Basic System Analysis

Preface
Contents
About the Author
1 Representation of Signals and Systems
1.1 Introduction
1.2 Terminologies Related to Signals and Systems
1.2.1 Signal
1.2.2 System
1.3 Continuous- and Discrete-Time Signals
1.4 Basic Continuous-Time Signals
1.4.1 Unit Impulse Function
1.4.2 Unit Step Function
1.4.3 Unit Ramp Function
1.4.4 Unit Parabolic Function
1.4.5 Unit Rectangular Pulse (or Gate) Function
1.4.6 Unit Area Triangular Function
1.4.7 Unit Signum Function
1.4.8 Unit Sinc Function
1.4.9 Sinusoidal Signal
1.4.10 Real Exponential Signal
1.4.11 Complex Exponential Signal
1.5 Basic Operations on Continuous-Time Signals
1.5.1 Addition of CT Signals
1.5.2 Multiplication of CT Signals
1.5.3 Amplitude Scaling of Signals
1.5.4 Time Scaling of CT Signals
1.5.5 Time Shifting of CT Signals
1.5.6 Signal Reflection or Folding
1.5.7 Inverted CT Signal
1.5.8 Multiple Transformation
1.6 Classification of Signals
1.6.1 Deterministic and Non-deterministic Continuous Signals
1.6.2 Periodic and Non-periodic Continuous Signals
1.6.3 Fundamental Period of Two Periodic Signals
1.6.4 Odd and Even Functions of Continuous-Time Signals
1.6.5 Energy and Power of Continuous-Time Signals
1.7 System
1.8 Linear Time Invariant Continuous (LTIC) Time System
1.9 Properties (Classification) of Continuous-Time System
1.9.1 Linear and Non-linear Systems
1.9.2 Time Invariant and Time Varying Systems
1.9.3 Static and Dynamic Systems (Memoryless and System with Memory)
1.9.4 Causal and Non-causal Systems
1.9.5 Stable and Unstable Systems
1.9.6 Invertibility and Inverse System
1.10 Modeling of Mechanical Systems
1.10.1 Dynamic Equations of Mechanical Translational System
1.11 Electrical Analogue
1.11.1 Force–Voltage Analogy (F–V analogy)
1.11.2 Force–Current Analogy (F–I Analogy)
1.12 Analysis of First- and Second-Order Linear Systems
1.13 First-Order Continuous-Time System
1.13.1 System Modeling
1.13.2 Time Response of First-Order System
1.13.3 Time Domain Specifications
1.14 Second-Order System Modeling
1.15 Time Response of a Second-Order System
1.15.1 Impulse Response
1.15.2 Step Response
1.15.3 Step Response of a Second-Order System
1.15.4 Time Domain Specifications of a Second-Order System
2 Fourier Series Analysis of Continuous-Time Signals
2.1 Introduction
2.2 Periodic Signal Representation by Fourier Series
2.3 Different Forms of Fourier Series Representation
2.3.1 Trigonometric Fourier Series
2.3.2 Complex Exponential Fourier Series
2.3.3 Polar or Harmonic Form Fourier Series
2.4 Properties of Fourier Series
2.4.1 Linearity
2.4.2 Time Shifting Property
2.4.3 Time Reversal Property
2.4.4 Time Scaling Property
2.4.5 Multiplication Property
2.4.6 Conjugation Property
2.4.7 Differentiation Property
2.4.8 Integration Property
2.4.9 Parseval's Theorem
2.5 Existence of Fourier Series—The Dirichlet Conditions
2.6 Convergence of Continuous-Time Fourier Series
2.7 Fourier Series Spectrum
3 Fourier Transform Analysis of Continuous Time Signals
3.1 Introduction
3.2 Representation of Aperiodic Signal by Fourier Integral–The Fourier Transform
3.3 Convergence of Fourier Transforms–The Dirichlet Conditions
3.4 Fourier Spectra
3.5 Connection Between the Fourier Transform and Laplace Transform
3.6 Properties of Fourier Transform
3.6.1 Linearity
3.6.2 Time Shifting
3.6.3 Conjugation and Conjugation Symmetry
3.6.4 Differentiation in Time
3.6.5 Differentiation in Frequency
3.6.6 Time Integration
3.6.7 Time Scaling
3.6.8 Frequency Shifting
3.6.9 Duality
3.6.10 The Convolution
3.6.11 Parseval's Theorem
3.7 Fourier Transform of Periodic Signal
