Contents

• 1. Introduction to Signals and Systems

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  Objectives, Introduction, Signals, Classification of Signals, Deterministic and Non-Deterministic Signals, Periodic and Aperiodic Signals, Even and Odd Signals, Energy and Power Signals, Some Basic Signals, Unit Impulse Function and Unit Step Function, Ramp Function, Exponential Signal, Sinusoidal Signal, Complex Exponential Signal, Signum Function, Sinc Function, Transformation of Independent Variable, Time Shifting, Time Scaling, Time Reversal/Folding, Steps for Transformation of The Form x(?t + ?), Transformation of Dependent Variable, Amplitude Shifting, Amplitude Scaling, Amplitude Reversal, System, Continuous Time System, Discrete Time System, Symbols Used to Represent a Discrete Time System, Adder, Constant Multiplier, Signal Multiplier, Unit Delay, Unit Advance, Basic System Properties, Causality, Time Invariance and Time Variance, Stability, Linearity, Static and Dynamic System (Memory Less and With Memory), Invertible and Inverse, Order of a System, Interconnection of Systems, Series or Cascade Interconnection of Subsystems, Parallel Interconnection of Subsystems, Series-Parallel Interconnection of Subsystems, Feedback Interconnection of Subsystems, Review Questions, Numerical Problems.
   
  2. Linear Time Invariant (LTI) Systems

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  Objectives, Introduction, Characterization of LTI Systems, The Continuous Time Unit Impulse Response and the Convolution Integral, Graphical Method for Solving Convolution, Convolution Sum for Discrete Time LTI Systems, Steps for Convolution Sum for two Discrete Signals, Properties of Linear Time Invariant (LTI) Systems, The Commutative Property, The Distributive Property, Associative Property of LTI Systems, Static and Dynamic LTI System, Invertibility of LTI Systems, Causality of LTI Systems, Stability of LTI Systems, Unit Step Response of an LTI System, Linear Constant Coefficient Differential Equations, Solution of Differential Equation, Linear Constant Coefficient Difference Equations, Classical Method, Iterative Method for Difference Equations, Basic Building Block or Elements of Continuous Time and Discrete Time LTI Systems, Review Questions, Numerical Problems.
   
  3. Fourier Series Representation of Signals

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  Objectives, Introduction, Fourier Series of Continuous Time Periodic Signal, Trigonometric Fourier Series, Dirichlet Conditions, Polar Fourier Series Representation, Exponential Fourier Series, Symmetry Conditions, Properties of Continuous Time Fourier Series (CTFS), Linearity, Time Shifting Property, Frequency Shifting Property, Time Reversal, Time Scaling, Time Differentiation, Time Convolution, Multiplication, Symmetry Properties, Parseval’s Relation For Continuous Time Periodic Signals, Fourier Series of Discrete Time Periodic Signal (DTFS), Properties of Discrete Time Fourier Series (DTFS), Linearity, Time Shifting Property, Frequency Shifting Property, Time Convolution, Multiplication, Symmetry Properties, Parseval’s Relation For Discrete Time Periodic Signals, Time Reversal, First Difference, Running Sum, Filters, Analog Filters, Digital Filters, Review Questions, Numerical Problems.
   
  4. The Continuous Time Fourier Transform (CTFT)

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  Objectives, Introduction, Comparison Between Various Transforms and Logarithms, Introduction to Fourier Transform, Development of Continuous Time Fourier Transform (CTFT), Continuous Time Fourier Transform (CTFT), Convergence of Continuous Time Fourier Transform (CTFT), Some Useful Functions and Their Fourier Transforms, Gate Function, Sampling Function or Interpolating Function or Sinc Function, Unit Impulse Function, Shifting Property of The Impulse Function, Rectangular Pulse, Gaussian Pulse, Signum Function, Triangular Pulse, CTFT of Continuous Time Periodic Signals, Properties of CTFT, Linearity, Time Shifting, Time Scaling, Time Differentiation, Time Integration, Frequency Shifting, Frequency Differentiation, Conjugation and Conjugate Symmetry, Duality Property, Time Convolution Property, Frequency Convolution or Time Multiplication, Modulation Property, Parseval’s Relation, Systems Characterized by Linear Differential Equations of Constant Coefficients, Review Questions, Numerical Problems.
   
