Zero Diagrams and Frequency Response, 385 8.13 MATLAB System Objects, 393 8.14 Summary of Important Points, 395 Exercises, 395 Exercises with Answers, 395 Laplace-Transform Definition, 395 Direct Form II System Realization, 396 Forward and Inverse Laplace Transforms, 396 Unilateral Laplace-Transform Integral, 399 Solving Differential Equations, 399 Exercises without Answers, 400 Region of Convergence, 400 Existence of the Laplace Transform, 400 Direct Form II System Realization, 400 Forward and Inverse Laplace Transforms, 401 Solution of Differential Equations, 403 Pole-Zero Diagrams and Frequency Response, 403 Chapter 9 The z Transform, 406 9.1 Introduction and Goals, 406 9.2 Generalizing the Discrete-Time Fourier Transform, 407 9.3 Complex Exponential Excitation and Response, 408 9.4 The Transfer Function, 408 9.5 Cascade-Connected Systems, 408 9.6 Direct Form II System Realization, 409 9.7 The Inverse z Transform, 410 9.8 Existence of the z Transform, 410 Time-Limited Signals, 410 Right- and Left-Sided Signals, 411 9.9 z-Transform Pairs, 413 9.10 z-Transform Properties, 416 9.11 Inverse z-Transform Methods, 417 Synthetic Division, 417 Partial-Fraction Expansion, 418 Examples of Forward and Inverse z Transforms, 418 9.12 The Unilateral z Transform, 423 Properties Unique to the Unilateral z Transform, 423 Solution of Difference Equations, 424 9.13 Pole-Zero Diagrams and Frequency Response, 425 9.14 MATLAB System Objects, 428 In MATLAB, 429 9.15 Transform Method Comparisons, 430 9.16 Summary of Important Points, 434
Contents viii Exercises, 435 Exercises with Answers, 435 Direct-Form II System Realization, 435 Existence of the z Transform, 435 Forward and Inverse z Transforms, 435 Unilateral z-Transform Properties, 438 Solution of Difference Equations, 438 Pole-Zero Diagrams and Frequency Response, 439 Exercises without Answers, 441 Direct Form II System Realization, 441 Existence of the z Transform, 441 Forward and Inverse z-Transforms, 441 Pole-Zero Diagrams and Frequency Response, 443 Chapter 10 Sampling and Signal Processing, 446 10.1 Introduction and Goals, 446 10.2 Continuous-Time Sampling, 447 Sampling Methods, 447 The Sampling Theorem, 449 Qualitative Concepts, 449 Sampling Theorem Derivation, 451 Aliasing, 454 Time-limited and Bandlimited Signals, 457 Interpolation, 458 Ideal Interpolation, 458 Practical Interpolation, 459 Zero-Order Hold, 460 First-Order Hold, 460 Sampling Bandpass Signals, 461 Sampling a Sinusoid, 464 Bandlimited Periodic Signals, 467 Signal Processing Using the DFT, 470 CTFT-DFT Relationship, 470 CTFT-DTFT Relationship, 471 Sampling and Periodic-Repetition Relationship, 474 Computing the CTFS Harmonic Function with the DFT, 478 Approximating the CTFT with the DFT, 478 Forward CTFT, 478 Inverse CTFT, 479 Approximating the DTFT with the DFT, 479 Approximating Continuous-Time Convolution with the DFT, 479 Aperiodic Convolution, 479 Periodic Convolution, 479 Discrete-Time Convolution with the DFT, 479 Aperiodic Convolution, 479 Periodic Convolution, 479 Summary of Signal Processing Using the DFT, 480 10.3 Discrete-Time Sampling, 481 Periodic-Impulse Sampling, 481 Interpolation, 483 10.4 Summary of Important Points, 486 Exercises, 487 Exercises with Answers, 487 Pulse Amplitude Modulation, 487 Sampling, 487 Impulse Sampling, 489 Nyquist Rates, 491 Time-Limited and Bandlimited Signals, 492 Interpolation, 493 Aliasing, 495 Bandlimited Periodic Signals, 495 CTFT-CTFS-DFT Relationships, 495 Windows, 497 DFT, 497 Exercises without Answers, 500 Sampling, 500 Impulse Sampling, 502 Nyquist Rates, 504 Aliasing, 505 Practical Sampling, 505 Bandlimited Periodic Signals, 505 DFT, 506 Discrete-Time Sampling, 508 Chapter 11 Frequency Response Analysis, 509 11.1 Introduction and Goals, 509 11.2 Frequency Response, 509 11.3 Continuous-Time Filters, 510 Examples of Filters, 510 Ideal Filters, 515 Distortion, 515 Filter Classifications, 516 Ideal Filter Frequency Responses, 516 Impulse Responses and Causality, 517 The Power Spectrum, 520 Noise Removal, 520 Bode Diagrams, 521
Contents ix The Decibel, 521 The One-Real-Pole System, 525 The One-Real-Zero System, 526 Integrators and Differentiators, 527 Frequency-Independent Gain, 527 Complex Pole and Zero Pairs, 530 Practical Filters, 532 Passive Filters, 532 The Lowpass Filter, 532 The Bandpass Filter, 535 Active Filters, 536 Operational Amplifiers, 537 The Integrator, 538 The Lowpass Filter, 538 11.4 Discrete-Time Filters, 546 Notation, 546 Ideal Filters, 547 Distortion, 547 Filter Classifications, 548 Frequency Responses, 548 Impulse Responses and Causality, 548 Filtering Images, 549 Practical Filters, 554 Comparison with Continuous-Time Filters, 554 Highpass, Bandpass, and Bandstop Filters, 556 The Moving Average Filter, 560 The Almost Ideal Lowpass Filter, 564 Advantages Compared to Continuous-Time Filters, 566 11.5 Summary of Important Points, 566 Exercises, 567 Exercises with Answers, 567 Continuous-Time Frequency Response, 567 Continuous-Time Ideal Filters, 567 Continuous-Time Causality, 567 Logarithmic Graphs, Bode Diagrams, and Decibels, 568 Continuous-Time Practical Passive Filters, 570 Continuous-Time Practical Active Filters, 574 Discrete-Time Frequency Response, 575 Discrete-Time Ideal Filters, 576 Discrete-Time Causality, 576 Discrete-Time Practical Filters, 577 Exercises without Answers, 579 Continuous-Time Frequency Response, 579 Continuous-Time Ideal Filters, 579 Continuous-Time Causality, 579 Bode Diagrams, 580 Continuous-Time Practical Passive Filters, 580 Continuous-Time Filters, 582 Continuous-Time Practical Active Filters, 582 Discrete-Time Causality, 586 Discrete-Time Filters, 587 Chapter 12 Laplace System Analysis, 592 12.1 Introduction and Goals, 592 12.2 System Representations, 592 12.3 System Stability, 596 12.4 System Connections, 599 Cascade and Parallel Connections, 599 The Feedback Connection, 599 Terminology and Basic Relationships, 599 Feedback Effects on Stability, 600 Beneficial Effects of Feedback, 601 Instability Caused by Feedback, 604 Stable Oscillation Using Feedback, 608 The Root-Locus Method, 612 Tracking Errors in Unity-Gain Feedback Systems, 618 12.5 System Analysis Using MATLAB, 621 12.6 System Responses to Standard Signals, 623 Unit-Step Response, 624 Sinusoid Response, 627 12.7 Standard Realizations of Systems, 630 Cascade Realization, 630 Parallel Realization, 632 12.8 Summary of Important Points, 632 Exercises, 633 Exercises with Answers, 633 Transfer Functions, 633 Stability, 634 Parallel, Cascade, and Feedback Connections, 635 Root Locus, 637 Tracking Errors in Unity-Gain Feedback Systems, 639 System Responses to Standard Signals, 640 System Realization, 641 Exercises without Answers, 642 Stability, 642 Transfer Functions, 642 Stability, 643
Contents x Parallel, Cascade, and Feedback Connections, 643 Root Locus, 646 Tracking Errors in Unity-Gain Feedback Systems, 647 Response to Standard Signals, 647 System Realization, 649 Chapter 13 z-Transform System Analysis, 650 13.1 Introduction and Goals, 650 13.2 System Models, 650 Difference Equations, 650 Block Diagrams, 651 13.3 System Stability, 651 13.4 System Connections, 652 13.5 System Responses to Standard Signals, 654 Unit-Sequence Response, 654 Response to a Causal Sinusoid, 657 13.6 Simulating Continuous-Time Systems with Discrete-Time Systems, 660 z-Transform-Laplace-Transform Relationships, 660 Impulse Invariance, 662 Sampled-Data Systems, 664 13.7 Standard Realizations of Systems, 670 Cascade Realization, 670 Parallel Realization, 670 13.8 Summary of Important Points, 671 Exercises, 672 Exercises with Answers, 672 Stability, 672 Parallel, Cascade, and Feedback Connections, 672 Response to Standard Signals, 673 Root Locus, 674 Laplace-Transform-z-Transform Relationship, 675 Sampled-Data Systems, 675 System Realization, 676 Exercises without Answers, 677 Stability, 677 Root Locus, 677 Parallel, Cascade, and Feedback Connections, 677 Response to Standard Signals, 677 Laplace-Transform-z-Transform Relationship, 679 Sampled-Data Systems, 679 System Realization, 679 General, 679 Chapter 14 Filter Analysis and Design, 680 14.1 Introduction and Goals, 680 14.2 Analog Filters, 680 Butterworth Filters, 681 Normalized Butterworth Filters, 681 Filter Transformations, 682 MATLAB Design Tools, 684 Chebyshev, Elliptic, and Bessel Filters, 686 14.3 Digital Filters, 689 Simulation of Analog Filters, 689 Filter Design Techniques, 689 IIR Filter Design, 689 Time-Domain Methods, 689 Impulse-Invariant Design, 689 Step-Invariant Design, 696 Finite-Difference Design, 698 Frequency-Domain Methods, 704 The Bilinear Method, 706 FIR Filter Design, 713 Truncated Ideal Impulse Response, 713 Optimal FIR Filter Design, 723 MATLAB Design Tools, 725 14.4 Summary of Important Points, 727 Exercises, 727 Exercises with Answers, 727 Continuous-Time Filters, 727 Finite-Difference Filter Design, 728 Matched-z Transform and Direct Substitution Filter Design, 729 Bilinear z-Transform Filter Design, 730 FIR Filter Design, 730 Digital Filter Design Method Comparison, 731 Exercises without Answers, 731 Analog Filter Design, 731 Impulse-Invariant and Step-Invariant Filter Design, 732 Finite-Difference Filter Design, 733 Matched z-Transform and Direct Substitution Filter Design, 733 Bilinear z-Transform Filter Design, 733 FIR Filter Design, 733 Digital Filter Design Method Comparison, 734
Contents xi Appendix I Useful Mathematical Relations, A-1     II Continuous-Time Fourier Series Pairs, A-4     III Discrete Fourier Transform Pairs, A-7     IV Continuous-Time Fourier Transform Pairs, A-10     V Discrete-Time Fourier Transform Pairs, A-17     VI Tables of Laplace Transform Pairs, A-22     VII z-Transform Pairs, A-24 Bibliography, B-1 Index, I-1
PREFACE MOTIVATION I wrote the first and second editions because I love the mathematical beauty of signal and system analysis. That has not changed. The motivation for the third edi- tion is to further refine the book structure in light of reviewers, comments, correct a few errors from the second edition and significantly rework the exercises. AUDIENCE This book is intended to cover a two-semester course sequence in the basics of signal and system analysis during the junior or senior year. It can also be used (as I have used it) as a book for a quick one-semester Master’s-level review of trans- form methods as applied to linear systems. CHANGES FROM THE SECOND EDITION 1. In response to reviewers, comments, two chapters from the second edition have been omitted: Communication Systems and State-Space Analysis. There seemed to be very little if any coverage of these topics in actual classes. 2. The second edition had 550 end-of-chapter exercises in 16 chapters. The third edition has 710 end-of-chapter exercises in 14 chapters. OVERVIEW Except for the omission of two chapters, the third edition structure is very similar to the second edition. The book begins with mathematical methods for describing signals and systems, in both continuous and discrete time. I introduce the idea of a transform with the continuous-time Fourier series, and from that base move to the Fourier trans- form as an extension of the Fourier series to aperiodic signals. Then I do the same for discrete-time signals. I introduce the Laplace transform both as a generalization of the continuous-time Fourier transform for unbounded signals and unstable systems and as a powerful tool in system analysis because of its very close association with the ei- genvalues and eigenfunctions of continuous-time linear systems. I take a similar path for discrete-time systems using the z transform. Then I address sampling, the relation between continuous and discrete time. The rest of the book is devoted to applications in frequency-response analysis, feedback systems, analog and digital filters. Through- out the book I present examples and introduce MATLAB functions and operations to implement the methods presented. A chapter-by-chapter summary follows. CHAPTER SUMMARIES CHAPTER 1 Chapter 1 is an introduction to the general concepts involved in signal and system analysis without any mathematical rigor. It is intended to motivate the student by xii
xiii Preface demonstrating the ubiquity of signals and systems in everyday life and the impor- tance of understanding them. CHAPTER 2 Chapter 2 is an exploration of methods of mathematically describing continuous- time signals of various kinds. It begins with familiar functions, sinusoids and exponentials and then extends the range of signal-describing functions to include continuous-time singularity functions (switching functions). Like most, if not all, signals and systems textbooks, I define the unit-step, the signum, the unit-impulse and the unit-ramp functions. In addition to these I define a unit rectangle and a unit periodic impulse function. The unit periodic impulse function, along with convolution, provides an especially compact way of mathematically describing arbitrary periodic signals. After introducing the new continuous-time signal functions, I cover the common types of signal transformations, amplitude scaling, time shifting, time scaling, differentiation and integration and apply them to the signal functions. Then I cover some characteristics of signals that make them invariant to certain transformations, evenness, oddness and periodicity, and some of the implications of these signal characteristics in signal analysis. The last section is on signal energy and power. CHAPTER 3 Chapter 3 follows a path similar to Chapter 2 except applied to discrete-time signals instead of continuous-time signals. I introduce the discrete-time sinu- soid and exponential and comment on the problems of determining period of a discrete-time sinusoid. This is the first exposure of the student to some of the implications of sampling. I define some discrete-time signal functions analo- gous to continuous-time singularity functions. Then I explore amplitude scaling, time shifting, time scaling, differencing and accumulation for discrete-time signal functions pointing out the unique implications and problems that occur, especially when time scaling discrete-time functions. The chapter ends with definitions and discussion of signal energy and power for discrete-time signals. CHAPTER 4 This chapter addresses the mathematical description of systems. First I cover the most common forms of classification of systems, homogeneity, additivity, linearity, time invariance, causality, memory, static nonlinearity and invertibility. By example I present various types of systems that have, or do not have, these properties and how to prove various properties from the mathematical description of the system. CHAPTER 5 This chapter introduces the concepts of impulse response and convolution as components in the systematic analysis of the response of linear, time-invariant systems. I present the mathematical properties of continuous-time convolution and a graphical method of understanding what the convolution integral says. I also show how the properties of convolution can be used to combine subsystems that are connected in cascade or parallel into one system and what the impulse response of the overall system must be. Then I introduce the idea of a transfer
xiv Preface function by finding the response of an LTI system to complex sinusoidal exci- tation. This section is followed by an analogous coverage of discrete-time impulse response and convolution. CHAPTER 6 This is the beginning of the student’s exposure to transform methods. I begin by graphically introducing the concept that any continuous-time periodic signal with engineering usefulness can be expressed by a linear combination of continuous-time sinusoids, real or complex. Then I formally derive the Fourier series using the concept of orthogonality to show where the signal description as a function of discrete harmonic number (the harmonic function) comes from. I mention the Dirichlet conditions to let the student know that the continuous-time Fourier series applies to all practical continuous-time signals, but not to all imaginable continuous-time signals. Then I explore the properties of the Fourier series. I have tried to make the Fourier series notation and properties as similar as possible and analogous to the Fourier transform, which comes later. The harmonic function forms a “Fourier series pair” with the time function. In the first edition I used a notation for har- monic function in which lower-case letters were used for time-domain quantities and upper-case letters for their harmonic functions. This unfortunately caused some confusion because continuous- and discrete-time harmonic functions looked the same. In this edition I have changed the harmonic function notation for continuous-time signals to make it easily distinguishable. I also have a section on the convergence of the Fourier series illustrating the Gibb’s phenomenon at function discontinuities. I encourage students to use tables and properties to find harmonic functions and this practice prepares them for a similar process in find- ing Fourier transforms and later Laplace and z transforms. The next major section of Chapter 6 extends the Fourier series to the Fourier transform. I introduce the concept by examining what happens to a continuous-time Fourier series as the period of the signal approaches infinity and then define and derive the continuous-time Fourier transform as a gener- alization of the continuous-time Fourier series. Following that I cover all the important properties of the continuous-time Fourier transform. I have taken an “ecumenical” approach to two different notational conventions that are commonly seen in books on signals and systems, control systems, digital signal processing, communication systems and other applications of Fourier methods such as image processing and Fourier optics: the use of either cyclic frequency, f or radian fre- quency, ω. I use both and emphasize that the two are simply related through a change of variable. I think this better prepares students for seeing both forms in other books in their college and professional careers. CHAPTER 7 This chapter introduces the discrete-time Fourier series (DTFS), the discrete Fou- rier transform (DFT) and the discrete-time Fourier transform (DTFT), deriving and defining them in a manner analogous to Chapter 6. The DTFS and the DFT are almost identical. I concentrate on the DFT because of its very wide use in digital signal processing. I emphasize the important differences caused by the differences between continuous- and discrete-time signals, especially the finite summation range of the DFT as opposed to the (generally) infinite summation range in the CTFS. I also point out the importance of the fact that the DFT relates
xv Preface a finite set of numbers to another finite set of numbers, making it amenable to direct numerical machine computation. I discuss the fast Fourier transform as a very efficient algorithm for computing the DFT. As in Chapter 6, I use both cyclic and radian frequency forms, emphasizing the relationships between them. I use F and Ω for discrete-time frequencies to distinguish them from f and ω, which were used in continuous time. Unfortunately, some authors reverse these symbols. My usage is more consistent with the majority of signals and systems texts. This is another example of the lack of standardization of notation in this area. The last major section is a comparison of the four Fourier methods. I emphasize particu- larly the duality between sampling in one domain and periodic repetition in the other domain. CHAPTER 8 This chapter introduces the Laplace transform. I approach the Laplace trans- form from two points of view, as a generalization of the Fourier transform to a larger class of signals and as result which naturally follows from the excitation of a linear, time-invariant system by a complex exponential signal. I begin by defining the bilateral Laplace transform and discussing significance of the re- gion of convergence. Then I define the unilateral Laplace transform. I derive all the important properties of the Laplace transform. I fully explore the method of partial-fraction expansion for finding inverse transforms and then show examples of solving differential equations with initial conditions using the uni- lateral form. CHAPTER 9 This chapter introduces the z transform. The development parallels the devel- opment of the Laplace transform except applied to discrete-time signals and systems. I initially define a bilateral transform and discuss the region of con- vergence. Then I define a unilateral transform. I derive all the important prop- erties and demonstrate the inverse transform using partial-fraction expansion and the solution of difference equations with initial conditions. I also show the relationship between the Laplace and z transforms, an important idea in the approximation of continuous-time systems by discrete-time systems in Chapter 14. CHAPTER 10 This is the first exploration of the correspondence between a continuous-time signal and a discrete-time signal formed by sampling it. The first section covers how sampling is usually done in real systems using a sample-and-hold and an A/D converter. The second section starts by asking the question of how many samples are enough to describe a continuous-time signal. Then the question is answered by deriving the sampling theorem. Then I discuss interpolation methods, theoret- ical and practical, the special properties of bandlimited periodic signals. I do a complete development of the relationship between the CTFT of a continuous-time signal and DFT of a finite-length set of samples taken from it. Then I show how the DFT can be used to approximate the CTFT of an energy signal or a periodic signal. The next major section explores the use of the DFT in numerically approx- imating various common signal-processing operations.
