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Fundamentals of Signals and Systems Chapter 1 - Introduction 1.1 Signals and Systems Defined 1-1 1.2 Types of Signals 1-2 Continuous-Time, Continuous-Value, Discrete-Time, Discrete- Value, Random 1.3 Examples of Systems 1-8 A Mechanical System Differential Equations A Fluid System A Discrete-Time System Difference Equations Feedback Systems Stability 1.4 A Familiar Signal and System Example 1-16 Power Spectrum, Signal-to-Noise Ratio 1.5 Use of MATLAB 1-21 Chapter 2 - Mathematical Description of Continuous-Time Signals 2.1 Introduction and Goals 2-1 2.2 Functions and Functional Notation 2-2 Domain, Range 2.3 Signal Functions 2-2 Continuous-Time Functions Complex Exponentials and Sinusoids Cyclic and Radian Frequency Functions with Discontinuities Singularity Functions and Related Functions The Unit Step Function The Signum Function The Unit Ramp Function The Unit Impulse Generalized Derivatives, Equivalence, Sampling and Scaling Properties The Periodic Impulse or Impulse Train A Coordinated Notation for Singularity Functions The Unit Rectangle Function The Unit Triangle Function The Unit Sinc Function The Dirichlet Function MATLAB m-Files for Some Singularity Functions and Related Functions 2.4 Functions and Combinations of Functions 2-24 Functional Notation - Conventions and Conversions Combinations of Functions 2.5 Scaling and Shifting 2-26 Amplitude Scaling Time Shifting Time Scaling Simultaneous Scaling and Shifting Significance of Sequence 2.6 Differentiation and Integration 2-39 Antiderivative, Indefinite Integral, Definite Integral, Cumulative Integral 2.7 Even and Odd Functions 2-42 Sums, Products, Differences, Quotients, Derivatives, Integrals 2.8 Periodic Functions 2-46 Least Common Multiple, Greatest Common Divisor 2.9 Signal Energy and Power 2-50 Energy Signals, Power Signals 2.10 Summary of Important Points 2-55 Exercises 2-56 Chapter 3 - Mathematical Description of Discrete-Time Signals 3.1 Introduction and Goals 3-1 3.2 Signal Functions 3-2 Sampling and Discrete Time Uniform Sampling, Sequential-State Machines Exponentials and Sinusoids Periods, Normalized Frequency Singularity Functions The Unit Impulse or Kronecker Delta Function The Unit Sequence The Signum Function The Rectangle Function The Unit Ramp Function The Periodic Impulse or Impulse Train 3.3 Scaling and Shifting 2-68 Amplitude Scaling Time Shifting Time Scaling Compression, Expansion, Decimation, Interpolation 3.4 Differencing and Accumulation 3-22 Approximation of Derivatives, Forward, Backward and Central Differences 3.5 Even and Odd Functions 3-26 Sums, Products, Differences and Quotients Accumulation 3.6 Periodic Functions 3-30 3.7 Signal Energy and Power 3-31 3.8 Summary of Important Points 3-34 Exercises 3-36 Chapter 4 - Properties of Continuous-Time Systems 4.1 Introduction and Goals 4-1 4.2 Block Diagrams and System Terminology 4-2 Components, Amplifiers, Summing Junctions, Integrators 4.3 System Modeling 4-7 Differential Equations, Model Complexity, Feedback and Stability 4.4 System Properties 4-14 Introductory Example Zero-Input Response, Zero-State Response Homogeneity Time Invariance Additivity Linearity and Superposition Linear, Time-Invariant Systems, Non-Linearity and Linearization Stability Bounded-Input-Bounded-Output Incremental Linearity Causality Memory Static and Dynamic Systems Static Non-Linearity Invertibility Inverse System 4.5 Eigenfunctions of LTI Systems 4-38 Natural Radian Frequency, Damping Factor, Critical Radian Frequency, Damping Ratio, Complex Exponential Excitation 4.6 Summary of Important Points 4-41 Exercises 4-42 Chapter 5 - Properties of Discrete-Time Systems 5.1 Introduction and Goals 5-1 5.2 Block Diagrams and System Terminology 5-1 Delay 5.3 System Modeling 5-2 Approximation of