---
title: "Periodic and aperiodic signals"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:periodic-aperiodic-examples"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/periodic-aperiodic-examples
site: BCT Engineering
---

# Periodic and aperiodic signals

Signals and Systems · Chapter 1: Signals

- Asked in: 2070 Bhadra, 4 marks
- Video: [Periodic and Non-Periodic Signals](https://www.youtube.com/watch?v=Gj9pDC_OBQU) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/periodic-aperiodic-examples
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

What do you understand by periodic and aperiodic signals? Explain with the help of examples.

## Short answer

A periodic signal repeats identically after a fixed period T, e.g., sin(2πt) with T=1. An aperiodic signal never repeats, e.g., e^{-2t}u(t) or cos(t)+cos(√2 t).

## Answer

#### Periodic signal
A signal is **periodic** if it repeats itself exactly after a fixed interval:

$$
x(t + T) = x(t) \ \ \forall t, \qquad x[n+N] = x[n] \ \ \forall n
$$

The smallest positive $T$ (or integer $N$) for which this holds is the **fundamental period**; $\omega_0 = 2\pi/T$ is the fundamental frequency.

Examples:
- $x(t) = \sin(2\pi t)$: period $T = 1$ s.
- A square wave or sawtooth wave repeating every $T$ seconds.
- $x[n] = \cos(\pi n/4)$: $\omega_0/2\pi = 1/8$, so $N = 8$.
- Sum $x(t) = \cos 2t + \sin 3t$: periods $\pi$ and $2\pi/3$, ratio $3/2$ is rational, so periodic with $T = 2\pi$.

#### Aperiodic signal
A signal that does not repeat for any finite $T$ (or $N$) is **aperiodic** (non-periodic). It can be seen as a periodic signal with $T \to \infty$.

Examples:
- $x(t) = e^{-2t}u(t)$, a single rectangular pulse, the unit step $u(t)$, the impulse $\delta(t)$.
- $x(t) = \cos t + \cos \sqrt{2}\,t$: ratio of periods $\sqrt{2}$ is irrational, so it never repeats.
- $x[n] = \cos(n/6)$: $\omega_0/2\pi = 1/12\pi$ is irrational, so it is aperiodic even though the continuous-time $\cos(t/6)$ is periodic.

```text
Periodic (repeats every T)       Aperiodic (one pulse)
  _    _    _    _                    ____
 | |  | |  | |  | |                  |    |
_| |__| |__| |__| |__ t          ____|    |_____ t
 |<-T->|                             0    2
```

| Point | Periodic | Aperiodic |
|---|---|---|
| Repetition | repeats every $T$ | never repeats |
| Analysis tool | Fourier series | Fourier transform |
| Spectrum | discrete lines | continuous |
| Usual type | power signal | often energy signal |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which defines a periodic CT signal?
   A. x(t) = 0 for t < 0
   B. x(-t) = x(t)
   C. x(t+T) = x(t) for all t, for some T > 0
   D. x(t) → 0 as t → ∞
   Answer: C. The signal repeats exactly after the interval T. (Key point: Periodic definition)

2. Why is cos t + cos √2 t aperiodic?
   A. Cosines are aperiodic
   B. The periods are equal
   C. The amplitudes differ
   D. The ratio of periods is √2, irrational
   Answer: D. A sum is periodic only if the ratio of periods is rational. (Key point: Aperiodic sum)

3. Why is x[n] = cos(n/6) aperiodic although cos(t/6) is periodic?
   A. ω0/2π = 1/(12π) is irrational, and DT needs an integer period
   B. n is negative
   C. DT cosines are never periodic
   D. The amplitude is too small
   Answer: A. CT periods can be any real number; DT periods must be integers. (Key point: DT vs CT)

4. Which tool is used for periodic signals as opposed to aperiodic ones?
   A. Fourier transform (discrete spectrum)
   B. Fourier series (discrete spectrum)
   C. Fourier transform for periodic, series for aperiodic
   D. Laplace only
   Answer: B. Periodic signals use the Fourier series with line spectra; aperiodic signals use the Fourier transform with continuous spectra. (Key point: Analysis tool)

## Related in Signals and Systems

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- [Parseval's theorem for DT aperiodic signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-parseval-aperiodic) — Signals and Systems, chapter 3, asked 3×
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- [Parseval's relation for CT periodic signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-parseval) — Signals and Systems, chapter 2, asked 4×

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