xvi CHAPTER 11 This chapter covers various aspects of the use of the CTFT and DTFT in fre- quency response analysis. The major topics are ideal filters, Bode diagrams, prac- tical passive and active continuous-time filters and basic discrete-time filters. CHAPTER 12 This chapter is on the application of the Laplace transform including block dia- gram representation of systems in the complex frequency domain, system stability, system interconnections, feedback systems including root locus, system responses to standard signals and lastly standard realizations of continuous-time systems. CHAPTER 13 This chapter is on the application of the z transform including block diagram representation of systems in the complex frequency domain, system stability, sys- tem interconnections, feedback systems including root-locus, system responses to standard signals, sampled-data systems and standard realizations of discrete-time systems. CHAPTER 14 This chapter covers the analysis and design of some of the most common types of practical analog and digital filters. The analog filter types are Butterworth, Chebyshev Types 1 and 2 and Elliptic (Cauer) filters. The section on digital filters covers the most common types of techniques for simulation of analog filters includ- ing, impulse- and step-invariant, finite difference, matched z transform, direct sub- stitution, bilinear z transform, truncated impulse response and Parks-McClellan numerical design. APPENDICES There are seven appendices on useful mathematical formulae, tables of the four Fourier transforms, Laplace transform tables and z transform tables. CONTINUITY The book is structured so as to facilitate skipping some topics without loss of continuity. Continuous-time and discrete-time topics are covered alternately and continuous-time analysis could be covered without reference to discrete time. Also, any or all of the last six chapters could be omitted in a shorter course. REVIEWS AND EDITING This book owes a lot to the reviewers, especially those who really took time and criticized and suggested improvements. I am indebted to them. I am also indebted to the many students who have endured my classes over the years. I believe that our relationship is more symbiotic than they realize. That is, they learn signal and system analysis from me and I learn how to teach signal and system analysis from them. I cannot count the number of times I have been asked a very perceptive question by a student that revealed not only that the students were not understand- ing a concept but that I did not understand it as well as I had previously thought. Preface
xvii WRITING STYLE Every author thinks he has found a better way to present material so that students can grasp it and I am no different. I have taught this material for many years and through the experience of grading tests have found what students generally do and do not grasp. I have spent countless hours in my office one-on-one with students explaining these concepts to them and, through that experience, I have found out what needs to be said. In my writing I have tried to simply speak directly to the reader in a straightforward conversational way, trying to avoid off-putting formality and, to the extent possible, anticipating the usual misconceptions and revealing the fallacies in them. Transform methods are not an obvious idea and, at first exposure, students can easily get bogged down in a bewildering morass of abstractions and lose sight of the goal, which is to analyze a system’s response to signals. I have tried (as every author does) to find the magic combination of ac- cessibility and mathematical rigor because both are important. I think my writing is clear and direct but you, the reader, will be the final judge of whether or not that is true. EXERCISES Each chapter has a group of exercises along with answers and a second group of exercises without answers. The first group is intended more or less as a set of “drill” exercises and the second group as a set of more challenging exercises. CONCLUDING REMARKS As I indicated in the preface to first and second editions, I welcome any and all criticism, corrections and suggestions. All comments, including ones I disagree with and ones which disagree with others, will have a constructive impact on the next edition because they point out a problem. If something does not seem right to you, it probably will bother others also and it is my task, as an author, to find a way to solve that problem. So I encourage you to be direct and clear in any re- marks about what you believe should be changed and not to hesitate to mention any errors you may find, from the most trivial to the most significant. Michael J. Roberts, Professor Emeritus Electrical and Computer Engineering University of Tennessee at Knoxville mjr@utk.edu Preface
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1 C H A P T E R 1 Introduction 1.1 SIGNALS AND SYSTEMS DEFINED Any time-varying physical phenomenon that is intended to convey information is a signal. Examples of signals are the human voice, sign language, Morse code, traffic signals, voltages on telephone wires, electric fields emanating from radio or television transmitters, and variations of light intensity in an optical fiber on a telephone or com- puter network. Noise is like a signal in that it is a time-varying physical phenomenon, but usually it does not carry useful information and is considered undesirable. Signals are operated on by systems. When one or more excitations or input signals are applied at one or more system inputs, the system produces one or more responses or output signals at its outputs. Figure 1.1 is a block diagram of a single-input, single-output system. System Input Output Excitation or Input Signal Response or Output Signal Figure 1.1 Block diagram of a single-input, single-output system Transmitter Channel Receiver Information Signal Noisy Information Signal Noise Noise Noise Figure 1.2 A communication system In a communication system, a transmitter produces a signal and a receiver acquires it. A channel is the path a signal takes from a transmitter to a receiver. Noise is inevitably introduced into the transmitter, channel and receiver, often at multiple points (Figure 1.2). The transmitter, channel and receiver are all components or subsystems of the overall system. Scientific instruments are systems that measure a physical phenom- enon (temperature, pressure, speed, etc.) and convert it to a voltage or current, a sig- nal. Commercial building control systems (Figure 1.3), industrial plant control systems (Figure 1.4), modern farm machinery (Figure 1.5), avionics in airplanes, ignition and fuel pumping controls in automobiles, and so on are all systems that operate on signals.
Ch ap ter 1  Introduction 2 Figure 1.3 Modern office buildings © Vol. 43 PhotoDisc/Getty Figure 1.4 Typical industrial plant control room © Royalty-Free/Punchstock
1.2 Types of Signals 3 The term system even encompasses things such as the stock market, government, weather, the human body and the like. They all respond when excited. Some systems are readily analyzed in detail, some can be analyzed approximately, but some are so complicated or difficult to measure that we hardly know enough to understand them. 1.2 TYPES OF SIGNALS There are several broad classifications of signals: continuous-time, discrete-time, continuous-value, discrete-value, random and nonrandom. A continuous-time sig- nal is defined at every instant of time over some time interval. Another common name for some continuous-time signals is analog signal, in which the variation of the signal with time is analogous (proportional) to some physical phenomenon. All analog sig- nals are continuous-time signals but not all continuous-time signals are analog signals (Figure 1.6 through Figure 1.8). Sampling a signal is acquiring values from a continuous-time signal at discrete points in time. The set of samples forms a discrete-time signal. A discrete-time signal Figure 1.5 Modern farm tractor with enclosed cab © Royalty-Free/Corbis Figure 1.6 Examples of continuous-time and discrete-time signals n x[n] Discrete-Time Continuous-Value Signal t x(t) Continuous-Time Continuous-Value Signal
Ch ap ter 1  Introduction 4 can also be created by an inherently discrete-time system that produces signal values only at discrete times (Figure 1.6). A continuous-value signal is one that may have any value within a continuum of allowed values. In a continuum any two values can be arbitrarily close together. The real numbers form a continuum with infinite extent. The real numbers between zero and one form a continuum with finite extent. Each is a set with infinitely many mem- bers (Figure 1.6 through Figure 1.8). A discrete-value signal can only have values taken from a discrete set. In a discrete set of values the magnitude of the difference between any two values is greater than some positive number. The set of integers is an example. Discrete-time signals are usually transmitted as digital signals, a sequence of values of a discrete-time signal in the form of digits in some encoded form. The term digital is also sometimes used loosely to refer to a discrete-value signal that has only two possible values. The digits in this type of digital signal are transmitted by signals that are continuous-time. In this case, the terms continuous-time and analog are not synonymous. A digital signal of this type is a continuous-time signal but not an analog signal because its variation of value with time is not directly analogous to a physical phenomenon (Figure 1.6 through Figure 1.8). A random signal cannot be predicted exactly and cannot be described by any math- ematical function. A deterministic signal can be mathematically described. A com- mon name for a random signal is noise (Figure 1.6 through Figure 1.8). In practical signal processing it is very common to acquire a signal for processing by a computer by sampling, quantizing and encoding it (Figure 1.9). The original signal is a continuous-value, continuous-time signal. Sampling acquires its values at discrete times and those values constitute a continuous-value, discrete-time signal. Quantization approximates each sample as the nearest member of a finite set of dis- crete values, producing a discrete-value, discrete-time signal. Each signal value in the set of discrete values at discrete times is converted to a sequence of rectangular pulses that encode it into a binary number, creating a discrete-value, continuous-time signal, commonly called a digital signal. The steps illustrated in Figure 1.9 are usually carried out by a single device called an analog-to-digital converter (ADC). Figure 1.8 Examples of noise and a noisy digital signal Noisy Digital Signal Continuous-Time Continuous-Value Random Signal t x(t) x(t) Noise t Figure 1.7 Examples of continuous-time, discrete-value signals Continuous-Time Discrete-Value Signal Continuous-Time Discrete-Value Signal t x(t) x(t) t Digital Signal
1.2 Types of Signals 5 Figure 1.9 Sampling, quantization and encoding of a signal to illustrate various signal types t kΔt (k+1)Δt (k+2)Δt (k–1)Δt kΔt (k+1)Δt (k+2)Δt (k–1)Δt xs[n] n k k+1 k+2 k–1 xsq[n] n k k+1 k+2 k–1 xsqe(t) t 111 001 111 011 Continuous-Value Continuous-Time Signal Continuous-Value Discrete-Time Signal Discrete-Value Discrete-Time Signal Discrete-Value Continuous-Time Signal Sampling Quantization Encoding x(t) Figure 1.10 Asynchronous serial binary ASCII-encoded voltage signal for the word SIGNAL 0 1 2 3 4 5 6 7 –1 0 1 2 3 4 5 6 Time, t (ms) Voltage, v(t) (V) Serial Binary Voltage Signal for the ASCII Message “SIGNAL” S I G N A L One common use of binary digital signals is to send text messages using the American Standard Code for Information Interchange (ASCII). The letters of the al- phabet, the digits 0–9, some punctuation characters and several nonprinting control characters, for a total of 128 characters, are all encoded into a sequence of 7 binary bits. The 7 bits are sent sequentially, preceded by a start bit and followed by 1 or 2 stop bits for synchronization purposes. Typically, in direct-wired connections between digital equipment, the bits are represented by a higher voltage (2 to 5V) for a 1 and a lower voltage level (around 0V) for a 0. In an asynchronous transmission using one start and one stop bit, sending the message SIGNAL, the voltage versus time would look as illustrated in Figure 1.10.
