---
title: "Even/odd decomposition of a signal"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:even-odd-decomposition"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/even-odd-decomposition
site: BCT Engineering
---

# Even/odd decomposition of a signal

Signals and Systems · Chapter 1: Signals

- Asked in: 2075 Bhadra, 4 marks; 2075 Baisakh, 4+4 marks (asked 2 times)
- Video: [Even and Odd Decomposition of a Signal](https://www.youtube.com/watch?v=YFjJfdSbbiY) — Adam Panagos
- 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-decomposition
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define even and odd signals. Develop the even/odd decomposition of a general signal x(t).

## Short answer

An even signal satisfies x(−t)=x(t); an odd signal satisfies x(−t)=−x(t). Any signal x(t) can be decomposed into even part x_e(t)=½[x(t)+x(−t)] and odd part x_o(t)=½[x(t)−x(−t)]. Example: e^t decomposes into cosh t (even) and sinh t (odd).

## Answer

- **Even signal:** symmetric about the vertical axis, $x(-t) = x(t)$ (DT: $x[-n] = x[n]$). Examples: $\cos\omega t$, $t^2$, $e^{-|t|}$.
- **Odd signal:** antisymmetric about the origin, $x(-t) = -x(t)$. An odd signal is always zero at $t = 0$. Examples: $\sin\omega t$, $t$, $t^3$.

#### Even/odd decomposition

Let any signal be written as the sum of an even part and an odd part:

$$
x(t) = x_e(t) + x_o(t) \qquad (1)
$$

where $x_e(-t) = x_e(t)$ and $x_o(-t) = -x_o(t)$. Replace $t$ by $-t$:

$$
x(-t) = x_e(-t) + x_o(-t) = x_e(t) - x_o(t) \qquad (2)
$$

Adding (1) and (2), and subtracting (2) from (1):

$$
\begin{aligned}
x_e(t) &= \frac{1}{2}\left[x(t) + x(-t)\right] \\
x_o(t) &= \frac{1}{2}\left[x(t) - x(-t)\right]
\end{aligned}
$$

These parts always exist for any $x(t)$, so **every signal can be decomposed into even and odd components**. The same holds for DT signals: $x_e[n] = \frac{1}{2}(x[n]+x[-n])$, $x_o[n] = \frac{1}{2}(x[n]-x[-n])$.

#### Example
$x(t) = e^{t}$:

$$
x_e(t) = \frac{e^{t}+e^{-t}}{2} = \cosh t, \qquad x_o(t) = \frac{e^{t}-e^{-t}}{2} = \sinh t
$$

Check: $\cosh t + \sinh t = e^{t}$.

Example 2: $x(t) = u(t)$ gives $x_e(t) = \frac{1}{2}$ for all $t \neq 0$ and $x_o(t) = \frac{1}{2}\,\text{sgn}(t)$.

**Useful properties:** even × even = even, odd × odd = even, even × odd = odd; the integral of an odd signal over $[-T, T]$ is zero.

## Check yourself

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

1. The even component of x(t) is:
   A. [x(t) - x(-t)]/2
   B. [x(t) + x(-t)]/2
   C. x(t) + x(-t)
   D. [x(t) x(-t)]/2
   Answer: B. Adding x(t) = xe + xo and x(-t) = xe - xo gives xe = [x(t) + x(-t)]/2. (Key point: Even part formula)

2. Which is true of an odd signal?
   A. x(-t) = x(t)
   B. It is zero at t = 0
   C. Its integral over [-T, T] is 2T
   D. It is symmetric about the vertical axis
   Answer: B. x(-t) = -x(t) at t = 0 gives x(0) = -x(0), so x(0) = 0. (Key point: Odd signal property)

3. The even and odd parts of e^t are:
   A. sinh t and cosh t
   B. cosh t and sinh t
   C. cos t and sin t
   D. e^t/2 and e^t/2
   Answer: B. xe = (e^t + e^-t)/2 = cosh t and xo = (e^t - e^-t)/2 = sinh t. (Key point: Decomposition of e^t)

4. The product of two odd signals is:
   A. Odd
   B. Even
   C. Zero
   D. Neither
   Answer: B. Odd x odd = even, since the two sign flips cancel. (Key point: Product properties)

## Related in Signals and Systems

- [Even/odd decomposition is unique: {2,3,4,5,6}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-uniqueness-sequence) — Signals and Systems, chapter 1
- [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 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
- [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
- [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

## Related in other subjects

- [Even and odd part of a signal](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aeven-odd-part-piecewise) — Digital Signal Analysis and Processing, chapter 1, asked 2×
- [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
- [Even and odd part from stem plot](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aeven-odd-part-stem-plot) — Digital Signal Analysis and Processing, chapter 1
- [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×