Signal Processing and Linear systems

Signal Processing and Linear Systems
Contents
Preface
Background
B.1 Complex Numbers
B1.1 A Historical Note
B.1.2 Algebra of Complex Numbers
B.2 Sinusoids
B.2.1 Addition of Sinusoids
B.2.2 Sinusoids in Terms of Exponentials
B.3 Sketching Signals
B.3.1 Monotonic Exponentials
B.3.2 The Exponentiality Varying Sinusoid
B.4 Cramer's Rule
B.5 Partial Fraction Expansion
B.5.1 Method of Clearing Fractions
B.5.2 The Heaviside "Cover-Up" Method
B.5.3 Repeated Factors of Q(x)
B.5.4 A Combination of Heaviside "Cover-Up" and Clearing Fractions
B.5.5 Improper F(x) with m = n
B.5.6 Modified Partial Fractions
B.6 Vectors and Matrices
B.6.1 Some Definitions and Properties
B.6.2 Matrix Algebra
B.7 MATLAB: Elementary Operations
B.7.1 MATLAB Overview
B.7.2 Calculator Operations
B.7.3 Vector Operations
B.7.4 Simple Plotting
B.7.5 Element-by-Element Operations
B.7.6 Matrix Operations
B.7.7 Partial Fraction Expansions
B.8 Appendix: Useful Mathematical Formulas
B.8.1 Some Useful Constants
B.8.2 Complex Numbers
B.8.3 Sums
B.8.4 Taylor and Maclaurin Series
B.8.5 Power Series
B.8.6 Trigonometric Functions
B.8.7 Common Derivative Formulas
B.8.8 Indefinite Integrals
B.8.9 L'Hopital's Rule
B.8.10 Solution of Quadratic and Cubic Equations
Problems
Chapter 1: Signals and Systems
1.1 Size of a Signal
1.1.1 Signal Energy
1.1.2 Signal Power
1.2 Some Useful Signal Operations
1.2.1 Time Shifting
1.2.2 Time Scaling
1.2.3 Time Reversal
1.2.4 Combined Operations
1.3 Classification of Signals
1.3.1 Continuous-Time and Discrete-Time Signals
1.3.2 Analog and Digital Signals
1.3.3 Periodic and Aperiodic Signals
1.3.4 Energy and Power Signals
1.3.5 Deterministic and Random Signals
1.4 Some Useful Signal Models
1.4.1 The Unit Step Function u(t)
1.4.2 The Unit Impulse Function delta(t)
1.4.3 The Exponential Function e^st
1.5 Even and Odd Functions
1.5.1 Some Properties of Even and Odd Functions
1.5.2 Even and Odd Components of a Signal
1.6 Systems
1.7 Classification of Systems
1.7.1 Linear and Nonlinear Systems
1.7.2 Time-Invariant and Time-Varying Systems
1.7.3 Instantaneous and Dynamic Systems
1.7.4 Causal and Noncausal Systems
1.7.5 Continuous-Time and Discrete-Time Systems
1.7.6 Analog and Digital Systems
1.7.7 Invertible and Noninvertible Systems
1.7.8 Stable and Unstable Systems
1.8 System Model: Input-Output Description
1.8.1 Electrical Systems
1.8.2 Mechanical Systems
1.8.3 Electromechanical Systems
1.9 Internal and External Descriptions of a System
1.10 Internal Description: The State Space Description
1.11 MATLAB: Working with Functions
1.11.1 Anonymous Functions
1.11.2 Relation Operators and the Unit Step Function
1.11.3 Visualizing Operations on the Independent Variable
1.11.4 Numerical Integration and Estimating Signal Energy
1.12 Summary
Problems
Chapter 2: Time-Domain Analysis of Continuous-Time Systems
2.1 Introduction
2.2 System Response to Internal Conditions: The Zero-Input Response
2.2.1 Some Insights into the Zero-Input Behavior of a System
2.3 The Unit Impulse Response h(t)
2.4 System Response to External Input: The Zero-State Response
2.4.1 The Convolution Integral
2.4.2 Graphical Understanding of Convolution Operation
2.4.3 Interconnected Systems
2.4.4 A Very Special Function for LTIC Systems: THe Everlasting Exponential e^st
2.4.5 Total Response
2.5 System Stability
2.5.1 External (BIBO) Stability
2.5.2 Internal (Asymptotic) Stability
2.5.3 Relationship Between BIBO and Asymptotic Stability
2.6 Intuitive Insights into System Behavior
2.6.1 Dependence of System Behavior on Characteristic Modes
2.6.2 Response Time of a System: The System Time Constant
2.6.3 Time Constant and Rise Time of a System
2.6.4 Time Constant and Filtering
2.6.5 Time Constant and Rate of Information Transmission