Ch ap ter 1  Introduction 6 In 1987 ASCII was extended to Unicode. In Unicode the number of bits used to represent a character can be 8, 16, 24 or 32 and more than 120,000 characters are cur- rently encoded in modern and historic language characters and multiple symbol sets. Digital signals are important in signal analysis because of the spread of digital systems. Digital signals often have better immunity to noise than analog signals. In binary signal communication the bits can be detected very cleanly until the noise gets very large. The detection of bit values in a stream of bits is usually done by comparing the signal value at a predetermined bit time with a threshold. If it is above the thresh- old it is declared a 1 and if it is below the threshold it is declared a 0. In Figure 1.11, the x’s mark the signal value at the detection time, and when this technique is applied to the noisy digital signal, one of the bits is incorrectly detected. But when the signal is processed by a filter, all the bits are correctly detected. The filtered digital signal does not look very clean in comparison with the noiseless digital signal, but the bits can still be detected with a very low probability of error. This is the basic reason that digital signals can have better noise immunity than analog signals. An introduction to the analysis and design of filters is presented in Chapters 11 and 15. In this text we will consider both continuous-time and discrete-time signals, but we will (mostly) ignore the effects of signal quantization and consider all signals to be continuous-value. Also, we will not directly consider the analysis of random signals, although random signals will sometimes be used in illustrations. The first signals we will study are continuous-time signals. Some continuous-time signals can be described by continuous functions of time. A signal x(t) might be described by a function x(t) = 50sin(200πt) of continuous time t. This is an exact description of the signal at every instant of time. The signal can also be described graphically (Figure 1.12). Many continuous-time signals are not as easy to describe mathematically. Consider the signal in Figure 1.13. Waveforms like the one in Figure 1.13 occur in various types of instrumentation and communication systems. With the definition of some signal functions and an operation called convolution, this signal can be compactly described, analyzed and manipulated mathematically. Continuous-time signals that can be described by math- ematical functions can be transformed into another domain called the frequency domain through the continuous-time Fourier transform. In this context, transformation means transformation of a signal to the frequency domain. This is an important tool in signal analysis, which allows certain characteristics of the signal to be more clearly observed Figure 1.11 Use of a filter to reduce bit error rate in a digital signal x(t) –1 2 Noiseless Digital Signal t 2.6 –1 2 t 2.6 –1 2 t 2.6 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 1 Bit Error Bit Detection Threshold xf(t) x(t) + n(t) Noisy Digital Signal Filtered Digital Signal
1.2 Types of Signals 7 and more easily manipulated than in the time domain. (In the frequency domain, signals are described in terms of the frequencies they contain.) Without frequency-domain analy- sis, design and analysis of many systems would be considerably more difficult. Discrete-time signals are only defined at discrete points in time. Figure 1.14 illustrates some discrete-time signals. Figure 1.12 A continuous-time signal described by a mathematical function ... ... –50 50 t = 10 ms t x(t) Figure 1.13 A second continuous-time signal ... ... 5 t = 20 μs t x(t) Figure 1.14 Some discrete-time signals n x[n] n x[n] n x[n] n x[n] So far all the signals we have considered have been described by functions of time. An important class of “signals” is functions of space instead of time: images. Most of the theories of signals, the information they convey and how they are processed by systems in this text will be based on signals that are a variation of a physical phenome- non with time. But the theories and methods so developed also apply, with only minor modifications, to the processing of images. Time signals are described by the variation of a physical phenomenon as a function of a single independent variable, time. Spa- tial signals, or images, are described by the variation of a physical phenomenon as a
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Signals and systems : analysis using transform methods and MATLAB Third Edition Michael J. Roberts
Signals and systems : analysis using transform methods and MATLAB Third Edition Michael J. Roberts
Signals and systems : analysis using transform methods and MATLAB Third Edition Michael J. Roberts
The Project Gutenberg eBook of The Union: Or, Select Scots and English Poems
This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: The Union: Or, Select Scots and English Poems Compiler: Thomas Warton Release date: August 8, 2012 [eBook #40444] Most recently updated: October 23, 2024 Language: English Credits: Produced by Margo von Romberg, Bryan Ness and the Online Distributed Proofreading Team at http://www.pgdp.net (This book was produced from scanned images of public domain material from the Google Print project.) *** START OF THE PROJECT GUTENBERG EBOOK THE UNION: OR, SELECT SCOTS AND ENGLISH POEMS ***
THE U N I O N : OR, SELECT SCOTS and ENGLISH P O E M S . THE SECOND EDITION. ----Dubiam facientia carmina palmam. Juv. LONDON: Printed for R. Baldwin, in Paternoster-Row. M.DCC.LIX.
Signals and systems : analysis using transform methods and MATLAB Third Edition Michael J. Roberts
PREFACE. As the mind of man is ever fond of variety, nothing seems better calculated to entertain, than a judicious collection of the smaller, though not on that account less laboured, productions of eminent poets: an entertainment, not unlike that which we receive from surveying a finished landschape, or well disposed piece of shell-work: where each particular object, tho' singly beautiful, and sufficiently striking by itself, receives an additional charm, thus, as Milton expresses it, sweetly interchanged. The first miscellaneous collection of poems, that ever appeared in Great-Britain with any reputation, is that published by Dryden: which was afterwards continued by Tonson. There are many pieces of the highest merit in this collection, by Dryden, Denham, Creech, Drayton, Garth, Marvell, and many others; yet the compilers, it is evident, were not always sufficiently scrupulous and cautious in their choice, as several pieces are admitted, among the rest, which would otherwise utterly have perished, and which had no other recommendation, than that they served to swell the volume. Since this, many miscellanies have been published both in Scotland and England: to enumerate which would be no less tedious than useless. It will be sufficient to remark, that through want of care or judgment in their respective editors, they are all forgotten or neglected. From these the miscellany known by the name of Mr. Pope perhaps ought to be excepted; tho' that, indeed, cannot properly be styled a collection of poems by different hands, which is such a one as we are speaking of at present, the greater part consisting of pieces by Mr. Pope only. The best miscellany at this day extant in our language, and the first complete one of the kind which we have seen, is that lately published by R. Dodsley, which boasts the greatest names of the present age among its contributors.
As to the poetical collection here exhibited to the public, we apprehend it challenges no small degree of regard, as it was made under the immediate inspection and conduct of several very ingenious gentlemen, whose names it would do us the highest honour to mention; and as it contains a variety not to be found even in the admirable collection last spoken of; I mean the Intermixture of poems both Scotch and English. Nor is this variety less agreeable than useful; as from it we have an opportunity of forming a comparison and estimate of the taste and genius of the two different nations, in their poetical compositions. It will be necessary to take notice, that our chief care has been to furnish out the following miscellany with those pieces, regard being first had to real merit, which have laid unknown and unobserved from their manner of publication; several of them having been printed by themselves, and so perished as it were for want of bulk, and others lost amid the rubbish of collections injudiciously made, and perhaps not easily to be met with. Nor will it be improper to mention, that in order to render our volume still more compleat, we have had the favour of some original poems, written by a late member of the university of Aberdeen, whose modesty would not permit us to prefix his name: one of which in this edition is printed with many improvements, from a corrected copy. And from these ingenious essays, the public may be enabled to form some judgment beforehand of a poem of a nobler and more important nature, which he is now preparing. Nor must we forget to return our public thanks to this gentleman, for the service he has been to us, not only in making this collection more excellent by his own contributions, but in selecting such pieces of others as were suitable to our design. It is hoped that the ancient Scottish poems (amongst which the thistle and the rose, and hardyknute are more particularly distinguished) will make no disagreeable figure amongst those of modern date; and that they will produce the same effect here, as Mr. Pope observes a moderate use of old words may have in a poem; which, adds he, is like working old abbey-stones into a modern building, and which I have sometimes seen practised with good success.
Upon the whole, as we have been favoured with the best assistance in compiling this volume, no further apology is necessary; and as the approbation of the public has been already secured to these poems separately, we hope they have no less reason to claim it, when thus published together.
CONTENTS. Page The Thistle and the Rose, by W. Dunbar 1 Verses on the Death of Queen Caroline. By Mr. Shipley 10 The Genealogy of Christ, by Mr. Lowth 13 A Fragment, by Mr. Mallet 24 The Eagle and Robin Red-Breast, a Fable, by Archibald Scott, written before the Year 1600. 28 Ode to Fancy, by Mr. Joseph Warton 31 Ode to Evening, by the same 37 Ode to Evening, by Mr. Collins 39 Isis, an Elegy, by Mr. Mason of Cambridge 42 The Triumph of Isis, by Mr. Thomas Warton of Oxford 47 A Love-Elegy, by Mr. Hammond 47 The Tears of Scotland, 1746. 62 An Elegy written in a country church-yard, by Mr. Grey 65 On the Death of Prince Frederic. Written at Paris, by David Lord Viscount Stormont 70 On the same, by Mr. James Clitherow of Oxford 75 Ode on the Approach of Summer, by a Gentleman formerly of the University of Aberdeen 81 A Pastoral in the manner of Spenser, from Theocritus, Idyll. 20. By the same 94 Inscribed on a beautiful Grotto near the Water 96 Love Elegy, by Mr. Smollet 97 A Panegyric on Oxford Ale, by a Gentleman of Trinity College 99 The Progress of Discontent, by the Same. 105 Ode to Arthur Onslow, Esq; 109 Job, Chapter XXXIX. By a Gentleman of Oxford 113
Ode on the Death of Mr. Thomson, by Mr. Collins 116 The Child-Birth, in the manner of Gay 119 On a Lady's presenting a Sprig of Myrtle to a Gentleman, by Mr. Hammond 125 To a Young Lady with Fontenelle's Plurality of Worlds 126 Ode on the Fifth of December, by Mr. Christopher Smart 128 Part of the Prologue to Sir David Lyndesay's Dream. Written in the Reign of King James V. 129 Hardyknute, a Fragment 132 Ode. By Dr. Akenside, on Lyric Poetry 147
A POEM IN HONOUR OF MARGARET DAUGHTER TO HENRY VII. OF ENGLAND, QUEEN TO JAMES IV. KING OF SCOTS. BY WILLIAM DUNBAR. The Thistle and the Rose, O'er flowers and herbage green, By Lady Nature chose, Brave King and lovely Queen. I. When March with varying winds was overpast, And sweet April had with his silver showers Ta'n leave of Nature with an orient blast, And lusty May, that mother is of flowers, Had made the birds begin by tymous hours; Among the tender odours red and white, Whose harmony to her was great delight.
II. In bed at morrow, sleeping as I lay, Methought Aurora with her ruby ene, In at my window looked by the day, And halsit me with visage pale and green; Upon her hand a lark sang frae the spleen, Lovers, awake out of your slumbering. See how the lusty morning does upspring. III. Methought fresh May before my bed upstood, In weed depainted of ilk diverse hue, Sober, benign, and full of mansuetude, In bright attire of flowers, all forged new, Of heavenly colour, white, red, brown and blue, Balmit in dew, and gilt with Phebus' beams, While all the house illumin'd with her leams. IV. Sluggard, she said, awake anon for shame, And in mine honour something thou go write; The lark has done, the merry day proclaim, Lovers to raise with comfort and delight; Will nought increase thy courage to indite, Whose heart sometime has glad and blissful been, Songs oft to make, under the branches green? V. Whereto, quoth I, shall I uprise at morrow,
For in thy month few birds have I heard sing, They have mare cause to weep and plain their sorrow: Thy air it is not wholsome nor benign, Lord Eolus does in thy season ring, So bousteous are the blasts of his shrill horn, Among thy boughs to walk I have forborn. VI. With that the lady soberly did smile, And said, uprise and do thy observance: Thou did promise in May's lusty while, Then to describe the ROSE of most pleasance Go see the birdis how they sing and dance, And how the skies illumined are bright, Enamell'd richly with new azure light. VII. When this was said, away then went the Queen, And enter'd in a lusty garden gent; And then methought, full hastily beseen, In sark and mantle after her I went Into this garth most dulce and redolent, Of herb and flower, and tender plants most sweet, And the green leaves doing of dew down fleit. VIII. The purple sun, with tender rayis red, In orient bright as angel did appear, Through golden skies advancing up his head,
Whose gilded tresses shone so wondrous clear, That all the world took comfort far and near, To look upon his fresh and blissful face, Doing all sable frae the Heavens chace. IX. And as the blissful sun drove up the sky, All nature sang through comfort of the light, The minstrels wing'd, with open voices cry, O Lovers now is fled the dully night, Come welcome day, that comforts ev'ry wight; Hail May! hail Flora! hail Aurora sheen, Hail Princess Nature! hail love's hartsome Queen! X. Dame Nature gave an inhibition there, To Neptune fierce, and Eolus the bold, Not to perturb the water or the air, That neither blashy shower, nor blasts more cold Should flowers affray nor fowls upon the fold. She bade eke Juno, Goddess of the sky, That she the heaven should keep amene and dry. XI. Also ordain'd that every bird and beast Before her Highness should anon compear; And every flower of virtue most and least, And every herb of fair field far and near,
As they had wont in May from year to year; To her their Queen to make obedience, Full low inclining with due reverence. XII. With that anon she sent the swift foot Roe, To bring in alkind beast from dale and down; The restless swallow order'd she to go, And fetch all fowl of great and small renown, And to gar flowers appear of all fassoun: Full craftily conjured she the Yarrow, Which did forth swirk as swift as any arrow. XIII. All brought in were in twinkling of an eye, Both beast and bird and flower before the Queen; And first the Lion, greatest of degree, Was summon'd there; and he, fair to be seen, With a full hardy countenance and keen, Before Dame Nature came, and did incline, With visage bold, and courage leonine. XIV. This awful beast was terrible of chear, Piercing of look, and stout of countenance, Right strong of corps, of fashion fair, but fear, Lusty of shape, light of deliverance, Red of his colour, as the ruby glance: In field of gold he stood full rampantly, With flower-de-lyces circled pleasantly.
XV. This Lady lifted up his claws so clear, And lute him listly lean upon her knee, And crowned him with diadem full dear, Of radious stones most royal there to see, Saying the King of all beasts make I thee; And the protector chief in woods and shaws, Go forth, and to thy lieges keep the laws. XVI. Justice exerce, with mercy and conscience, And let no small beast suffer skaith or scorns Of greater beasts, that been of more puissance; Do law alike to Apes and Unicorns, And let no Bugle with his bousteous horns Oppress the meek plough Ox, for all his pride, But in the yoke go quietly him beside. XVII. When this was said, with noise and sound of joy, All kind of Quadrupeds in their degree, At once cry'd laud, and then vive le roy, Then at his feet fell with humility; To him they all paid homage and fealty; And he did them receive with princely laits, Whose noble ire his greatness mitigates. XVIII. Then crowned she the Eagle King of fowls;
And sharp as darts of steel she made his pens, And bade him be as just to Whawps and Owls, As unto Peacocks, Papingoes, or Cranes, And make one law for Wicht Fowls, and for Wrens, And let no fowl of rapine do affray, Nor birds devour, but his own proper prey. XIX. Then called she all flowers grew in the field, Describing all their fashions and effeirs, Upon the awful THISTLE she beheld. And saw him guarded with a bush of spears, Considering him so able for the wars, A radiant crown of rubies she him gave, And said, in field go forth, and fend the laif. XX. And since thou art a King, be thou discreet, Herb without value hold not of such price, As herb of virtue and of odour sweet; And let no nettle vile, and full of vice, Her fellow with the goodly Flower-de-lyce; Nor let no wild weed full of churlishness, Compare her to the Lilly's nobleness. XXI. Nor hold none other flower in such dainty As the fresh ROSE, of colour red and white; For if thou dost, hurt is thine honesty, Considering that no flower is so perfyte, So full of pleasaunce, virtue and delight;
So full of blissful angelic beauty, Imperial birth, honour and dignity. XXII. Then to the ROSE she did her visage turn, And said, O lusty daughter most benign, Above the Lilly thou art illustrious born, From royal lineage rising fresh and young, But any spot, or macul doing sprung; Come bloom of joy, with richest gems becrown'd, For o'er the laif thy beauty is renown'd. XXIII. A costly crown with stones clarified bright, This comely Queen did in her head inclose, While all the land illumined of light; Wherefore methought, the flowers did all rejoyce, Crying at once, Hail to the fragrant ROSE! Hail Empress of the herbs! fresh Queen of flowers! To thee be glore and honour at all hours. XXIV. Then all the birds they sang with voice on height, Whose mirthful sound was marvellous to hear: The Mavys sang, Hail ROSE most rich and right, That does upflourish under Phebus' sphere, Hail plant of youth, hail Prince's daughter dear,
Hail blossom breaking out of blood royal, Whose precious virtue is imperial. XXV. The Merle she sang, Hail ROSE of most delight, Hail of all flowers the sweet and sovereign Queen: The lark she sang, hail ROSE both red and white, Most pleasant flower of mighty colours[1] twain: Nightingals sang, hail Natures suffragan, In beauty, nurture, and each nobleness, In rich array, renown, and gentleness. XXVI. The common voice uprose of warblers small, Upon this wise, O blessed be the hour That thou wast chose to be our principal, Welcome to be our Princess crown'd with pow'r, Our pearl, our pleasance, and our paramour, Our peace, our play, our plain felicity: Christ thee conserve from all adversity. XXVII. Then all the concert sang with such a shout, That I anon awaken'd where I lay, And with a braid I turned me about To see this court, but all were gone away;
Then up I lean'd me, halflings in affray, Call'd to my Muse, and for my subject chose To sing the royal THISTLE and the ROSE. FOOTNOTES: [1] Alluding to the Houses of york and lancaster, which were distinguished by the white and red rose, and united in the person of Queen margaret. VERSES ON THE DEATH OF QUEEN
CAROLINE. BY MR. SHIPLEY. Oblivion wraps not in her silent shade All human labours. Virtue blooms a flower, That Time's rough hand shall never violate. Still CAROLINE shall live in faithful verse, Sweet nurse of Memory, and in the voice Of grateful Britain. These shall testify How well her calm impartial rule supplied A monarch's absence; these commemorate Her soul contemplative of peaceful Truth And nature, mindful midst the pomp of Courts Of wise retirement, and the silent grove. She stretch'd thro' length'ning shades thy spacious walks, Delightful Richmond, and the terrass rais'd Of regal grandeur, whence the eye discerns Fair Thames with copious waters winding slow Midst pastures, spreading herds, and villages Of aspect neat, and villas wrapt in shades: Fair scene of chearful peace! the lovely sight Frequent she view'd, and bless'd the honour'd reign Of her great Consort, provident and mild. Now wander'd musing thro' the darkening depth Of thickest woods, friendly to solemn thought: Now o'er broad lawns fair opening to the sun. Nor midst her rural plans disdain'd to mix The useful arable, and waving corn
With soft turf border'd, and the lowly cot, That half appears, in branching elms obscur'd. Here beauty dwells, assembled from the scenes Of various nature; such as oft inflam'd With rapture Grecian bards, in that fair vale, Thessalian Tempe, or thy favorite soil, Arcadia, erst by awe-struck fancy fill'd With wand'ring forms, the woodland Deities, Light Nymphs and wanton Satyrs, faintly seen Quick glancing thro' the shade at close of eve, Great Pan, and old Silenus. Hither led By solitary grief shall GEORGE recall Th' endearing manners, the soft speech, that flow'd From his lov'd Consort, virtue mix'd with love, Prudence, and mild insinuating sense: But chief her thoughtful breast of counsels deep Capacious, nor unequal to the weight Of Government. Such was the royal mind Of wise ELIZA, name of loveliest sound To British ears, and pattern fair to Kings: Or she who rules the Scepter of the North Illustrious, spreading o'er a barbarous world The light of arts and manners, and with arms Infests th' astonish'd Sultan, hardly now With scatter'd troops resisting; she drives on The heavy war, and shakes th' Imperial Throne Of old Byzantium. Latest time shall sound The praise of female genius. Oft shall GEORGE Pay the kind tear, and grief of tender words To CAROLINE, thus oft lamenting sad. Hail sacred shade! by me with endless woe Still honour'd! ever in my Breast shall dwell Thy image, ever present to my soul Thy faithful love, in length of years mature: O skill'd t'enliven time, to soften care
With looks and smiles and friendship's chearful voice! Anxious, of Thee bereft, a solitude I feel, that not the fond condoling cares Of our sad offspring can remove. Ev'n now With lonely steps I trace the gloomy groves, Thy lov'd recesses, studious to recall The vanish'd bliss, and cheat my wand'ring thoughts With sweet illusion. Yet I not accuse Heav'n's dispensation. Prosperous and long Have been my days, and not unknown to fame, That dwells with virtue. But 'tis hard to part The league of ancient friendship, to resign The home-felt fondness, the secure delight, That reason nourish'd, and fair fame approv'd.
