Chapter 1 Introduction to Signals
1.1 A Signal
1.2 Signal Modeling
1.3 Continuous-Time, Discrete-Time and Digital Signals
1.3.1 Continuous-Time Signal
1.3.2 Discrete-Time Signal
1.4 Elementary Continuous Time Signals
1.4.1 Unit Step Function
1.4.2 Unit Ramp Function
1.4.3 Unit Parabolic Function
1.4.4 Impulse Function
1.4.5 Rectangular Pulse Function
1.4.6 Triangular Pulse Function
1.4.7 Signum Function
1.4.8 Sinc Function
1.4.9 Gaussian Function
1.4.10 Sinusoidal Signal
1.4.11 Real Exponential Signals
1.4.12 Complex Exponential Signal
1.5 Representation of Discrete-Time Signals
1.5.1 Graphical Representation
1.5.2 Functional Representation
1.5.3 Tabular Representation
1.5.4 Sequence Representation
1.6 Elementary Discrete-Time Signals
1.6.1 Unit Step Sequence
1.6.2 Unit Ramp Sequence
1.6.3 Unit-Sample Sequence (Unit Impulse Sequence)
1.6.4 Exponential Sequence
1.6.5 Sinusoidal Signal
1.6.6 Complex Exponential Signal
1.7 Basic Operations on Signals
1.7.1 Time Shifting
1.7.2 Time Reversal
1.7.3 Amplitude Scaling
1.7.4 Time Scaling
1.7.5 Signal Addition
1.7.6 Signal Multiplication
1.8 Classification of Signals
1.8.1 Continuous-Time and Discrete-Time Signals
1.8.2 Deterministic and Random Signals
1.8.3 Periodic and Aperiodic Signals
1.8.4 Symmetric (Even) and Anti-symmetric (Odd) Signals
1.8.5 Energy and Power Signals
1.8.6 Causal and Non-Causal Signals
1.9 Signals and Vectors
1.9.1 Vector
1.9.2 Vector Addition
1.9.3 Scalar Multiplication
1.9.4 Dot Product (Inner Product)
1.9.5 Norm (Length) in Rn
1.9.6 Distance
1.9.7 Angle
1.9.8 Projection
1.10 Vector Space
1.10.1 Subspace
1.10.2 Linear Dependence and Independence
1.10.3 Spanning a Subspace
1.10.4 Basis
1.10.5 Orthogonality in Vectors
1.10.6 Orthonormal Vectors
1.10.7 Orthogonal Subspace
1.10.8 Orthogonal Bases
1.11 Orthogonal Vectors Space
1.12 Orthogonality in Real Signals
1.13 Orthogonality in Complex Signal
1.14 Orthogonal Signal Space
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 2 Introduction to Systems
2.1 A System
2.2 Classification of Systems
2.2.1 Continuous-Time and Discrete-Time Systems
2.2.2 Lumped-Parameter and Distributed-Parameter Systems
2.2.3 Static and Dynamic Systems
2.2.4 Causal and Non-Causal Systems
2.2.5 Linear and Non-Linear Systems
2.2.6 Time-Invariant and Time-Variant Systems
2.2.7 Stable and Unstable Systems
2.3 System Modeling
2.3.1 Modeling of Mechanical and Electrical Elements
2.3.2 Examples of Discrete-Time System Models
2.4 Invertibility and Inverse Systems
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 3 Time Domain Analysis of Discrete-Time Systems
3.1 Introduction
3.2 Solution of Difference Equations
3.3 Natural Response (Zero-Input Response)
3.4 Forced Response (Zero State Response)
3.5 Total Response
3.6 Impulse Response
3.7 Representation of Discrete-Time Signals in Terms of Impulses
3.8 Impulse Response and Convolution Sum
3.9 Properties of Convolution
3.9.1 The Distributive Property
3.9.2 The Associative Property
3.9.3 Commutative Property
3.9.4 The Shifting Property
3.9.5 The Convolution with an Impulse
3.9.6 Convolution with a Shifted Impulse
3.10 Convolution of Two Sequences
3.10.1 Matrix Convolution
3.11 Causality
3.12 FIR and IIR Systems
3.13 Stability
3.14 BIBO Stability Criterion
3.15 Step Response
3.16 Correlation of Two Sequences
3.16.1 Cross-Correlation
3.16.2 Auto-Correlation
