Signals & Systems

Welcome

Signals and Systems is one of the most foundational courses in electrical, electronics, and communication engineering. It provides the mathematical framework for understanding how information-carrying signals interact with the systems that process them. From music streaming to medical imaging, from cellular networks to autonomous vehicles — every technology handling information relies on these principles.

What This Course Covers

This comprehensive course guides you through the complete landscape of signal processing theory and practice, organized into interconnected modules that build upon each other systematically.

Signal Fundamentals

We begin with the building blocks — understanding what signals are, how to classify them (continuous vs discrete, periodic vs aperiodic, energy vs power), and the basic signals that form the vocabulary of the field: impulse, step, ramp, exponential, sinusoidal, rectangular, and triangular signals. You will learn operations like shifting, scaling, reversal, addition, multiplication, and decomposition.

System Properties

Next, we study systems — mathematical operators that transform input signals into output signals. You will learn to classify systems by linearity, time-invariance, causality, stability, memory, and invertibility. The Linear Time-Invariant (LTI) system emerges as the central object of study, fully characterized by its impulse response.

Convolution and LTI Analysis

The convolution operation allows computing the output of any LTI system for any input. We cover both continuous-time convolution (integral) and discrete-time convolution (sum), along with graphical methods and computational techniques.

Fourier Analysis

The Fourier series decomposes periodic signals into harmonics, while the Fourier transform handles aperiodic signals. Together they provide complete frequency-domain analysis — understanding bandwidth, filtering, and spectral content. The DFT and FFT bring these tools to computational reality.

Laplace and Z-Transforms

The Laplace transform generalizes Fourier analysis to the complex s-plane, enabling stability analysis, transfer function characterization, and circuit analysis. The Z-transform does the same for discrete-time systems, forming the basis of digital filter design.

Sampling Theory

The sampling theorem bridges continuous and discrete worlds — establishing when and how analog signals can be perfectly represented digitally. Aliasing, quantization, and reconstruction complete the analog-digital pipeline.

Digital Signal Processing

Building on all previous topics, DSP covers digital filter design (FIR and IIR), practical spectral analysis, and the implementation of signal processing algorithms on digital hardware.

Applications

Real-world applications demonstrate how theory becomes practice: audio processing, communications, biomedical signals, radar, image processing, control systems, and more.

Course Structure

1. Introduction — Signals, systems, and classification
2. Basic Signals — Fundamental waveforms and building blocks
3. Signal Operations — Shifting, scaling, reversal, decomposition
4. System Basics — Properties and classification
5. LTI Systems — Convolution and impulse response
6. Fourier Series — Periodic signal decomposition
7. Fourier Transform — Aperiodic signal frequency analysis
8. Laplace Transform — Complex frequency domain
9. Z-Transform — Discrete-time transform analysis
10. Sampling — Analog-digital conversion
11. DSP Basics — Digital filters and processing
12. Frequency Analysis — Correlation and spectral density
13. Applications — Real-world systems
14. Interview Preparation — GATE problems and cheatsheets

Prerequisites

• Calculus (integration, differentiation, infinite series)
• Complex numbers and Euler's formula
• Basic linear algebra
• Circuit analysis fundamentals (helpful but not mandatory)

How to Study

Start with the Introduction and Basic Signals sections to build vocabulary and intuition. Progress through Signal Operations and System Basics before tackling LTI systems and convolution. The transform chapters (Fourier, Laplace, Z) build upon each other — study them sequentially. Finally, explore Applications and Interview Preparation to solidify understanding.

Each topic includes clear explanations, mathematical derivations, worked examples, real-world applications, and key takeaways. Focus on understanding WHY formulas work, not just memorizing them.