THE GENEALOGY OF CHRIST, AS IT IS REPRESENTED ON THE EAST WINDOW OF WINCHESTER COLL. CHAPEL. WRITTEN AT WINTON SCHOOL, BY DR. LOWTHE. At once to raise our rev'rence and delight, To elevate the mind, and please the sight, To pour in virtue at th' attentive eye, And waft the soul on wings of extacy; For this the painter's art with nature vies, And bids the visionary saint arise; Who views the sacred forms in thought aspires, Catches pure zeal, and as he gazes, fires; Feels the same ardour to his breast convey'd, Is what he sees, and emulates the shade. Thy strokes, great Artist, so sublime appear, They check our pleasure with an awful fear; While, thro' the mortal line, the God you trace, Author himself, and Heir of Jesse's race; In raptures we admire thy bold design, And, as the subject, own the hand divine. While thro' thy work the rising day shall stream, So long shall last thine honour, praise and name. And may thy labours to the Muse impart Some emanation from her sister art, To animate the verse, and bid it shine In colours easy, bright, and strong, as Thine. Supine on earth an awful figure lies,
While softest slumbers seem to seal his eyes; The hoary sire Heav'ns guardian care demands, And at his feet the watchful angel stands. The form august and large, the mien divine Betray the [2]founder of Messiah's line. Lo! from his loins the promis'd stem ascends, And high to Heaven its sacred Boughs extends: Each limb productive of some hero springs, And blooms luxuriant with a race of kings. Th' eternal plant wide spreads its arms around, And with the mighty branch the mystic top is crown'd. And lo! the glories of th' illustrious line At their first dawn with ripen'd splendors shine, In DAVID all express'd; the good, the great, The king, the hero, and the man compleat. Serene he sits, and sweeps the golden lyre, And blends the prophet's with the poet's fire. See! with what art he strikes the vocal strings, The God, his theme, inspiring what he sings! Hark—or our ears delude us—from his tongue Sweet flows, or seems to flow, some heav'nly song. Oh! could thine art arrest the flitting sound, And paint the voice in magic numbers bound; Could the warm sun, as erst when Memnon play'd Wake with his rising beam the vocal shade: Then might he draw th' attentive angels down, Bending to hear the lay, so sweet, so like their own. On either side the monarch's offspring shine, And some adorn, and some disgrace their line. Here Ammon glories; proud, incestuous lord! This hand sustains the robe, and that the sword. Frowning and fierce, with haughty strides he tow'rs, And on his horrid brow defiance low'rs.
There Absalom the ravish'd sceptre sways, And his stol'n honour all his shame displays: The base usurper Youth! who joins in one The rebel subject, and th' ungrateful son. Amid the royal race, see Nathan stand: Fervent he seems to speak, and lift his hand; His looks th' emotion of his soul disclose, And eloquence from every gesture flows. Such, and so stern he came, ordain'd to bring Th' ungrateful mandate to the guilty King: When, at his dreadful voice, a sudden smart Shot thro' the trembling monarch's conscious heart; From his own lips condemn'd; severe decree! Had his God prov'd so stern a Judge as He. But man with frailty is allay'd by birth; Consummate purity ne'er dwelt on earth: Thro' all the soul tho' virtue holds the rein, Beats at the heart, and springs in ev'ry vein: Yet ever from the clearest source have ran Some gross allay, some tincture of the man. But who is he——deep-musing——in his mind, He seems to weigh, in reason's scales, mankind; Fix'd contemplation holds his steady eyes—— I know the sage[3]; the wisest of the wise. Blest with all man could wish, or prince obtain, Yet his great heart pronounc'd those blessings vain. And lo! bright glitt'ring in his sacred hands, In miniature the glorious temple stands. Effulgent frame! stupendous to behold! Gold the strong valves, the roof of burnish'd gold. The wand'ring ark, in that bright dome enshrin'd, Spreads the strong light, eternal, unconfin'd! Above th' unutterable glory plays Presence divine! and the full-streaming rays Pour thro' reluctant clouds intolerable blaze.
But stern oppression rends Reboam's reign; See the gay prince, injurious, proud and vain! Th' imperial sceptre totters in his hand, And proud rebellion triumphs in the land. Curs'd with corruption's ever-fruitful spring, A beardless Senate, and a haughty King. There Asa, good and great, the sceptre bears, Justice attends his peace, success his wars: While virtue was his sword, and Heaven his shield, Without controul the warrior swept the field; Loaded with spoils, triumphant he return'd, And half her swarthy Sons sad Ethiopia mourn'd. But since thy flagging piety decay'd, And barter'd God's defence for human aid; See their fair laurels wither on thy brow, Nor herbs, nor healthful arts avail thee now, Nor is heav'n chang'd, apostate prince, but Thou. No mean atonement does this lapse require; But see the Son, you must forgive the Sire: He, [4]the just prince—with ev'ry virtue bless'd, He reign'd, and goodness all the man possess'd, Around his throne, fair happiness and peace Smooth'd ev'ry brow, and smil'd in ev'ry face. As when along the burning waste he stray'd, Where no pure streams in bubbling mazes play'd, Where drought incumbent on the thirsty ground, Long since had breath'd her scorching blasts around; The [5]Prophet calls, th' obedient floods repair To the parch'd fields, for Josaphat was there. The new-sprung waves, in many a gurgling vein, Trickle luxurious through the sucking plain; Fresh honours the reviving fields adorn, And o'er the desart plenty pours her horn. So, from the throne his influence he sheds,
And bids the virtues raise their languid heads: Where'er he goes, attending Truth prevails, Oppression flies, and Justice lifts her scales. See, on his arm, the royal eagle stand, Great type of conquest and supreme command; Th' exulting bird distinguish'd triumph brings, And greets the Monarch with expanded wings. Fierce Moab's sons prevent th' impending blow, Rush on themselves, and fall without the foe. The pious hero vanquish'd Heav'n by pray'r; His faith an army, and his vows a war. Thee too, Ozias, fates indulgent blest And thy days shone, in fairest actions drest; Till that rash hand, by some blind frenzy sway'd, Unclean, the sacred office durst invade. Quick o'er thy limbs the scurfy venom ran, And hoary filth besprinkled all the man. Transmissive worth adorns the pious [6]Son, The father's virtues with the father's throne. Lo! there he stands: he who the rage subdu'd Of Ammon's sons, and drench'd his sword in blood, And dost thou, Ahaz, Judah's scourge, disgrace, With thy base front, the glories of thy race? See the vile King his iron sceptre bear—— His only praise attends the pious [7]Heir; He, in whose soul the virtues all conspire, The best good son, from the worst wicked sire. And lo! in Hezekiah's golden reign, Long-exil'd piety returns again; Again, in genuine purity she shines, And with her presence gilds the long-neglected shrines. Ill-starr'd does proud Assyria's impious [8]Lord Bid Heav'n to arms, and vaunt his dreadful sword; His own vain threats th' insulting King o'erthrow,
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  • 6. Signals and Systems Analysis Using Transform Methods and MATLAB® Michael J. Roberts Professor Emeritus, Department of Electrical and Computer Engineering University of Tennessee Third Edition
  • 7. SIGNALS AND SYSTEMS: ANALYSIS USING TRANSFORM METHODS AND MATLAB® , THIRD EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2018 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2012, and 2004. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 QVS 22 21 20 19 18 17 ISBN 978-0-07-802812-0 MHID 0-07-802812-4 Chief Product Officer, SVP Products & Markets: G. Scott Virkler Vice President, General Manager, Products & Markets: Marty Lange Vice President, Content Design & Delivery: Betsy Whalen Managing Director: Thomas Timp Brand Manager: Raghothaman Srinivasan/Thomas Scaife, Ph.D. Director, Product Development: Rose Koos Product Developer: Christine Bower Marketing Manager: Shannon O’Donnell Director of Digital Content: Chelsea Haupt, Ph.D. Director, Content Design & Delivery: Linda Avenarius Program Manager: Lora Neyens Content Project Managers: Jeni McAtee; Emily Windelborn; Sandy Schnee Buyer: Jennifer Pickel Content Licensing Specialists: Carrie Burger, photo; Lorraine Buczek, text Cover Image: © Lauree Feldman/Getty Images Compositor: MPS Limited Printer: Quad Versailles All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Roberts, Michael J., Dr. Signals and systems : analysis using transform methods and MATLAB / Michael J. Roberts, professor, Department of Electrical and Computer Engineering, University of Tennessee. Third edition. | New York, NY : McGraw-Hill Education, [2018] | Includes bibliographical references (p. 786–787) and index. LCCN 2016043890 | ISBN 9780078028120 (alk. paper) LCSH: Signal processing. | System analysis. | MATLAB. LCC TK5102.9 .R63 2018 | DDC 621.382/2—dc23 LC record available at https://lccn.loc.gov/2016043890 The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. mheducation.com/highered
  • 8. To my wife Barbara for giving me the time and space to complete this effort and to the memory of my parents, Bertie Ellen Pinkerton and Jesse Watts Roberts, for their early emphasis on the importance of education.