  5. Discrete Time Fourier Transform (DTFT)

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  Objectives, Introduction, Discrete Time Fourier Transform (DTFT) from Discrete Time Fourier Series, Convergence of the DTFT, Fourier Transform for Discrete Time Periodic Signals, Properties of Discrete Time Fourier Transform (DTFT), Periodicity, Linearity, Time Shifting Property, Frequency Shifting Property, Multiplication by n: Frequency Differentiation, Complex Conjugation and Conjugate Symmetry, Time Reversal, Convolution Property, Scaling, Multiplication in Time Domain, Parseval’s Relation, Energy Density Spectrum of Discrete Time Aperiodic Signals, Duality, Differencing and Accumulation in Time, Discrete Time LTI Systems Characterized by Linear Constant Coefficient Difference Equations, Review Questions, Numerical Problems.
   
  6. Laplace Transform

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  Objectives, Introduction, Definition of Laplace Transform, Relationship between Laplace and Fourier Transform, Region of Convergence (ROC), Properties of ROCs for Laplace Transforms, Inverse Laplace Transform, Properties of the Laplace Transform, Linearity of the Laplace Transform, Time Shifting of a Continuous-Time Signal, Shifting of Laplace Transform X(s) in the s-Domain, Time Scaling of Continuous Time Signal, Complex Conjugation, Convolution Property for Laplace Transform, Differentiation of a Continuous Time Signal in Time Domain, Differentiation in s-Domain, Integration of a Continuous Time Signal in the Time Domain, Initial Value Theorem for Laplace Transform, Final Value Theorem for Laplace Transform, Applications of Laplace Transform to System Analysis, Causality of a Continuous-Time LTI System, Stability of a Continuous Time LTI System, Characterization of Continuous Time LTI Systems by Linear Constant Coefficient Differential Equations, Review Questions, Numerical Problems.
   
  7. The z-Transform

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  Objectives, Introduction, Definition of z-Transform, Two-Sided or Bilateral z-Transform, Relationship between z-Transform and DTFT, One-Sided or Unilateral z-Transform, Region of Convergence (ROC), Properties of ROC for the z-Transform, Properties of z-Transform, Linearity, Time Shifting, Scaling in the z-Domain, Time Reversal, Time Expansion (Time Scaling), Differentiation in The z-Domain, Convolution of Two Discrete Time Sequences, Correlation of Two Discrete Time Sequences, Multiplication, Parseval’s Theorem or Relation, Conjugation of a Complex Sequence, Initial Value Theorem, Final Value Theorem, The Inverse z-Transform, The Inverse z-Transform by Contour Integration Method, The Inverse z-Transform by Power Series Expansion Method, The Inverse z-Transform form by Long Division Method, The Inverse z-Transform by Partial Fraction Expansion Method, Applications of z-Transform in Analysis of Discrete Time LTI Systems, Causality of Discrete Time LTI System, Stability Criteria for a Discrete Time LTI Systems, Causal and Stable Discrete Time LTI Systems, The Characterization of Discrete Time LTI System by Linear Constant Coefficient Difference Equation, Determination of Poles and Zeros of Rational z-Transform, Block Diagram Representation for Discrete Time LTI System, Transfer Function of Inter-Connection of Discrete Time LTI Systems, Review Questions, Numerical Problems.
   
  8. Sampling

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  Objectives, Introduction, Mathematical Theory of Sampling, Sampling, Impulse Train Sampling of a Continuous Time Signal (Ideal Sampling), Sampling Theorem, Sampling Techniques, Real Sampling, Sampling with a Zero-Order Hold, Natural Sampling, Reconstruction of Signal, Interpolation Function (Linear Interpolation) Ideal Reconstruction, Interpolation with Zero-order Hold (ZOH), Interpolation with First Order Hold (FOH), Aliasing, Sampling of Discrete Time Signals, Impulse Train Sampling of Discrete Time Signals, Discrete Time Decimation (Down Sampling), Review Questions, Numerical Problems.
   
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