2.6.7 The Resonance Phenomenon
2.7 MATLAB: M-Files
2.7.1 Script M-Files
2.7.2 Function M-Files
2.7.3 For-Loops
2.7.4 Graphical Understanding of Convolution
2.8 Appendix: Determining the Impulse Response
2.9 Summary
Problems
Chapter 3: Signal Representation By Fourier Series
3.1 Signals as Vectors
3.1.1 Component of a Vector
3.1.2 Component of a Signal
3.1.3 Extension to Complex Signals
3.2 Signal Comparison: Correlation
3.2.1 Application to Signal Detection
3.2.2 Correlation Functions
3.3 Signal Representation by and Orthgonal Signal Set
3.3.1 Orthogonal Vector Space
3.3.2 Orthogonal Signal Space
3.4 Trigonometric Fourier Series
3.4.1 The Effect of Symmetry
3.4.2 Determining the Fundamental Frequency and Period
3.5 Existence and Convergence of the Fourier Series
3.5.1 Convergence of a Series
3.5.2 The Role of Amplitude and Phase Spectra in Waveshaping
3.6 Exponential Fourier Series
3.6.1 Exponential Fourier Spectra
3.6.2 Parseval's Theorem
3.6.3 Making Life Easier: Fourier Series Properties
3.7 LTIC System Response to Periodic Inputs
3.8 Numerical Computation of D_n
3.9 MATLAB: Fourier Series Applications
3.9.1 Periodic Functions and the Gibbs Phenomenon
3.9.2 Optimization and Phase Spectra
3.10 Summary
Problems
Chapter 4: Continuous-Time Signal Analysis: The Fourier Transform
4.1 Aperiodic Signal Representation by the Fourier Integral
4.1.1 Physical Appreciation of the Fourier Transform
4.1.2 LTIC System Response Using the Fourier Transform
4.2 Transforms of Some Useful Functions
4.3 Some Properties of the Fourier Transform
Time-Frequency Duality in the Transform Operations
Linearity
Conjugation and Conjugate Symmetry
Duality
The Scaling Property
Reciprocity of Signal Duration and Its Bandwidth
The Time-Shifting Property
The Frequency-Shifting Property
4.4 Signal Transmission Through LTIC Systems
4.5 Ideal and Practical Filters
4.6 Signal Energy
4.7 Application to Communications: Amplitude Modulation
4.7.1 Double-Sideband, Suppressed Carrier (DSB-SC) Modulation
4.7.2 Amplitude Modulation (AM)
4.7.3 Single-Sideband Modulation (SSB)
4.8 Angle Modulation
4.8.1 The Concept of Instantaneous Frequency
4.8.2 Bandwidth of Angle-Modulated Signals
4.8.3 Generation and Demodulation of Angle-Modulated Signals
4.8.4 Frequency-Division Multiplexing
4.9 Data Truncation: Window Functions
4.9.1 Using Windows in Filter Design
4.10 MATLAB: FOurier Transform Topics
4.10.1 The Sinc Function and the Scaling Property
4.10.2 Parseval's Theorem and Essential Bandwidth
4.10.3 Spectral Sampling
4.10.4 Kaiser Window Functions
4.11 Summary
Problems
Chapter 5: Sampling
5.1 The Sampling Theorem
5.1.1 Practical Sampling
5.2 Signal Reconstruction
5.2.1 Practical Difficulties in Signal Reconstruction
5.2.2 Some Applications of the Sampling Theorem
5.3 Analog-to-Digital (A/D) Conversion
5.4 Dual of Time Sampling: Spectral Sampling
5.5 Numerical Computation of the Fourier Transform: The Discrete Fourier Transform
5.5.1 Some Properties of the DFT
5.5.2 Some Applications of the DFT
5.6 The Fast Fourier Transform (FFT)
5.7 MATLAB: The Discrete Fourier Transform
5.7.1 Computing the Discrete Fourier Transform
5.7.2 Improving the Picture with Zero Padding
5.7.3 Quantization
5.8 Summary
Problems
Chapter 6: Continuous-Time System Analysis Using the Laplace Transform
6.1 The Laplace Transform
6.1.1 An Intuitive Undesrtanding of the Laplace Transform
6.1.2 Analytical Development of the Bilateral Laplace Transform
6.1.3 Finding the Inverse Transform
6.2 Some Properties of the Laplace Transform
6.2.1 Time Shifting
6.2.2 Frequency Shifting
6.2.3 The Time-Differentiation Property
6.2.4 The Time-Integration Property
6.2.5 The Scaling Property
6.2.6 Time Convolution and Frequency Convolution
