Linear Systems and Signals

Linear Systems and Signals, Third Edition, is a textbook for the required junior-year signals and systems course in the typical Electrical Engineering department curriculum. The book's success lies in its thorough, inclusive presentation of key concepts supported by a unique, bottom-up explanation of the theories and reasoning behind the material. The heuristic approach is a trademark of Dr. Lathi's books and a vital reason why instructors who adopt the text stick with it.

This text shares many topics and features with Dr. Lathi's Signal Processing and Linear System, but differs in its sequence of topics, particularly in the placement of the Laplace transform ahead of the Fourier Transform. The two books cover much the same ground but their different organizations mirror the two different approaches in wide use by instructors.

B.P. Lathi is Professor Emeritus at California State University, Sacramento. He is author of Signals and Systems, Linear Systems and Signal Processing, Modern Digital and Analog Communication Systems, and Digital Signal Processing.

Roger Green is Associate Professor of Electrical Engineering at North Dakota State University. He has published numerous scholarly articles and given presentations on MATLAB, Signal Processing, and Fourier Analysis as a member of both the IEEE and ASEE. Along with four colleagues, he is the proud owner of a patent for a Vector Calibration System, designed to identify vector mismatch between a plurality of signal paths and frequencies.

1 BACKGROUND

1.1 Complex Numbers
1.1-1 A Historical Note
1.1-2 Algebra of Complex Numbers
1.2 Sinusoids and Exponentials
1.2-1 Addition of Sinusoids
1.2-2 Sinusoids in Terms of Exponentials
1.2-3 Monotonic Exponentials
1.2-4 The Exponentially Varying Sinusoid
1.3 Cramer's Rule
1.4 Partial Fraction Expansion
1.4-1 Method of Clearing Fractions
1.4-2 The Heaviside "Cover-Up" Method
1.4-3 Repeated Factors of Q(x)
1.4-4 A Combination of Heaviside "Cover-Up" and Clearing Fractions
1.4-6 Modified Partial Fractions
1.5 Vectors and Matrices
1.5-1 Some Definitions and Properties
1.5-2 Matrix Algebra
1.6 MATLAB: Elementary Operations
1.6-1 MATLAB Overview
1.6-2 Calculator Operations
1.6-3 Vector Operations
1.6-4 Simple Plotting
1.6-5 Element-by-Element Operations
1.6-6 Matrix Operations
1.6-7 Partial Fraction Expansions
1.7 Appendix: Useful Mathematical Formulas
1.7-1 Some Useful Constants
1.7-2 Complex Numbers
1.7-3 Sums
1.7-4 Taylor and Maclaurin Series
1.7-5 Power Series
1.7-6 Trigonometric Identities
1.7-7 Common Derivative Formulas
1.7-8 Indefinite Integrals
1.7-9 L'Hôpital's Rule
1.7-10 Solution of Quadratic and Cubic Equations
References
Problems

2 SIGNALS AND SYSTEMS
2.1 Size of a Signals
2.1-1 Signal Energy
2.1-2 Signal Power
2.2 Some Useful Signal Operations
2.2-1 Time Shifting
2.2-2 Time Scaling
2.2-3 Time Reversal
2.2-4 Combined Operations
2.3 Classification of Signals
2.3-1 Continuous-Time and Discrete-Time Signals
2.3-2 Analogue and Digital Signals
2.3-3 Periodic and Aperiodic Signals
2.3-4 Energy and Power Signals
2.3-5 Deterministic and Random Signals
2.4 Some Useful Signal Models
2.4-1 The Unit Step Function u(t)
2.4-2 The Unit Impulse Function ?(t)
2.4-3 The Exponential Function est
2.5 Even and Odd Functions
2.5-1 Some Properties of Even and Odd Functions
2.5-2 Even and Odd Components of a Signal
2.6 Systems and System Classification
2.6-1 Classification of Systems
2.6-2 Linear and Nonlinear Systems
2.6-3 Time-Invariant and Time-Varying Systems
2.6-4 Instantaneous and Dynamic Systems
2.6-5 Causal and Noncausal Systems
2.6-6 Continuous-Time and Discrete-Time Systems
2.6-7 Analogue and Digital Systems
2.6-8 Invertible and Noninvertible Systems
2.6-9 Stable and Unstable Systems
2.7 System Model: Input-Output Description
2.7-1 Electrical Systems
2.7-2 Mechanical Systems
2.7-3 Electromechanical Systems
2.8 Internal and External Descriptions of a System
2.8-1 Internal Description: The State-Space Description
2.9 MATLAB: Working with Functions
2.9-1 Anonymous Functions
2.9-2 Relational Operators and the Unit Step Functions
2.9-3 Visualising Operations on the Independent Variable
2.9-4 Numerical Integration and Estimating Signal Energy
2.10 Summary
References
Problems