Differential Equations with Difference Equations, Feedback, Oscillation, Stability 5.4 System Properties 5-10 Stability Criterion 5.5 Eigenfunctions of LTI Systems 5-14 Complex Exponential Excitation 5.6 Summary of Important Points 5-14 Exercises 5-15 Chapter 6 - Time-Domain Analysis of Continuous-Time Systems 6.1 Introduction and Goals 6-1 6.2 The Convolution Integral 6-1 Impulse Response Calculation from Differential Equations Convolution Derivation from Rectangular-Pulse Approximation Convolution Properties Convolution with an Impulse, Graphical Convolution Methods, Time Shifting, Commutativity, Associativity, Distributivity, Differentiation, Area, Scaling, Convergence of Convolution Integral System Interconnections Cascade, Parallel Stability and Impulse Response Responses of Systems to Standard Signals Unit-Step Response, Complex Exponential Response, Transfer Function, Frequency Response 6.3 Block-Diagram Realization of Differential Equations 6-31 Preference for Integrators over Differentiators, Direct Forms 1 and 2 6.4 Summary of Important Points 6-37 Exercises 6-39 Chapter 7 - Time-Domain Analysis of Discrete-Time Systems 7.1 Introduction and Goals 7-1 7.2 The Convolution Sum 7-1 Impulse Response Calculation from Difference Equations Convolution Development, Graphical Method, Array Method Convolution Properties Convolution with an Impulse, Time Shifting, Commutativity, Associativity, Distributivity, Differencing, Sum, Convergence of Convolution Sum Numerical Convolution Approximating the Convolution Integral System Interconnections Cascade, Parallel Stability and Impulse Response Responses to Standard Signals Unit-Sequence Response, Complex Exponential Response, Transfer Function, Frequency Response 7.3 Block-Diagram Realization of Difference Equations 7-31 Direct Forms 1 and 2 7.4 Summary of Important Points 7-35 Exercises 7-36 Chapter 8 - The Continuous-Time Fourier Series 8.1 Introduction and Goals 8-1 8.2 Periodic Excitation and Response of LTI Systems 8-2 Inadequacy of Convolution 8.3 Basic Concepts and Development of the Fourier Series 8-5 Linearity and Complex-Exponential Excitation Illustrative Example, Trigonometric and Complex Forms Derivation of the Fourier Series Inner Products and Orthogonality, Limitations on Functions Representable by a Fourier Series The Dirichlet Conditions The Trigonometric Fourier Series Relation to Complex Form Cyclic-Frequency and Radian-Frequency Forms Periodicity of Fourier Series Representations 8.4 Calculation of the Fourier Series 8-19 Sinusoidal Signals Non-sinusoidal Signals The Fourier Series of Periodic Signals over an Integer Number of Fundamental Periods Minimum Mean-Squared Error of Fourier-Series Partial Sums The Fourier Transform of Even and Odd Periodic Signals 8.5 Numerical Computation of the Fourier Series 8-32 The Discrete Fourier Transform (DFT) 8.6 Convergence of the Fourier Series 8-38 Continuous Signals Signals with Discontinuities and The Gibbs Phenomenon 8.7 Properties of the Fourier Series 8-42 Linearity Time Shifting Frequency Shifting Time Reversal Time Scaling Change of Period Time Differentiation Time Integration Effect of Average Value Multiplication-Convolution Duality Periodic Convolution Integral Conjugation Parseval?s Theorem 8.8 Use of Tables and Properties 8-60 8.9 Bandlimited Signals 8-63 8.10 Responses of LTI Systems with Periodic Excitation 8-68 Harmonic Response 8.11 Summary of Important Points 8-70 Exercises 8-71 Chapter 9 - The Discrete-Time Fourier Series 9.1 Introduction and Goals 9-1 9.2 Periodic Excitation and Response of LTI Systems 9-1 Inadequacy of Convolution 9.3 Basic Concepts and Development of the Fourier Series 9-3 Linearity and Complex-Exponential Excitation Illustrative Example, Trigonometric and Complex Forms Derivation of the Fourier Series Completeness