Related Concepts: 1.02 Elementary Operations on Signals | 1.03 Signal Classification I — Theory & Periodicity | 1.04 Signal Classification II — Symmetry, Energy & Power

1.01 Signals, Systems & Singularity Functions

Course Focus & Fundamentals

These concepts form the absolute prerequisite for all time-domain and frequency-domain transformations. Master these definitions, properties, and standard shapes for guaranteed exam marks.

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1. Introduction to Signals and Systems

1.1 What is a Signal?

Definition: Signal

A signal is defined as any physical quantity that carries information and varies with time, space, or any other independent variable or variables.

$x(t)\text{or}x[n]$

• Mathematical Context: A signal is represented as a function of one or more independent variables. For example, $x(t)$ represents a continuous-time signal varying with time $t$.
• Physical Examples: voltage or current waveforms, sound/pressure waves, or temperature measurements.
• Event mapping: Event $\to$ Sequence $\to$ Time $\to$ Phase $\to$ Amplitude. (Frequency matters first, then phase, then amplitude, but ultimately everything depends on time).

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1.2 What is a System?

Definition: System

A system is a physical device, process, or mathematical algorithm that manipulates or operates on an input signal to produce a desired output signal.

$y(t)=\mathbb{T}[x(t)]$

• Mathematical Representation:
  • $x(t)$ = Input (Excitation / Independent variable)
  • $y(t)$ = Output (Response / Dependent variable)
  • $\mathbb{T}[\cdot ]$ = The mathematical operator or process acting on the input.

graph LR
    x["Input Signal x(t)"] --> System["System Operator T[•]"]
    System --> y["Output Signal y(t) = T[x(t)]"]

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1.3 Importance and Real-World Applications

Signals and systems are the backbone of communication engineering, allowing us to transmit, filter, and extract data from noise.

• Key Applications: Speech processing, television broadcast transmission, image/video compression, physiological signals (ECG/EEG), radar/sonar echoes, seismic vibration detection, GPS satellite tracking, and genetic sequence analysis.

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Past Year Questions (PYQs)

• [PYQ 2025]: What is signal & system? (04 Marks)
• [PYQ 2024]: What are signal and system? What are the importance of these in communication engineering? (06 Marks) - missed-in-note
• [PYQ 2023]: What are meant by signal and system? (04 Marks)
• [PYQ 2020]: Define: (i) Signal, (ii) System.
• [PYQ 2024, 2023, 2019, 2018, 2016]: Define system. What are the properties of a system? (08 Marks) (Note: Properties are covered in Chapter 2)
• [PYQ 2021]: Define signal. Classify signals based on major criteria. (06 Marks) (Note: Classifications are covered in Note 1.03 & 1.04)

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2. Basic Continuous-Time Singularity Functions

Singularity functions are mathematical idealizations used as basic building blocks to construct or analyze more complex signals.

2.1 The Unit Step Function, $u(t)$

Unit Step Function

The unit step function represents a signal that turns “on” at $t=0$ and stays at a constant amplitude of 1. It models a DC switch closing.

$u(t)=\{\begin{array}{ll}1,& t>0\\ 0,& t<0\end{array}$

left = -4; right = 4
top = 2; bottom = -1
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y = 1 | x > 0 | BLUE | SOLID
y = 0 | x < 0 | BLUE | SOLID
(0, 1) | label:u(t) | BLUE
(0, 0) | open | BLUE

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2.2 The Unit Impulse Function (Dirac Delta), $\delta (t)$

Unit Impulse Function

An idealized pulse that is infinitely narrow, infinitely tall, but contains a total area (weight) of exactly 1. It is the derivative of the unit step function.

$\delta (t)=0\text{for }t\ne 0$ ${\int }_{-\infty }^{\infty }\delta (t)dt=1$

left = -2; right = 2
top = 2; bottom = -0.5
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x = 0 | 0 <= y <= 1 | BLUE | SOLID
(0, 1) | label:δ(t) (Area = 1) | BLUE

Note: In physical drawings, the impulse $\delta (t)$ is represented by an upward-pointing arrow at the origin with parentheses denoting its area/weight.

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2.3 The Ramp Function, $r(t)$

Ramp Function

A signal that grows linearly with time for $t\ge 0$. It is the integral of the unit step function.

$r(t)=\{\begin{array}{ll}t,& t>0\\ 0,& t\le 0\end{array}⟹r(t)=t\cdot u(t)$

left = -2; right = 4
top = 4; bottom = -1
---
y = x | x > 0 | BLUE | SOLID
y = 0 | x <= 0 | BLUE | SOLID
(0, 0) | BLUE

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2.4 The Signum Function, $\text{sgn}(t)$

Signum Function

A function that extracts the mathematical sign of the time variable $t$.

$\text{sgn}(t)=\{\begin{array}{ll}1,& t>0\\ 0,& t=0\\ -1,& t<0\end{array}⟹\text{sgn}(t)=u(t)-u(-t)$

left = -4; right = 4
top = 2; bottom = -2
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y = 1 | x > 0 | BLUE | SOLID
y = -1 | x < 0 | BLUE | SOLID
(0, 0) | label:sgn(t) = 0 | BLUE
(0, 1) | open | BLUE
(0, -1) | open | BLUE

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2.5 The Rectangular Pulse Function, $\Pi (t/\tau )$

Rectangular Pulse Function

A symmetrical gate pulse centered at the origin with a total duration (width) of $\tau$ and an amplitude of 1.

$\Pi \left(\frac{t}{\tau }\right)=\{\begin{array}{ll}1,& -\frac{\tau }{2}<t<\frac{\tau }{2}\\ 0,& \text{otherwise}\end{array}$

For example, a unit rectangular pulse with width $\tau =2$ is plotted below:

left = -3; right = 3
top = 2; bottom = -1
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y = 1 | -1 < x < 1 | BLUE | SOLID
y = 0 | x < -1 | BLUE | SOLID
y = 0 | x > 1 | BLUE | SOLID
(-1, 0) | open | BLUE
(-1, 1) | open | BLUE
(1, 0) | open | BLUE
(1, 1) | open | BLUE

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Past Year Questions (PYQs)

• [PYQ 2022]: Define: (i) Unit step function, (ii) Unit impulse function, (iii) Ramp function, and (iv) Signum function. (08 Marks)