03

Here is Note 3, specifically engineered to cover the conceptual classifications and the high-yield periodicity numericals. This note synthesizes your textbook theory, your senior’s intuitive class notes, and the exact traps examiners set in the PYQs.

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Note 3: Signal Classification Part I (Theory & Periodicity)

Focus: Understanding the fundamental categories of signals and mastering the mathematical test for periodicity. The periodicity test is a highly repeated numerical in Section A of your exams.

3.1 Continuous-Time vs. Discrete-Time Signals

This classification is based strictly on the independent variable (usually time).

• Continuous-Time (CT) Signal: Defined for a continuous, unbroken range of time. The independent variable $t$ is continuous.
• Discrete-Time (DT) Signal: Defined only at specific, isolated time instances. The independent variable is an integer index $n$, meaning the signal only exists at discrete sets of amplitude values. [Graph: A smooth bell-shaped continuous curve x(t) plotted against continuous time t. Beside it, a discrete-time signal x[n] represented as stems/lollipop samples strictly at integer values of n]

3.2 Analog vs. Digital Signals

This classification is based strictly on the dependent variable (amplitude).

• Analog Signal: The amplitude can take on an infinite number of continuous values within a given range.
• Digital Signal: The amplitude is restricted to a finite, discrete set of quantized values.
• Concept Note: The process of converting a discrete-time signal (continuous amplitude) into a digital signal (discrete amplitude) is called quantization.

3.3 Deterministic vs. Random (Nondeterministic) Signals

• Deterministic Signal: A signal whose future values are completely specified and predictable for any given time. It can be perfectly described by a known mathematical formula (e.g., $x(t)=\sin (2\pi t)$).
• Random Signal: A signal that is unpredictable and takes random, statistically characterized values. It lacks a specific predictable frequency ($f$), amplitude ($A$), or phase. Noise is the most common example of a random signal. [Graph: A clean, perfectly predictable repeating sine wave for the Deterministic signal. Next to it, a jagged, chaotic, unpredictable noise waveform for the Random signal]

🔥 Exact PYQs for Classifications:

• What is signal? What are the major classifications of signals? (2021, 2019, 2018, 2016, 2015)
• Classify discrete time signals. (2020)
• Distinguish between (i) deterministic and nondeterministic signals; (ii) even and odd signals. (2018)

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3.4 Periodic and Aperiodic Signals

• Periodic Signal: A signal $x(t)$ is periodic if it exactly repeats its pattern after a specific, constant time interval $T$.
• Mathematical Condition: $x(t+T)=x(t)$ for all values of $t$, where $T$ is a positive non-zero value. The smallest value of $T$ that satisfies this is called the Fundamental Period ($T_0$).
• Aperiodic Signal: Any signal that does not satisfy the periodicity condition. It does not repeat. [Graph: A repeating sine or sawtooth wave showing periodicity T. Next to it, a signal that rises and then flattens out infinitely, representing an aperiodic signal]

3.5 The Periodicity Test for Composite Signals (A+ MUST-KNOW)

Examiners rarely ask you to find the period of a single sine wave. They will give you a sum of two or more sine waves and ask if the combined signal is periodic.

The Golden Rule: Let a continuous-time signal be $x(t)=x_1(t)+x_2(t)$, with fundamental periods $T_1$ and $T_2$. The composite signal $x(t)$ is periodic if and only if the ratio of their periods $\left(\frac{T_1}{T_2}\right)$ is a Rational Number (a simple fraction of integers, like $3/4$ or $2/5$).

Steps to Solve:

1. Find angular frequencies ${\omega }_1$ and ${\omega }_2$.
2. Calculate individual periods: $T_1=\frac{2\pi }{{\omega }_1}$ and $T_2=\frac{2\pi }{{\omega }_2}$.
3. Check the ratio $\frac{T_1}{T_2}$.
   • If it contains an un-canceled $\pi$, $\sqrt{2}$, or any irrational number, stop here. The signal is Aperiodic.
   • If the ratio is a rational fraction, the signal is Periodic.
4. If periodic, the fundamental period $T_0$ is the Least Common Multiple (LCM) of $T_1$ and $T_2$.
   • Sir’s Math Shortcut for fractions: $LCM\left(\frac{a}{b},\frac{c}{d}\right)=\frac{LCM(a,c)}{HCF(b,d)}$ (HCF = Highest Common Factor).