of a Finite Sum, Vector and Matrix Methods, Orthogonality, Projections, Summation Forms, Relation to the Discrete Fourier Transform (DFT), Complex and Trigonometric Forms Representing Periodic Functions with the DTFS Conservation of Information 9.4 Properties of the Fourier Series 9-30 Linearity, Time Shifting, Frequency Shifting, Conjugation and Time Reversal Time Scaling Restrictions Stemming from Discrete Time Change of Period Multiplication-Convolution Duality Periodic Convolution Sum First Backward Difference Accumulation Even and Odd Signals Parseval?s Theorem 9.5 Convergence of the Fourier Series 9-49 9.6 Responses of LTI Systems with Periodic Excitation 9-52 Harmonic Response 9.7 Summary of Important Points 9-57 Exercises 9-58 Chapter 10 - The Continuous-Time Fourier Transform 10.1 Introduction and Goals 10-1 10.2 Aperiodic Excitation and Response of LTI Systems 10-1 Inadequacy of the Fourier Series 10.3 Basic Concepts and Development of the Fourier Transform 10-3 The Transition from the Fourier Series to the Fourier Transform Normalization and Notation Changes Definition of the Fourier Transform Cyclic and Radian Frequency Forms, Time and Frequency Domains, Physical Interpretation 10.4 Convergence and the Generalized Fourier Transform 10-14 Convergence Factor, Negative Frequency, 10.5 Numerical Computation of the Fourier Transform 10-21 10.6 Properties of the Fourier Transform 10-25 Linearity Time Shifting and Frequency Shifting Time Scaling and Frequency Scaling The Uncertainty Principle Transform of a Conjugate Even Magnitude and Odd Phase Multiplication-Convolution Duality System Interconnection, Frequency Response Time Differentiation Transforms of Periodic Signals Parseval?s Theorem Integral Definition of an Impulse Duality Total-Area Integral Using Fourier Transforms Time Integration 10.7 Summary of Important Points 10-48 Exercises 10-50 Chapter 11 - The Discrete-Time Fourier Transform 11.1 Introduction and Goals 11-1 11.2 Basic Concepts and Development of the Fourier Transform 11-1 Graphical Illustration Analytical Derivation Definition of the Fourier Transform Cyclic and Radian Frequency Forms, Four Fourier Methods Comparison Summary 11.3 Convergence of the Fourier Transform 11-6 11.4 Numerical Computation of the Fourier Transform 11-6 11.5 Properties of the Fourier Transform 11-7 Linearity Time Shifting and Frequency Shifting Time Scaling and Frequency Scaling Limitations Due to Discrete Time Transform of a Conjugate Differencing and Accumulation Time Reversal Multiplication-Convolution Duality System Interconnection, Frequency Response Accumulation Definition of a Periodic Impulse Function Parseval?s Theorem 11.6 Relations Among Fourier Methods 11-24 Discrete-Periodic Duality CTFT and CTFS Information Equivalence, Periodic Extension CTFT and DTFT Sampling and Impulse Sampling DTFT and DTFS Information Equivalence, Periodic Extension Method Comparison Examples 11.7 Summary of Important Points 11-45 Exercises 11-46 Chapter 12 - Continuous-Time Fourier Transform Analysis of Signals and Systems 12.1 Introduction and Goals 12-1 12.2 Frequency Response 12-1 Cascade and Parallel Connections, Power Spectrum, Audio Amplifiers and Filters, Graphic Equalizer 12.3 Ideal Filters 12-9 Distortion Distortionless System Filter Classifications Lowpass, Highpass, Bandpass, Bandstop Ideal Filter Frequency Responses Distortionless in Passband Bandwidth Impulse Responses and Causality Impulse and Frequency Responses of Causal Filters The Power Spectrum Noise Removal 12.4 Practical Passive Filters 12-18 The RC Lowpass Filter Impedance and Voltage Division The RLC Bandpass Filter Resonance Examples of Filters Automobile Suspension 12.5 Log-Magnitude Frequency-Response Graphs and Bode Diagrams 12-27 Component Diagrams Poles and Zeros One-Real-Pole System