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# Signals and Systems: Even and Odd Signals

## Even Signal

A signal which is symmetrical about the vertical axis or time origin is known as **even signal** or **even function**. Therefore, the even signals are also called the **symmetrical signals**. Cosine wave is an example of even signal.

## Continuous-time Even Signal

A continuous-time signal x(t) is called the even signal or symmetrical signal if it satisfies the following condition,

𝑥(𝑡) = 𝑥(−𝑡); for − ∞ < 𝑡 < ∞

Some examples of continuous-time even signals are shown in Figure-1.

## Discrete-time Even Signal

A discrete-time signal x(n) is said to be even signal or symmetrical signal if it satisfies the condition,

𝑥(𝑛) = 𝑥(−𝑛); for − ∞ < 𝑛 < ∞

Examples of discrete-time even signals are shown in Figure-2.

## Properties of Even Signals

The properties of the even signals are given as follows −

- The even signals are symmetrical about the vertical axis.
- The value of an even signal at time (t) is same as at time (-t).
- The even signal is identical with its reflection about the origin.
- Area under the even signal is two time of its one side area.

## Odd Signal

A signal that is anti-symmetrical about the vertical axis is known as odd signal or **odd function**. Therefore, the odd signals are also called the **antisymmetric signals**. Sine wave is an example of odd signal.

## Continuous-time Odd Signal

A continuous time signal x(t) is called an odd signal or antisymmetric signal if it satisfies the following condition,

𝑥(−𝑡) = −𝑥(𝑡); for − ∞ < 𝑡 < ∞

Examples of continuous time odd signals or antisymmetric signals are shown in Figure-3.

## Discrete-time Odd Signal

A discrete time signal x(n) is said to be an odd signal or antisymmetric signal, if it satisfies the following condition,

𝑥(−𝑛) = −𝑥(𝑛); for − ∞ < 𝑛 < ∞

Examples of discrete-time odd signals are shown in Figure-4.

## Properties of Odd Signals

Following are the properties of the odd signals −

The odd signal is antisymmetric about the origin.

The value of odd signal at time (t) is negative of its value at time (-t) for all t, i.e., −∞ < 𝑡 < ∞.

The odd signal must necessarily be zero at time t = 0 to hold 𝑥(0) = −𝑥(0).

Area under the odd signal is always zero.

**Note** – A continuous-time signal is said to be **neither even nor odd** if it does not satisfy the condition of the even signal and that of the odd signal. Some examples of such signals (neither even nor odd) are shown in Figure-5.

## Numerical Example

Find whether the signals are even or odd.

𝑥(𝑡) = 𝑒

^{−5𝑡}𝑥(𝑡) = sin 2𝑡

𝑥(𝑡) = cos 5𝑡

### Solution

Given signal is,

𝑥(𝑡) = 𝑒

^{−5𝑡}𝑥(−𝑡) = 𝑒

^{5𝑡}−𝑥(𝑡) = −𝑒

^{−5𝑡}It is clear that 𝑥(𝑡) ≠ 𝑥(−𝑡) and 𝑥(−𝑡) ≠ −𝑥(𝑡), thus the given signal is neither even signal nor odd signal.

The given signal is,

𝑥(𝑡) = sin 2𝑡

𝑥(−𝑡) = −sin 2𝑡

−𝑥(𝑡) = −sin 2𝑡

Hence, 𝑥(𝑡) ≠ 𝑥(−𝑡); but 𝑥(−𝑡) = −𝑥(𝑡), thus the given signal is an odd signal.

Given signal is,

𝑥(𝑡) = cos 5𝑡

𝑥(−𝑡) = cos 5𝑡

−𝑥(𝑡) = −cos 5𝑡

Therefore, 𝑥(𝑡) = 𝑥(−𝑡) and 𝑥(−𝑡) ≠ −𝑥(𝑡), thus the given signal is an even signal.

- Related Articles
- Signals and Systems – Properties of Even and Odd Signals
- Signals and Systems – What is Odd Symmetry?
- Signals and Systems – What is Even Symmetry?
- Signals and Systems: Periodic and Aperiodic Signals
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