3 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS
3.1 Introduction
3.2 System Response to Internal Conditions: The Zero-Input Response
3.2-1 Some Insights into the Zero-Input Behaviour of a System
3.3 The Unit Impulse Response h(t)
3.4 System Response to External Input: The Zero-State Response
3.4-1 The Convolution Integral
3.4-2 Graphical Understanding of Convolution Operation
3.4-3 Interconnected Systems
3.4-4 A Very Special Function for LTIC Systems:
The Everlasting Exponential est
3.4-5 Total Response
3.5 System Stability
3.5-1 External (BIBO) Stability
3.5-2 Internal (Asymptotic) Stability
3.5-3 Relationship Between BIBO and Asymptotic Stability
3.6 Intuitive Insights into System Behaviour
3.6-1 Dependence of System Behaviour on Characteristic Modes
3.6-2 Response Time of a System: The System Time Constant
3.6-3 Time Constant and Rise Time of a System
3.6-4 Time Constant and Filtering
3.6-5 Time Constant and Pulse Dispersion (Spreading)
3.6-6 Time Constant and Rate of Information Transmission
3.6-7 The Resonance Phenomenon
3.7 MATLAB: M-Files
3.7-1 Script M-Files
3.7-2 Function M-Files
3.7-3 For-Loops
3.7-4 Graphical Understanding of Convolution
3.8 Appendix: Determining the Impulse Response
3.9 Summary
References
Problems

4 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS
4.1 Introduction
4.1-1 Size of a Discrete-Time Signal
4.1-2 Useful Signal Operations
4.2 Some Useful Discrete-Time Signal Models
4.2-1 Discrete-Time Impulse Function ?[n]
4.2-2 Discrete-Time Unit Step Function u[n]
4.2-3 Discrete-Time Exponential ? n
4.2-4 Discrete-Time Sinusoid cos(\'04n+?)
4.2-5 Discrete-Time Complex Exponential ej\'04n
4.3 Examples of Discrete-Time Systems
4.3-1 Classification of Discrete-Time Systems
4.4 Discrete-Time System Equations
4.4-1 Recursive (Iterative) Solution of Difference Equation
4.5 System Response to Internal Conditions: The Zero-Input Response
4.6 The Unit Impulse Response h[n]
4.6-1 The Closed-Form Solution of h[n]
4.7 System Response to External Input: The Zero-State Response
4.7-1 Graphical Procedure for the Convolution Sum
4.7-2 Interconnected Systems
4.7-3 Total Response
4.8 System Stability and Behaviour
4.8-1 External (BIBO) Stability
4.8-2 Internal (Asymptotic) Stability
4.8-3 Relationship Between BIBO and Asymptotic Stability
4.8-4 Intuitive Insights into System Behaviour
4.9 MATLAB: Discrete-Time Signals and Systems
4.9-1 Discrete-Time Functions and Stem Plots
4.9-2 System Responses Through Filtering
4.9-3 A Custom Filter Function
4.9-4 Discrete-Time Convolution
4.10 Appendix: Impulse Response for a Special Case
4.11 Summary
Problems

