Signals And Systems

Singnals

Introduction

      Continuous Time Signals
      Discrete Time Signals
      Representation

Elementary Signals/Basic Signals

      Unit Step Function
      Unit Ramp Function
      Unit Parabolic Function
      Unit Impulse Function
        Properties of Continuous Time Impulse signal
        Properties of Discrete Time Impulse signals
      Real Exponential Function
      Complex Exponential Function
      Rectangular Pulse Function
      Triangular Pulse Function
      Signum_Function
      Gaussian_Function
      Sinc_Function

Operation on Signals

      Time Shifting
      Time Reversal
      Time Scaling
      Amplitude Scaling
      Signal Addition
      Signal Multiplication

Classification of Signals

      Periodic and Non-Periodic Signals
      Even And Ood Signals
      Causal And Non Causal Signals
      Energy And Power Signals
Systems

Classification of Systems

      Continuous Time Systems
      Discrete Time Systems
      Lumped Parameter And Distributed Parameter Systems
      Static And Dynamic Systems
      Causal And Non Causal Systems
      Linear And Non Linear Systems
      Time Invarient And Time Variant Systems
      Stable And Unstable Systems
      FIR And IIR System
SIGNAL ANALYSIS

Analogy between vectors and signals

      Orthogonality in vectors
      Orthogonality in signals
      Problems

Orthogonal space

      Mean square error

Orthogonality in complex functions

Problems

FOURIER SERIES

Representation of Fourier Series

Existence of Fourier Series

Evaluation of fourier Coefficients

Trigonometric Form of Fourier Series

Cosine Representation (Alternate Form of the Trigonometric Representation)

Wave Symmetry

      Even or Mirror Symmetry
      Odd or Rotation Symmetry
      Half Wave Symmetry
      Quarter Wave Symmetry

Exponential Fourier Series

Determination of the Coefficients of Exponential Fourier Series

Cosine Fourier Series from EFS

Fourier Spectrum

Properties of Continuous-Time Fourier Series

      Linearity Property
      Time Shifting Property
      Time Reversal Property
      Time Scaling Property
      Time Differentiation Property
      Time Integration Property
      Convolution Theorem or Property
      Modulation or Multiplication Property
      Conjugation and Conjugate Symmetry Property
      Parseval’s Relation or Theorem or Property
Fourier Transforms

• Introduction
• Limitation Of FS
• Need Of FT
• Derivation of the FT of a Non-periodic Signal from the FS of a Periodic Signal
• Magnitude and Phase Representation of FT
• Existence of FT
• FT of Standard Signals
  • Impulse Function
  • Single-sided Real Exponential Function
  • Double-sided Real Exponential Function
  • Complex Exponential Function
  • Constant Amplitude
  • Signum Function
  • Unit Step Function
  • Rectangular Pulse
  • Triangular Pulse
  • Cosine Wave
  • Sine Wave
• Properties of Continuous Time FT
  • Linearity Property
  • Time Shifting Property
  • Frequency Shifting Property (Multiplication by an Exponential)
  • Time Reversal Property
  • Time Scaling Property
  • Differentiation in Time Domain Property
  • Differentiation in Frequency Domain Property
  • Time Integration Property
  • Convolution Property or Theorem
  • Multiplication Property or Theorem
  • Duality (Symmetry) Property
  • Modulation Property
  • Conjugation Property
  • Autocorrelation Property
  • Parseval’s Relation or Theorem or Property
  • Area under the Curve
  • Fourier Transform of Complex and Real Functions
• FT of a Periodic Signal

Signal Transmission Through Linear System

Signal Transmission Through Linear System

LTV Systems

RESPONSE_OF_A_LINEAR_SYSTEMS

• Impulse Response
• Step Response

Properties of Linear Time Invariant Systems

• The Commutative Property
• The Distributive Property
• The Associative Property
• Causality
• Stability

Transfer Function of an LTI System

• Frequency Domain
• S Domain Or Laplace Domain

Filter Characteristics of Linear Systems

• Low Pass Filter
• High Pass Filter
• Band Pass Filter
• Band Reject Filter
• All Pass Filter

Distortionless Transmission through a System

Signal Bandwidth

System Bandwidth

Ideal Filter Characteristics

Causality and Paley-Wiener Criterion for Physical Realization

Relationship between Bandwidth and Rise Time

Convolution And Correlation Of Signal

Convolution And Correlation Of Signal

Properties of Convolution

Convolution Theorems

• Time Convolution Theorem
• Frequency Convolution Theorem

Problems

Laplace Transforms

Introduction

Fourier_Transform

Laplace Transform

Region of Convergence

• Properties Of ROC

Existence of Laplace Transform

Relation between Fourier Transform and Laplace Transform

One-sided (Unilateral) Laplace Transform of Some Commonly Used Signals

• Impulse Function
• Step Function
• Ramp Function
• Parabolic Function
• Real Exponential Function
• Complex Exponential Function
• Sine and Cosine Functions
• Hyperbolic sine and cosine Functions
• Damped sine and cosine Functions
• Damped Hyperbolic sine and cosine Functions

Properties and Theorems of Laplace Transform

• Linearity Property
• Time Shift Property
• Time Scaling Property
• Time Reversal Property
• Differentiation_Time_domain
• Differentiation in s-domain Property
• Integration_in_Time_Domain
• Frequency Shift Property
• Conjugation_and_Conjugate_Symmetry_Property
• Time Convolution Property
• Multiplication or Modulation or Convolution in s-domain Property
• Parseval’s Relation or Theorem or Property
• Initial Value Theorem
• Final Value Theorem
• Laplace_Transform_Of_Periodic_Function
• Partial_Fraction_Expansions

Inverse_Laplace_Transform

Solution of Differential Equations Using Laplace Transform

Z-Transforms

Introduction

Definition Of Z-Transform

ROC_For_Z-Transform

Relation between DTFT and Z-Transform

Problems

Z-Transforms And ROC Of Finite Duration sequences

Z-Transforms of Some Common Sequences

• The Unit-impulse Sequence
• The Unit-step Sequence
• The Unit-ramp Sequence
• The Exponential Sequence
• The Sinusoidal Sequence
• The Cosinusoidal Sequence

Inverse Z-Transform

• Long Division Method
• Partial Fraction Expansion Method
• Residue Method
• Convolution Method

Solution of Difference Equations Using Z-Transforms