6.3 Solution of Differential and Integro-Differential Equations
6.3.1 Comments on Initial Conditions at 0- and at 0+
6.3.2 Zero-State Response
6.3.3 Stability
6.4 Analysis of Electrical Networks: The Transformed Network
6.4.1 Analysis of Active Circuits
6.5 Block Diagrams
6.6 System Realization
6.6.1 Direct Form I Realization
6.6.2 Direct Form II Realization
6.6.3 Cascade and Parallel Realization
6.6.4 Transposed Realization
6.6.5 Using Operational Amplifiers for System Realization
6.7 Application to Feedback and Controls
6.7.1 Analysis of a Simple Control System
6.7.2 Analysis of a Second-Order System
6.7.3 Root Locus
6.7.4 Steady-State Errors
6.7.5 Compensation
6.7.6 Stability Considerations
6.8 The Bilarteral Laplace Transform
6.8.1 Properties of Bilateral Laplace Transform
6.8.2 Using the Bilateral Transform for Linear System Analysis
6.9 Summary
Problems
Chapter 7: Frequency Response and Analog Filters
7.1 Frequency Response of an LTIC system
7.1.1 Steady-State Response to Causal Sinusoidal Inputs
7.2 Bode Plots
7.2.1 Constant (k * a1 * a2) / (b1 * b3)
7.2.2 Pole (or Zero) at the origin
7.2.3 First-Order Pole (or Zero)
7.2.4 Second-Order Pole (or Zero)
7.2.5 The Transfer Function from the Frequency Response
7.3 Control System Design Using Frequency Response
7.3.1 Relative Stability: Gain and Phase Margins
7.3.2 Transient Performance in Terms of Frequency Response
7.4 Filter Design by Placement of Poles and Zeros of H(s)
7.4.1 Dependence of Frequency Response on Poles and Zeros of H(s)
7.4.2 Lowpass Filters
7.4.3 Bandpass Filters
7.4.4 Notch (Bandstop) Filters
7.4.5 Practical Filters and Their Specifications
7.5 Butterworth Filters
7.6 Chebyshev FIlters
7.6.1 Inverse Chebyshev Filters
7.6.2 Elliptic Filters
7.7 Frequency Transformations
7.7.1 Highpass Filters
7.7.2 Bandpass Filters
7.7.3 Bandstop Filters
7.8 Filters to Satisfy Distortionless Transmission Conditions
7.9 MATLAB: Continuous-Time Filters
7.9.1 Frequency Response and Polynomial Evaluation
7.9.2 Butterworth Filters and the Find Command
7.9.3 Using Cascaded Second-Order Sections for Butterworth FIlter Realization
7.10 Summary
Problems
Chapter 8: Discrete-Time Signals and Systems
8.1 Introduction
8.1.1 Size of a Discrete-Time Signal
8.2 Useful Signal Operations
8.3 Some Useful Discrete-Time Signal Models
8.3.1 Discrete-Time IMpulse Function delta[n]
8.3.2 Discrete-Time Unit Step Function u[n]
8.3.3 Discrete-Time Exponential y^n
8.3.4 Discrete-Time Complex Exponential e^j(ohm)(n)
8.3.5 Discrete-Time Sinusoid cos((ohm)(n) + (theta))
8.4 Aliasing and Sampling Rate
8.5 Examples of Discrete-Time Systems
8.6 MATLAB: Representing, Manipulating, and Plotting Discrete-Time Signals
8.6.1 Discrete-Time Functions and Stem Plots
8.7 Summary
Problems
Chapter 9: Time-Domain Analysis of Discrete-Time Systems
9.1 Classification of Discrete-Time Systems
9.2 Discrete-Time System Equations
9.2.1 Recursive (Iterative) Solution of Difference Equation
9.3 System Response to Internal Conditions: The Zero-Input Response
9.4 The Unit Impulse Response h[n]
9.4.1 The Closed-Form Solution of h[n]
9.5 System Response to External Input: The Zero-State Response
9.5.1 Graphical Procedure for the Convolution Sum
9.5.2 Interconnected Systems
9.5.3 Total Response
9.6 System Stability
9.6.1 External (BIBO) Stability
9.6.2 Internal (Asymptotic) Stability
9.6.3 Relationship between BIBO and Asymptotic Stability
9.7 Intuitive Insights into System Behavior
9.8 MATLAB: Discrete Time Systems
9.8.1 System Responses Through Filtering
9.8.2 A Custom Filter Function
9.8.3 Discrete-Time Convolution
9.9 Appendix: Impulse Response for a Special Case
9.10 Summary
Problems
Chapter 10: Fourier Analysis of Discrete-Time Signals
10.1 Periodic Signal Representation by Discrete-Time Fourier Series