Asymptotes, Octaves, Decades One-Real-Zero-System Integrators and Differentiators Frequency-Independent Gain Complex Pole and Zero Pairs Natural Radian Frequency, Damping Ratio 12.6 Practical Active Filters 12-44 Operational Amplifiers Inverting and Non-Inverting Amplifiers, Integrator, Lowpass Filter 12.7 Communication Systems 12-56 Modulation Problems of Communicating at a Distance, Frequency Multiplexing Double-Sideband Suppressed-Carrier Modulation Sidebands, Demodulation, Synchronous Demodulation, Local Oscillator Double-Sideband Transmitted-Carrier Modulation Envelope Detector, Asynchronous Demodulation, Overmodulation Single-Sideband Modulation and Demodulation Pulse Amplitude Modulation 12.8 Impulse Sampling 12-70 Aliases, The Sampling Theorem, Intrerpolation 12.9 Summary of Important Points 12-76 Exercises 12-78 Chapter 13 - Discrete-Time Fourier Transform Analysis of Signals and Systems 13.1 Introduction and Goals 13-1 13.2 Ideal Filters 13-1 Distortion Distortionless System Filter Classifications Lowpass, Highpass, Bandpass, Bandstop Ideal Filter Frequency Responses Distortionless in Passband Impulse Responses and Causality Impulse and Frequency Responses of Causal Filters Filtering Images Pixels, Discrete Space, Causality, 13.3 Practical Filters 13-12 Lowpass, Highpass, Bandpass, Bandstop, Responses to Standard Signals, Moving Average Filters, Advantages over Continuous-Time Filters 13.4 Summary of Important Points 13-25 Exercises 13-26 Chapter 14 - Sampling and the Discrete Fourier Transform 14.1 Introduction and Goals 14-1 14.2 Representing a Continuous-Time Signal by Samples 14-2 Qualitative Concepts The Sampling Theorem Aliases, Nyquist Rate, Oversampling, Undersampling Aliasing Anti-Aliasing Filter Timelimited and Bandlimited Signals Interpolation Sinc Function, Zero-Order Hold, First-Order Hold Sampling a Sinusoid 14.3 Bandlimited Periodic Signals 14-22 Described by a Finite Set of Numbers 14.4 The Discrete Fourier Transform and its Relation to Other Fourier Methods 14-27 Sampling, Windowing, Frequency-Domain Sampling, Periodic Repetition Approximating the CTFT with the DFT Computing the CTFS with the DFT Computing the DTFS with the DFT Approximating the DTFT with the DFT Approximating Continuous-Time Convolution with the DFT Approximating Discrete-Time Convolution with the DFT Approximating Continuous-Time Periodic Convolution with the DFT Computing Discrete-Time Periodic Convolution with the DFT 14.5 The Fast Fourier Transform 14-36 Number of Complex Multiplications and Additions, Importance of an Integer Power of 2 as Number of Points 14.6 Summary of Important Points 14-38 Exercises 14-40 Chapter 15 - The Laplace Transform 15.1 Introduction and Goals 15-1 15.2 Development of the Laplace Transform 15-2 Derivation and Definition Generalization of Fourier Transform, Natural Result of Complex- Exponential Excitation Region of Convergence Relation to Causality The Unilateral Laplace Transform 15.3 Properties of the Laplace Transform 15-16 Linearity Time Shifting Complex-Frequency Shifting Time Scaling Frequency Scaling First Time Derivative Nth Time Derivative Complex-Frequency Differentiation Multiplication-Convolution Duality Integration Initial Value Theorem Final Value Theorem Conditions for Validity 15.4 The Inverse Laplace Transform Using Partial-Fraction Expansion 15-25 Proper Rational Functions of s, Handling Repeated Poles 15.5 Laplace Transform - Fourier Transform Equivalence 15-38 15.6 Solution of Differential Equations with Initial Conditions 15-39 15.7 Transfer Functions from Circuit and System Diagrams 15-41 15.8 System Stability 15-47 Pole Locations, Marginal Stability 15.9 Parallel, Cascade and Feedback Connections 15-50 Loop Transfer Function, Beneficial Effects 15.10 System Responses to Standard Signals 15-56 Unit Step