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Signals and Systems

Book Description

Signals and Systems provides comprehensive coverage of all topics within the signals and systems' paper offered to undergraduates of electrical and electronics engineering.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Dedication
  5. Preface
  6. Mathematical Preliminaries
    1. Matrices
      1. Equality of Two Matrices
      2. Vector
      3. Square Matrix
      4. Diagonal Matrix
      5. Identity Matrix or Unity Matrix
      6. Zero Matrix
      7. Determinant of a Matrix
      8. Singular Matrix
      9. Transpose
      10. Symmetric Matrix
      11. Conjugate Matrix
      12. Conjugate Transpose
      13. Hermitian Matrix
      14. Addition of Matrices
      15. Multiplication of a Matrix by a Scalar
      16. Multiplication of a Matrix by a Matrix
      17. Power of a Matrix
      18. Rank of a Matrix
      19. Minor of a Matrix
      20. Co-factor
      21. Adjoint Matrix
      22. Inverse of a Matrix
      23. More Properties of Matrices
    2. Trigonometry Formulae
    3. Calculus
  7. 1. Fundamentals of Signals and Systems
    1. 1.1 Signals and Systems
    2. 1.2 Classification of Signals
    3. 1.3 Continuous Time Signals
      1. 1.3.1 Classification of Continuous Time Signals
    4. 1.4 Basic Continuous Time Signals
      1. 1.4.1 Unit Step Function
      2. 1.4.2 Ramp Function
      3. 1.4.3 Unit Impulse Function
      4. 1.4.4 Complex Exponential Signals
      5. 1.4.5 General Complex Exponential Function
      6. 1.4.6 Real Exponential Function
      7. 1.4.7 Sinusoidal Signal
    5. 1.5 Classification of Continuous Time Systems
      1. 1.5.1 Static and Dynamic Systems
      2. 1.5.2 Causal and Non-Causal Systems
      3. 1.5.3 Time Invariant and Time Variant Systems
      4. 1.5.4 Linear and Non-Linear Systems
      5. 1.5.5 Linear Time Invariant System
      6. 1.5.6 Stable and Unstable Systems
      7. 1.5.7 Invertible and Non-Invertible Systems
      8. 1.5.8 Feedback System
    6. 1.6 Discrete Time Signals
    7. 1.7 Concept of Frequency in Discrete Time Signals
    8. 1.8 Standard Discrete Time Signals
      1. 1.8.1 Unit Sample Sequence
      2. 1.8.2 Unit Step Sequence
      3. 1.8.3 Unit Ramp Sequence
      4. 1.8.4 Exponential Sequence
    9. 1.9 Classification of Discrete Time Signals
      1. 1.9.1 Even and Odd Signals
      2. 1.9.2 Periodic Signals and Non-Periodic Signals
      3. 1.9.3 Deterministic and Random Signals
      4. 1.9.4 Energy Signals and Power Signals
      5. 1.9.5 Multichannel and Multidimensional Signals
    10. 1.10 Discrete Time Systems
    11. 1.11 Representation of Discrete Time Systems
      1. 1.11.1 Adder
      2. 1.11.2 Constant Multiplier
      3. 1.11.3 Signal Multiplier
      4. 1.11.4 Unit Delay Block
      5. 1.11.5 Unit Advance Block
    12. 1.12 Classifications of Discrete Time Systems
      1. 1.12.1 Static and Dynamic Systems
      2. 1.12.2 Causal and Non-Causal Systems
      3. 1.12.3 Time Invariant and Time Variant Systems
      4. 1.12.4 Linear and Non-Linear Systems
      5. 1.12.5 Stable and Unstable Systems
    13. 1.13 Nyquist Rate
    14. 1.14 Sampling Theorem
    15. 1.15 Aliasing
    16. 1.16 Convolution
    17. 1.17 Correlation
