---
title: "Even/odd DT signals and periodicity"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:even-odd-dt-periodicity"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/even-odd-dt-periodicity
site: BCT Engineering
---

# Even/odd DT signals and periodicity

Signals and Systems · Chapter 1: Signals

- Asked in: 2072 Magh, 2+3 marks
- Video: [Even and Odd Discrete Time Signals](https://www.youtube.com/watch?v=kP9gWdLu8YM) — 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/even-odd-dt-periodicity
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define even and odd discrete time signal. Derive the periodicity condition for discrete time signal.

## Short answer

Even: x[-n]=x[n]; odd: x[-n]=-x[n]. A DT signal is periodic with integer period N if x[n+N]=x[n]. For e^(jω0 n), periodicity requires ω0/(2π) rational; if ω0/(2π)=m/N in lowest terms, the fundamental period is N.

## Answer

#### Even and odd DT signals
- **Even:** $x[-n] = x[n]$ for all $n$ (symmetric about $n = 0$). Examples: $\cos(\pi n/3)$, $\{2, 1, \underline{5}, 1, 2\}$.
- **Odd:** $x[-n] = -x[n]$ for all $n$ (antisymmetric, $x[0] = 0$). Examples: $\sin(\pi n/3)$, $\{-1, \underline{0}, 1\}$.

Any sequence has $x_e[n] = \frac{1}{2}(x[n]+x[-n])$ and $x_o[n] = \frac{1}{2}(x[n]-x[-n])$.

#### Periodicity condition
A discrete-time signal is periodic with period $N$ (a positive integer) if $x[n+N] = x[n]$ for all $n$.

For $x[n] = e^{j\omega_0 n}$:

$$
\begin{aligned}
x[n+N] &= e^{j\omega_0 (n+N)} = e^{j\omega_0 n}\,e^{j\omega_0 N}
\end{aligned}
$$

For $x[n+N] = x[n]$ we need

$$
e^{j\omega_0 N} = 1 \;\Rightarrow\; \omega_0 N = 2\pi m, \quad m \text{ an integer}
$$

$$
\frac{\omega_0}{2\pi} = \frac{m}{N} = \text{rational number}
$$

So **$e^{j\omega_0 n}$ is periodic only if $\omega_0/2\pi$ is a rational number.** The fundamental period is the smallest positive integer $N = 2\pi m/\omega_0$, taking $m$ as the smallest integer that makes $N$ an integer (with $m$ and $N$ having no common factor). The same condition holds for $\cos(\omega_0 n+\theta)$ and $\sin(\omega_0 n+\theta)$, because they are sums of such exponentials.

This differs from continuous time, where $e^{j\omega_0 t}$ is periodic for **every** $\omega_0 \neq 0$ (period $2\pi/|\omega_0|$). In discrete time $n$ takes only integer values, so $N$ must be an integer.

## Check yourself

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

1. Which condition defines an even DT signal?
   A. x[-n] = -x[n]
   B. x[-n] = x[n]
   C. x[n+N] = x[n]
   D. x[n] = 0 for n < 0
   Answer: B. Even sequences are symmetric about n = 0. (Key point: Even DT signal)

2. What is x[0] for an odd DT signal?
   A. 1
   B. Maximum
   C. 0
   D. Undefined
   Answer: C. x[0] = -x[0] forces x[0] = 0. (Key point: Odd DT signal at zero)

3. What is the condition for e^(jωn) to be periodic?
   A. ω is rational
   B. ω is an integer
   C. ω/2π is irrational
   D. ω/2π is rational
   Answer: D. e^(jωN) = 1 requires ωN = 2πm, so ω/2π = m/N. (Key point: Periodicity condition)

4. Why does DT periodicity require a rational ω/2π, unlike CT?
   A. The period N must be an integer since n is an integer
   B. DT signals have no frequency
   C. DT exponentials are real
   D. The period is always 2π
   Answer: A. CT time is continuous so any ω0 works; DT needs an integer N. (Key point: Why N integer)

## Related in Signals and Systems

- [Even and odd signals in CT and DT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-ct-dt-examples) — Signals and Systems, chapter 1
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