3.16.3 Properties of Cross-Correlation and Auto-Correlation Sequences
3.16.4 Computation of Correlation
3.16.5 Correlation of Power and Periodic Signals
3.17 Inverse System and Deconvolution
3.17.1 Inverse System
3.17.2 Deconvolution
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 4 Time Domain Analysis of Continuous-Time Systems
4.1 Introduction
4.2 Solution of Differential Equations (Classical Method)
4.2.1 Natural Response
4.2.2 Forced Response
4.2.3 Total Response
4.3 Representation of a Continuous-Time Signal
4.4 Convolution Integral
4.5 Properties of Convolution
4.6 Impulse Response of Interconnected Systems
4.6.1 Systems in Parallel
4.6.2 Systems in Cascade
4.7 Causality
4.8 Graphical Procedure to Perform Convolution
4.9 Stability
4.10 Step Response
4.11 Correlation
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 5 Fourier Series Analysis of Continuous-Time Periodic Signals
5.1 Introduction
5.2 Fourier Series Representation of Periodic Signals
5.3 Evaluation of Fourier Coefficients
5.4 Symmetry Conditions
5.4.1 Half Wave Symmetry
5.5 Cosine Representation
5.6 Exponential Fourier Series
5.7 Existence of Fourier Series
5.8 Properties of Continuous-Time Fourier Series
5.8.1 Linearity
5.8.2 Time Shifting
5.8.3 Time Reversal
5.8.4 Time Scaling
5.8.5 Multiplication
5.8.6 Convolution
5.8.7 Conjugation
5.8.8 Parseval’s Theorem
5.9 Power Representation using the Fourier Series
5.10 Fourier Spectrum
5.11 Gibb’s Phenomenon
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 6 The Continuous-Time Fourier Transform
6.1 Introduction
6.2 Development of Fourier Transform
6.3 Existence of Fourier Transform
6.4 Fourier Transform of Some Standard Signals
6.4.1 Rectangular Pulse
6.4.2 Triangular Pulse
6.5 Properties of Fourier Transform
6.5.1 Linearity
6.5.2 Time Shifting
6.5.3 Time Reversal
6.5.4 Frequency Shifting Property
6.5.5 Time Scaling
6.5.6 Differentiation in Time
6.5.7 Differentiation in Frequency
6.5.8 Time Integration
6.5.9 Conjugation
6.5.10 Fourier Transform of Complex and Real Functions
6.5.11 Auto-Correlation
6.5.12 Duality
6.5.13 Convolution
6.5.14 Multiplication Property
6.6 Fourier Transform of a Periodic Signal
6.7 Modulation
6.8 System Analysis with Fourier Transform
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 7 Signal and System Analysis using the Laplace Transform
7.1 Introduction
7.2 Convergence of the Laplace Transform
7.3 s - Plane
7.4 The Unilateral Laplace Transform
7.5 Properties of Unilateral Laplace Transform
7.5.1 Linearity
7.5.2 Transform of Derivatives
7.5.3 Transform of the Integrals
7.5.4 Scaling Property
7.5.5 Time Shift
7.5.6 Frequency Shift
7.5.7 Differentiation in the s-Domain
7.5.8 Time Convolution
7.5.9 Frequency Convolution
7.5.10 Initial Value Theorem
7.5.11 Final Value Theorem
7.6 Inversion of Unilateral Laplace Transform
7.6.1 Distinct Poles
7.6.2 Multiple Poles
7.6.3 Complex Roots
7.7 Inversion of the Bilateral Laplace Transform
7.8 Solution of Differential Equations using Laplace Transform
7.9 Analysis of Electrical Networks using Laplace Transform
7.9.1 Initial Conditions
7.9.2 Transformed Form of Elements
7.10 Stability
7.11 Block Diagram Representation
7.11.1 Summer
7.11.2 Gain
7.11.3 Feed Back
7.11.4 Integrator
7.11.5 Cascade Connection of Blocks
7.11.6 Parallel Connection of Blocks
7.12 Signal-Flow Graph
7.13 System Realization
7.13.1 Direct Form-I
7.13.2 Direct Form-II
