---
title: "Periodic signals; condition for cos(2πf₀n + θ)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:periodic-cos-condition"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/periodic-cos-condition
site: BCT Engineering
---

# Periodic signals; condition for cos(2πf₀n + θ)

Signals and Systems · Chapter 1: Signals

- Asked in: 2082 Chaitra, 2+2 marks
- Video: [Periodic Discrete Time Signals](https://www.youtube.com/watch?v=bX8OEmOlD2U) — 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-cos-condition
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

What is periodic and aperiodic signal? Find the necessary condition for the signal x[n] = cos(2πf₀n + θ) to be periodic.

## Short answer

A periodic signal repeats after a fixed interval; aperiodic does not. For cos(2πf0 n+θ) to be periodic, we need x[n+N]=x[n] ⇒ 2πf0 N = 2πk (k integer). Hence f0 = k/N must be a rational number. The fundamental period N is the denominator in lowest terms.

## Answer

- **Periodic signal:** repeats itself after a fixed interval. CT: $x(t+T) = x(t)$ for all $t$; DT: $x[n+N] = x[n]$ for all $n$, $N$ a positive integer. The smallest such $T$ or $N$ is the **fundamental period**. Examples: $\sin t$, $\cos(\pi n/4)$.
- **Aperiodic signal:** does not repeat for any finite period. Examples: $e^{-t}u(t)$, $\delta[n]$, $\cos(2n)$.

#### Condition for $x[n] = \cos(2\pi f_0 n + \theta)$ to be periodic

For period $N$:

$$
\begin{aligned}
x[n+N] &= \cos(2\pi f_0 n + 2\pi f_0 N + \theta)
\end{aligned}
$$

This equals $x[n]$ for all $n$ only if the extra angle is a multiple of $2\pi$:

$$
2\pi f_0 N = 2\pi k \;\Rightarrow\; f_0 = \frac{k}{N}, \quad k, N \text{ integers}
$$

So **$x[n]$ is periodic only if $f_0$ is a rational number.** If $f_0 = k/N$ in lowest terms, the fundamental period is $N$.

**Example:** $f_0 = 3/10$ gives $N = 10$. $f_0 = 1/(2\pi)$ (i.e. $\cos n$) is irrational, so the signal is aperiodic.

## Check yourself

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

1. A DT signal is periodic with period N if:
   A. x[n+N] = x[n] for all n, N a positive integer
   B. x[n+N] = -x[n]
   C. x[N] = x[0] only
   D. x[n] = x[-n]
   Answer: A. Periodicity requires repetition for all n with an integer period. (Key point: Periodic definition)

2. cos(2 pi f0 n + theta) is periodic only if:
   A. f0 is an integer
   B. f0 is rational
   C. f0 is irrational
   D. theta = 0
   Answer: B. 2 pi f0 N = 2 pi k gives f0 = k/N, a rational number. (Key point: Condition on f0)

3. For f0 = 3/10 (lowest terms), the fundamental period is:
   A. 3
   B. 10
   C. 30
   D. 13
   Answer: B. With f0 = k/N in lowest terms, N = 10. (Key point: Fundamental period)

4. Why is cos(n) aperiodic?
   A. Its amplitude is 1
   B. f0 = 1/(2 pi) is irrational
   C. It is real
   D. It has no phase
   Answer: B. No integer N makes N/(2 pi) an integer, so it never repeats. (Key point: Aperiodic example)

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