      1. 1.17.1 Cross-correlation and Auto-correlation
    18. Additional Solved Examples
    19. Significant Points
    20. Short Questions and Answers
    21. Exercises
    22. Multiple Choice Questions
    23. Answers
  8. 2. Fourier Series
    1. 2.1 Fourier Series
    2. 2.2 Dirichlet Conditions
    3. 2.3 Determination of Fourier Co-efficients
    4. 2.4 Wave Symmetry
      1. 2.4.1 Even or Mirror Symmetry
      2. 2.4.2 Odd or Rotation Symmetry
      3. 2.4.3 Half Wave Symmetry
      4. 2.4.4 Quarter Wave Symmetry
    5. 2.5 Exponential Form of Fourier Series
    6. Additional Solved Examples
    7. Significant Points
    8. Short Questions and Answers
    9. Exercises
    10. Multiple Choice Questions
    11. Answers
  9. 3. Fourier Transform
    1. 3.1 Fourier Transform
    2. 3.2 Condition for the Existence of Fourier Integral
    3. 3.3 Fourier Transform of Some Functions
      1. 3.3.1 Fourier Transform of Gate Function
      2. 3.3.2 Fourier Transform of Impulse Function
      3. 3.3.3 Fourier Transform of Shifted Impulse Function
      4. 3.3.4 Fourier Transform of One-Sided Exponential Function
      5. 3.4.1 Fourier Transform of Two-Sided Exponential Function
      6. 3.3.1 Fourier Transform of sgn(t) e–a(t)
      7. 3.3.2 Fourier Transform of Signum Function
      8. 3.3.3 Fourier Transform of f (t)=1
      9. 3.3.4 Fourier Transform of u (t)
    4. 3.4 Fourier Transformation Theorem
      1. 3.4.1 Linearity
      2. 3.4.2 Time Scaling
      3. 3.4.3 Time Differentiation
      4. 3.4.4 Time Shifting Property
      5. 3.4.5 Translation in the Frequency Domain
      6. 3.4.6 Modulation Theorem
      7. 3.4.7 Symmetry or Duality Property
      8. 3.4.8 Time Convolution Property
      9. 3.4.9 Frequency Convolution
      10. 3.4.10 Frequency Differentiation
      11. 3.4.11 Time Integration
      12. 3.4.12 Fourier Transform of f (–t)
      13. 3.4.13 Symmetry Properties of Fourier Transform
    5. 3.5 Fourier Transform of Periodic Signals
      1. 3.5.1 Fourier Transform in Terms of Fourier Series Co-efficients
    6. 3.6 Energy Density and Power Spectral Density
      1. 3.6.1 Energy in a Signal
      2. 3.6.2 Power in a Signal
      3. 3.6.3 Energy Spectrum and Energy Spectral Density
      4. 3.6.4 Parseval’s Relation for Energy Signals
      5. 3.6.5 Power Spectral Density
      6. 3.6.7 Power Spectrum for Periodic Signal
      7. 3.6.8 Parseval’s Relation for Periodic Signals
    7. 3.7 Nyquist Theorem
      1. 3.7.1 Proof of Nyquist Theorem
      2. 3.7.2 Practical Implementation
    8. 3.8 System Analysis Using Fourier Transform
    9. 3.9 Relation between Differential Equation and System Function
    10. Additional Solved Examples
    11. Significant Points
    12. Short Questions and Answers
    13. Exercises
    14. Multiple Choice Questions
    15. Answers
  10. 4. Laplace Transform
    1. 4.1 Laplace Transform
    2. 4.2 Region of Convergence (ROC)
    3. 4.3 Inverse Laplace Transformation
    4. 4.4 Basic Properties of Laplace Transforms
    5. 4.5 Laplace Transform of a Derivative
    6. 4.6 Laplace Transform of an Integral ∫ f (t) dt
    7. 4.7 Laplace Transform of Some Common Time Function
      1. 4.7.1 Unit Step Function
      2. 4.7.2 Impulse Function
      3. 4.7.3 Ramp Function
