Signals and Systems: Analysis Using Transform Methods and MATLAB®
M. J. Roberts,
University of Tennessee
ISBN: 0072499427 Copyright year: 2004
Table of Contents
Chapter 1 Introduction 1
1.1 Signals and Systems Defined 1
1.2 Types of Signals 2
1.3 A Signal and System Example 10
1.4 Use of MATLAB 16 Chapter 2 Mathematical Description of Signals 17
2.1 Introduction and Goals 17
2.2 Continuous-Time versus Discrete-Time
Functions 18 Continuous-Time Functions 18 Sampling and Discrete Time 19
2.3 Continuous-Time Signal Functions 22 Complex Exponentials and Sinusoids 22 Functions with Discontinuities 23 Singularity Functions and Related Functions 23 MATLAB® Files for Some Singularity Functions
and Related Functions 38
2.4 Functions and Combinations of Functions 39 Combinations of Functions41
2.5 Continuous-Time Scaling and Shifting
Transformations 43 Amplitude Scaling 43 Time Shifting 44 Time Scaling 45 Multiple Transformations 48
2.6 Differentiation and Integration 54
2.7 Continuous-Time Even and Odd
Functions 56 Sums, Products, Differences, and Quotients 57
2.8 Continuous-Time Periodic
Functions 63
2.9 Discrete-Time Signal Functions 67 Discrete-Time Singularity Functions 71
2.10 Discrete-Time Scaling and Shifting
Transformations 75 Time Shifting 75 Time Scaling 76
2.11 Differencing and Accumulation 81
2.12 Discrete-Time Even and Odd Functions 85 Sums, Products, Differences, and Quotients 86 Accumulation 86
2.13 Discrete-Time Periodic Functions 88
2.14 Signal Energy and Power 90
2.15 Summary of Important Points 98
Exercises with Answers 98
Exercises without Answers 112 Chapter 3 Description and Analysis of Systems 125
3.1 Introduction and Goals 125 Block Diagrams and System Terminology 126 Discrete-Time versus Continuous-Time Systems 129
3.2 System Characteristics 129 Homogeneity 132 Time Invariance 133 Additivity 134 Linearity and Superposition 135 Stability 138 Incremental Linearity 138 Causality 141 Memory 142 Static Nonlinearity 142 Invertibility 146
3.3 Eigenfunctions of LTI Systems 148 Continuous-Time Systems 148 Discrete-Time Systems 150
3.4 Analogies 150
3.5 The Convolution Sum 151 Unit Impulse Response 151 Convolution 153 Convolution Properties 158 System Interconnections 163 Stability and Impulse Response 164 Responses of Systems to Standard Signals 165
3.6 The Convolution Integral 170 Impulse Response 170 Convolution 174 Convolution Properties 178 An Exploration of Impulse Properties
Using Convolution 183 System Interconnections 185 Stability and Impulse Response 186 Responses of Systems to Standard Signals 187
3.7 Block Diagram Simulation of Differential
or Difference Equations 189
3.8 Summary of Important Points 190
Exercises with Answers 191
Exercises without Answers 199 Chapter 4 The Fourier Series 207
4.1 Introduction and Goals 207
4.2 The Continuous-Time Fourier
Series (CTFS) 208 Linearity and Complex Exponential Excitation 208 Definition of the Continuous–Time Fourier Series 212
4.3 Calculation of the Continuous-Time
Fourier Series 219 Sinusoidal Signals 219 Nonsinusoidal Signals 229 The Continuous-Time Fourier Series of Periodic
Signals over a Noninteger Number of
Fundamental Periods 230 The Continuous-Time Fourier Series of Periodic
Signals over an Integer Number of
Fundamental Periods 233 The CTFS of Even and Odd Periodic Signals 234 Cyclic Frequency and Radian Frequency Forms 235 The Continuous-Time Fourier Series
of a Random Signal 236
4.4 Properties of the Continuous-Time
Fourier Series 238 Linearity 238 Time Shifting 239 Frequency Shifting 241 Time Reversal 241 Time Scaling 243 Change of Representation Period 245 Time Differentiation 246 Time Integration 247 Multiplication–Convolution Duality 248 Conjugation 250 Parseval’s Theorem 251 Summary of CTFS Properties 251
4.5 Use of Tables and Properties 252
4.6 Band-Limited Signals 256
4.7 Convergence of the Continuous-Time
