Signals & Systems Mastery | Part 1: Fundamental Signals,
System Properties & Sampling
The complete guide to Singularity Functions, Linearity, Causality, and Nyquist Rates
Course: Signals and Systems
Institution: University of Engineering & Technology, Lahore Reference
Texts: Signals and Systems (Oppenheim & Willsky)
What's Inside This Part:
• Fundamental Signals & Transformations: Plain-English definitions for singularity
functions and a foolproof "Order of Operations" for time shifting, scaling, and reversal.
• The "Proof Cookbook": Step-by-step algorithms for proving Linearity, Time-
Invariance, Causality, and Stability on an exam without guessing.
• Continuous & Discrete Classifications: Clear mathematical tests to find the Even/Odd
parts of a signal and to determine if a signal holds Energy or Power.
• Nyquist & Aliasing Demystified: A straightforward breakdown of the Sampling
Theorem and Analog-to-Digital conversion.
About These Notes:
These materials were carefully curated, organized, and synthesized by a Biomedical
Engineering student. They are designed to cut through dense, intimidating mathematical
jargon and provide practical, algorithmic methods for solving the exact types of problems you
will see on your midterms and finals.
Stop memorizing slides. Start learning how to pass.
, Signals & Systems: Part 1
Chapter 1: Fundamental Signal Types (The Building Blocks)
1. The "Big Three" Singularity Functions
These are the most common standard signals. Understanding how they operate and relate to each
other is critical for passing this course.
Signal
Notation Mathematical Definition Plain English
Name
Unit 𝛿(𝑡) = 0 for 𝑡 ≠ 0, and A sudden, instantaneous spike of energy at
𝛿(𝑡)
Impulse 1 for t = 0. exactly time zero.
𝑢(𝑡) = 0 for t < 0, and 1 A switch that "turns on" (jumps to 1) exactly
Unit Step 𝑢(𝑡)
for 𝑡 ≥ 0. at t=0 and stays on forever.
Unit 𝑟(𝑡) = 0 for t < 0, and 𝑡 A signal that is zero when time is negative,
$r(t)$
Ramp for 𝑡 ≥ 0. and then grows linearly over time.
Exam Pro-Tip: The Integration Ladder Professors rarely ask you just to define these signals; they will
test if you know their calculus relationship. Memorize this sequence:
1. Integrating an Impulse 𝛿(𝑡) gives you a Step 𝑢(𝑡).
2. Integrating a Step 𝑢(𝑡) gives you a Ramp 𝑟(𝑡).
𝑡2
3. Integrating a Ramp 𝑟(𝑡) gives you a Parabola 𝑥(𝑡) = 2
. (Conversely, taking the derivative
moves you backward down the ladder!)
2. Other Standard Functions
• Signum Function 𝒔𝒈𝒏(𝒕): Outputs 1 for 𝑡 > 0, -1 for t < 0, and 0 at exactly t = 0. It is
commonly written in terms of unit steps: 𝑠𝑔𝑛(𝑡) = 2𝑢(𝑡) − 1.
• Exponential Function 𝑥(𝑡) = 𝑒 𝛽𝑡 : * If 𝛽 = 0: It is a constant flat line.
o If 𝛽 is positive: It grows infinitely.
o If 𝛽 is negative: It decays toward zero.
𝑡
• Rectangular Function 𝑥(𝑡) = 𝐴~𝑟𝑒𝑐𝑡 [𝑇] : A pulse with an amplitude of A that exists strictly
between −𝑇/2 𝑎𝑛𝑑 𝑇/2.
Page 1 of 6
System Properties & Sampling
The complete guide to Singularity Functions, Linearity, Causality, and Nyquist Rates
Course: Signals and Systems
Institution: University of Engineering & Technology, Lahore Reference
Texts: Signals and Systems (Oppenheim & Willsky)
What's Inside This Part:
• Fundamental Signals & Transformations: Plain-English definitions for singularity
functions and a foolproof "Order of Operations" for time shifting, scaling, and reversal.
• The "Proof Cookbook": Step-by-step algorithms for proving Linearity, Time-
Invariance, Causality, and Stability on an exam without guessing.
• Continuous & Discrete Classifications: Clear mathematical tests to find the Even/Odd
parts of a signal and to determine if a signal holds Energy or Power.
• Nyquist & Aliasing Demystified: A straightforward breakdown of the Sampling
Theorem and Analog-to-Digital conversion.
About These Notes:
These materials were carefully curated, organized, and synthesized by a Biomedical
Engineering student. They are designed to cut through dense, intimidating mathematical
jargon and provide practical, algorithmic methods for solving the exact types of problems you
will see on your midterms and finals.
Stop memorizing slides. Start learning how to pass.
, Signals & Systems: Part 1
Chapter 1: Fundamental Signal Types (The Building Blocks)
1. The "Big Three" Singularity Functions
These are the most common standard signals. Understanding how they operate and relate to each
other is critical for passing this course.
Signal
Notation Mathematical Definition Plain English
Name
Unit 𝛿(𝑡) = 0 for 𝑡 ≠ 0, and A sudden, instantaneous spike of energy at
𝛿(𝑡)
Impulse 1 for t = 0. exactly time zero.
𝑢(𝑡) = 0 for t < 0, and 1 A switch that "turns on" (jumps to 1) exactly
Unit Step 𝑢(𝑡)
for 𝑡 ≥ 0. at t=0 and stays on forever.
Unit 𝑟(𝑡) = 0 for t < 0, and 𝑡 A signal that is zero when time is negative,
$r(t)$
Ramp for 𝑡 ≥ 0. and then grows linearly over time.
Exam Pro-Tip: The Integration Ladder Professors rarely ask you just to define these signals; they will
test if you know their calculus relationship. Memorize this sequence:
1. Integrating an Impulse 𝛿(𝑡) gives you a Step 𝑢(𝑡).
2. Integrating a Step 𝑢(𝑡) gives you a Ramp 𝑟(𝑡).
𝑡2
3. Integrating a Ramp 𝑟(𝑡) gives you a Parabola 𝑥(𝑡) = 2
. (Conversely, taking the derivative
moves you backward down the ladder!)
2. Other Standard Functions
• Signum Function 𝒔𝒈𝒏(𝒕): Outputs 1 for 𝑡 > 0, -1 for t < 0, and 0 at exactly t = 0. It is
commonly written in terms of unit steps: 𝑠𝑔𝑛(𝑡) = 2𝑢(𝑡) − 1.
• Exponential Function 𝑥(𝑡) = 𝑒 𝛽𝑡 : * If 𝛽 = 0: It is a constant flat line.
o If 𝛽 is positive: It grows infinitely.
o If 𝛽 is negative: It decays toward zero.
𝑡
• Rectangular Function 𝑥(𝑡) = 𝐴~𝑟𝑒𝑐𝑡 [𝑇] : A pulse with an amplitude of A that exists strictly
between −𝑇/2 𝑎𝑛𝑑 𝑇/2.
Page 1 of 6