  • 9. Preface, xii Chapter 1 Introduction, 1 1.1 Signals and Systems Defined, 1 1.2 Types of Signals, 3 1.3 Examples of Systems, 8 A Mechanical System, 9 A Fluid System, 9 A Discrete-Time System, 11 Feedback Systems, 12 1.4 A Familiar Signal and System Example, 14 1.5 Use of MATLAB® , 18 Chapter 2 Mathematical Description of Continuous-Time Signals, 19 2.1 Introduction and Goals, 19 2.2 Functional Notation, 20 2.3 Continuous-Time Signal Functions, 20 Complex Exponentials and Sinusoids, 21 Functions with Discontinuities, 23 The Signum Function, 24 The Unit-Step Function, 24 The Unit-Ramp Function, 26 The Unit Impulse, 27 The Impulse, the Unit Step, and Generalized Derivatives, 29 The Equivalence Property of the Impulse, 30 The Sampling Property of the Impulse, 31 The Scaling Property of the Impulse, 31 The Unit Periodic Impulse or Impulse Train, 32 A Coordinated Notation for Singularity Functions, 33 The Unit-Rectangle Function, 33 2.4 Combinations of Functions, 34 2.5 Shifting and Scaling, 36 Amplitude Scaling, 36 Time Shifting, 37 Time Scaling, 39 Simultaneous Shifting and Scaling, 43 2.6 Differentiation and Integration, 47 2.7 Even and Odd Signals, 49 Combinations of Even and Odd Signals, 51 Derivatives and Integrals of Even and Odd Signals, 53 2.8 Periodic Signals, 53 2.9 Signal Energy and Power, 56 Signal Energy, 56 Signal Power, 58 2.10 Summary of Important Points, 60 Exercises, 61 Exercises with Answers, 61 Signal Functions, 61 Shifting and Scaling, 62 Derivatives and Integrals of Functions, 66 Generalized Derivative, 67 Even and Odd Functions, 67 Periodic Signals, 69 Signal Energy and Power of Signals, 70 Exercises without Answers, 71 Signal Functions, 71 Scaling and Shifting, 71 Generalized Derivative, 76 Derivatives and Integrals of Functions, 76 Even and Odd Functions, 76 Periodic Functions, 77 Signal Energy and Power of Signals, 77 Chapter 3 Discrete-Time Signal Description, 79 3.1 Introduction and Goals, 79 3.2 Sampling and Discrete Time, 80 3.3 Sinusoids and Exponentials, 82 Sinusoids, 82 Exponentials, 85 3.4 Singularity Functions, 86 The Unit-Impulse Function, 86 The Unit-Sequence Function, 87 CONTENTS iv
  • 10. Contents v The Signum Function, 87 The Unit-Ramp Function, 88 The Unit Periodic Impulse Function or Impulse Train, 88 3.5 Shifting and Scaling, 89 Amplitude Scaling, 89 Time Shifting, 89 Time Scaling, 89 Time Compression, 90 Time Expansion, 90 3.6 Differencing and Accumulation, 94 3.7 Even and Odd Signals, 98 Combinations of Even and Odd Signals, 100 Symmetrical Finite Summation of Even and Odd Signals, 100 3.8 Periodic Signals, 101 3.9 Signal Energy and Power, 102 Signal Energy, 102 Signal Power, 103 3.10 Summary of Important Points, 105 Exercises, 105 Exercises with Answers, 105 Functions, 105 Scaling and Shifting Functions, 107 Differencing and Accumulation, 109 Even and Odd Functions, 110 Periodic Functions, 111 Signal Energy and Power, 112 Exercises without Answers, 113 Signal Functions, 113 Shifting and Scaling Functions, 113 Differencing and Accumulation, 114 Even and Odd Functions, 114 Periodic Signals, 115 Signal Energy and Power, 116 Chapter 4 Description of Systems, 118 4.1 Introduction and Goals, 118 4.2 Continuous-Time Systems, 119 System Modeling, 119 Differential Equations, 120 Block Diagrams, 124 System Properties, 127 Introductory Example, 127 Homogeneity, 131 Time Invariance, 132 Additivity, 133 Linearity and Superposition, 134 LTI Systems, 134 Stability, 138 Causality, 139 Memory, 139 Static Nonlinearity, 140 Invertibility, 142 Dynamics of Second-Order Systems, 143 Complex Sinusoid Excitation, 145 4.3 Discrete-Time Systems, 145 System Modeling, 145 Block Diagrams, 145 Difference Equations, 146 System Properties, 152 4.4 Summary of Important Points, 155 Exercises, 156 Exercises with Answers, 156 System Models, 156 Block Diagrams, 157 System Properties, 158 Exercises without Answers, 160 System Models, 160 System Properties, 162 Chapter 5 Time-Domain System Analysis, 164 5.1 Introduction and Goals, 164 5.2 Continuous Time, 164 Impulse Response, 164 Continuous-Time Convolution, 169 Derivation, 169 Graphical and Analytical Examples of Convolution, 173 Convolution Properties, 178 System Connections, 181 Step Response and Impulse Response, 181 Stability and Impulse Response, 181 Complex Exponential Excitation and the Transfer Function, 182 Frequency Response, 184 5.3 Discrete Time, 186 Impulse Response, 186 Discrete-Time Convolution, 189
  • 11. Contents vi Derivation, 189 Graphical and Analytical Examples of Convolution, 192 Convolution Properties, 196 Numerical Convolution, 196 Discrete-Time Numerical Convolution, 196 Continuous-Time Numerical Convolution, 198 Stability and Impulse Response, 200 System Connections, 200 Unit-Sequence Response and Impulse Response, 201 Complex Exponential Excitation and the Transfer Function, 203 Frequency Response, 204 5.4 Summary of Important Points, 207 Exercises, 207 Exercises with Answers, 207 Continuous Time, 207 Impulse Response, 207 Convolution, 209 Stability, 213 Frequency Response, 214 Discrete Time, 214 Impulse Response, 214 Convolution, 215 Stability, 219 Exercises without Answers, 221 Continuous Time, 221 Impulse Response, 221 Convolution, 222 Stability, 224 Discrete Time, 225 Impulse Response, 225 Convolution, 225 Stability, 228 Chapter 6 Continuous-Time Fourier Methods, 229 6.1 Introduction and Goals, 229 6.2 The Continuous-Time Fourier Series, 230 Conceptual Basis, 230 Orthogonality and the Harmonic Function, 234 The Compact Trigonometric Fourier Series, 237 Convergence, 239 Continuous Signals, 239 Discontinuous Signals, 240 Minimum Error of Fourier-Series Partial Sums, 242 The Fourier Series of Even and Odd Periodic Functions, 243 Fourier-Series Tables and Properties, 244 Numerical Computation of the Fourier Series, 248 6.3 The Continuous-Time Fourier Transform, 255 Extending the Fourier Series to Aperiodic Signals, 255 The Generalized Fourier Transform, 260 Fourier Transform Properties, 265 Numerical Computation of the Fourier Transform, 273 6.4 Summary of Important Points, 281 Exercises, 281 Exercises with Answers, 281 Fourier Series, 281 Orthogonality, 282 Forward and Inverse Fourier Transforms, 286 Relation of CTFS to CTFT, 293 Numerical CTFT, 294 System Response, 294 Exercises without Answers, 294 Fourier Series, 294 Forward and Inverse Fourier Transforms, 300 System Response, 305 Relation of CTFS to CTFT, 306 Chapter 7 Discrete-Time Fourier Methods, 307 7.1 Introduction and Goals, 307 7.2 The Discrete-Time Fourier Series and the Discrete Fourier Transform, 307 Linearity and Complex-Exponential Excitation, 307 Orthogonality and the Harmonic Function, 311 Discrete Fourier Transform Properties, 315 The Fast Fourier Transform, 321 7.3 The Discrete-Time Fourier Transform, 323 Extending the Discrete Fourier Transform to Aperiodic Signals, 323 Derivation and Definition, 324 The Generalized DTFT, 326 Convergence of the Discrete-Time Fourier Transform, 327 DTFT Properties, 327
  • 12. Contents vii Numerical Computation of the Discrete-Time Fourier Transform, 334 7.4 Fourier Method Comparisons, 340 7.5 Summary of Important Points, 341 Exercises, 342 Exercises with Answers, 342 Orthogonality, 342 Discrete Fourier Transform, 342 Discrete-Time Fourier Transform Definition, 344 Forward and Inverse Discrete-Time Fourier Transforms, 345 Exercises without Answers, 348 Discrete Fourier Transform, 348 Forward and Inverse Discrete-Time Fourier Transforms, 352 Chapter 8 The Laplace Transform, 354 8.1 Introduction and Goals, 354 8.2 Development of the Laplace Transform, 355 Generalizing the Fourier Transform, 355 Complex Exponential Excitation and Response, 357 8.3 The Transfer Function, 358 8.4 Cascade-Connected Systems, 358 8.5 Direct Form II Realization, 359 8.6 The Inverse Laplace Transform, 360 8.7 Existence of the Laplace Transform, 360 Time-Limited Signals, 361 Right- and Left-Sided Signals, 361 8.8 Laplace-Transform Pairs, 362 8.9 Partial-Fraction Expansion, 367 8.10 Laplace-Transform Properties, 377 8.11 The Unilateral Laplace Transform, 379 Definition, 379 Properties Unique to the Unilateral Laplace Transform, 381 Solution of Differential Equations with Initial Conditions, 383 8.12 Pole-Zero Diagrams and Frequency Response, 385 8.13 MATLAB System Objects, 393 8.14 Summary of Important Points, 395 Exercises, 395 Exercises with Answers, 395 Laplace-Transform Definition, 395 Direct Form II System Realization, 396 Forward and Inverse Laplace Transforms, 396 Unilateral Laplace-Transform Integral, 399 Solving Differential Equations, 399 Exercises without Answers, 400 Region of Convergence, 400 Existence of the Laplace Transform, 400 Direct Form II System Realization, 400 Forward and Inverse Laplace Transforms, 401 Solution of Differential Equations, 403 Pole-Zero Diagrams and Frequency Response, 403 Chapter 9 The z Transform, 406 9.1 Introduction and Goals, 406 9.2 Generalizing the Discrete-Time Fourier Transform, 407 9.3 Complex Exponential Excitation and Response, 408 9.4 The Transfer Function, 408 9.5 Cascade-Connected Systems, 408 9.6 Direct Form II System Realization, 409 9.7 The Inverse z Transform, 410 9.8 Existence of the z Transform, 410 Time-Limited Signals, 410 Right- and Left-Sided Signals, 411 9.9 z-Transform Pairs, 413 9.10 z-Transform Properties, 416 9.11 Inverse z-Transform Methods, 417 Synthetic Division, 417 Partial-Fraction Expansion, 418 Examples of Forward and Inverse z Transforms, 418 9.12 The Unilateral z Transform, 423 Properties Unique to the Unilateral z Transform, 423 Solution of Difference Equations, 424 9.13 Pole-Zero Diagrams and Frequency Response, 425 9.14 MATLAB System Objects, 428 In MATLAB, 429 9.15 Transform Method Comparisons, 430 9.16 Summary of Important Points, 434
  • 13. Contents viii Exercises, 435 Exercises with Answers, 435 Direct-Form II System Realization, 435 Existence of the z Transform, 435 Forward and Inverse z Transforms, 435 Unilateral z-Transform Properties, 438 Solution of Difference Equations, 438 Pole-Zero Diagrams and Frequency Response, 439 Exercises without Answers, 441 Direct Form II System Realization, 441 Existence of the z Transform, 441 Forward and Inverse z-Transforms, 441 Pole-Zero Diagrams and Frequency Response, 443 Chapter 10 Sampling and Signal Processing, 446 10.1 Introduction and Goals, 446 10.2 Continuous-Time Sampling, 447 Sampling Methods, 447 The Sampling Theorem, 449 Qualitative Concepts, 449 Sampling Theorem Derivation, 451 Aliasing, 454 Time-limited and Bandlimited Signals, 457 Interpolation, 458 Ideal Interpolation, 458 Practical Interpolation, 459 Zero-Order Hold, 460 First-Order Hold, 460 Sampling Bandpass Signals, 461 Sampling a Sinusoid, 464 Bandlimited Periodic Signals, 467 Signal Processing Using the DFT, 470 CTFT-DFT Relationship, 470 CTFT-DTFT Relationship, 471 Sampling and Periodic-Repetition Relationship, 474 Computing the CTFS Harmonic Function with the DFT, 478 Approximating the CTFT with the DFT, 478 Forward CTFT, 478 Inverse CTFT, 479 Approximating the DTFT with the DFT, 479 Approximating Continuous-Time Convolution with the DFT, 479 Aperiodic Convolution, 479 Periodic Convolution, 479 Discrete-Time Convolution with the DFT, 479 Aperiodic Convolution, 479 Periodic Convolution, 479 Summary of Signal Processing Using the DFT, 480 10.3 Discrete-Time Sampling, 481 Periodic-Impulse Sampling, 481 Interpolation, 483 10.4 Summary of Important Points, 486 Exercises, 487 Exercises with Answers, 487 Pulse Amplitude Modulation, 487 Sampling, 487 Impulse Sampling, 489 Nyquist Rates, 491 Time-Limited and Bandlimited Signals, 492 Interpolation, 493 Aliasing, 495 Bandlimited Periodic Signals, 495 CTFT-CTFS-DFT Relationships, 495 Windows, 497 DFT, 497 Exercises without Answers, 500 Sampling, 500 Impulse Sampling, 502 Nyquist Rates, 504 Aliasing, 505 Practical Sampling, 505 Bandlimited Periodic Signals, 505 DFT, 506 Discrete-Time Sampling, 508 Chapter 11 Frequency Response Analysis, 509 11.1 Introduction and Goals, 509 11.2 Frequency Response, 509 11.3 Continuous-Time Filters, 510 Examples of Filters, 510 Ideal Filters, 515 Distortion, 515 Filter Classifications, 516 Ideal Filter Frequency Responses, 516 Impulse Responses and Causality, 517 The Power Spectrum, 520 Noise Removal, 520 Bode Diagrams, 521
  • 14. Contents ix The Decibel, 521 The One-Real-Pole System, 525 The One-Real-Zero System, 526 Integrators and Differentiators, 527 Frequency-Independent Gain, 527 Complex Pole and Zero Pairs, 530 Practical Filters, 532 Passive Filters, 532 The Lowpass Filter, 532 The Bandpass Filter, 535 Active Filters, 536 Operational Amplifiers, 537 The Integrator, 538 The Lowpass Filter, 538 11.4 Discrete-Time Filters, 546 Notation, 546 Ideal Filters, 547 Distortion, 547 Filter Classifications, 548 Frequency Responses, 548 Impulse Responses and Causality, 548 Filtering Images, 549 Practical Filters, 554 Comparison with Continuous-Time Filters, 554 Highpass, Bandpass, and Bandstop Filters, 556 The Moving Average Filter, 560 The Almost Ideal Lowpass Filter, 564 Advantages Compared to Continuous-Time Filters, 566 11.5 Summary of Important Points, 566 Exercises, 567 Exercises with Answers, 567 Continuous-Time Frequency Response, 567 Continuous-Time Ideal Filters, 567 Continuous-Time Causality, 567 Logarithmic Graphs, Bode Diagrams, and Decibels, 568 Continuous-Time Practical Passive Filters, 570 Continuous-Time Practical Active Filters, 574 Discrete-Time Frequency Response, 575 Discrete-Time Ideal Filters, 576 Discrete-Time Causality, 576 Discrete-Time Practical Filters, 577 Exercises without Answers, 579 Continuous-Time Frequency Response, 579 Continuous-Time Ideal Filters, 579 Continuous-Time Causality, 579 Bode Diagrams, 580 Continuous-Time Practical Passive Filters, 580 Continuous-Time Filters, 582 Continuous-Time Practical Active Filters, 582 Discrete-Time Causality, 586 Discrete-Time Filters, 587 Chapter 12 Laplace System Analysis, 592 12.1 Introduction and Goals, 592 12.2 System Representations, 592 12.3 System Stability, 596 12.4 System Connections, 599 Cascade and Parallel Connections, 599 The Feedback Connection, 599 Terminology and Basic Relationships, 599 Feedback Effects on Stability, 600 Beneficial Effects of Feedback, 601 Instability Caused by Feedback, 604 Stable Oscillation Using Feedback, 608 The Root-Locus Method, 612 Tracking Errors in Unity-Gain Feedback Systems, 618 12.5 System Analysis Using MATLAB, 621 12.6 System Responses to Standard Signals, 623 Unit-Step Response, 624 Sinusoid Response, 627 12.7 Standard Realizations of Systems, 630 Cascade Realization, 630 Parallel Realization, 632 12.8 Summary of Important Points, 632 Exercises, 633 Exercises with Answers, 633 Transfer Functions, 633 Stability, 634 Parallel, Cascade, and Feedback Connections, 635 Root Locus, 637 Tracking Errors in Unity-Gain Feedback Systems, 639 System Responses to Standard Signals, 640 System Realization, 641 Exercises without Answers, 642 Stability, 642 Transfer Functions, 642 Stability, 643
  • 15. Contents x Parallel, Cascade, and Feedback Connections, 643 Root Locus, 646 Tracking Errors in Unity-Gain Feedback Systems, 647 Response to Standard Signals, 647 System Realization, 649 Chapter 13 z-Transform System Analysis, 650 13.1 Introduction and Goals, 650 13.2 System Models, 650 Difference Equations, 650 Block Diagrams, 651 13.3 System Stability, 651 13.4 System Connections, 652 13.5 System Responses to Standard Signals, 654 Unit-Sequence Response, 654 Response to a Causal Sinusoid, 657 13.6 Simulating Continuous-Time Systems with Discrete-Time Systems, 660 z-Transform-Laplace-Transform Relationships, 660 Impulse Invariance, 662 Sampled-Data Systems, 664 13.7 Standard Realizations of Systems, 670 Cascade Realization, 670 Parallel Realization, 670 13.8 Summary of Important Points, 671 Exercises, 672 Exercises with Answers, 672 Stability, 672 Parallel, Cascade, and Feedback Connections, 672 Response to Standard Signals, 673 Root Locus, 674 Laplace-Transform-z-Transform Relationship, 675 Sampled-Data Systems, 675 System Realization, 676 Exercises without Answers, 677 Stability, 677 Root Locus, 677 Parallel, Cascade, and Feedback Connections, 677 Response to Standard Signals, 677 Laplace-Transform-z-Transform Relationship, 679 Sampled-Data Systems, 679 System Realization, 679 General, 679 Chapter 14 Filter Analysis and Design, 680 14.1 Introduction and Goals, 680 14.2 Analog Filters, 680 Butterworth Filters, 681 Normalized Butterworth Filters, 681 Filter Transformations, 682 MATLAB Design Tools, 684 Chebyshev, Elliptic, and Bessel Filters, 686 14.3 Digital Filters, 689 Simulation of Analog Filters, 689 Filter Design Techniques, 689 IIR Filter Design, 689 Time-Domain Methods, 689 Impulse-Invariant Design, 689 Step-Invariant Design, 696 Finite-Difference Design, 698 Frequency-Domain Methods, 704 The Bilinear Method, 706 FIR Filter Design, 713 Truncated Ideal Impulse Response, 713 Optimal FIR Filter Design, 723 MATLAB Design Tools, 725 14.4 Summary of Important Points, 727 Exercises, 727 Exercises with Answers, 727 Continuous-Time Filters, 727 Finite-Difference Filter Design, 728 Matched-z Transform and Direct Substitution Filter Design, 729 Bilinear z-Transform Filter Design, 730 FIR Filter Design, 730 Digital Filter Design Method Comparison, 731 Exercises without Answers, 731 Analog Filter Design, 731 Impulse-Invariant and Step-Invariant Filter Design, 732 Finite-Difference Filter Design, 733 Matched z-Transform and Direct Substitution Filter Design, 733 Bilinear z-Transform Filter Design, 733 FIR Filter Design, 733 Digital Filter Design Method Comparison, 734
  • 16. Contents xi Appendix I Useful Mathematical Relations, A-1     II Continuous-Time Fourier Series Pairs, A-4     III Discrete Fourier Transform Pairs, A-7     IV Continuous-Time Fourier Transform Pairs, A-10     V Discrete-Time Fourier Transform Pairs, A-17     VI Tables of Laplace Transform Pairs, A-22     VII z-Transform Pairs, A-24 Bibliography, B-1 Index, I-1
  • 17. PREFACE MOTIVATION I wrote the first and second editions because I love the mathematical beauty of signal and system analysis. That has not changed. The motivation for the third edi- tion is to further refine the book structure in light of reviewers, comments, correct a few errors from the second edition and significantly rework the exercises. AUDIENCE This book is intended to cover a two-semester course sequence in the basics of signal and system analysis during the junior or senior year. It can also be used (as I have used it) as a book for a quick one-semester Master’s-level review of trans- form methods as applied to linear systems. CHANGES FROM THE SECOND EDITION 1. In response to reviewers, comments, two chapters from the second edition have been omitted: Communication Systems and State-Space Analysis. There seemed to be very little if any coverage of these topics in actual classes. 2. The second edition had 550 end-of-chapter exercises in 16 chapters. The third edition has 710 end-of-chapter exercises in 14 chapters. OVERVIEW Except for the omission of two chapters, the third edition structure is very similar to the second edition. The book begins with mathematical methods for describing signals and systems, in both continuous and discrete time. I introduce the idea of a transform with the continuous-time Fourier series, and from that base move to the Fourier trans- form as an extension of the Fourier series to aperiodic signals. Then I do the same for discrete-time signals. I introduce the Laplace transform both as a generalization of the continuous-time Fourier transform for unbounded signals and unstable systems and as a powerful tool in system analysis because of its very close association with the ei- genvalues and eigenfunctions of continuous-time linear systems. I take a similar path for discrete-time systems using the z transform. Then I address sampling, the relation between continuous and discrete time. The rest of the book is devoted to applications in frequency-response analysis, feedback systems, analog and digital filters. Through- out the book I present examples and introduce MATLAB functions and operations to implement the methods presented. A chapter-by-chapter summary follows. CHAPTER SUMMARIES CHAPTER 1 Chapter 1 is an introduction to the general concepts involved in signal and system analysis without any mathematical rigor. It is intended to motivate the student by xii
