Related Concepts: 1.01 Signals, Systems & Singularity Functions | 1.02 Elementary Operations on Signals | 1.04 Signal Classification II — Symmetry, Energy & Power

1.03 Signal Classification I — Theory & Periodicity

Overview

Signals are classified based on properties of their independent variables (time), dependent variables (amplitude), predictability, and repetition. Periodicity calculations are highly repeated numericals in Section A of examinations.

graph TD
    S["Signal Classifications"] --> IV["By Independent Variable"]
    S --> DV["By Dependent Variable"]
    S --> PR["By Predictability"]
    S --> PER["By Repetition"]
    
    IV --> CT["Continuous-Time x(t)"]
    IV --> DT["Discrete-Time x[n]"]
    
    DV --> AN["Analog"]
    DV --> DIG["Digital"]
    
    PR --> DET["Deterministic"]
    PR --> RAN["Random / Noise"]
    
    PER --> PERIODIC["Periodic"]
    PER --> APERIODIC["Aperiodic"]

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1. Fundamental Classifications

1.1 Continuous-Time vs. Discrete-Time Signals

This classification is based strictly on the independent variable (domain of definition).

• Continuous-Time (CT) Signal: Defined for a continuous, unbroken range of time. The independent variable $t$ is continuous.
  • Example: $x(t)=\cos (\omega t)$
• Discrete-Time (DT) Signal: Defined only at specific, isolated integer time instances. The independent variable is an integer index $n$.
  • Example: $x[n]=\{1,-2,3,0\}$

Continuous-Time x(t) [Smooth Curve]:         Discrete-Time x[n] [Lollipop/Stem Plot]:
      ^                                            ^
    1 |    _--_                                  1 |      o   o
      |  _-    -_                                  |    o | o |
  ----+-----------> t                          ----+----+---+-+-> n
      | -1                                         |   0  1   2

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1.2 Analog vs. Digital Signals

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

• 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 mapping a continuous amplitude discrete-time signal into a quantized digital signal is called quantization.

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1.3 Deterministic vs. Random (Nondeterministic) Signals

• Deterministic Signal: A signal whose future values are completely specified and predictable. It is defined by a known mathematical formula.
  • Example: $x(t)=5\sin (100\pi t)$
• Random Signal: A signal that is unpredictable, containing random, statistically characterized values. It lacks a specific predictable frequency, amplitude, or phase. Noise is the most common example.

left = -2; right = 10
top = 1.5; bottom = -1.5
---
y = sin(2*x) | BLUE | SOLID
(2.5, 1.2) | label:Deterministic Sine wave | BLUE

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1.4 Periodic vs. 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: $x(t+T)=x(t)\text{for all }t$ where the smallest positive non-zero value $T_0$ satisfying this is the Fundamental Period.
• Aperiodic Signal: Any signal that does not satisfy the periodicity condition (does not repeat).

Below is a periodic sine wave (blue) and an aperiodic decaying sine wave (red):

left = -1; right = 10
top = 1.5; bottom = -1.5
---
y = sin(2*x) | BLUE | SOLID
y = exp(-0.4*x)*sin(2*x) | x >= 0 | RED | SOLID
(4, 1.2) | label:Periodic | BLUE
(4.5, 0.4) | label:Aperiodic Decaying | RED

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2. Periodicity Test for Composite Signals

Examiners frequently ask whether a sum of two or more continuous-time signals is periodic, and if so, what its fundamental period is.

2.1 The Rationality 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 individual periods is a Rational Number (a ratio of two integers).

$\frac{T_1}{T_2}=\frac{p}{q}(p,q\in \mathbb{Z})$

If the ratio is rational, the fundamental period $T_0$ is the Least Common Multiple (LCM) of $T_1$ and $T_2$.

Sir's Math Shortcut for Fractional LCM

To find the LCM of fractional values:

$\text{LCM}\left(\frac{a}{b},\frac{c}{d}\right)=\frac{\text{LCM}(a,c)}{\text{HCF}(b,d)}$

where $\text{HCF}$ is the Highest Common Factor (Greatest Common Divisor).