5 CONTINUOUS-TIME SYSTEM ANALYSIS USING THE
LAPLACE TRANSFORM
5.1 The Laplace Transform
5.2 Some Properties of the Laplace Transform
5.2-1 Time Shifting
5.2-2 Frequency Shifting
5.2-3 The Time-Differentiation Property
5.2-4 The Time-Integration Property
5.2-5 The Scaling Property
5.2-6 Time Convolution and Frequency Convolution
5.3 Solution of Differential and Integro-Differential Equations
5.3-1 Comments on Initial Conditions at 0? and at 0+
5.3-2 Zero-State Response
5.3-3 Stability
5.3-4 Inverse Systems
5.4 Analysis of Electrical Networks: The Transformed Network
5.4-1 Analysis of Active Circuits
5.5 Block Diagrams and System Realisations
5.5-1 Direct Form I Realisation
5.5-2 Direct Form II Realisation
5.5-3 Cascade and Parallel Realisations
5.5-4 Transposed Realisation
5.5-5 Using Operational Amplifiers for System Realisation
5.5-6 Application to Feedback and Controls
5.5-7 Analysis of a Simple Control System
5.6 Frequency Response of an LTIC System
5.6-1 Steady-State Response to Causal Sinusoidal Inputs
5.7 Bode Plots
5.7-1 Constant Ka1a2/b1b3
5.7-2 Pole (or Zero) at the Origin
5.7-3 First-Order Pole (or Zero)
5.7-4 Second-Order Pole (or Zero)
5.7-5 The Transfer Function from the Frequency Response
5.8 Filter Design by Placement of Poles and Zeros of H(s)
5.8-1 Dependence of Frequency Response on Poles and Zeros of H(s)
5.8-2 Lowpass Filters
5.8-3 Bandpass Filters
5.8-4 Notch (Bandstop) Filters
5.8-5 Practical Filters and Their Specifications
5.9 The Bilateral Laplace Transform
5.9-1 Properties of the Bilateral Laplace Transform
5.9-2 Using the Bilateral Transform for Linear System Analysis
5.10 MATLAB: Continuous-Time Filters
5.10-1 Frequency Response and Polynomial Evaluation
5.10-2 Butterworth Filters and the Find Command
5.10-3 Using Cascaded Second-Order Sections for Butterworth Filter Realisation
5.10-4 Chebyshev Filters
5.11 Summary
References
Problems

6 DISCRETE-TIME SYSTEM ANALYSIS USING THE Z-TRANSFORM
6.1 The z-Transform
6.1-1 Inverse Transform by Partial Fraction Expansion and Tables
6.1-2 Inverse z-Transform by Power Series Expansion
6.2 Some Properties of the z-Transform
6.2-1 Time-Shifting Properties
6.2-2 z-Domain Scaling Property (Multiplication by ? n)
6.2-3 z-Domain Differentiation Property (Multiplication by n)
6.2-4 Time-Reversal Property
6.2-5 Convolution Property
6.3 z-Transform Solution of Linear Difference Equations
6.3-1 Zero-State Response of LTID Systems: The Transfer Function
6.3-2 Stability
6.3-3 Inverse Systems
6.4 System Realisation
6.5 Frequency Response of Discrete-Time Systems
6.5-1 The Periodic Nature of Frequency Response
6.5-2 Aliasing and Sampling Rate
6.5-3 Frequency Response from Pole-Zero Locations
6.6 Digital Processing of Analogue Signals
6.7 The Bilateral z-Transform
6.7-1 Properties of the Bilateral z-Transform
6.7-2 Using the Bilateral z-Transform for Analysis of LTID Systems
6.7-3 Connecting the Laplace and z-Transforms
6.8 MATLAB: Discrete-Time IIR Filters
6.8-1 Frequency Response and Pole-Zero Plots
6.8-2 Transformation Basics
6.8-3 Transformation by First-Order Backward Difference
6.8-4 Bilinear Transformation
6.8-5 Bilinear Transformation with Prewarping
6.8-6 Example: Butterworth Filter Transformation
6.8-7 Problems Finding Polynomial Roots
6.8-8 Using Cascaded Second-Order Sections to Improve Design
6.9 Summary
References
Problems

7 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES
7.1 Periodic Signal Representation by Trigonometric Fourier Series
7.1-1 The Fourier Spectrum
7.1-2 The Effect of Symmetry
7.1-3 Determining the Fundamental Frequency and Period
7.2 Existence and Convergence of the Fourier Series
7.2-1 Convergence of a Series
7.2-2 The Role of Amplitude and Phase Spectra in Waveshaping
7.3 Exponential Fourier Series
7.3-1 Exponential Fourier Spectra
7.3-2 Parseval’s Theorem
7.3-3 Properties of the Fourier Series
7.4 LTIC System Response to Periodic Inputs
7.5 Generalised Fourier Series: Signals as Vectors
7.5-1 Component of a Vector
7.5-2 Signal Comparison and Component of a Signal
7.5-3 Extension to Complex Signals
7.5-4 Signal Representation by an Orthogonal Signal Set
7.6 MATLAB: Fourier Series Applications
7.6-1 Numerical Computation of Dn
7.6-2 Periodic Functions and the Gibbs Phenomenon
7.6-3 Optimisation and Phase Spectra
7.7 Summary
References
Problems