Fourier Series 256 Continuous Signals 256 Signals with Discontinuities and the Gibbs
Phenomenon 257
4.8 The Discrete-Time Fourier Series (DTFS) 259 Mathematical Development 259
4.9 Properties of the Discrete-Time
Fourier Series 267 Linearity 268 Time Shifting 268 Frequency Shifting 269 Conjugation 269 Time Reversal 269 Time Scaling 269 Change of Period 270 Multiplication-Convolution Duality 274 First Backward Difference 276 Accumulation 276 Even and Odd Signals 277 Parseval’s Theorem 277 Summary of DTFS Properties 278
4.10 Convergence of the Discrete-Time
Fourier Series 279
4.11 Frequency Response of LTI Systems
with Periodic Excitation 283
4.12 Summary of Important Points 287
Exercises with Answers 288
Exercises without Answers 294 Chapter 5 The Fourier Transform 299
5.1 Introduction and Goals 299
5.2 The Continuous-Time Fourier Transform 299 The Transition from the Continuous-Time Fourier
Series to the Continuous-Time Fourier Transform 299 Definition of the Continuous-Time
Fourier Transform 303
5.3 Convergence and the Generalized Fourier
Transform 309
5.4 Comparisons between the Continuous-Time
Fourier Series and the Continuous-Time Fourier
Transform 311
5.5 Properties of the Continuous-Time
Fourier Transform 314 Linearity 314 Time Shifting and Frequency Shifting 314 Time Scaling and Frequency Scaling 316 Transform of a Conjugate 318 Multiplication–Convolution Duality 319 Time Differentiation 321 Modulation 322 Transforms of Periodic Signals 323 Parseval’s Theorem 323 Integral Definition of an Impulse 325 Duality 326 Total-Area Integral Using Fourier Transforms 327 Integration 329 Summary of CTFT Properties 333 Use of Tables and Properties 334
5.6 The Discrete-Time Fourier Transform 338 Graphical Illustration 338 Analytical Derivation 341 Definition of the Discrete-Time Fourier Transform 341
5.7 Convergence of the Discrete-Time
Fourier Transform 342
5.8 Properties of the Discrete-Time
Fourier Transform 342 Linearity 343 Time Shifting and Frequency Shifting 343 Transform of a Conjugate 344 Differencing and Accumulation 345 Time Reversal 346 Multiplication–Convolution Duality 346 Accumulation Definition of a Comb Function 348 Parseval’s Theorem 350 Summary of DTFT Properties 351
5.9 Relations Among Fourier Methods 353 CTFT and CTFS 356 CTFT and DTFT 361 DTFT and DTFS 362 Method Comparison Examples 366
5.10 Summary of Important Points 370
Exercises with Answers 371
Exercises without Answers 386 Chapter 6 Fourier Transform Analysis of Signals
and Systems 391
6.1 Introduction and Goals 391
6.2 Frequency Response 392
6.3 Ideal Filters 395 Distortion 395 Filter Classifications 398 Ideal-Filter Frequency Responses 398 Bandwidth 398 Impulse Responses and Causality 400 The Power Spectrum 409 Noise Removal 410
6.4 Practical Passive Filters 412 The RC Lowpass Filter 412 The RLC Bandpass Filter 415
6.5 Log-Magnitude Frequency-Response Plots
and Bode Diagrams 417 Component Diagrams 423 Complex Pole and Zero Pairs 428
6.6 Practical Active Filters 430 Operational Amplifiers 430 Filters 431
6.7 Discrete-Time Filters 438
6.8 Filter Specifications and Figures of Merit 443
6.9 Communication Systems 448 Modulation 450 Phase and Group Delay 459 Pulse-Amplitude Modulation 464
6.10 Spectral Analysis 468
6.11 Summary of Important Points 471
Exercises with Answers 472
Exercises without Answers 483 Chapter 7 Sampling and the Discrete
Fourier Transform 491
7.1 Introduction and Goals 491
7.2 Sampling Methods 492
7.3 Representing a Continuous-Time Signal
by Samples 496 Qualitative Concepts 496 Shannon’s Sampling Theorem 497 Aliasing 503 Time-Limited and Band-Limited Signals 507 Sampling Bandpass Signals 508 Interpolation 509 Sampling a Sinusoid 513
7.4 Sampling Discrete-Time Signals 516
7.5 Band-Limited Periodic Signals 522
7.6 The Discrete Fourier Transform and Its Relation
to Other Fourier Methods 525
