Signals and Systems Course

Lesson 1 • Standard Signal Models

Introduces unit step, impulse, ramp, and complex exponential signals. These models serve as building blocks for system analysis.

Lesson 2 • Interconnection of Systems

Analyzes series, parallel, and feedback interconnections of subsystems. Prepares students to model complex systems as combinations of simpler blocks.

Lesson 3 • Elementary Signal Operations

Covers time shifting, scaling, and reversal applied to signals. These operations underpin convolution and transform analysis later.

Lesson 4 • Introduction to Signal Representations

Defines continuous-time and discrete-time signals with amplitude, frequency, and phase. Establishes vocabulary used throughout the course.

Lesson 1 • Continuous-Time Convolution

Derives the convolution integral and computes it analytically and graphically. Mastery here is essential for Fourier and Laplace analysis.

Lesson 2 • Impulse Response of LTI Systems

Defines the impulse response h(t) and h[n] as the system's fingerprint. Connects impulse response to system properties established in Chapter 1.

Lesson 3 • Discrete-Time Convolution

Develops the convolution sum for discrete-time LTI systems. Parallels continuous-time treatment to reinforce unified LTI theory.

Lesson 4 • Differential and Difference Equations

Models LTI systems with constant-coefficient equations and links them to impulse responses. Provides the time-domain foundation for transform methods.

Lesson 1 • LTI System Response to Periodic Inputs

Applies Fourier series to find steady-state output of LTI systems driven by periodic signals. Introduces frequency response as a filter concept.

Lesson 2 • Properties of Fourier Series

Covers linearity, time shift, differentiation, and Parseval's theorem for Fourier series. Properties accelerate spectrum computation without re-deriving coefficients.

Lesson 3 • Continuous-Time Fourier Series

Derives synthesis and analysis equations for CT Fourier series. Students compute spectra of standard periodic waveforms.

Lesson 4 • Discrete-Time Fourier Series

Develops the DTFS for periodic discrete-time signals with finite harmonic content. Highlights differences from the continuous-time case.

Lesson 1 • Properties of the Fourier Transform

Covers linearity, duality, time/frequency shift, scaling, differentiation, and convolution. Properties reduce complex transforms to table lookups.

Lesson 2 • Fourier Transform of Standard Signals

Computes transforms of impulse, step, sinusoid, Gaussian, and exponential signals. Builds a reference table students use in system analysis.

Lesson 3 • From Fourier Series to Fourier Transform

Derives the CTFT as the limit of Fourier series as period approaches infinity. Motivates the transform as a natural extension of spectral analysis.

Lesson 4 • Parseval's Theorem and Energy Spectra

Relates signal energy to the magnitude-squared spectrum via Parseval's theorem. Introduces energy spectral density for signal characterization.

Lesson 1 • Properties of the DTFT

Covers linearity, time shift, frequency shift, convolution, and Parseval's theorem for the DTFT. Mirrors CTFT properties to reinforce unified transform theory.

Lesson 2 • Discrete-Time Fourier Transform

Defines the DTFT and its inverse for aperiodic discrete-time sequences. Establishes the 2π-periodic nature of discrete-time spectra.

Lesson 3 • Discrete Fourier Transform

Defines the N-point DFT as a sampled DTFT and derives its matrix form. Enables practical spectral computation on finite-length sequences.

Lesson 4 • Spectral Leakage and Windowing

Explains leakage from finite observation windows and introduces window functions to reduce it. Prepares students for practical spectral estimation.

Lesson 1 • Stability and System Design in s-Domain

Applies Routh-Hurwitz criterion and pole placement to assess and design stable systems. Connects s-domain analysis to practical control and filter design.

Lesson 2 • Inverse Laplace Transform

Computes inverse transforms via partial fraction expansion and table lookup. Handles repeated and complex poles systematically.

Lesson 3 • Properties of the Laplace Transform

Covers linearity, time shift, s-domain shift, differentiation, integration, and convolution. Properties convert differential equations to algebraic equations.

Lesson 4 • Laplace Transform Fundamentals

Defines the bilateral and unilateral Laplace transform and its region of convergence. ROC determines signal properties and system stability.

Lesson 1 • Transfer Functions and Stability in z-Domain

Defines H(z) from difference equations and maps stability to pole locations inside the unit circle. Parallels s-domain analysis for DT systems.

Lesson 2 • Digital Filter Structures

Implements IIR and FIR filters using direct-form, cascade, and parallel structures. Structural choices affect numerical precision and computational cost.

Lesson 3 • Z-Transform Fundamentals

Defines the bilateral z-transform, its ROC, and the relationship to the DTFT. ROC encodes convergence and determines signal-side properties.

Lesson 4 • Properties of the Z-Transform

Covers linearity, time shift, z-domain scaling, convolution, and Parseval's theorem. Properties simplify analysis of difference equations.

Lesson 1 • Controllability and Observability

Defines and tests controllability and observability using rank conditions on Gramian matrices. These properties determine whether a system can be steered and monitored.

Lesson 2 • Solution of State Equations

Solves CT state equations using the matrix exponential and DT equations using matrix powers. Connects state-space solutions to impulse response.

Lesson 3 • Eigenvalues, Modes, and Stability

Links system poles to eigenvalues of the state matrix and modal decomposition. Eigenvalue locations determine stability and transient modes.

Lesson 4 • State-Space Formulation

Introduces state variables, state equations, and output equations for CT and DT systems. State-space unifies SISO and MIMO system descriptions.