Signals and Systems - Electrical Engineering

Part 1 Introduction ACKNOWLEDGMENTS xvi

CHAPTER 0 From the Ground Up!.............................................................................

0.1 Signals and Systems and Digital Technologies

0.2 Examples of Signal Processing Applications
- 0.2.1 Compact-Disc Player
- 0.2.2 Software-Defined Radio and Cognitive Radio...............................
- 0.2.3 Computer-Controlled Systems

0.3 Analog or Discrete?
- 0.3.1 Continuous-Time and Discrete-Time Representations
- 0.3.2 Derivatives and Finite Differences
- 0.3.3 Integrals and Summations.........................................................
- 0.3.4 Differential and Difference Equations

0.4 Complex or Real?
- 0.4.1 Complex Numbers and Vectors
- 0.4.2 Functions of a Complex Variable
- 0.4.3 Phasors and Sinusoidal Steady State
- 0.4.4 Phasor Connection

0.5 Soft Introduction to MATLAB
- 0.5.1 Numerical Computations
- 0.5.2 Symbolic Computations
- Problems............................................................................................

Signals and Systems Part 2 Theory and Application of Continuous-Time

CHAPTER 1 Continuous-Time Signals.........................................................................

1.1 Introduction

1.2 Classification of Time-Dependent Signals...............................................

1.3 Continuous-Time Signals Contents v
- 1.3.1 Basic Signal Operations—Time Shifting and Reversal
- 1.3.2 Even and Odd Signals
- 1.3.3 Periodic and Aperiodic Signals
- 1.3.4 Finite-Energy and Finite Power Signals

1.4 Representation Using Basic Signals.......................................................
- 1.4.1 Complex Exponentials
- 1.4.2 Unit-Step, Unit-Impulse, and Ramp Signals
- 1.4.3 Special Signals—the Sampling Signal and the Sinc
  - and Windowing 1.4.4 Basic Signal Operations—Time Scaling, Frequency Shifting,
- 1.4.5 Generic Representation of Signals..............................................

1.5 What Have We Accomplished? Where Do We Go from Here?....................
- Problems............................................................................................

CHAPTER 2 Continuous-Time Systems.......................................................................

2.1 Introduction

2.2 System Concept
- 2.2.1 System Classification................................................................

2.3 LTI Continuous-Time Systems
- 2.3.1 Linearity
- 2.3.2 Time Invariance
- 2.3.3 Representation of Systems by Differential Equations
- 2.3.4 Application of Superposition and Time Invariance
- 2.3.5 Convolution Integral.................................................................
- 2.3.6 Causality
- 2.3.7 Graphical Computation of Convolution Integral
- 2.3.8 Interconnection of Systems—Block Diagrams
- 2.3.9 Bounded-Input Bounded-Output Stability

- Problems............................................................................................

CHAPTER 3 The Laplace Transform............................................................................

3.1 Introduction

3.2 The Two-Sided Laplace Transform
- 3.2.1 Eigenfunctions of LTI Systems
- 3.2.2 Poles and Zeros and Region of Convergence

3.3 The One-Sided Laplace Transform
- 3.3.1 Linearity
- 3.3.2 Differentiation
- 3.3.3 Integration
- 3.3.4 Time Shifting...........................................................................
- 3.3.5 Convolution Integral.................................................................

3.4 Inverse Laplace Transform vi Contents
- 3.4.1 Inverse of One-Sided Laplace Transforms
- 3.4.2 Inverse of Functions Containinge−ρsTerms
- 3.4.3 Inverse of Two-Sided Laplace Transforms

3.5 Analysis of LTI Systems
- 3.5.1 LTI Systems Represented by Ordinary Differential Equations
- 3.5.2 Computation of the Convolution Integral

- Problems............................................................................................

CHAPTER 4 Frequency Analysis: The Fourier Series..................................................