7.7 Examples of the Use of the Discrete
Fourier Transform 536
7.8 The Fast Fourier Transform 551
7.9 Summary of Important Points 555
Exercises with Answers 555
Exercises without Answers 568 Chapter 8 Correlation, Energy Spectral Density,
and Power Spectral Density 571
8.1 Introduction and Goals 571
8.2 Correlation and the Correlogram 572
8.3 The Correlation Function 578 Conceptual Basis 578 Energy Signals 579 Power Signals 582
8.4 Autocorrelation 589 Relation to Signal Energy and Signal Power 589 Properties of Autocorrelation 590 Autocorrelation Examples 591
8.5 Cross Correlation 601 Properties of Cross Correlation 601 Examples of Cross Correlation 601
8.6 Correlation and the Fourier Series 605
8.7 Energy Spectral Density (ESD) 605 Definition and Derivation of Energy
Spectral Density 606 Effects of Systems on ESD 606 The ESD Concept 607 Relation of ESD to Autocorrelation 607
8.8 Power Spectral Density (PSD) 609 Definition and Derivation of Power
Spectral Density 609 Effects of Systems on PSD 609 The PSD Concept 610 Relation of PSD to Autocorrelation 612
8.9 Summary of Important Points 615
Exercises with Answers 615
Exercises without Answers 620 Chapter 9 The Laplace Transform 623
9.1 Introduction and Goals 623
9.2 Development of the Laplace Transform 624 Derivation and Definition 624 Region of Convergence 627 The Unilateral Laplace Transform 631
9.3 Properties of the Laplace Transform 635 Linearity 636 Time Shifting 636 Complex-Frequency Shifting 636 Time Scaling 637 Frequency Scaling 638 Time Differentiation Once 638 Time Differentiation Twice 639 Complex-Frequency Differentiation 640 Multiplication–Convolution Duality 640 Integration 642 Initial Value Theorem 642 Final Value Theorem 643 Summary of Properties of the Unilateral
Laplace Transform 645
9.4 The Inverse Laplace Transform Using
Partial-Fraction Expansion 645
9.5 Laplace Transform–Fourier Transform
Equivalence 655
9.6 Solution of Differential Equations
with Initial Conditions 655
9.7 The Bilateral Laplace Transform 657 Calculation Using the Unilateral
Laplace Transform 658 Properties 660
9.8 Summary of Important Points 663
Exercises with Answers 664
Exercises without Answers 668 Chapter 10 Laplace Transform Analysis of Signals
and Systems 671
10.1 Introduction and Goals 671
10.2 Transfer Functions from Circuits
and System Diagrams 671
10.3 System Stability 675
10.4 Parallel, Cascade, and Feedback
Connections 677
10.5 Analysis of Feedback Systems 681 Beneficial Effects of Feedback 681 Instability Caused by Feedback 685 Stable Oscillation Using Feedback 689 The Routh–Hurwitz Stability Test 693 The Root-Locus Method 695 Gain-Margin and Phase-Margin Analysis
of System Stability 698 Steady-State Tracking Errors in Unity-Gain
Feedback Systems 699
10.6 Block-Diagram Reduction and Mason’s
Theorem 703
10.7 System Responses to Standard Signals 708 Unit Step Response 709 Response to a Suddenly Applied Sinusoid 713
10.8 Pole–Zero Diagrams and Graphical Calculation
of Frequency Response 716
10.9 Butterworth Filters 720
10.10 Frequency Transformations 721
10.11 Analog Filter Design with MATLAB 725
10.12 Standard Realizations of Systems 727
10.13 State-Space Signal and System
Analysis 731
10.14 Summary of Important Points 740
Exercises with Answers 741
Exercises without Answers 753 Chapter 11 The z Transform 761
11.1 Introduction and Goals 761
11.2 Development of the z Transform 761 Derivation and Definition 762 Region of Convergence 764 The Unilateral z Transform 767
11.3 Properties of the z Transform 768 Linearity 768 Time Shifting 769 Change of Scale 772 Initial Value Theorem 774 z-Domain Differentiation 774 Convolution in Discrete Time 775 Differencing 775 Accumulation 776 Final Value Theorem 777 Summary of z-Transform Properties 778
11.4 The Inverse z Transform 778
11.5 Solution of Difference Equations
with Initial Conditions 781