      4. 4.7.4 Parabolic Function
      5. 4.7.5 f(t)=eatu(t)with a > 0
      6. 4.7.6 f(t)=e–atu(t)
      7. 4.7.7 Sinusoidal Function
      8. 4.7.8 Cosine Function
      9. 4.7.9 Hyperbolic Sine and Cosine Functions
      10. 4.7.10 Damped Sine and Cosine Functions
      11. 4.7.11 Damped Hyperbolic Sine and Cosine Function
      12. 4.7.12 Laplace Transform of tn
    8. 4.8 Laplace Transform of Two-Sided Functions (BLT)
      1. 4.8.1 ROC for f(t)=ebtu(–t)
      2. 4.8.2 ROC for f(t)=eatu(t)
      3. 4.8.3 ROC for Two-Sided Function
    9. 4.9 Initial Value Theorem
    10. 4.10 Final Value Theorem
    11. 4.11 Partial Fraction Expansions
    12. 4.12 Relation between Step Response and Impulse Response
    13. 4.13 Application of Laplace Transforms in Circuit
    14. 4.14 Pure Resistive Element
    15. 4.15 Pure Inductive Element
    16. 4.16 Pure Capacitive Element
    17. 4.17 Step Response of Series R-L Circuit
    18. 4.18 Step Response of Series R-C Circuit
    19. 4.19 Step Response of Series R-L-C Circuit
    20. 4.20 Impulse Response of Series R-L Circuit
    21. 4.21 Impulse Response of Series R-C Circuit
    22. 4.22 Pulse Response of Series R-L Circuit
    23. 4.23 Pulse Response of Series R-C Circuit
    24. Additional Solved Examples
    25. Significant Points
    26. Short Questions and Answers
    27. Exercises
    28. Multiple Choice Questions
    29. Answers
  11. 5. System Modelling
    1. 5.1 Transfer Function
    2. 5.2 Impulse Response and Transfer Function
    3. 5.3 Properties of Transfer Function (TF)
    4. 5.4 Definition of Basic Elements of Block Diagram
    5. 5.5 Basic Definition of Signal Flow Graph (SFG)
    6. 5.6 Mason’s Gain Formula
    7. 5.7 Modelling of Mechanical Systems
      1. 5.7.1 Translational Motion
      2. 5.7.2 Rotational Motion
    8. 5.8 Modelling of Electrical Systems
      1. 5.8.1 Resistor
      2. 5.8.2 Inductor
      3. 5.8.3 Capacitor
    9. 5.9 Analogous Systems
      1. 5.9.1 Force-Voltage (f-v)Analogy
      2. 5.9.2 Force-Current (f-i)Analogy
    10. 5.10 Representation by Nodal Method
    11. Additional Solved Examples
    12. Significant Points
    13. Short Questions and Answers
    14. Exercises
    15. Multiple Choice Questions
    16. Answers
  12. 6. z-Transform
    1. 6.1 z-Transform
    2. 6.2 Region of Convergence (ROC)
    3. 6.3 Properties of z-transform
      1. 6.3.1 Linearity
      2. 6.3.2 Time Shifting
      3. 6.3.3 Scaling in z-domain
      4. 6.3.4 Time Reversal
      5. 6.3.5 Differentiation in z-domain
      6. 6.3.6 Convolution in Time Domain
      7. 6.3.7 Correlation of Two Sequences
      8. 6.3.8 Multiplication of Two Sequences
      9. 6.3.9 Conjugate of a Complex Sequence
      10. 6.3.10 Real Part of a Sequence
      11. 6.3.11 Imaginary Part of a Sequence
      12. 6.3.12 Initial Value Theorem
      13. 6.3.13 Final Value Theorem
      14. 6.3.14 Partial Sum
      15. 6.3.15 Parseval’s Theorem
    4. 6.4 z-Transform of Right-Sided Exponential Sequences
    5. 6.5 z-Transform of Left-Sided Exponential Sequences
    6. 6.6 Finite Length Sequence
    7. 6.7 z-Transform of Unit Sample Sequence
    8. 6.8 z-Transform of Delayed Unit Sample Sequence
    9. 6.9 z-Transform of Unit Step Sequence