  • 18. xiii Preface demonstrating the ubiquity of signals and systems in everyday life and the impor- tance of understanding them. CHAPTER 2 Chapter 2 is an exploration of methods of mathematically describing continuous- time signals of various kinds. It begins with familiar functions, sinusoids and exponentials and then extends the range of signal-describing functions to include continuous-time singularity functions (switching functions). Like most, if not all, signals and systems textbooks, I define the unit-step, the signum, the unit-impulse and the unit-ramp functions. In addition to these I define a unit rectangle and a unit periodic impulse function. The unit periodic impulse function, along with convolution, provides an especially compact way of mathematically describing arbitrary periodic signals. After introducing the new continuous-time signal functions, I cover the common types of signal transformations, amplitude scaling, time shifting, time scaling, differentiation and integration and apply them to the signal functions. Then I cover some characteristics of signals that make them invariant to certain transformations, evenness, oddness and periodicity, and some of the implications of these signal characteristics in signal analysis. The last section is on signal energy and power. CHAPTER 3 Chapter 3 follows a path similar to Chapter 2 except applied to discrete-time signals instead of continuous-time signals. I introduce the discrete-time sinu- soid and exponential and comment on the problems of determining period of a discrete-time sinusoid. This is the first exposure of the student to some of the implications of sampling. I define some discrete-time signal functions analo- gous to continuous-time singularity functions. Then I explore amplitude scaling, time shifting, time scaling, differencing and accumulation for discrete-time signal functions pointing out the unique implications and problems that occur, especially when time scaling discrete-time functions. The chapter ends with definitions and discussion of signal energy and power for discrete-time signals. CHAPTER 4 This chapter addresses the mathematical description of systems. First I cover the most common forms of classification of systems, homogeneity, additivity, linearity, time invariance, causality, memory, static nonlinearity and invertibility. By example I present various types of systems that have, or do not have, these properties and how to prove various properties from the mathematical description of the system. CHAPTER 5 This chapter introduces the concepts of impulse response and convolution as components in the systematic analysis of the response of linear, time-invariant systems. I present the mathematical properties of continuous-time convolution and a graphical method of understanding what the convolution integral says. I also show how the properties of convolution can be used to combine subsystems that are connected in cascade or parallel into one system and what the impulse response of the overall system must be. Then I introduce the idea of a transfer
  • 19. xiv Preface function by finding the response of an LTI system to complex sinusoidal exci- tation. This section is followed by an analogous coverage of discrete-time impulse response and convolution. CHAPTER 6 This is the beginning of the student’s exposure to transform methods. I begin by graphically introducing the concept that any continuous-time periodic signal with engineering usefulness can be expressed by a linear combination of continuous-time sinusoids, real or complex. Then I formally derive the Fourier series using the concept of orthogonality to show where the signal description as a function of discrete harmonic number (the harmonic function) comes from. I mention the Dirichlet conditions to let the student know that the continuous-time Fourier series applies to all practical continuous-time signals, but not to all imaginable continuous-time signals. Then I explore the properties of the Fourier series. I have tried to make the Fourier series notation and properties as similar as possible and analogous to the Fourier transform, which comes later. The harmonic function forms a “Fourier series pair” with the time function. In the first edition I used a notation for har- monic function in which lower-case letters were used for time-domain quantities and upper-case letters for their harmonic functions. This unfortunately caused some confusion because continuous- and discrete-time harmonic functions looked the same. In this edition I have changed the harmonic function notation for continuous-time signals to make it easily distinguishable. I also have a section on the convergence of the Fourier series illustrating the Gibb’s phenomenon at function discontinuities. I encourage students to use tables and properties to find harmonic functions and this practice prepares them for a similar process in find- ing Fourier transforms and later Laplace and z transforms. The next major section of Chapter 6 extends the Fourier series to the Fourier transform. I introduce the concept by examining what happens to a continuous-time Fourier series as the period of the signal approaches infinity and then define and derive the continuous-time Fourier transform as a gener- alization of the continuous-time Fourier series. Following that I cover all the important properties of the continuous-time Fourier transform. I have taken an “ecumenical” approach to two different notational conventions that are commonly seen in books on signals and systems, control systems, digital signal processing, communication systems and other applications of Fourier methods such as image processing and Fourier optics: the use of either cyclic frequency, f or radian fre- quency, ω. I use both and emphasize that the two are simply related through a change of variable. I think this better prepares students for seeing both forms in other books in their college and professional careers. CHAPTER 7 This chapter introduces the discrete-time Fourier series (DTFS), the discrete Fou- rier transform (DFT) and the discrete-time Fourier transform (DTFT), deriving and defining them in a manner analogous to Chapter 6. The DTFS and the DFT are almost identical. I concentrate on the DFT because of its very wide use in digital signal processing. I emphasize the important differences caused by the differences between continuous- and discrete-time signals, especially the finite summation range of the DFT as opposed to the (generally) infinite summation range in the CTFS. I also point out the importance of the fact that the DFT relates
  • 20. xv Preface a finite set of numbers to another finite set of numbers, making it amenable to direct numerical machine computation. I discuss the fast Fourier transform as a very efficient algorithm for computing the DFT. As in Chapter 6, I use both cyclic and radian frequency forms, emphasizing the relationships between them. I use F and Ω for discrete-time frequencies to distinguish them from f and ω, which were used in continuous time. Unfortunately, some authors reverse these symbols. My usage is more consistent with the majority of signals and systems texts. This is another example of the lack of standardization of notation in this area. The last major section is a comparison of the four Fourier methods. I emphasize particu- larly the duality between sampling in one domain and periodic repetition in the other domain. CHAPTER 8 This chapter introduces the Laplace transform. I approach the Laplace trans- form from two points of view, as a generalization of the Fourier transform to a larger class of signals and as result which naturally follows from the excitation of a linear, time-invariant system by a complex exponential signal. I begin by defining the bilateral Laplace transform and discussing significance of the re- gion of convergence. Then I define the unilateral Laplace transform. I derive all the important properties of the Laplace transform. I fully explore the method of partial-fraction expansion for finding inverse transforms and then show examples of solving differential equations with initial conditions using the uni- lateral form. CHAPTER 9 This chapter introduces the z transform. The development parallels the devel- opment of the Laplace transform except applied to discrete-time signals and systems. I initially define a bilateral transform and discuss the region of con- vergence. Then I define a unilateral transform. I derive all the important prop- erties and demonstrate the inverse transform using partial-fraction expansion and the solution of difference equations with initial conditions. I also show the relationship between the Laplace and z transforms, an important idea in the approximation of continuous-time systems by discrete-time systems in Chapter 14. CHAPTER 10 This is the first exploration of the correspondence between a continuous-time signal and a discrete-time signal formed by sampling it. The first section covers how sampling is usually done in real systems using a sample-and-hold and an A/D converter. The second section starts by asking the question of how many samples are enough to describe a continuous-time signal. Then the question is answered by deriving the sampling theorem. Then I discuss interpolation methods, theoret- ical and practical, the special properties of bandlimited periodic signals. I do a complete development of the relationship between the CTFT of a continuous-time signal and DFT of a finite-length set of samples taken from it. Then I show how the DFT can be used to approximate the CTFT of an energy signal or a periodic signal. The next major section explores the use of the DFT in numerically approx- imating various common signal-processing operations.
  • 21. xvi CHAPTER 11 This chapter covers various aspects of the use of the CTFT and DTFT in fre- quency response analysis. The major topics are ideal filters, Bode diagrams, prac- tical passive and active continuous-time filters and basic discrete-time filters. CHAPTER 12 This chapter is on the application of the Laplace transform including block dia- gram representation of systems in the complex frequency domain, system stability, system interconnections, feedback systems including root locus, system responses to standard signals and lastly standard realizations of continuous-time systems. CHAPTER 13 This chapter is on the application of the z transform including block diagram representation of systems in the complex frequency domain, system stability, sys- tem interconnections, feedback systems including root-locus, system responses to standard signals, sampled-data systems and standard realizations of discrete-time systems. CHAPTER 14 This chapter covers the analysis and design of some of the most common types of practical analog and digital filters. The analog filter types are Butterworth, Chebyshev Types 1 and 2 and Elliptic (Cauer) filters. The section on digital filters covers the most common types of techniques for simulation of analog filters includ- ing, impulse- and step-invariant, finite difference, matched z transform, direct sub- stitution, bilinear z transform, truncated impulse response and Parks-McClellan numerical design. APPENDICES There are seven appendices on useful mathematical formulae, tables of the four Fourier transforms, Laplace transform tables and z transform tables. CONTINUITY The book is structured so as to facilitate skipping some topics without loss of continuity. Continuous-time and discrete-time topics are covered alternately and continuous-time analysis could be covered without reference to discrete time. Also, any or all of the last six chapters could be omitted in a shorter course. REVIEWS AND EDITING This book owes a lot to the reviewers, especially those who really took time and criticized and suggested improvements. I am indebted to them. I am also indebted to the many students who have endured my classes over the years. I believe that our relationship is more symbiotic than they realize. That is, they learn signal and system analysis from me and I learn how to teach signal and system analysis from them. I cannot count the number of times I have been asked a very perceptive question by a student that revealed not only that the students were not understand- ing a concept but that I did not understand it as well as I had previously thought. Preface
  • 22. xvii WRITING STYLE Every author thinks he has found a better way to present material so that students can grasp it and I am no different. I have taught this material for many years and through the experience of grading tests have found what students generally do and do not grasp. I have spent countless hours in my office one-on-one with students explaining these concepts to them and, through that experience, I have found out what needs to be said. In my writing I have tried to simply speak directly to the reader in a straightforward conversational way, trying to avoid off-putting formality and, to the extent possible, anticipating the usual misconceptions and revealing the fallacies in them. Transform methods are not an obvious idea and, at first exposure, students can easily get bogged down in a bewildering morass of abstractions and lose sight of the goal, which is to analyze a system’s response to signals. I have tried (as every author does) to find the magic combination of ac- cessibility and mathematical rigor because both are important. I think my writing is clear and direct but you, the reader, will be the final judge of whether or not that is true. EXERCISES Each chapter has a group of exercises along with answers and a second group of exercises without answers. The first group is intended more or less as a set of “drill” exercises and the second group as a set of more challenging exercises. CONCLUDING REMARKS As I indicated in the preface to first and second editions, I welcome any and all criticism, corrections and suggestions. All comments, including ones I disagree with and ones which disagree with others, will have a constructive impact on the next edition because they point out a problem. If something does not seem right to you, it probably will bother others also and it is my task, as an author, to find a way to solve that problem. So I encourage you to be direct and clear in any re- marks about what you believe should be changed and not to hesitate to mention any errors you may find, from the most trivial to the most significant. Michael J. Roberts, Professor Emeritus Electrical and Computer Engineering University of Tennessee at Knoxville mjr@utk.edu Preface
  • 23. Required=Results McGraw-Hill Connect® Learn Without Limits Connect is a teaching and learning platform that is proven to deliver better results for students and instructors. Connect empowers students by continually adapting to deliver precisely what they need, when they need it and how they need it, so your class time is more engaging and effective. Connect Insight® Connect Insight is Connect’s new one- of-a-kind visual analytics dashboard that provides at-a-glance information regarding student performance, which is immediately actionable. By presenting assignment, assessment and topical performance results together with a time metric that is easily visible for aggregate or individual results, Connect Insight gives the user the ability to take a just-in-time approach to teaching and learning, which was never before available. Connect Insight presents data that helps instructors improve class performance in a way that is efficient and effective. 73% of instructors who use Connect require it; instructor satisfaction increases by 28% when Connect is required. Analytics ©Getty Images/iStockphoto Using Connect improves passing rates by 12.7% and retention by 19.8%.
  • 24. SmartBook® Proven to help students improve grades and study more efficiently, SmartBook contains the same content within the print book, but actively tailors that content to the needs of the individual. SmartBook’s adaptive technology provides precise, personalized instruction on what the student should do next, guiding the student to master and remember key concepts, targeting gaps in knowledge and offering customized feedback and driving the student toward comprehension and retention of the subject matter. Available on smartphones and tablets, SmartBook puts learning at the student’s fingertips—anywhere, anytime. Adaptive Over 5.7 billion questions have been answered, making McGraw-Hill Education products more intelligent, reliable and precise. THE ADAPTIVE READING EXPERIENCE DESIGNED TO TRANSFORM THE WAY STUDENTS READ More students earn A’s and B’s when they use McGraw-Hill Education Adaptive products. www.mheducation.com ©Getty Images/iStockphoto
  • 26. 1 C H A P T E R 1 Introduction 1.1 SIGNALS AND SYSTEMS DEFINED Any time-varying physical phenomenon that is intended to convey information is a signal. Examples of signals are the human voice, sign language, Morse code, traffic signals, voltages on telephone wires, electric fields emanating from radio or television transmitters, and variations of light intensity in an optical fiber on a telephone or com- puter network. Noise is like a signal in that it is a time-varying physical phenomenon, but usually it does not carry useful information and is considered undesirable. Signals are operated on by systems. When one or more excitations or input signals are applied at one or more system inputs, the system produces one or more responses or output signals at its outputs. Figure 1.1 is a block diagram of a single-input, single-output system. System Input Output Excitation or Input Signal Response or Output Signal Figure 1.1 Block diagram of a single-input, single-output system Transmitter Channel Receiver Information Signal Noisy Information Signal Noise Noise Noise Figure 1.2 A communication system In a communication system, a transmitter produces a signal and a receiver acquires it. A channel is the path a signal takes from a transmitter to a receiver. Noise is inevitably introduced into the transmitter, channel and receiver, often at multiple points (Figure 1.2). The transmitter, channel and receiver are all components or subsystems of the overall system. Scientific instruments are systems that measure a physical phenom- enon (temperature, pressure, speed, etc.) and convert it to a voltage or current, a sig- nal. Commercial building control systems (Figure 1.3), industrial plant control systems (Figure 1.4), modern farm machinery (Figure 1.5), avionics in airplanes, ignition and fuel pumping controls in automobiles, and so on are all systems that operate on signals.