Response Stable and Unstable First and Second Order Systems, Damping, Resonance Response to a Sinusoid Applied at a Finite Time Steady-State or Forced Response 15.11 Pole-Zero Diagrams and Graphical Calculation of Frequency Response 15-66 Vector Interpretation, Influences of Poles and Zeros, Lowpass, Bandpass, Highpass Examples 15.12 Standard Realizations of Systems 15-77 Canonical Direct-Form 2, Cascade, Parallel 15.13 Summary of Important Points 15-82 Exercises 15-84 Chapter 16 - The z Transform 16.1 Introduction and Goals 16-1 16.2 Development of the z Transform 16-1 Definition and Derivation Generalization of Discrete-Time Fourier Transform, Natural Result of Complex-Exponential Excitation Region of Convergence The Unilateral z Transform 16.3 Properties of the z Transform 16-8 Linearity Time Shifting Delay and Advance, Relation to Block Diagrams Change of Scale Relation to Frequency Shifting Initial Value Theorem z-Domain Differentiation Convolution in Discrete Time Differencing Accumulation Final Value Theorem Conditions for Validity 16.4 The Inverse z Transform 16-17 Synthetic Division, Partial-Fraction Expansion 16.5 Solution of Difference Equations with Initial Conditions 16-23 16.6 The Relationships Among the z Transform, the Discrete-Time Fourier Transform and the Laplace Transform 16-25 s-Plane-to-z-Plane Mapping, Relation to Aliasing 16.7 Transfer Functions 16-29 16.8 System Stability 16-31 Pole Locations, Marginal Stability 16.9 Parallel, Cascade and Feedback Connections 16-32 Loop Transfer Function 16.10 System Responses to Standard Signals 16-33 Unit Sequence Response First and Second Order Systems Response to a Sinusoid Applied at a Finite Time Steady-State or Forced Response 16.11 Pole-Zero Diagrams and the Graphical Calculation of Frequency Response 16-41 Vector Interpretation, Influences of Poles and Zeros 16.12 Simulating Continuous-Time Systems with Discrete-Time Systems 16-47 Discretization, Impulse-Invariant Design 16.13 Sampled-Data Systems 16-49 Analog-to-Digital and Digital-to-Analog Converters, Zero-Order Hold, Sampling-Rate Effects 16.14 Standard Realizations of Systems 16-57 Canonical Direct-Form 2, Cascade, Parallel 16.15 Summary of Important Points 16-60 Exercises 16-62 Appendix A - Useful Mathematical Relations Appendix B - The Continuous-Time Fourier Series Pairs Appendix C - Discrete-Time Fourier Series Pairs Appendix D - Continuous-Time Fourier Transform Pairs Appendix E - Discrete-Time Fourier Transform Pairs Appendix F - Laplace Transform Pairs Appendix G - z Transform Pairs Web Appendix A - Introduction to MATLAB Web Appendix B - Method for Finding Least Common Multiples Web Appendix C - Example of Linearizing and Small Signal Analysis Web Appendix D - Derivations of Convolution Properties Web Appendix E - An Exploration of Impulse Properties using Convolution Web Appendix F - Derivations of the Properties of the Continuous- Time Fourier Series Web Appendix G - Derivations of the Properties of the Discrete- Time Fourier Series Web Appendix H - Derivations of the Properties of the Continuous- Time Fourier Transform Web Appendix I - Derivations of the Properties of the Discrete- Time Fourier Transform Web Appendix J - Frequency Response of an Ideal Lowpass Filter Web Appendix K - Sampling Methods Web Appendix L - The DFT in Relation to the Other Fourier Methods - With Examples Web Appendix M - The Fast Fourier Transform Web Appendix N - Derivations of the Properties of the Laplace Transform Web Appendix O - Derivations of the Properties of the z Transform Web Appendix P - Complex Numbers and Complex Functions Web Appendix Q - Differential and Difference Equations Web Appendix R - Vectors and Matrices
Library of Congress Subject Headings for this publication:
Signal processing -- Textbooks.
System analysis -- Textbooks.