11.6 The Relationship Between the z
and Laplace Transforms 783
11.7 The Bilateral z Transform 785 Properties 786
11.8 Summary of Important Points 790
Exercises with Answers 790
Exercises without Answers 795 Chapter 12 z-Transform Analysis of Signals and Systems 797
12.1 Introduction and Goals 797
12.2 Transfer Functions 797
12.3 System Stability 799
12.4 Parallel, Cascade, and Feedback
Connections 800
12.5 System Responses to Standard Signals 801 Unit Sequence Response 801 Response to a Suddenly Applied Sinusoid 805
12.6 Pole–Zero Diagrams and the Graphical
Calculation of Frequency
Response 809
12.7 Discrete-Time Systems with Feedback 812 The Jury Stability Test 812 The Root-Locus Method 814
12.8 Simulating Continuous-Time Systems
with Discrete-Time Systems 816
12.9 Sampled-Data Systems 817
12.10 Digital Filters 824 Digital Filter Design Methods 824 Impulse- and Step-Invariant Design 825 Difference Equation Approximation
to Differential Equations 832 Direct Substitution and the Matched
z Transform 837 The Bilinear Transformation 839 FIR Filter Design 848
12.11 Digital Filter Design and Implementation
with MATLAB 860
12.12 Standard Realizations of Systems 863
12.13 State-Space Signal and System
Analysis 865
12.14 Summary of Important Points 871
Exercises with Answers 872
Exercises without Answers 881 Appendix A Useful Mathematical Relations 886 Appendix B Introduction to MATLAB 889
B.1 Numbers, Variables, and Matrices 889
B.2 Operators 890
B.3 Scripts and Functions 899
B.4 MATLAB Functions and Commands 900 General-Purpose Commands 901 Programming Language Flow Control 903 Elementary Matrices and Matrix Manipulation 906 Elementary Math Functions 908 Specialized Math Functions 910 Matrix Functions and Numerical Linear Algebra 911 Data Analysis and Fourier Transforms 911 Interpolation and Polynomials 914 Two-Dimensional Graphs 915 Three-Dimensional Graphs 922 Specialized Graphs 924 Handle Graphics 925 Graphical User Interface Tools 931 Character Strings 931 File Input-Output 933 Time and Dates 934 Data Types and Structures 934 Appendix C Method for Finding Least Common
Multiples 935 Appendix D Convolution Properties 937
D.1 DT Convolution Properties 937 Commutativity 937 Associativity 937 Distributivity 938
D.2 CT Convolution Properties 938 Commutativity 938 Associativity 938 Distributivity 940 Appendix E Table of Fourier Pairs 941
E.1 Fourier Series 941 Continuous-Time Fourier Series
(CTFS) 941 Discrete-Time Fourier Series (DTFS) 943
E.2 Fourier Transform 946 Continuous-Time Fourier
Transform (CTFT) 946 Discrete-Time Fourier Transform
(DTFT) 956 Appendix F Table of Laplace Transform Pairs 964
F.1 Causal Functions 964
F.2 Anticausal Functions 965
F.3 Noncausal Functions 965 Appendix G Table of z Transforms 966
G.1 Causal Functions 966
G.2 Anticausal Functions 967
G.3 Noncausal Functions 967 Appendix H Complex Numbers and Complex
Functions 968
H.1 Basic Properties of Complex
Numbers 968
H.2 The Polar Form 971
H.3 Functions of a Complex Variable 975
H.4 Complex Functions of a Real
Variable 978 Appendix I Differential and Difference Equations 990
I.1 Introduction 990
I.2 Homogeneous Constant-Coefficient Linear
Differential Equations 990 Inhomogeneous Constant-Coefficient Linear
Differential Equations 992 Systems of Linear Differential Equations 997
I.3 Linear Ordinary Difference Equations 1002 Finite-Difference Approximations
to a Derivative 1002 Homogeneous Linear Constant-Coefficient
Difference Equations 1006 Inhomogeneous Linear Constant-Coefficient Difference
Equations 1008 Systems of Linear Difference Equations 1010 Appendix J Vectors and Matrices 1018
J.1 Definitions and Operations 1018
J.2 Determinants, Cramer’s Rule, and
the Matrix Inverse 1024
J.3 Derivatives and Differences 1028
J.4 Eigenvalues and Eigenvectors 1029 Bibliography 1040 Index 1043
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