    10. 6.10 z-Transform of Folded Unit Step Sequence
    11. 6.11 z-Transform of the Signal x(n)=Nanu(n)
    12. 6.12 z-Transform of Unit Ramp Sequence
    13. 6.13 z-Transform of Causal cosine Sequence
    14. 6.14 z-Transform of Causal sine Sequence
    15. 6.15 z-Transform of ancos(nω)u(n)
    16. 6.16 z-Transform of ansin (nω) u(n)
    17. 6.17 Inverse z-transform
    18. 6.18 Inverse z-transform Using Partial Fraction Expansion
    19. 6.19 Inverse z-transform Using Power Series Expansion
    20. 6.20 System Function and Pole-Zero Plots from z-transform
    21. 6.21 Pole-Zero Plot
    22. 6.22 System Function of the LTI System
    23. 6.23 Causality and Stability in Terms of z-transform
    24. Additional Solved Examples
    25. Significant Points
    26. Short Questions and Answers
    27. Exercises
    28. Multiple Choice Questions
    29. Answers
  13. 7. Convolution
    1. 7.1 Convolution Theorem for Continuous System
    2. 7.2 Properties of Linear Convolution
    3. 7.3 Graphical Convolution
    4. 7.4 Discrete Convolution
      1. 7.4.1 Properties of Discrete Convolution
    5. 7.5 Important Properties of Systems
    6. 7.6 Convolution Theorem in z-transform
    7. 7.7 Circular or Periodic Convolution
    8. Additional Solved Examples
    9. Significant Points
    10. Short Questions and Answers
    11. Exercises
    12. Multiple Choice Questions
    13. Answers
  14. 8. Stability
    1. 8.1 Effect of Location of Poles on Stability
    2. 8.2 Routh-Hurwitz Criterion
      1. 8.2.1 Hurwitz’s Criterion
      2. 8.2.2 Routh’s Stability Criterion
    3. Significant Points
    4. Short Questions and Answers
    5. Exercises
    6. Multiple Choice Questions
    7. Answers
  15. 9. State Variable Approach (Continuous Systems)
    1. 9.1 Advantages and Disadvantages of Modern Control Theory
    2. 9.2 Concepts of State, State Variables and State Model
    3. 9.3 State Model
    4. 9.4 Non-Uniqueness of the State Model
    5. 9.5 Different Representations of a State Model
      1. 9.5.1 State Space Representation Using Phase Variables Controllable Conical Form (CCF)
      2. 9.5.2 Phase Variable CCF Form for Numerator Terms
      3. 9.5.3 Phase Variable OCF Form
      4. 9.5.4 Cascade Decomposition
      5. 9.5.5 Parallel Decomposition
      6. 9.5.6 Jordan’s Canonical Form
    6. 9.6 Eigen Value
    7. 9.7 Transfer Function Derivation from the State Model
    8. 9.8 Solution of the State Equation
    9. 9.8.1 Solution of Homogeneous State Equation
    10. 9.8.2 State Transition Matrix
    11. 9.8.3 Properties of STM, φ (t)
    12. 9.8.4 Solution of Non-homogeneous State Equation
    13. 9.9 Controllability
    14. 9.10 Observability
    15. Additional Solved Examples
    16. Significant Points
    17. Short Questions and Answers
    18. Exercises
    19. Multiple Choice Questions
    20. Answers
  16. 10. State Variable Methods (Discrete Case)
    1. 10.1 Delay Elements
    2. 10.2 State Model of a First-Order System
    3. 10.3 State Model of a Second-Order System
    4. 10.4 Non-Uniqueness State Model
    5. 10.5 Canonical Form of State Model
    6. 10.6 Transfer Function from State Model
    7. 10.7 Solution of State Equation
      1. 10.7.1 z-Transform Method
      2. 10.7.2 Series Expansion Method
    8. Additional Solved Examples