4.1 Introduction

4.2 Eigenfunctions Revisited

4.3 Complex Exponential Fourier Series

4.4 Line Spectra
- 4.4.1 Parseval’s Theorem—Power Distribution over Frequency
- 4.4.2 Symmetry of Line Spectra

4.5 Trigonometric Fourier Series

4.6 Fourier Coefficients from Laplace

4.7 Convergence of the Fourier Series

4.8 Time and Frequency Shifting................................................................

4.9 Response of LTI Systems to Periodic Signals...........................................
- 4.9.1 Sinusoidal Steady State.............................................................
- 4.9.2 Filtering of Periodic Signals

4.10 Other Properties of the Fourier Series
- 4.10.1 Reflection and Even and Odd Periodic Signals
- 4.10.2 Linearity of Fourier Series—Addition of Periodic Signals
- 4.10.3 Multiplication of Periodic Signals
- 4.10.4 Derivatives and Integrals of Periodic Signals

- Problems............................................................................................

CHAPTER 5 Frequency Analysis: The Fourier Transform...........................................

5.1 Introduction

5.2 From the Fourier Series to the Fourier Transform

5.3 Existence of the Fourier Transform

5.4 Fourier Transforms from the Laplace Transform

5.5 Linearity, Inverse Proportionality, and Duality
- 5.5.1 Linearity
- 5.5.2 Inverse Proportionality of Time and Frequency
- 5.5.3 Duality
- 5.6 Spectral Representation Contents vii
  - 5.6.1 Signal Modulation
  - 5.6.2 Fourier Transform of Periodic Signals
  - 5.6.3 Parseval’s Energy Conservation
  - 5.6.4 Symmetry of Spectral Representations........................................
- 5.7 Convolution and Filtering.....................................................................
  - 5.7.1 Basics of Filtering
  - 5.7.2 Ideal Filters
  - 5.7.3 Frequency Response from Poles and Zeros..................................
  - 5.7.4 Spectrum Analyzer...................................................................
- 5.8 Additional Properties
  - 5.8.1 Time Shifting
  - 5.8.2 Differentiation and Integration
- 5.9 What Have We Accomplished? What Is Next?
  - Problems............................................................................................

CHAPTER 6 Application to Control and Communications...........................................

6.1 Introduction

6.2 System Connections and Block Diagrams

6.3 Application to Classic Control...............................................................
- 6.3.1 Stability and Stabilization
- 6.3.2 Transient Analysis of First- and Second-Order Control Systems

6.4 Application to Communications
- 6.4.1 AM with Suppressed Carrier
- 6.4.2 Commercial AM
- 6.4.3 AM Single Sideband
- 6.4.4 Quadrature AM and Frequency-Division Multiplexing
- 6.4.5 Angle Modulation

6.5 Analog Filtering
- 6.5.1 Filtering Basics
- 6.5.2 Butterworth Low-Pass Filter Design
- 6.5.3 Chebyshev Low-Pass Filter Design
- 6.5.4 Frequency Transformations
- 6.5.5 Filter Design with MATLAB

6.6 What Have We Accomplished? What Is Next?
- Problems............................................................................................

Signals and Systems Part 3 Theory and Application of Discrete-Time

CHAPTER 7 Sampling Theory......................................................................................

7.1 Introduction

7.2 Uniform Sampling viii Contents
- 7.2.1 Pulse Amplitude Modulation
- 7.2.2 Ideal Impulse Sampling
- 7.2.3 Reconstruction of the Original Continuous-Time Signal
- 7.2.4 Signal Reconstruction from Sinc Interpolation..............................
- 7.2.5 Sampling Simulation with MATLAB

7.3 The Nyquist-Shannon Sampling Theorem
- 7.3.1 Sampling of Modulated Signals

7.4 Practical Aspects of Sampling
- 7.4.1 Sample-and-Hold Sampling
- 7.4.2 Quantization and Coding
- 7.4.3 Sampling, Quantizing, and Coding with MATLAB

- Problems............................................................................................

CHAPTER 8 Discrete-Time Signals and Systems.........................................................

8.1 Introduction.....................................................................................

8.2 Discrete-Time Signals
- 8.2.1 Periodic and Aperiodic Signals
- 8.2.2 Finite-Energy and Finite-Power Discrete-Time Signals
- 8.2.3 Even and Odd Signals
- 8.2.4 Basic Discrete-Time Signals

8.3 Discrete-Time Systems
- 8.3.1 Recursive and Nonrecursive Discrete-Time Systems.....................
  - Equations 8.3.2 Discrete-Time Systems Represented by Difference
- 8.3.3 The Convolution Sum
- 8.3.4 Linear and Nonlinear Filtering with MATLAB..............................
- 8.3.5 Causality and Stability of Discrete-Time Systems

- Problems............................................................................................