7.13.3 Cascade Form
7.13.4 Parallel Form Realization
7.14 State Space Analysis
7.14.1 State and State Variable of a System
7.14.2 Procedure for Developing State Equations for RLC Networks
7.14.3 State-Space Representation from System’s Transfer Function
7.14.4 Derivation of Transfer Function from State Model
7.14.5 Time Domain Solution of State Equation
7.14.6 Laplace Transform Solution of State Equation
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 8 Fourier Analysis of Discrete-Time Signals
8.1 Introduction
8.2 Discrete Frequency Spectrum and Frequency Range
8.2.1 Properties of Discrete Fourier Series
8.3 Discrete-Time Fourier Transform
8.3.1 Existence of Discrete-Time Fourier Transform
8.3.2 Properties of Discrete-Time Fourier Transform
8.4 Frequency Response of Discrete-Time Systems
8.4.1 Frequency Response of Second Order System
8.5 Transfer Function
8.6 The Discrete Fourier Transform (DFT)
8.6.1 Introduction
8.7 Zero Padding
8.8 Properties of the DFT
8.8.1 Circular Convolution of Two Sequences
8.9 Fast Fourier Transform (FFT)
8.9.1 Decimation-in-Time Algorithm
8.10 Summary of Steps of Radix - 2 DIT-FFT Algorithm
8.11 Decimation-in-Frequency Algorithm
8.12 Summary of Steps for Radix - 2 DIF-FFT Algorithm
8.13 Differences and Similarities between DIT and DIF Algorithms
8.14 IDFT using FFT Algorithm
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 9 Sampling
9.1 Introduction
9.2 Analog to Digital Conversion
9.3 Sampling and Aliasing
9.4 Impulse Sampling
9.5 Sampling Theorem
9.6 Anti Aliasing Filter
9.7 Pulse Sampling
9.8 Flat-Top Sampling
9.9 Signal Reconstruction
9.10 Bandpass Signals
9.11 Sampling Bandpass Signals
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 10 Signal and System Analysis using the Z-Transform
10.1 Introduction
10.2 The z-Transform
10.3 z-Transform and ROC of Finite Duration Sequences
10.3.1 Right Hand Sequence
10.3.2 Left Hand Sequence
10.3.3 Two Sided Sequence
10.4 Properties of Region of Convergence
10.5 Properties of z-Transform
10.5.1 Linearity
10.5.2 Time Shifting
10.5.3 Multiplication by an Exponential Sequence
10.5.4 Time Reversal
10.5.5 Multiplication by n
10.5.6 Convolution
10.5.7 Time Expansion
10.5.8 Conjugation
10.5.9 Complex Convolution Theorem
10.5.10 Parseval’s Relation
10.5.11 Correlation
10.5.12 Initial Value Theorem
10.5.13 Final Value Theorem
10.6 The Inverse z-Transform
10.6.1 Long Division Method
10.6.2 Partial Fraction Expansion Method
10.6.3 Residue Method
10.6.4 Convolution Method
10.7 The System Function
10.8 Relationship between z-Transform and DTFT
10.9 Stability Criterion
10.10 Solution of Difference Equations using z-Transform
10.11 Relationship between s-plane and z-plane
10.12 Block Diagram Representation
10.12.1 Direct Form I Realization
10.12.2 Direct Form II
10.12.3 Cascade Form
10.12.4 Parallel Form Realization
10.13 State Variable Model for Discrete-Time Systems
10.14 Deconvolution using z-Transform
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Chapter 11 Signal Transmission through Linear Systems
11.1 Introduction
11.2 Distortionless Transmission through a System
11.3 Linear Phase Systems
11.4 Ideal Filters
11.5 Signal Bandwidth
11.6 System Bandwidth
11.7 Relationship between Bandwidth and Rise Time
Short Questions and Answers
Multiple Choice Questions
Answers to Multiple Choice Questions
Appendix A
Appendix B
Index
Bibliography