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3. PYQ Step-by-Step Solutions

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. Find its fundamental period if periodic. (2019, 2015)

Solution:

1. Find individual frequencies and periods:
   • Term 1: ${\omega }_1=\sqrt{2}\pi ⟹T_1=\frac{2\pi }{\sqrt{2}\pi }=\sqrt{2}\text{ s}$
   • Term 2: ${\omega }_2=15\pi ⟹T_2=\frac{2\pi }{15\pi }=\frac{2}{15}\text{ s}$
2. Determine the ratio: $\frac{T_1}{T_2}=\frac{\sqrt{2}}{\frac{2}{15}}=\frac{15\sqrt{2}}{2}$
3. Because $\sqrt{2}$ is an irrational number, this ratio is irrational.
4. Conclusion: The signal is Aperiodic (does not have a fundamental period).

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Trap 2: LCM of Fractions

Question: Determine whether the signal $x(t)=\sin (20\pi t)+\sin (5\pi t)$ is periodic. If so, find its fundamental period. (2017)

Solution:

1. Find individual frequencies and periods:
   • Term 1: ${\omega }_1=20\pi ⟹T_1=\frac{2\pi }{20\pi }=\frac{1}{10}\text{ s}$
   • Term 2: ${\omega }_2=5\pi ⟹T_2=\frac{2\pi }{5\pi }=\frac{2}{5}\text{ s}$
2. Determine the ratio: $\frac{T_1}{T_2}=\frac{\frac{1}{10}}{\frac{2}{5}}=\frac{5}{20}=\frac{1}{4}$ Since $\frac{1}{4}$ is a rational fraction, the signal is Periodic.
3. Find the fundamental period $T_0$: $T_0=\text{LCM}\left(\frac{1}{10},\frac{2}{5}\right)=\frac{\text{LCM}(1,2)}{\text{HCF}(10,5)}=\frac{2}{5}=0.4\text{ seconds}$

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Trap 3: Single Term Irrational Frequency

Question: Consider a signal $x(t)=10\sin (\sqrt{5}\pi t)$. Is it periodic? Find its fundamental period. (2016)

Solution:

• Exam Warning: The rational ratio rule only applies when adding multiple signals together. A single sinusoidal term is always periodic, even if its frequency is irrational.
• Fundamental frequency: $\omega =\sqrt{5}\pi \text{ rad/s}$
• Fundamental period: $T_0=\frac{2\pi }{\omega }=\frac{2\pi }{\sqrt{5}\pi }=\frac{2}{\sqrt{5}}\text{ seconds}$

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Trap 4: Mixed Sines & Cosines

Question: Determine if $x(t)=2\cos (4\pi t)+3\sin (3\pi t)$ is periodic or aperiodic. If periodic, find the period. (2025)

Solution:

1. Term 1: ${\omega }_1=4\pi ⟹T_1=\frac{2\pi }{4\pi }=\frac{1}{2}\text{ s}$
2. Term 2: ${\omega }_2=3\pi ⟹T_2=\frac{2\pi }{3\pi }=\frac{2}{3}\text{ s}$
3. Ratio: $\frac{T_1}{T_2}=\frac{\frac{1}{2}}{\frac{2}{3}}=\frac{3}{4}(\text{Rational }⟹\text{Periodic})$
4. Fundamental period: $T_0=\text{LCM}\left(\frac{1}{2},\frac{2}{3}\right)=\frac{\text{LCM}(1,2)}{\text{HCF}(2,3)}=\frac{2}{1}=2\text{ seconds}$

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Past Year Questions (PYQs)

• [PYQ 2021, 2019, 2018, 2016, 2015]: What is signal? What are the major classifications of signals? (06 to 10 Marks)
• [PYQ 2020]: Classify discrete time signals.
• [PYQ 2018]: Distinguish between deterministic and nondeterministic signals.