CHAPTER 9 The Z-Transform......................................................................................

9.1 Introduction

9.2 Laplace Transform of Sampled Signals...................................................

9.3 Two-Sided Z-Transform
- 9.3.1 Region of Convergence

9.4 One-Sided Z-Transform........................................................................
- 9.4.1 Computing the Z-Transform with Symbolic MATLAB
- 9.4.2 Signal Behavior and Poles
- 9.4.3 Convolution Sum and Transfer Function
- 9.4.4 Interconnection of Discrete-Time Systems................................... Contents ix
- 9.4.5 Initial and Final Value Properties

9.5 One-Sided Z-Transform Inverse
- 9.5.1 Long-Division Method
- 9.5.2 Partial Fraction Expansion
- 9.5.3 Inverse Z-Transform with MATLAB............................................
- 9.5.4 Solution of Difference Equations
- 9.5.5 Inverse of Two-Sided Z-Transforms

- Problems............................................................................................

CHAPTER 10 Fourier Analysis of Discrete-Time Signals and Systems...........................

10.1 Introduction

10.2 Discrete-Time Fourier Transform
- 10.2.1 Sampling, Z-Transform, Eigenfunctions, and the DTFT
- 10.2.2 Duality in Time and Frequency
- 10.2.3 Computation of the DTFT Using MATLAB
- 10.2.4 Time and Frequency Supports
- 10.2.5 Parseval’s Energy Result...........................................................
- 10.2.6 Time and Frequency Shifts........................................................
- 10.2.7 Symmetry
- 10.2.8 Convolution Sum

10.3 Fourier Series of Discrete-Time Periodic Signals......................................
- 10.3.1 Complex Exponential Discrete Fourier Series
- 10.3.2 Connection with the Z-Transform
- 10.3.3 DTFT of Periodic Signals
- 10.3.4 Response of LTI Systems to Periodic Signals
- 10.3.5 Circular Shifting and Periodic Convolution

10.4 Discrete Fourier Transform
- 10.4.1 DFT of Periodic Discrete-Time Signals
- 10.4.2 DFT of Aperiodic Discrete-Time Signals
- 10.4.3 Computation of the DFT via the FFT
- 10.4.4 Linear and Circular Convolution Sums

- Problems............................................................................................

CHAPTER 11 Introduction to the Design of Discrete Filters..........................................

11.1 Introduction

11.2 Frequency-Selective Discrete Filters
- 11.2.1 Linear Phase
- 11.2.2 IIR and FIR Discrete Filters

11.3 Filter Specifications x Contents
- 11.3.1 Frequency-Domain Specifications
- 11.3.2 Time-Domain Specifications

11.4 IIR Filter Design
- 11.4.1 Transformation Design of IIR Discrete Filters
- 11.4.2 Design of Butterworth Low-Pass Discrete Filters
- 11.4.3 Design of Chebyshev Low-Pass Discrete Filters
- 11.4.4 Rational Frequency Transformations
- 11.4.5 General IIR Filter Design with MATLAB

11.5 FIR Filter Design
- 11.5.1 Window Design Method
- 11.5.2 Window Functions

11.6 Realization of Discrete Filters
- 11.6.1 Realization of IIR Filters
- 11.6.2 Realization of FIR Filters

- Problems............................................................................................

CHAPTER 12 Applications of Discrete-Time Signals and Systems.................................

12.1 Introduction

12.2 Application to Digital Signal Processing
- 12.2.1 Fast Fourier Transform
- 12.2.2 Computation of the Inverse DFT
- 12.2.3 General Approach of FFT Algorithms

12.3 Application to Sampled-Data and Digital Control Systems
- 12.3.1 Open-Loop Sampled-Data System
- 12.3.2 Closed-Loop Sampled-Data System

12.4 Application to Digital Communications
- 12.4.1 Pulse Code Modulation
- 12.4.2 Time-Division Multiplexing
  - Multiplexing............................................................................ 12.4.3 Spread Spectrum and Orthogonal Frequency-Division

APPENDIX Useful Formulas.......................................................................................

BIBLIOGRAPHY

INDEX

Signals and Systems - Electrical Engineering

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