10.1.1 Fourier Spctra of a Periodic Signal x[n]
10.2 Aperiodic Signal Representation by Fourier Integral
10.3 Properties of the DTFT
10.4 DTFT Connection with the CTFT
10.5 LTI Discrete-Time System Analysis by DTFT
10.5.1 Distortionless Transmission
10.5.2 Ideal and Practical Filters
10.7 Generalization of the DTFT to the z-Transform
10.6 Signal Processing by the DFT and FFT
10.6.1 Computation of the Discrete-Time Fourier Series (DTFS)
10.6.2 Computation of the DTFT and Its Inverse
10.6.3 Discrete-Time Filtering (Convolution) Using the DFT
10.6.4 Block Convolution
10.8 MATLAB: Working with the DTFS and the DTFT
10.8.1 Computing the Discrete-Time Fourier Series
10.8.2 Measuring Code Performance
10.9 Summary
Problems
Chapter 11: Discrete-Time System Analysis Using the z-Transform
11.1 The z-Transform
11.1.1 Inverse Transform by Partial Fraction Expansion and Tables
11.1.2 Inverser z-Transform by Power Series Expansion
11.2 Some Properties of the z-Transform
11.2.1 Time-Shifting Properties
11.2.2 z-Domain Scaling Property (Multiplication by y^n)
11.2.3 z-Domain Differentiation Property (Multiplication by n)
11.2.4 Time-Reversal Property
11.2.5 Convolution Property
11.3 z-Transform Solution of Linear Difference Equations
11.3.1 Zero-State Response of LTID Systems: The Transfer Function
11.3.2 Stability
11.3.3 Inverse Systems
11.4 System Realization
11.5 Connecting the Laplace and z-Transforms
11.6 Sampled-Data (Hybrid) Systems
11.7 The Bilateral z-Transform
11.7.1 Properties of the Bilateral z-Transform
11.7.2 Using the Bilateral z-Transform for Analysis of LTID Systems
11.8 Summary
Problems
Chapter 12: Frequency Response and Digital Filters
12.1 Frequency Response of Discrete-Time Systems
12.1.1 The Periodic Nature of Frequency Response
12.2 Frequency Response From Pole-Zero Locations
12.3 Digital Filters
12.4 Filter Design Camera
12.4.1 Time-Domain Equivalence Criterion
12.4.2 Frequency-Domain Equivalence Criterion
12.5 Recursive Filter Design by the Time-Domain Criterion: The Impulse Invariance Method
12.6 Recursive Filter Design By The Frequency-Domain Criterion: The BIlinear Transformation Method
12.6.1 Bilinear Transformation Method with Prewarping
12.7 Nonrecursive Filters
12.7.1 Symmetry Conditions for LInear-Phase Response
12.8 Nonrecursive Filter Design
12.8.1 Time-Domain Equivalence Method of FIR Filter Design
12.8.2 Nonrecursive Filter Design by the Frequency-Domain Criterion: The Frequency Sampling Method
12.9 MATLAB: Designing High-Order Filters
12.9.1 IIR Filter Design Using the Bilinear Transform
12.9.2 FIR Filter Design Using Frequency Sampling
12.10 Summary
Problems
Chapter 13: State-Space Analysis
13.1 Mathematical Preliminaries
13.1.1 Derivatives and Integrals of a Matrix
13.1.2 The Characteristic Equation of a Matrix: The Cayley-Hamilton Theorem
13.1.3 Computation of an Exponential and a Power of a Matrix
13.2 Introduction of State Space
13.3 A Systematic Procedure to Determine State Equations
13.3.1 Electrical Circuits
13.3.2 State Equations from a Transfer Function
13.4 Solution of State Equations
13.4.1 Laplace Transform Solution of State Equations
13.5 Linear Transformation of a State Vector
13.5.1 Diagonalization of Matrix A
13.6 Controllability and Observability
13.6.1 Inadequacy of the Transfer Function Description of a System
13.7 State-Space Analysis of Discrete-Time Systems
13.7.1 Solution in State Space
13.8 MATLAB: Toolboxes and State-Space Analysis
13.8.1 z-Transform Solutions to Discrete=Time, State-Space Systems
13.8.2 Transfer Functions from State-Space Representations
13.8.3 Controllability and Observability of Discrete-Time Systems
13.9 Summary
Problems
Index
A - B
B - C
C - D
D - F
F - I
I - M
M - N
N - R
R - S
S - T
U - Z