  • 27. Ch ap ter 1  Introduction 2 Figure 1.3 Modern office buildings © Vol. 43 PhotoDisc/Getty Figure 1.4 Typical industrial plant control room © Royalty-Free/Punchstock
  • 28. 1.2 Types of Signals 3 The term system even encompasses things such as the stock market, government, weather, the human body and the like. They all respond when excited. Some systems are readily analyzed in detail, some can be analyzed approximately, but some are so complicated or difficult to measure that we hardly know enough to understand them. 1.2 TYPES OF SIGNALS There are several broad classifications of signals: continuous-time, discrete-time, continuous-value, discrete-value, random and nonrandom. A continuous-time sig- nal is defined at every instant of time over some time interval. Another common name for some continuous-time signals is analog signal, in which the variation of the signal with time is analogous (proportional) to some physical phenomenon. All analog sig- nals are continuous-time signals but not all continuous-time signals are analog signals (Figure 1.6 through Figure 1.8). Sampling a signal is acquiring values from a continuous-time signal at discrete points in time. The set of samples forms a discrete-time signal. A discrete-time signal Figure 1.5 Modern farm tractor with enclosed cab © Royalty-Free/Corbis Figure 1.6 Examples of continuous-time and discrete-time signals n x[n] Discrete-Time Continuous-Value Signal t x(t) Continuous-Time Continuous-Value Signal
  • 29. Ch ap ter 1  Introduction 4 can also be created by an inherently discrete-time system that produces signal values only at discrete times (Figure 1.6). A continuous-value signal is one that may have any value within a continuum of allowed values. In a continuum any two values can be arbitrarily close together. The real numbers form a continuum with infinite extent. The real numbers between zero and one form a continuum with finite extent. Each is a set with infinitely many mem- bers (Figure 1.6 through Figure 1.8). A discrete-value signal can only have values taken from a discrete set. In a discrete set of values the magnitude of the difference between any two values is greater than some positive number. The set of integers is an example. Discrete-time signals are usually transmitted as digital signals, a sequence of values of a discrete-time signal in the form of digits in some encoded form. The term digital is also sometimes used loosely to refer to a discrete-value signal that has only two possible values. The digits in this type of digital signal are transmitted by signals that are continuous-time. In this case, the terms continuous-time and analog are not synonymous. A digital signal of this type is a continuous-time signal but not an analog signal because its variation of value with time is not directly analogous to a physical phenomenon (Figure 1.6 through Figure 1.8). A random signal cannot be predicted exactly and cannot be described by any math- ematical function. A deterministic signal can be mathematically described. A com- mon name for a random signal is noise (Figure 1.6 through Figure 1.8). In practical signal processing it is very common to acquire a signal for processing by a computer by sampling, quantizing and encoding it (Figure 1.9). The original signal is a continuous-value, continuous-time signal. Sampling acquires its values at discrete times and those values constitute a continuous-value, discrete-time signal. Quantization approximates each sample as the nearest member of a finite set of dis- crete values, producing a discrete-value, discrete-time signal. Each signal value in the set of discrete values at discrete times is converted to a sequence of rectangular pulses that encode it into a binary number, creating a discrete-value, continuous-time signal, commonly called a digital signal. The steps illustrated in Figure 1.9 are usually carried out by a single device called an analog-to-digital converter (ADC). Figure 1.8 Examples of noise and a noisy digital signal Noisy Digital Signal Continuous-Time Continuous-Value Random Signal t x(t) x(t) Noise t Figure 1.7 Examples of continuous-time, discrete-value signals Continuous-Time Discrete-Value Signal Continuous-Time Discrete-Value Signal t x(t) x(t) t Digital Signal
  • 30. 1.2 Types of Signals 5 Figure 1.9 Sampling, quantization and encoding of a signal to illustrate various signal types t kΔt (k+1)Δt (k+2)Δt (k–1)Δt kΔt (k+1)Δt (k+2)Δt (k–1)Δt xs[n] n k k+1 k+2 k–1 xsq[n] n k k+1 k+2 k–1 xsqe(t) t 111 001 111 011 Continuous-Value Continuous-Time Signal Continuous-Value Discrete-Time Signal Discrete-Value Discrete-Time Signal Discrete-Value Continuous-Time Signal Sampling Quantization Encoding x(t) Figure 1.10 Asynchronous serial binary ASCII-encoded voltage signal for the word SIGNAL 0 1 2 3 4 5 6 7 –1 0 1 2 3 4 5 6 Time, t (ms) Voltage, v(t) (V) Serial Binary Voltage Signal for the ASCII Message “SIGNAL” S I G N A L One common use of binary digital signals is to send text messages using the American Standard Code for Information Interchange (ASCII). The letters of the al- phabet, the digits 0–9, some punctuation characters and several nonprinting control characters, for a total of 128 characters, are all encoded into a sequence of 7 binary bits. The 7 bits are sent sequentially, preceded by a start bit and followed by 1 or 2 stop bits for synchronization purposes. Typically, in direct-wired connections between digital equipment, the bits are represented by a higher voltage (2 to 5V) for a 1 and a lower voltage level (around 0V) for a 0. In an asynchronous transmission using one start and one stop bit, sending the message SIGNAL, the voltage versus time would look as illustrated in Figure 1.10.
  • 31. Ch ap ter 1  Introduction 6 In 1987 ASCII was extended to Unicode. In Unicode the number of bits used to represent a character can be 8, 16, 24 or 32 and more than 120,000 characters are cur- rently encoded in modern and historic language characters and multiple symbol sets. Digital signals are important in signal analysis because of the spread of digital systems. Digital signals often have better immunity to noise than analog signals. In binary signal communication the bits can be detected very cleanly until the noise gets very large. The detection of bit values in a stream of bits is usually done by comparing the signal value at a predetermined bit time with a threshold. If it is above the thresh- old it is declared a 1 and if it is below the threshold it is declared a 0. In Figure 1.11, the x’s mark the signal value at the detection time, and when this technique is applied to the noisy digital signal, one of the bits is incorrectly detected. But when the signal is processed by a filter, all the bits are correctly detected. The filtered digital signal does not look very clean in comparison with the noiseless digital signal, but the bits can still be detected with a very low probability of error. This is the basic reason that digital signals can have better noise immunity than analog signals. An introduction to the analysis and design of filters is presented in Chapters 11 and 15. In this text we will consider both continuous-time and discrete-time signals, but we will (mostly) ignore the effects of signal quantization and consider all signals to be continuous-value. Also, we will not directly consider the analysis of random signals, although random signals will sometimes be used in illustrations. The first signals we will study are continuous-time signals. Some continuous-time signals can be described by continuous functions of time. A signal x(t) might be described by a function x(t) = 50sin(200πt) of continuous time t. This is an exact description of the signal at every instant of time. The signal can also be described graphically (Figure 1.12). Many continuous-time signals are not as easy to describe mathematically. Consider the signal in Figure 1.13. Waveforms like the one in Figure 1.13 occur in various types of instrumentation and communication systems. With the definition of some signal functions and an operation called convolution, this signal can be compactly described, analyzed and manipulated mathematically. Continuous-time signals that can be described by math- ematical functions can be transformed into another domain called the frequency domain through the continuous-time Fourier transform. In this context, transformation means transformation of a signal to the frequency domain. This is an important tool in signal analysis, which allows certain characteristics of the signal to be more clearly observed Figure 1.11 Use of a filter to reduce bit error rate in a digital signal x(t) –1 2 Noiseless Digital Signal t 2.6 –1 2 t 2.6 –1 2 t 2.6 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 1 Bit Error Bit Detection Threshold xf(t) x(t) + n(t) Noisy Digital Signal Filtered Digital Signal
  • 32. 1.2 Types of Signals 7 and more easily manipulated than in the time domain. (In the frequency domain, signals are described in terms of the frequencies they contain.) Without frequency-domain analy- sis, design and analysis of many systems would be considerably more difficult. Discrete-time signals are only defined at discrete points in time. Figure 1.14 illustrates some discrete-time signals. Figure 1.12 A continuous-time signal described by a mathematical function ... ... –50 50 t = 10 ms t x(t) Figure 1.13 A second continuous-time signal ... ... 5 t = 20 μs t x(t) Figure 1.14 Some discrete-time signals n x[n] n x[n] n x[n] n x[n] So far all the signals we have considered have been described by functions of time. An important class of “signals” is functions of space instead of time: images. Most of the theories of signals, the information they convey and how they are processed by systems in this text will be based on signals that are a variation of a physical phenome- non with time. But the theories and methods so developed also apply, with only minor modifications, to the processing of images. Time signals are described by the variation of a physical phenomenon as a function of a single independent variable, time. Spa- tial signals, or images, are described by the variation of a physical phenomenon as a
  • 33. Another Random Scribd Document with Unrelated Content
  • 37. The Project Gutenberg eBook of The Union: Or, Select Scots and English Poems
  • 38. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: The Union: Or, Select Scots and English Poems Compiler: Thomas Warton Release date: August 8, 2012 [eBook #40444] Most recently updated: October 23, 2024 Language: English Credits: Produced by Margo von Romberg, Bryan Ness and the Online Distributed Proofreading Team at http://www.pgdp.net (This book was produced from scanned images of public domain material from the Google Print project.) *** START OF THE PROJECT GUTENBERG EBOOK THE UNION: OR, SELECT SCOTS AND ENGLISH POEMS ***
  • 39. THE U N I O N : OR, SELECT SCOTS and ENGLISH P O E M S . THE SECOND EDITION. ----Dubiam facientia carmina palmam. Juv. LONDON: Printed for R. Baldwin, in Paternoster-Row. M.DCC.LIX.
  • 41. PREFACE. As the mind of man is ever fond of variety, nothing seems better calculated to entertain, than a judicious collection of the smaller, though not on that account less laboured, productions of eminent poets: an entertainment, not unlike that which we receive from surveying a finished landschape, or well disposed piece of shell-work: where each particular object, tho' singly beautiful, and sufficiently striking by itself, receives an additional charm, thus, as Milton expresses it, sweetly interchanged. The first miscellaneous collection of poems, that ever appeared in Great-Britain with any reputation, is that published by Dryden: which was afterwards continued by Tonson. There are many pieces of the highest merit in this collection, by Dryden, Denham, Creech, Drayton, Garth, Marvell, and many others; yet the compilers, it is evident, were not always sufficiently scrupulous and cautious in their choice, as several pieces are admitted, among the rest, which would otherwise utterly have perished, and which had no other recommendation, than that they served to swell the volume. Since this, many miscellanies have been published both in Scotland and England: to enumerate which would be no less tedious than useless. It will be sufficient to remark, that through want of care or judgment in their respective editors, they are all forgotten or neglected. From these the miscellany known by the name of Mr. Pope perhaps ought to be excepted; tho' that, indeed, cannot properly be styled a collection of poems by different hands, which is such a one as we are speaking of at present, the greater part consisting of pieces by Mr. Pope only. The best miscellany at this day extant in our language, and the first complete one of the kind which we have seen, is that lately published by R. Dodsley, which boasts the greatest names of the present age among its contributors.
  • 42. As to the poetical collection here exhibited to the public, we apprehend it challenges no small degree of regard, as it was made under the immediate inspection and conduct of several very ingenious gentlemen, whose names it would do us the highest honour to mention; and as it contains a variety not to be found even in the admirable collection last spoken of; I mean the Intermixture of poems both Scotch and English. Nor is this variety less agreeable than useful; as from it we have an opportunity of forming a comparison and estimate of the taste and genius of the two different nations, in their poetical compositions. It will be necessary to take notice, that our chief care has been to furnish out the following miscellany with those pieces, regard being first had to real merit, which have laid unknown and unobserved from their manner of publication; several of them having been printed by themselves, and so perished as it were for want of bulk, and others lost amid the rubbish of collections injudiciously made, and perhaps not easily to be met with. Nor will it be improper to mention, that in order to render our volume still more compleat, we have had the favour of some original poems, written by a late member of the university of Aberdeen, whose modesty would not permit us to prefix his name: one of which in this edition is printed with many improvements, from a corrected copy. And from these ingenious essays, the public may be enabled to form some judgment beforehand of a poem of a nobler and more important nature, which he is now preparing. Nor must we forget to return our public thanks to this gentleman, for the service he has been to us, not only in making this collection more excellent by his own contributions, but in selecting such pieces of others as were suitable to our design. It is hoped that the ancient Scottish poems (amongst which the thistle and the rose, and hardyknute are more particularly distinguished) will make no disagreeable figure amongst those of modern date; and that they will produce the same effect here, as Mr. Pope observes a moderate use of old words may have in a poem; which, adds he, is like working old abbey-stones into a modern building, and which I have sometimes seen practised with good success.
  • 43. Upon the whole, as we have been favoured with the best assistance in compiling this volume, no further apology is necessary; and as the approbation of the public has been already secured to these poems separately, we hope they have no less reason to claim it, when thus published together.
  • 44. CONTENTS. Page The Thistle and the Rose, by W. Dunbar 1 Verses on the Death of Queen Caroline. By Mr. Shipley 10 The Genealogy of Christ, by Mr. Lowth 13 A Fragment, by Mr. Mallet 24 The Eagle and Robin Red-Breast, a Fable, by Archibald Scott, written before the Year 1600. 28 Ode to Fancy, by Mr. Joseph Warton 31 Ode to Evening, by the same 37 Ode to Evening, by Mr. Collins 39 Isis, an Elegy, by Mr. Mason of Cambridge 42 The Triumph of Isis, by Mr. Thomas Warton of Oxford 47 A Love-Elegy, by Mr. Hammond 47 The Tears of Scotland, 1746. 62 An Elegy written in a country church-yard, by Mr. Grey 65 On the Death of Prince Frederic. Written at Paris, by David Lord Viscount Stormont 70 On the same, by Mr. James Clitherow of Oxford 75 Ode on the Approach of Summer, by a Gentleman formerly of the University of Aberdeen 81 A Pastoral in the manner of Spenser, from Theocritus, Idyll. 20. By the same 94 Inscribed on a beautiful Grotto near the Water 96 Love Elegy, by Mr. Smollet 97 A Panegyric on Oxford Ale, by a Gentleman of Trinity College 99 The Progress of Discontent, by the Same. 105 Ode to Arthur Onslow, Esq; 109 Job, Chapter XXXIX. By a Gentleman of Oxford 113
  • 45. Ode on the Death of Mr. Thomson, by Mr. Collins 116 The Child-Birth, in the manner of Gay 119 On a Lady's presenting a Sprig of Myrtle to a Gentleman, by Mr. Hammond 125 To a Young Lady with Fontenelle's Plurality of Worlds 126 Ode on the Fifth of December, by Mr. Christopher Smart 128 Part of the Prologue to Sir David Lyndesay's Dream. Written in the Reign of King James V. 129 Hardyknute, a Fragment 132 Ode. By Dr. Akenside, on Lyric Poetry 147
  • 46. A POEM IN HONOUR OF MARGARET DAUGHTER TO HENRY VII. OF ENGLAND, QUEEN TO JAMES IV. KING OF SCOTS. BY WILLIAM DUNBAR. The Thistle and the Rose, O'er flowers and herbage green, By Lady Nature chose, Brave King and lovely Queen. I. When March with varying winds was overpast, And sweet April had with his silver showers Ta'n leave of Nature with an orient blast, And lusty May, that mother is of flowers, Had made the birds begin by tymous hours; Among the tender odours red and white, Whose harmony to her was great delight.