8 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER TRANSFORM
8.1 Aperiodic Signal Representation by the Fourier Integral
8.1-1 Physical Appreciation of the Fourier Transform
8.2 Transforms of Some Useful Functions
8.2-1 Connection Between the Fourier and Laplace Transforms
8.3 Some Properties of the Fourier Transform
8.4 Signal Transmission Through LTIC Systems
8.4-1 Signal Distortion During Transmission
8.4-2 Bandpass Systems and Group Delay
8.4-3 Ideal and Practical Filters
8.5 Signal Energy
8.6 Application to Communications: Amplitude Modulation
8.6-1 Double-Sideband, Suppressed-Carrier (DSB-SC) Modulation
8.6-2 Amplitude Modulation (AM)
8.6-3 Single-Sideband Modulation (SSB)
8.6-4 Frequency-Division Multiplexing
8.7 Data Truncation: Window Functions
8.7-1 Using Windows in Filter Design
8.8 MATLAB: Fourier Transform Topics
8.8-1 The Sinc Function and the Scaling Property
8.8-2 Parseval’s Theorem and Essential Bandwidth
8.8-3 Spectral Sampling
8.8-4 Kaiser Window Functions
8.9 Summary
References
Problems

9 SAMPLING: THE BRIDGE FROM CONTINUOUS TO DISCRETE
9.1 The Sampling Theorem
9.1-1 Practical Sampling
9.2 Signal Reconstruction
9.2-1 Practical Difficulties in Signal Reconstruction
9.2-2 Some Applications of the Sampling Theorem
9.3 Analogue-to-Digital (A/D) Conversion
9.4 Dual of Time Sampling: Spectral Sampling
9.5 Numerical Computation of the Fourier Transform: The Discrete
Fourier Transform
9.5-1 Some Properties of the DFT
9.5-2 Some Applications of the DFT
9.5-3 The Fast Fourier Transform (FFT)
9.6 MATLAB: The Discrete Fourier Transform
9.6-1 Computing the Discrete Fourier Transform
9.6-2 Improving the Picture with Zero Padding
9.6-3 Quantisation
9.7 Summary
References
Problems

10 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS
10.1 Discrete-Time Fourier Series (DTFS)
10.1-1 Periodic Signal Representation by Discrete-Time Fourier Series
10.1-2 Fourier Spectra of a Periodic Signal x[n]
10.2 Aperiodic Signal Representation by Fourier Integral
10.2-1 Nature of Fourier Spectra
10.2-2 Connection Between the DTFT and the z-Transform
10.3 Properties of the DTFT
10.4 LTI Discrete-Time System Analysis by DTFT
10.4-1 Distortionless Transmission
10.4-2 Ideal and Practical Filters
10.5 DTFT Connection with the CTFT
10.5-1 Use of DFT and FFT for Numerical Computation of the DTFT
10.5-2 Generalisation of the DTFT to the z-Transform
10.6 MATLAB: Working with the DTFS and the DTFT
10.6-1 Computing the Discrete-Time Fourier Series
10.6-2 Measuring Code Performance
10.6-3 FIR Filter Design by Frequency Sampling
10.7 Summary
Reference
Problems

11 STATE-SPACE ANALYSIS
11.1 Mathematical Preliminaries
11.1-1 Derivatives and Integrals of a Matrix
11.1-2 The Characteristic Equation of a Matrix: The Cayley-Hamilton Theorem
11.1-3 Computation of an Exponential and a Power of a Matrix
11.2 Introduction to State Space
11.3 A Systematic Procedure to Determine State Equations
11.3-1 Electrical Circuits
11.3-2 State Equations from a Transfer Function
11.4 Solution of State Equations
11.4-1 Laplace Transform Solution of State Equations
11.4-2 Time-Domain Solution of State Equations
11.5 Linear Transformation of a State Vector
11.5-1 Diagonalisation of Matrix A
11.6 Controllability and Observability
11.6-1 Inadequacy of the Transfer Function Description of a System
11.7 State-Space Analysis of Discrete-Time Systems
11.7-1 Solution in State Space
11.7-2 The z-Transform Solution
11.8 MATLAB: Toolboxes and State-Space Analysis
11.8-1 z-Transform Solutions to Discrete-Time, State-Space Systems
11.8-2 Transfer Functions from State-Space Representations
11.8-3 Controllability and Observability of Discrete-Time Systems
11.8-4 Matrix Exponentiation and the Matrix Exponential
11.9 Summary
References
Problems

Index