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

# Even and odd signals in CT and DT

Signals and Systems · Chapter 1: Signals

- Asked in: 2076 Baisakh, 4 marks
- Video: [Even and Odd Signals](https://www.youtube.com/watch?v=c_YLkvk_ZXI) — 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-ct-dt-examples
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define even and odd signal in both continuous time and discrete time with examples.

## Short answer

A signal is even if x(-t)=x(t) (CT) or x[-n]=x[n] (DT); odd if x(-t)=-x(t) or x[-n]=-x[n]. CT even example: cosωt; odd: sinωt. DT even: {1,2,3,2,1}; odd: {-2,-1,0,1,2}. Odd signals must be zero at the origin.

## Answer

#### Even signal
A signal that is symmetric about the vertical axis (time origin).
- CT: $x(-t) = x(t)$ for all $t$. Examples: $\cos\omega t$, $t^2$, $e^{-|t|}$, rectangular pulse centred at 0.
- DT: $x[-n] = x[n]$ for all $n$. Examples: $\cos(\pi n/4)$, $\delta[n]$, $\{1, 2, \underline{3}, 2, 1\}$.

#### Odd signal
A signal that is antisymmetric about the origin.
- CT: $x(-t) = -x(t)$ for all $t$. Examples: $\sin\omega t$, $t$, $t^3$, $\text{sgn}(t)$.
- DT: $x[-n] = -x[n]$ for all $n$. Examples: $\sin(\pi n/4)$, $n$, $\{-2, -1, \underline{0}, 1, 2\}$.

An odd signal must be zero at the origin: $x(0) = -x(0) \Rightarrow x(0) = 0$.

```text
 Even: {1, 2, 3, 2, 1}
        o
     o  |  o
  o  |  |  |  o
--+--+--+--+--+-> n
 -2 -1  0  1  2

 Odd: {-2, -1, 0, 1, 2}
              o
           o  |
--+--+--o--+--+-> n
  |  o
  o
 -2 -1  0  1  2
```

**Decomposition:** any signal can be written as $x_e + x_o$ with

$$
x_e(t) = \frac{x(t)+x(-t)}{2}, \qquad x_o(t) = \frac{x(t)-x(-t)}{2}
$$

**Properties:** even × even = even, odd × odd = even, even × odd = odd; $\sum_{n=-N}^{N}x_o[n] = 0$.

## Check yourself

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

1. Which condition defines an even CT signal?
   A. x(-t) = -x(t) for all t
   B. x(t+T) = x(t) for all t
   C. x(t) = 0 for t < 0
   D. x(-t) = x(t) for all t
   Answer: D. Even signals are symmetric about the vertical axis, so reflecting in time leaves them unchanged. (Key point: Even signal definition)

2. What must be true of an odd signal at the origin?
   A. x(0) = 0
   B. x(0) = 1
   C. x(0) is the maximum value
   D. x(0) can be any value
   Answer: A. From x(0) = -x(0) it follows that x(0) = 0. (Key point: Odd signal at origin)

3. Which list contains only odd signals?
   A. cos ωt, t^2, e^-|t|
   B. sin ωt, t, t^3, sgn(t)
   C. sin ωt, t^2, sgn(t)
   D. t, t^3, cos ωt
   Answer: B. sin ωt, t, t^3 and sgn(t) satisfy x(-t) = -x(t); cos ωt, t^2 and e^-|t| are even. (Key point: Classifying examples)

4. What is the product of an odd signal and an even signal?
   A. Even
   B. Zero
   C. Odd
   D. Neither, depends on the signals
   Answer: C. Even × odd = odd; odd × odd = even; even × even = even. (Key point: Product properties)

## Related in Signals and Systems

- [Even/odd DT signals and periodicity](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-dt-periodicity) — Signals and Systems, chapter 1
- [Continuous-time vs discrete-time signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-vs-dt-signals) — Signals and Systems, chapter 1, asked 3×
- [Even/odd signals; energy or power check](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-energy-power-check) — Signals and Systems, chapter 1
- [Even/odd parts; time scaling and folding](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-parts-scaling-folding) — Signals and Systems, chapter 1
- [Even/odd decomposition of a signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-decomposition) — Signals and Systems, chapter 1, asked 2×
- [CT vs DT signals; u[n] = Σδ[n−k]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-dt-signals-step-impulse-sum) — Signals and Systems, chapter 1

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- [Even and odd parts of a signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aeven-odd-parts) — Data Communication, chapter 2
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- [Handshaking theorem; even number of odd vertices](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Ahandshaking-even-odd-degree-proof) — Discrete Structure, chapter 4, asked 3×