  • 47. II. In bed at morrow, sleeping as I lay, Methought Aurora with her ruby ene, In at my window looked by the day, And halsit me with visage pale and green; Upon her hand a lark sang frae the spleen, Lovers, awake out of your slumbering. See how the lusty morning does upspring. III. Methought fresh May before my bed upstood, In weed depainted of ilk diverse hue, Sober, benign, and full of mansuetude, In bright attire of flowers, all forged new, Of heavenly colour, white, red, brown and blue, Balmit in dew, and gilt with Phebus' beams, While all the house illumin'd with her leams. IV. Sluggard, she said, awake anon for shame, And in mine honour something thou go write; The lark has done, the merry day proclaim, Lovers to raise with comfort and delight; Will nought increase thy courage to indite, Whose heart sometime has glad and blissful been, Songs oft to make, under the branches green? V. Whereto, quoth I, shall I uprise at morrow,
  • 48. For in thy month few birds have I heard sing, They have mare cause to weep and plain their sorrow: Thy air it is not wholsome nor benign, Lord Eolus does in thy season ring, So bousteous are the blasts of his shrill horn, Among thy boughs to walk I have forborn. VI. With that the lady soberly did smile, And said, uprise and do thy observance: Thou did promise in May's lusty while, Then to describe the ROSE of most pleasance Go see the birdis how they sing and dance, And how the skies illumined are bright, Enamell'd richly with new azure light. VII. When this was said, away then went the Queen, And enter'd in a lusty garden gent; And then methought, full hastily beseen, In sark and mantle after her I went Into this garth most dulce and redolent, Of herb and flower, and tender plants most sweet, And the green leaves doing of dew down fleit. VIII. The purple sun, with tender rayis red, In orient bright as angel did appear, Through golden skies advancing up his head,
  • 49. Whose gilded tresses shone so wondrous clear, That all the world took comfort far and near, To look upon his fresh and blissful face, Doing all sable frae the Heavens chace. IX. And as the blissful sun drove up the sky, All nature sang through comfort of the light, The minstrels wing'd, with open voices cry, O Lovers now is fled the dully night, Come welcome day, that comforts ev'ry wight; Hail May! hail Flora! hail Aurora sheen, Hail Princess Nature! hail love's hartsome Queen! X. Dame Nature gave an inhibition there, To Neptune fierce, and Eolus the bold, Not to perturb the water or the air, That neither blashy shower, nor blasts more cold Should flowers affray nor fowls upon the fold. She bade eke Juno, Goddess of the sky, That she the heaven should keep amene and dry. XI. Also ordain'd that every bird and beast Before her Highness should anon compear; And every flower of virtue most and least, And every herb of fair field far and near,
  • 50. As they had wont in May from year to year; To her their Queen to make obedience, Full low inclining with due reverence. XII. With that anon she sent the swift foot Roe, To bring in alkind beast from dale and down; The restless swallow order'd she to go, And fetch all fowl of great and small renown, And to gar flowers appear of all fassoun: Full craftily conjured she the Yarrow, Which did forth swirk as swift as any arrow. XIII. All brought in were in twinkling of an eye, Both beast and bird and flower before the Queen; And first the Lion, greatest of degree, Was summon'd there; and he, fair to be seen, With a full hardy countenance and keen, Before Dame Nature came, and did incline, With visage bold, and courage leonine. XIV. This awful beast was terrible of chear, Piercing of look, and stout of countenance, Right strong of corps, of fashion fair, but fear, Lusty of shape, light of deliverance, Red of his colour, as the ruby glance: In field of gold he stood full rampantly, With flower-de-lyces circled pleasantly.
  • 51. XV. This Lady lifted up his claws so clear, And lute him listly lean upon her knee, And crowned him with diadem full dear, Of radious stones most royal there to see, Saying the King of all beasts make I thee; And the protector chief in woods and shaws, Go forth, and to thy lieges keep the laws. XVI. Justice exerce, with mercy and conscience, And let no small beast suffer skaith or scorns Of greater beasts, that been of more puissance; Do law alike to Apes and Unicorns, And let no Bugle with his bousteous horns Oppress the meek plough Ox, for all his pride, But in the yoke go quietly him beside. XVII. When this was said, with noise and sound of joy, All kind of Quadrupeds in their degree, At once cry'd laud, and then vive le roy, Then at his feet fell with humility; To him they all paid homage and fealty; And he did them receive with princely laits, Whose noble ire his greatness mitigates. XVIII. Then crowned she the Eagle King of fowls;
  • 52. And sharp as darts of steel she made his pens, And bade him be as just to Whawps and Owls, As unto Peacocks, Papingoes, or Cranes, And make one law for Wicht Fowls, and for Wrens, And let no fowl of rapine do affray, Nor birds devour, but his own proper prey. XIX. Then called she all flowers grew in the field, Describing all their fashions and effeirs, Upon the awful THISTLE she beheld. And saw him guarded with a bush of spears, Considering him so able for the wars, A radiant crown of rubies she him gave, And said, in field go forth, and fend the laif. XX. And since thou art a King, be thou discreet, Herb without value hold not of such price, As herb of virtue and of odour sweet; And let no nettle vile, and full of vice, Her fellow with the goodly Flower-de-lyce; Nor let no wild weed full of churlishness, Compare her to the Lilly's nobleness. XXI. Nor hold none other flower in such dainty As the fresh ROSE, of colour red and white; For if thou dost, hurt is thine honesty, Considering that no flower is so perfyte, So full of pleasaunce, virtue and delight;
  • 53. So full of blissful angelic beauty, Imperial birth, honour and dignity. XXII. Then to the ROSE she did her visage turn, And said, O lusty daughter most benign, Above the Lilly thou art illustrious born, From royal lineage rising fresh and young, But any spot, or macul doing sprung; Come bloom of joy, with richest gems becrown'd, For o'er the laif thy beauty is renown'd. XXIII. A costly crown with stones clarified bright, This comely Queen did in her head inclose, While all the land illumined of light; Wherefore methought, the flowers did all rejoyce, Crying at once, Hail to the fragrant ROSE! Hail Empress of the herbs! fresh Queen of flowers! To thee be glore and honour at all hours. XXIV. Then all the birds they sang with voice on height, Whose mirthful sound was marvellous to hear: The Mavys sang, Hail ROSE most rich and right, That does upflourish under Phebus' sphere, Hail plant of youth, hail Prince's daughter dear,
  • 54. Hail blossom breaking out of blood royal, Whose precious virtue is imperial. XXV. The Merle she sang, Hail ROSE of most delight, Hail of all flowers the sweet and sovereign Queen: The lark she sang, hail ROSE both red and white, Most pleasant flower of mighty colours[1] twain: Nightingals sang, hail Natures suffragan, In beauty, nurture, and each nobleness, In rich array, renown, and gentleness. XXVI. The common voice uprose of warblers small, Upon this wise, O blessed be the hour That thou wast chose to be our principal, Welcome to be our Princess crown'd with pow'r, Our pearl, our pleasance, and our paramour, Our peace, our play, our plain felicity: Christ thee conserve from all adversity. XXVII. Then all the concert sang with such a shout, That I anon awaken'd where I lay, And with a braid I turned me about To see this court, but all were gone away;
  • 55. Then up I lean'd me, halflings in affray, Call'd to my Muse, and for my subject chose To sing the royal THISTLE and the ROSE. FOOTNOTES: [1] Alluding to the Houses of york and lancaster, which were distinguished by the white and red rose, and united in the person of Queen margaret. VERSES ON THE DEATH OF QUEEN
  • 56. CAROLINE. BY MR. SHIPLEY. Oblivion wraps not in her silent shade All human labours. Virtue blooms a flower, That Time's rough hand shall never violate. Still CAROLINE shall live in faithful verse, Sweet nurse of Memory, and in the voice Of grateful Britain. These shall testify How well her calm impartial rule supplied A monarch's absence; these commemorate Her soul contemplative of peaceful Truth And nature, mindful midst the pomp of Courts Of wise retirement, and the silent grove. She stretch'd thro' length'ning shades thy spacious walks, Delightful Richmond, and the terrass rais'd Of regal grandeur, whence the eye discerns Fair Thames with copious waters winding slow Midst pastures, spreading herds, and villages Of aspect neat, and villas wrapt in shades: Fair scene of chearful peace! the lovely sight Frequent she view'd, and bless'd the honour'd reign Of her great Consort, provident and mild. Now wander'd musing thro' the darkening depth Of thickest woods, friendly to solemn thought: Now o'er broad lawns fair opening to the sun. Nor midst her rural plans disdain'd to mix The useful arable, and waving corn
  • 57. With soft turf border'd, and the lowly cot, That half appears, in branching elms obscur'd. Here beauty dwells, assembled from the scenes Of various nature; such as oft inflam'd With rapture Grecian bards, in that fair vale, Thessalian Tempe, or thy favorite soil, Arcadia, erst by awe-struck fancy fill'd With wand'ring forms, the woodland Deities, Light Nymphs and wanton Satyrs, faintly seen Quick glancing thro' the shade at close of eve, Great Pan, and old Silenus. Hither led By solitary grief shall GEORGE recall Th' endearing manners, the soft speech, that flow'd From his lov'd Consort, virtue mix'd with love, Prudence, and mild insinuating sense: But chief her thoughtful breast of counsels deep Capacious, nor unequal to the weight Of Government. Such was the royal mind Of wise ELIZA, name of loveliest sound To British ears, and pattern fair to Kings: Or she who rules the Scepter of the North Illustrious, spreading o'er a barbarous world The light of arts and manners, and with arms Infests th' astonish'd Sultan, hardly now With scatter'd troops resisting; she drives on The heavy war, and shakes th' Imperial Throne Of old Byzantium. Latest time shall sound The praise of female genius. Oft shall GEORGE Pay the kind tear, and grief of tender words To CAROLINE, thus oft lamenting sad. Hail sacred shade! by me with endless woe Still honour'd! ever in my Breast shall dwell Thy image, ever present to my soul Thy faithful love, in length of years mature: O skill'd t'enliven time, to soften care
  • 58. With looks and smiles and friendship's chearful voice! Anxious, of Thee bereft, a solitude I feel, that not the fond condoling cares Of our sad offspring can remove. Ev'n now With lonely steps I trace the gloomy groves, Thy lov'd recesses, studious to recall The vanish'd bliss, and cheat my wand'ring thoughts With sweet illusion. Yet I not accuse Heav'n's dispensation. Prosperous and long Have been my days, and not unknown to fame, That dwells with virtue. But 'tis hard to part The league of ancient friendship, to resign The home-felt fondness, the secure delight, That reason nourish'd, and fair fame approv'd.
  • 59. THE GENEALOGY OF CHRIST, AS IT IS REPRESENTED ON THE EAST WINDOW OF WINCHESTER COLL. CHAPEL. WRITTEN AT WINTON SCHOOL, BY DR. LOWTHE. At once to raise our rev'rence and delight, To elevate the mind, and please the sight, To pour in virtue at th' attentive eye, And waft the soul on wings of extacy; For this the painter's art with nature vies, And bids the visionary saint arise; Who views the sacred forms in thought aspires, Catches pure zeal, and as he gazes, fires; Feels the same ardour to his breast convey'd, Is what he sees, and emulates the shade. Thy strokes, great Artist, so sublime appear, They check our pleasure with an awful fear; While, thro' the mortal line, the God you trace, Author himself, and Heir of Jesse's race; In raptures we admire thy bold design, And, as the subject, own the hand divine. While thro' thy work the rising day shall stream, So long shall last thine honour, praise and name. And may thy labours to the Muse impart Some emanation from her sister art, To animate the verse, and bid it shine In colours easy, bright, and strong, as Thine. Supine on earth an awful figure lies,
  • 60. While softest slumbers seem to seal his eyes; The hoary sire Heav'ns guardian care demands, And at his feet the watchful angel stands. The form august and large, the mien divine Betray the [2]founder of Messiah's line. Lo! from his loins the promis'd stem ascends, And high to Heaven its sacred Boughs extends: Each limb productive of some hero springs, And blooms luxuriant with a race of kings. Th' eternal plant wide spreads its arms around, And with the mighty branch the mystic top is crown'd. And lo! the glories of th' illustrious line At their first dawn with ripen'd splendors shine, In DAVID all express'd; the good, the great, The king, the hero, and the man compleat. Serene he sits, and sweeps the golden lyre, And blends the prophet's with the poet's fire. See! with what art he strikes the vocal strings, The God, his theme, inspiring what he sings! Hark—or our ears delude us—from his tongue Sweet flows, or seems to flow, some heav'nly song. Oh! could thine art arrest the flitting sound, And paint the voice in magic numbers bound; Could the warm sun, as erst when Memnon play'd Wake with his rising beam the vocal shade: Then might he draw th' attentive angels down, Bending to hear the lay, so sweet, so like their own. On either side the monarch's offspring shine, And some adorn, and some disgrace their line. Here Ammon glories; proud, incestuous lord! This hand sustains the robe, and that the sword. Frowning and fierce, with haughty strides he tow'rs, And on his horrid brow defiance low'rs.
  • 61. There Absalom the ravish'd sceptre sways, And his stol'n honour all his shame displays: The base usurper Youth! who joins in one The rebel subject, and th' ungrateful son. Amid the royal race, see Nathan stand: Fervent he seems to speak, and lift his hand; His looks th' emotion of his soul disclose, And eloquence from every gesture flows. Such, and so stern he came, ordain'd to bring Th' ungrateful mandate to the guilty King: When, at his dreadful voice, a sudden smart Shot thro' the trembling monarch's conscious heart; From his own lips condemn'd; severe decree! Had his God prov'd so stern a Judge as He. But man with frailty is allay'd by birth; Consummate purity ne'er dwelt on earth: Thro' all the soul tho' virtue holds the rein, Beats at the heart, and springs in ev'ry vein: Yet ever from the clearest source have ran Some gross allay, some tincture of the man. But who is he——deep-musing——in his mind, He seems to weigh, in reason's scales, mankind; Fix'd contemplation holds his steady eyes—— I know the sage[3]; the wisest of the wise. Blest with all man could wish, or prince obtain, Yet his great heart pronounc'd those blessings vain. And lo! bright glitt'ring in his sacred hands, In miniature the glorious temple stands. Effulgent frame! stupendous to behold! Gold the strong valves, the roof of burnish'd gold. The wand'ring ark, in that bright dome enshrin'd, Spreads the strong light, eternal, unconfin'd! Above th' unutterable glory plays Presence divine! and the full-streaming rays Pour thro' reluctant clouds intolerable blaze.
  • 62. But stern oppression rends Reboam's reign; See the gay prince, injurious, proud and vain! Th' imperial sceptre totters in his hand, And proud rebellion triumphs in the land. Curs'd with corruption's ever-fruitful spring, A beardless Senate, and a haughty King. There Asa, good and great, the sceptre bears, Justice attends his peace, success his wars: While virtue was his sword, and Heaven his shield, Without controul the warrior swept the field; Loaded with spoils, triumphant he return'd, And half her swarthy Sons sad Ethiopia mourn'd. But since thy flagging piety decay'd, And barter'd God's defence for human aid; See their fair laurels wither on thy brow, Nor herbs, nor healthful arts avail thee now, Nor is heav'n chang'd, apostate prince, but Thou. No mean atonement does this lapse require; But see the Son, you must forgive the Sire: He, [4]the just prince—with ev'ry virtue bless'd, He reign'd, and goodness all the man possess'd, Around his throne, fair happiness and peace Smooth'd ev'ry brow, and smil'd in ev'ry face. As when along the burning waste he stray'd, Where no pure streams in bubbling mazes play'd, Where drought incumbent on the thirsty ground, Long since had breath'd her scorching blasts around; The [5]Prophet calls, th' obedient floods repair To the parch'd fields, for Josaphat was there. The new-sprung waves, in many a gurgling vein, Trickle luxurious through the sucking plain; Fresh honours the reviving fields adorn, And o'er the desart plenty pours her horn. So, from the throne his influence he sheds,
  • 63. And bids the virtues raise their languid heads: Where'er he goes, attending Truth prevails, Oppression flies, and Justice lifts her scales. See, on his arm, the royal eagle stand, Great type of conquest and supreme command; Th' exulting bird distinguish'd triumph brings, And greets the Monarch with expanded wings. Fierce Moab's sons prevent th' impending blow, Rush on themselves, and fall without the foe. The pious hero vanquish'd Heav'n by pray'r; His faith an army, and his vows a war. Thee too, Ozias, fates indulgent blest And thy days shone, in fairest actions drest; Till that rash hand, by some blind frenzy sway'd, Unclean, the sacred office durst invade. Quick o'er thy limbs the scurfy venom ran, And hoary filth besprinkled all the man. Transmissive worth adorns the pious [6]Son, The father's virtues with the father's throne. Lo! there he stands: he who the rage subdu'd Of Ammon's sons, and drench'd his sword in blood, And dost thou, Ahaz, Judah's scourge, disgrace, With thy base front, the glories of thy race? See the vile King his iron sceptre bear—— His only praise attends the pious [7]Heir; He, in whose soul the virtues all conspire, The best good son, from the worst wicked sire. And lo! in Hezekiah's golden reign, Long-exil'd piety returns again; Again, in genuine purity she shines, And with her presence gilds the long-neglected shrines. Ill-starr'd does proud Assyria's impious [8]Lord Bid Heav'n to arms, and vaunt his dreadful sword; His own vain threats th' insulting King o'erthrow,
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