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3.6 PYQ Step-by-Step Solutions (Periodicity)

🔥 Exam Trap 1: The Irrational Ratio Question: Determine whether the signal $x(t)=\sin (\sqrt{2}\pi t)+\sin (15\pi t)$ is periodic or not. Then find its fundamental period. (2019, 2015) Solution:

1. For $x_1(t)=\sin (\sqrt{2}\pi t)$, ${\omega }_1=\sqrt{2}\pi$. Period $T_1=\frac{2\pi }{\sqrt{2}\pi }=\sqrt{2}$ sec.
2. For $x_2(t)=\sin (15\pi t)$, ${\omega }_2=15\pi$. Period $T_2=\frac{2\pi }{15\pi }=\frac{2}{15}$ sec.
3. Check ratio: $\frac{T_1}{T_2}=\frac{\sqrt{2}}{2/15}=\frac{15\sqrt{2}}{2}$.
4. Because $\sqrt{2}$ is an irrational number, the ratio is irrational. Therefore, the signal is Aperiodic (it does not have a fundamental period).

🔥 Exam Trap 2: The LCM of Fractions Question: Determine whether the signal $x(t)=\sin (20\pi t)+\sin (5\pi t)$ is periodic or not. If it is periodic, then find the fundamental period. (2017) Solution:

1. For $x_1(t)=\sin (20\pi t)$, ${\omega }_1=20\pi$. Period $T_1=\frac{2\pi }{20\pi }=\frac{1}{10}$ sec.
2. For $x_2(t)=\sin (5\pi t)$, ${\omega }_2=5\pi$. Period $T_2=\frac{2\pi }{5\pi }=\frac{2}{5}$ sec.
3. Check ratio: $\frac{T_1}{T_2}=\frac{1/10}{2/5}=\frac{5}{20}=\frac{1}{4}$. This is a rational number. The signal is Periodic.
4. Find Fundamental Period $T_0=LCM(T_1,T_2)=LCM\left(\frac{1}{10},\frac{2}{5}\right)$.
   • $T_0=\frac{LCM(1,2)}{HCF(10,5)}=\frac{2}{5}=0.4$ seconds.

🔥 Exam Trap 3: Single Term Irrational Frequency Question: Consider a signal: $x(t)=10\sin (\sqrt{5}\pi t)$. Is it periodic? If it is, find its fundamental period. (2016) Solution: Yes, a single sinusoidal term is always periodic, even if its frequency involves an irrational number (the rational ratio rule only applies when adding multiple signals together).

• $\omega =\sqrt{5}\pi$.
• $T_0=\frac{2\pi }{\sqrt{5}\pi }=\frac{2}{\sqrt{5}}$ seconds.

🔥 Exam Trap 4: Mixed sines and cosines Question: For the following signal, determine whether it is periodic or aperiodic. If periodic, find period $x(t)=2\cos (4\pi t)+3\sin (3\pi t)$. (2025) Solution:

1. For $x_1(t)=2\cos (4\pi t)$, $T_1=\frac{2\pi }{4\pi }=\frac{1}{2}$ sec.
2. For $x_2(t)=3\sin (3\pi t)$, $T_2=\frac{2\pi }{3\pi }=\frac{2}{3}$ sec.
3. Ratio: $\frac{T_1}{T_2}=\frac{1/2}{2/3}=\frac{3}{4}$ (Rational $\to$ Periodic).
4. $T_0=LCM\left(\frac{1}{2},\frac{2}{3}\right)=\frac{LCM(1,2)}{HCF(2,3)}=\frac{2}{1}=2$ seconds.