Signal and System

Signal and System#

Course Title:#

Signal and Systems

Course Description:#

This course provides an in-depth understanding of the fundamental concepts in signals and systems, focusing on both continuous-time and discrete-time signals and systems. The course covers the mathematical tools and techniques necessary to analyze, represent, and understand the behavior of systems in various domains, including time, frequency, and Laplace.

Course Objectives:#

  • To understand the basic concepts of signals and systems.

  • To learn the mathematical methods used in analyzing signals and systems.

  • To study the properties and behaviors of linear, time-invariant systems.

  • To develop an understanding of Fourier series, Fourier transform, and their applications.

  • To explore the representation of systems using differential equations.

Lecture Outline:#

Lecture 1: Introduction#

  • Overview of Signals and Systems

  • Types of Signals: Continuous-time and Discrete-time

  • Types of Systems: Linear vs. Non-linear, Time-Invariant vs. Time-Variant

  • Basic Operations on Signals (e.g., scaling, shifting, etc.)

Lecture 2: Signals and Systems: Part I#

  • Definitions and Classifications of Signals

  • Elementary Signals (e.g., unit step, unit impulse, exponential, sinusoidal)

  • Basic System Properties: Linearity, Time-Invariance, Causality, Stability

  • Introduction to System Representation and Block Diagrams

Lecture 3: Signals and Systems: Part II#

  • More on System Properties and Their Implications

  • System Response to Elementary Signals

  • System Characterization using Impulse Response

  • Convolution Integral for Continuous-Time Systems

  • Convolution Sum for Discrete-Time Systems

Lecture 4: Convolution#

  • Definition and Concept of Convolution

  • Convolution in Continuous-Time Systems

  • Convolution in Discrete-Time Systems

  • Properties of Convolution

  • Applications of Convolution in Signal Processing

Lecture 5: Properties of Linear, Time-Invariant (LTI) Systems#

  • Time-Domain Analysis of LTI Systems

  • Impulse Response and its Significance

  • Step Response of LTI Systems

  • Stability and Causality in LTI Systems

  • Invariance Properties of LTI Systems

Lecture 6: Systems Represented by Differential Equations#

  • Modeling Physical Systems with Differential Equations

  • Differential Equations in Continuous-Time Systems

  • Difference Equations in Discrete-Time Systems

  • Solutions to Differential Equations: Zero Input and Zero State Responses

  • Initial Conditions and Their Effects on System Behavior

Lecture 7: Continuous-Time Fourier Series#

  • Introduction to Fourier Series Representation

  • Representation of Periodic Signals using Fourier Series

  • Trigonometric and Exponential Forms of Fourier Series

  • Convergence of Fourier Series

  • Applications of Fourier Series in Signal Analysis

Lecture 8: Continuous-Time Fourier Transform#

  • Introduction to Fourier Transform

  • Fourier Transform of Common Signals

  • Properties of the Fourier Transform (e.g., linearity, time-shifting, frequency-shifting)

  • Signal Reconstruction from Fourier Transform

  • Applications of Fourier Transform in Communication Systems

Lecture 9: Fourier Transform Properties#

  • Deep Dive into Fourier Transform Properties

  • Parseval’s Theorem and Energy Spectral Density

  • Duality, Convolution, and Modulation Properties

  • Fourier Transform of Periodic Signals

  • Filtering and Frequency Response Analysis using Fourier Transform

Lecture 10: Discrete-Time Fourier Series#

  • Discrete-Time Fourier Series (DTFS) Representation

  • Properties of DTFS

  • DTFS of Periodic Discrete-Time Signals

  • Comparison between Continuous-Time and Discrete-Time Fourier Series

  • Applications of DTFS in Digital Signal Processing

Evaluation:#

  • Assignments: 20%

  • Quizzes: 15%

  • Midterm Exam: 30%

  • Final Exam: 35%

Prerequisites:#

  • Basic Calculus

  • Introduction to Linear Algebra