    9. Significant Points
    10. Short Questions and Answers
    11. Exercises
    12. Multiple Choice Questions
    13. Answers
  17. 11. Discrete Fourier Transform and Fast Fourier Transform
    1. 11.1 Fourier Transform of Discrete Time Signals
    2. 11.2 Properties of Fourier Transform
    3. 11.3 Inverse Fourier Transform
    4. 11.4 Magnitude/Phase Transfer Functions Using Fourier Transform
    5. 11.5 Discrete Fourier Transform
    6. 11.6 Properties of Discrete Fourier Transform
      1. 11.6.1 Periodicity
      2. 11.6.2 Linearity
      3. 11.6.3 Circular Symmetries of a Sequence
      4. 11.6.4 Symmetry Properties
      5. 11.6.5 Circular Convolution
      6. 11.6.6 Time Reversal of a Sequence
      7. 11.6.7 Circular Time Shift Sequence
      8. 11.6.8 Circular Frequency Shift of a Sequence
      9. 11.6.9 Circular Correlation of Two Sequences
      10. 11.6.10 Multiplication of Two Sequences
      11. 11.6.11 Parseval’s Theorem
    7. 11.7 Relationship Between DFT and z-Transform
    8. 11.8 Fast Fourier Transform (FFT)
    9. 11.8.1 Properties of FFT
    10. 11.9 Radix-2 FFT Algorithm
    11. 11.10 Radix-2 DIT-FFT Algorithm
    12. 11.11 Computational Complexity Compared to Direct Computation
    13. 11.12 Memory Requirement and In Place Computations
    14. 11.13 Bit Reversal
    15. 11.14 Radix-2 DIF FFT Algorithm
    16. Significant Points
    17. Short Questions and Answers
    18. Exercises
    19. Multiple Choice Questions
    20. Answers
  18. 12. Structures and design of digital filters
    1. 12.1 Classification of Filters
      1. 12.1.1 Analog Filter
      2. 12.1.2 Digital Filter
    2. 12.2 Review of Analog Filters
    3. 12.3 Specification of the Frequency Response Characteristics of the Filters
      1. 12.3.1 Low Pass Filter
      2. 12.3.2 High Pass Filter
      3. 12.3.3 Band Pass Filter
      4. 12.3.4 Band Stop Filter
    4. 12.4 Specifications of Phase Response
    5. 12.5 Structure of Digital Filter
    6. 12.6 Describing Equation
    7. 12.7 Structure of FIR Filter
      1. 12.7.1 Direct Form Structure of FIR Filter
      2. 12.7.2 Cascade Form Structure of FIR Filter
      3. 12.7.3 Frequency Sampling Structure for FIR Filter
      4. 12.7.4 Lattice Structure for FIR Filter
    8. 12.8 Structure for IIR Filter
      1. 12.8.1 Direct Form Structure for IIR Filter
      2. 12.8.2 Cascade Structure for IIR Filter
      3. 12.8.3 Parallel Form Structure for IIR Filter
      4. 12.8.4 Lattice Structure of IIR Filter
    9. 12.9 Realization Procedure for Digital Filter
      1. 12.9.1 Recursive Realization
      2. 12.9.2 Non-Recursive Realization
      3. 12.9.3 FFT Realization
    10. 12.10 Notch Filter
    11. 12.11 Comb Filter
    12. 12.12 All-pass Filter
    13. 12.13 Design of an IIR Filter
      1. 12.13.1 Impulse Response Invariance Method
      2. 12.13.2 Bilinear Transformation Method
      3. 12.13.3 Digital Butterworth Filter
      4. 12.13.4 Digital Chebyshev Filter
    14. 12.14 Design of FIR Filters
      1. 12.14.1 Fourier Series Method for Design of FIR Filters
    15. Significant Points
    16. Short Questions and Answers
    17. Exercises
    18. Multiple Choice Questions
    19. Answers
  19. Appendix
  20. Acknowledgements
  21. Copyright