4 The Laplace Transform Method for the Analysis of Continuous-Time Signals and Systems
4.1 Introduction
4.2 Definition and Derivations of the LT
4.2.1 LT of Causal and Non-causal Systems
4.3 The Existence of LT
4.4 The Region of Convergence
4.4.1 Properties of ROCs for LT
4.5 The Unilateral Laplace Transform
4.6 Properties of Laplace Transform
4.6.1 Linearity
4.6.2 Time Shifting
4.6.3 Frequency Shifting
4.6.4 Time Scaling
4.6.5 Frequency Scaling
4.6.6 Time Differentiation
4.6.7 Time Integration
4.6.8 Time Convolution
4.6.9 Complex Frequency Differentiation
4.6.10 Complex Frequency Shifting
4.6.11 Conjugation Property
4.6.12 Initial Value Theorem
4.6.13 Final Value Theorem
4.7 Laplace Transform of Periodic Signal
4.8 Inverse Laplace Transform
4.8.1 Graphical Method of Determining the Residues
4.9 Solving Differential Equation
4.9.1 Solving Differential Equation Without Initial Conditions
4.9.2 Solving Differential Equation with the Initial Conditions
4.9.3 Zero Input and Zero State Response
4.9.4 Natural and Forced Response Using LT
4.10 Time Convolution Property of the Laplace Transform
4.11 Network Analysis Using Laplace Transform
4.11.1 Mathematical Description of R-L-C-Elements
4.11.2 Transfer Function and Pole-Zero Location
4.12 Connection between Laplace Transform and Fourier Transform
4.13 Causality of Continuous-Time Invariant System
4.14 Stability of Linear Time Invariant Continuous System
4.15 The Bilateral Laplace Transform
4.15.1 Representation of Causal and Anti-causal Signals
4.15.2 ROC of Bilateral Laplace Transform
5 The z-Transform Analysis of Discrete Time Signals and Systems
5.1 Introduction
5.2 The z-Transform
5.3 Existence of the z-Transform
5.4 Connection Between Laplace Transform, z-Transform and Fourier Transform
5.5 The Region of Convergence (ROC)
5.6 Properties of the ROC
5.7 Properties of z-Transform
5.7.1 Linearity
5.7.2 Time Shifting
5.7.3 Time Reversal
5.7.4 Multiplication by n
5.7.5 Multiplication by an Exponential
5.7.6 Time Expansion
5.7.7 Convolution Theorem
5.7.8 Initial Value Theorem
5.7.9 Final Value Theorem
5.8 Inverse z-Transform
5.8.1 Partial Fraction Method
5.8.2 Inverse z-Transform using Power Series Expansion
5.8.3 Inverse z-Transform using Contour Integration or the Method of Residue
5.9 The System Function of DT Systems
5.10 Causality of DT Systems
5.11 Stability of DT System
5.12 Causality and Stability of DT System
5.13 z-Transform Solution of Linear Difference Equations
5.13.1 Right Shift (Delay)
5.13.2 Left Shift (Advance)
5.14 Zero-Input and Zero State Response
5.15 Natural and Forced Response
5.16 Difference Equation from System Function
6 State Space Modeling and Analysis
6.1 Introduction
6.2 The State of a System and State Equation of Continuous-Time System
6.3 Vector-Matrix Differential Equation of Continuous-Time System
6.3.1 State Equations for Mechanical Systems
6.3.2 State Equations for Electrical Circuits
6.4 State Equations from Transfer Function
6.4.1 General Case of Representation
6.4.2 Step by Step Procedure to Determine A, B and C Matrices
6.5 Transfer Function of Continuous-Time System from State Equations
6.6 Solution of State Equations
6.6.1 Laplace Transform Solution of State Equations
6.6.2 Time Domain Solution to State Equations
6.6.3 Determination of eAt—The Cayley–Hamilton Theorem
6.7 State Equations of A Discrete-Time System
6.7.1 Canonical Form II Model
6.7.2 Canonical Form I Model
6.7.3 Diagonal Form (Parallel Form) Model
7 Application of MATLAB and Python Programs to Solve Problems
7.1 Application of MATLAB Program
7.2 Application of Python Program to Solve Engineering Problems
Index