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Note 1: Core Fundamentals, Applications, and Basic Signals

Focus: The absolute basics and foundational vocabulary. These concepts form the prerequisite for all time-domain and frequency-domain transformations. Master these for guaranteed short-note marks.

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Topic 1: Introduction to Signals and Systems

1.1 What is a Signal?

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

• Mathematical Context: A signal is represented as a function of one or more independent variables. For example, $x(t)$ represents a signal varying with time $t$.
• Event mapping: Event $\to$ Sequence $\to$ Time $\to$ Phase $\to$ Amplitude. (Frequency matters first, then phase, then amplitude, but ultimately everything depends on time).

1.2 What is a 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.

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

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 (Memorize these keywords): Speech, TV, images, heart & brain pulses (ECG/EEG), generator outputs, radar and sonar echoes, seismic vibrations, GPS satellite signals, and human gene signals.

🔥 Exact PYQs for Topic 1:

• What is signal & system? (2025)
• What are signal and system? What are the importance of these in communication engineering? (2024)
• What are meant by signal and system? (2023)
• Define: (i) Signal, (ii) System, (iii) Shanon-Nyquist Sampling Theorem. (2020) (Note: Sampling theorem is covered in Note 5)
• Define system. What are properties of a system? (2025) (Note: Properties are covered in Note 3)
• Define system. What are the properties of a system? (2024, 2023, 2019, 2018)
• What is system? What are the properties of a system? (2016)
• Define signal. Classify signals based on major criterias. (2021) (Note: Classifications are covered in Note 3)
• What is signal? What are the major classifications of signal? Explain the operations performed on the independent variables in the processing of discrete signals mathematically & graphically. (2018, 2015) (Note: Operations covered in Note 2)
• What is signal? Explain the operations performed on the independent variables in the processing of discrete signals. (2017)
• What is signal? What are the major classifications of signals? (2016)

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Topic 2: Basic Continuous-Time Singularity Functions

Singularity functions are elementary signals used as building blocks to construct or analyze more complex signals. You must know their mathematical definitions and be able to draw them instantly.

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

The unit step function represents a signal that turns “on” at $t=0$ and stays at a constant amplitude of 1. It is widely used to model a DC switch closing. Mathematical Definition: $u(t)=\{\begin{array}{llc}1,& t>00,& t<0\end{array}$ [Graph: u(t) with a horizontal line at y=1 for x>0, and y=0 for x<0. Discontinuity at x=0]

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

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. Mathematical Definition: $\delta (t)=0,\text{ for }t\ne 0$ ${\int }_{-\infty }^{\infty }\delta (t)dt=1$ [Graph: \delta(t) represented as an upward pointing arrow at the origin (x=0) with the number '1' written next to the arrow head indicating its area]

2.3 The Ramp Function, $r(t)$

A signal that grows linearly with time for $t\ge 0$. It can be mathematically generated by integrating the unit step function or multiplying time $t$ by $u(t)$. Mathematical Definition: $r(t)=\{\begin{array}{llc}t,& t>00,& t\le 0\end{array}$ Alternatively: $r(t)=t\cdot u(t)$ [Graph: r(t) showing a straight diagonal line starting from origin with a slope of 1 for x>0, and y=0 for x<0]

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

A function that extracts the sign of the time variable. It is $+1$ for positive time and $-1$ for negative time. Mathematical Definition: $\text{sgn}(t)=\{\begin{array}{llcc}1,& t>00,& t=0-1,& t<0\end{array}$ Alternatively, it can be written using unit step functions: $\text{sgn}(t)=u(t)-u(-t)$ [Graph: \text{sgn}(t) showing a horizontal line at y=1 for x>0, a horizontal line at y=-1 for x<0, and passing through origin]

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

A symmetrical gate pulse centered at the origin with a total width of $\tau$ and an amplitude of 1. Mathematical Definition: $\Pi \left(\frac{t}{\tau }\right)=\{\begin{array}{llc}1,& -\frac{\tau }{2}<t<\frac{\tau }{2}0,& \text{otherwise}\end{array}$ [Graph: \Pi(t/\tau) showing a symmetrical rectangular block of height 1 spanning from x=-\tau/2 to x=+\tau/2]

🔥 Exact PYQs for Topic 2:

• Define: (i) Unit step function, (ii) Unit impulse function, (iii) Ramp function, and (iv) Signum function. (2022)